NIKHEF
Gravity Geometry and Physics
J W van Holten
NIKHEF P O Box
DB Amsterdam NL
c
Lectures presented at the Summerschool of Theoretical Physics Saalburg Germany Sept
Abstract
Geometry is present in physics at many levels most prominently in the theory of
gravity In these introductory lectures geometrical concepts like manifolds geo desics
curvature and top ology are introduced as a to ol to describ e and interpret physical
phenomena in space time including the gravitational eld itself The fo cus is not only
on p ossible static structures e g black holes but also on dynamical e ects like waves
and traveling domain walls Although quantum gravity itself is outside the scop e of
these lectures it is brie y discussed how geometry can b e used even in that context
to characterise con gurations dominating the path integral in various circumstances
including typical quantum pro cesses like vacuum tunneling Many of these geometrical
metho ds can also b e used in other branches of physics by linking the dynamics of a system with the geometry of its con guration space some examples are mentioned
Contents
Gravity and Geometry
The gravitational force
Fields
Geometrical interpretation of gravity
Curvature
The Einstein equations
The action principle
Geo desics
Curves and geo desics
Canonical formulation
Action principles
Symmetries and Killing vectors
Phase space symmetries and conservation laws
Example the rigid rotor
Dynamics of space time
Classical solutions of the gravitational eld equations
Plane fronted waves
Nature of the space time
Scattering of test particles
Symmetry breaking as a source of gravitational waves
Coupling to the electro magnetic eld
Black holes
Horizons
The Schwarzschild solution
Discussion
The interior of the Schwarzschild sphere
Geo desics
Extended Schwarzschild geometry
Charged black holes
Spinning black holes i
The Kerr singularity
Black holes and thermo dynamics
Topological invariants and self duality
Topology and top ological invariants
Topological invariants in gravity
Self dual solutions of the Einstein equations
Runge Lenz vector and Yano tensors ii
Chapter
Gravity and Geometry
The gravitational force
Gravity is the most universal force in nature As far as we can tell from ob
servations and exp eriments every ob ject every particle in the universe attracts
any other one by a force prop ortional to its mass For slow moving b o dies at
large distances this is a central force inversely prop ortional to the square of the
distance As the action is recipro cal and since according to Newton action and
reaction forces are equal in magnitude the expression for the gravitational force
b etween two ob jects of mass M and M at a distance R is then determined to
have the unique form
M M
F G
R
The constant of prop ortionality Newton s constant of gravity has dimensions
of acceleration p er unit of mass times an area Therefore its numerical value
obviously dep ends on the choice of units In the MKS system this is
G m kg s
It is also p ossible and sometimes convenient to x the unit of mass in such
a way that Newtons constant has the numerical value G In the natural
system of units in which also the velocity of light and Planck s constant are
unity c h this unit of mass is the Planck mass m
P
q
m h c G kg
P
GeV c
Newton s law of gravity is valid for any two massive b o dies as long as they
are far apart and do not move to o fast with resp ect to one another In particular
it describ es the motions of celestial b o dies like the mo on circling the earth or the
planets orbiting the sun as well as those of terrestrial ob jects like apples falling
from a tree or canon balls in free ight Ever since Newton this uni cation of
celestial and terrestial mechanics has continued to impress p eople and has had a
tremendous impact on our view of the universe It is the origin and basis for the
b elief in the general validity of physical laws indep endent of time and place
Fields
Although as a force gravity is universal Newton s law itself has only limited
validity Like Coulomb s law for the electrostatic force b etween two xed charges
Newton s law holds strictly sp eaking only for static forces b etween b o dies at rest
Moreover and unlike the electric forces there are mo di cations at smaller nite
distances which can b e observed exp erimentally
For example if the gravitational force would have a pure R dep endence
the orbits of particles around a very heavy central b o dy would b e conic sections
ellipses parab ola s or hyperb ola s dep ending on the energy and angular mo
mentum in accordance with Kepler s laws The observation of an excess in the
precession of the p erihelion of the orbit of Mercury around the sun by LeVerrier
in and improved by Newcomb in was one of the rst clear indica
tions that this is actually not the case and that the gravitational force is more
complicated
The exact form of the gravitational forces exerted by moving b o dies is a
problem with many similarities to the analogous problem in electro dynamics
The understanding of electro dynamical phenomena greatly improved with the
introduction of the concept of lo cal eld of force This concept refers to the
following characteristics of electro dynamical forces
the in uence of electric charges and currents and of magnetic p oles extends
throughout empty space
the force on a standard test charge or test magnet at any given time dep ends
on its lo cation with resp ect to the source of the eld and the relative state
of motion
changes in the sources of the elds give rise to changes in the force on test
ob jects at a later time dep ending on the distance the sp eed of propagation
of disturbances in empty space is nite
In the case of electro dynamics this sp eed turned out to b e a universal constant
the sp eed of light c One of the most striking consequences of these prop erties
which follow directly from the mathematical description of the elds as expressed
by Maxwell s equations is the existence of electromagnetic waves lo cal variations
in the elds which transp ort energy momentum and angular momentum over
large distances through empty space Maxwell s predictions were magni cently
veri ed in the exp eriments of Hertz con rming the reality and characteristics of
electromagnetic waves From these results it b ecame clear for example that
light is just a sp ecial variety of such waves Therefore optical and electromagnetic
phenomena have a common origin and can b e describ ed in a single theoretical
framework
The concept of eld has found other applications in physics for example in the
phenomenological description of uids and gases Treating them as continua one
can describ e the lo cal pressure and temp erature as elds with lo cal variations in
these quantities propagating through the medium as sound waves or heat waves
As in electro dynamics also in gravity the concept of eld has taken a central
place What Maxwell achieved for electromagnetism was accomplished by Ein
stein in the case of gravity to obtain a complete description of the forces
in terms of space and time dep endent elds This eld theory is known as the
general theory of relativity One of the most remarkable asp ects of this theory
is that it provides an interpretation of gravitational phenomena in terms of the
geometry of space time Many detailed presentations of the geometrical descrip
tion of space time can b e found in the literature including a number of b o oks
included in the references at the end of these lecture notes A general
outline of the structure of the theory is presented in the following sections
Geometrical interpretation of gravity
The rst step in describing a lo cal eld b oth in electro dynamics and gravity
is to sp ecify the p otentials The force on a test particle at a certain lo cation
is then computed from the gradients of these p otentials In the case of gravity
these quantities have not only a dynamical interpretation but also a geometrical
one Whereas the electromagnetic eld is describ ed in terms of one scalar and
one vector p otential together forming the comp onents of a four vector A x
the gravitational eld is describ ed by p otentials which can b e assembled in
a symmetric four tensor g x This tensor has a geometrical interpretation as
the metric of space time determining invariant space time intervals ds in terms
of lo cal co ordinates x
ds g x dx dx
In the absence of gravity space time ob eys the rules of Minkowski geometry well
known from sp ecial relativity In cartesian co ordinates ct x y z the metric of
Minkowski space is
diag
1
We use greek letters taking values to denote the space time comp onents of four vectors and tensors
Then the invariant ds takes the form
ds dx dx c dt dx dy dz
For a test particle there is a simple physical interpretation of invariant space
time intervals measured along its world line in terms of prop er time measured
by a clo ck in the lo cal rest frame of the particle As world lines are time like
any interval along a world line satisi es ds and the prop er time interval d
measured by a clo ck at rest with resp ect to the particle is given by
c d ds
The lo cal matrix inverse of the metric is denoted by g x
g x g x
where is the Kronecker delta and the Einstein summation convention has
b een used This convention implies that one should automatically p erform a full
summation over any rep eated upp er and lower indices like on the left hand
side of this equation
From the gradients of the p otentials i e of the metric tensor one constructs a
quantity known as the connection with comp onents given by the expression
g g g g
At this p oint this expression may simply b e considered a de nition It is a useful
de nition b ecause this combination of gradients of the metric o ccurs very often
in equations in general relativity The deep er mathematical reason for this stems
from the transformation prop erties of expressions for physical quantities under
lo cal co ordinate transformations However it is not necessary to discuss these
asp ects at this p oint
A physical argument for the imp ortance of the connection is that it gives
directly the gravitational acceleration of a test particle in a given co ordinate
system by the following prescription due to Einstein the world lines of p oint
like test particles in general relativity are time like geo desics of space time the
latter b eing considered as a continuous pseudo euclidean space with metric g
Time like geo desics are lines along which the total prop er time b etween two xed
R
events d is an extremum such world lines are solutions of the equation of
motion
2
The term pseudo euclidean here refers to the fact that the metric is not p ositive de nite as
it would b e in a true euclidean space space time has a lorentzian signature which
implies that a general metric g has one negative time like and three p ositive space like
eigenvalues
d x dx dx
d d d
A geometrical interpretation of the connection is therefore that it de nes geo desics
As concerns its dynamical interpretation there is some similarity of eq with
the Lorentz force in electro dynamics in that the prop er acceleration dep ends on
the four velocity x we use an overdot to denote a derivative w r t prop er time
However whereas the Lorentz force is linear in the four velocity the gravitational
acceleration dep ends quadratically on the four velocity The co e cients of this
quadratic term are given by the connection which therefore directly determines
the force Notice that the gravitational acceleration do es not involve the mass
of the test particle This is the mathematical content of the famous equivalence
principle which states that in a xed gravitational eld all b o dies fall with the
same acceleration indep endent of their mass
Eqs and express the force on a test particle in terms of a eld
the connection and the eld in terms of p otentials the metric This provides
the basis for the eld theory of gravity formulated by Einstein in his theory of
general relativity
What is still lacking here of course is a set of equations determining the elds
or p otentials in terms of given sources like Maxwell s equations allow one to
compute the electromagnetic elds in terms of charges and currents General
relativity provides such equations they are known as the Einstein equations
and they determine the gravitational elds arising from given distributions of
mass energy and momentum These distributions the sources of the gravita
tional elds are describ ed by a symmetric tensor T x with indep endent
comp onents equal to the number of indep endent comp onents of the metric T
is called the energy momentum tensor In order to give a concise formulation of
Einstein s equations it is useful to rst introduce one more geometrical concept
that of curvature
Curvature
Curvature is a imp ortant concept b ecause it characterizes the space time geom
etry in a way which is basically co ordinate indep endent In order to illustrate
the idea we rst present a heuristic discussion Then we turn to mathematical
de nitions which allow us to write Einsteins equations in a compact way with a
direct and elegant geometrical interpretation
Curvature is de ned by the prop erties of surfaces it is a measure of their
deviation from a at plane Curvature can b e expressed conveniently in terms of
the change of a vector when transp orted parallel to itself around a lo op enclosing
a two dimensional surface element In a plane displacement of a vector parallel
to itself will always result in the same vector when you return to the starting
p oint there is absolute parallelism but no curvature In a curved surface this is
no longer true Consider the example of a sphere a surface of constant curvature
If a vector is moved southward from the north p ole parallel to itself along a great
circle it ends up on the equator cutting it at a right angle Now transp ort it
along the equator over an arc of angular extension always keeping at a right
angle Finally move it back to the north p ole along a great circle keeping it
parallel to the great circle and to itself It then ends up at the north p ole rotated
by an angle with resp ect to its original orientation Thus parallel transp ort of a
vector along a closed lo op on a curved surface do es not leave the vector invariant
in general
In the example of the sphere the rotation of the vector is prop ortional to the
distance traveled along the equator as measured by the angle This is also
prop ortional to the area enclosed by the lo op consisting of the two sections of
great circles through the north p ole and the arc of the equator For a sphere with
radius R the change dV in a vector V when the lo op encloses an area dA R d
is
V
dV V d dA K V dA
R
The constant of prop ortionality K R b etween the relative change in the
vector dV V and the change in the area dA is a measure of the intrinsic curva
ture of the surface known as the Gaussian curvature Note that the Gaussian
curvature is large when the radius of the sphere is small and that it vanishes in
the limit R as one would exp ect intuitively
This idea can now b e generalized to situations in higher dimensional spaces Of
course in a space of three or more dimensions there are many surfaces one can
draw through a given p oint Each of these surfaces may have its own curva
ture Our rst conclusion is therefore that in general the curvature has many
comp onents
Fortunately it is only necessary to consider curvature comp onents in surfaces
which are linearly indep endent To this end let us choose a lo cally non degenerate
co ordinate system x and consider in particular the surfaces de ned by keeping
all co ordinates constant except two say x and x
Let d d b e the area of an oriented surface element in the x x
plane which contains the p oint with co ordinates x Any vector with comp o
nents V transp orted parallel to itself around the lo op de ned by the circumfer
ence of this surface element now changes to rst order by an amount
dV R x V d
The constants of prop ortionality R x de ne a tensor known as the Riemann
curvature tensor at the lo cation x A comparison with eq shows that the
Riemann tensor is a higher dimensional generalization of the Gaussian curvature
From eq it follows immediately that a space time is completely at if and
only if all comp onents of the Riemann curvature tensor vanish
De ning parallel transp ort of a vector along a line element dx in terms of
the Riemann Christo el connection
V dV dx
a direct computation for an in nitesimal closed lo op gives the result that the
comp onents of the Riemann tensor can b e expressed as
R
where the square brackets denote an ordinary matrix commutator of two connec
tion comp onents considered as matrices the rst one with
the second one with
Using eq for the connection we can ultimately express the curvature in
terms of the metric and its rst and second derivatives It may then b e checked
from this explicit expression that it identically satis es the relation
R R R
This relation is known as the Bianchi identity It is analogous to the homogeneous
Maxwell equation which implies that the electric and magnetic elds can b e
obtained from a four vector p otential A In a similar way the Bianchi identity
for the curvature tensor implies that the curvature tensor can b e obtained from
a symmetric tensor g
We wish to emphasize here that the curvature of space time is not a measure
for the strength of the gravitational elds or forces rather it is a measure for
the variation of the gravitational elds in space and time the gradients
The Einstein equations
The Einstein equations are second order non linear partial di erential equations
for the metric of space time in terms of a given energy momentum distribution
As a result these equations describ e variations of the elds in space and time
rather than the elds themselves In view of the role of curvature as a measure for
such variations and the geometrical interpetation of gravitational forces it is quite
natural that the Einstein equations should admit an interpretation as equations
for the curvature of space time However rather than the full Riemann tensor it
turns out that these equations involve only its contracted forms the Ricci tensor
R and the curvature scalar R
R R R R g R
Note that we adhere to the general convention that contraction with an inverse
metric can b e used to raise an index of a vector or tensor turning a covariant
comp onent into a contravariant one similarly contraction of an upp er index on
a vector or tensor with the metric itself is used to lower it to make a covariant
comp onent out of a contravariant one
The reason that only the contracted forms of the Riemann curvature app ear
in the Einstein equations is that in four dimensional space time the Riemann
curvature tensor has to o many comp onents twice as many as the metric There
fore xing the metric comp onents in terms of the full Riemann tensor would
generally lead to an overdetermined system of equations In contrast the Ricci
tensor is symmetric R R and has indep endent comp onents precisely
equal to the number of metric comp onents to b e solved for It is therefore to b e
exp ected that the Einstein equations take the form of an equation for the Ricci
tensor Indeed Einstein s equations for the gravitational elds can b e written in
the simple form
G
R T g R
c
or equivalently
G
R T g T
c
g T Notice the app earance of Newton s constant on the where T T
right hand side of the equations Of course in empty space time T the
second version of Einstein s equations then implies that the Ricci tensor
vanishes
R
Space time geometries which satisfy these conditions are some times called Ricci
at The most interesting asp ect of these equations is that an empty space time
can still supp ort non trivial gravitational elds the Riemann curvature tensor
can have non zero comp onents even if its contracted form the Ricci tensor is
zero everywhere
This observation is imp ortant b ecause it lies at the heart of the theoretical
arguments for the existence of gravitational waves Indeed as we discuss in
more detail later on one can nd non trivial solutions of the Einstein equations
representing lumps in the gravitational eld which travel at the sp eed of light
and transp ort a nite amount of energy and momentum p er unit of volume from
one at part of empty space to another at part of empty space
There is a mo di ed version of Einstein s equations in which the gravitational
eld itself creates a non vanishing energy momentum tensor even in the absence
of matter This gravitational eld b ehaves like a homogeneous continuum with
constant density and pressure one of them negative satisfying the equations
G
g R
c
with a constant Such an equation arises if the energy momentum tensor is
T g
The constant is known as the cosmological constant Contraction of eq
with the inverse metric gives
G
R
c
This shows that in the presence of a cosmological constant empty space time has
a constant non zero scalar curvature Dep ending on the sign of the cosmological
constant eq then admits solutions of Einstein s equations corresp onding
to empty space times in continuous expansion or contraction This mo di ed
version of Einstein s equations is esp ecially interesting with regard to cosmological
applications
Of course in the general case there is a contribution to the energy momentum
tensor from all matter and energy which happ ens to b e present in the part of
the universe we wish to describ e The precise form of the energy momentum
tensor dep ends on the kind of matter or radiation which constitutes the source
of the gravitational eld However it is very often justi ed and convenient to
treat matter and radiation as an ideal uid the energy momentum tensor of
which dep ends only on the mass and pressure density and its ow as decrib ed
by the lo cal four velocity of an element of uid at lo cation with co ordinates x
Denoting the mass density by x the pressure density by p x and the lo cal
four velocity by u x the energy momentum tensor then takes the form
T pg p u u
Comparison with eq shows that the cosmological constant can b e reob
tained from an ideal uid with p
The action principle
The Einstein equations can b e obtained from a variational principle that
is they follow by requiring an action S to b e stationary under variations g of
the metric S The action for the gravitational eld equations including a
coupling to material energy momentum is
Z
p
c
S g R g x d
c G
Here x represents the sp eci c source term but generally it is not the energy
momentum tensor itself rather the energy momentum tensor is given in terms
by of and its covariant trace
g T
However we consider to b e the indep endent variable hence by at there
is no contribution from this tensor under variations with resp ect to the metric
Moreover in eq g x det g x and the minus sign is necessary b ecause
of the lorentzian signature of the metric which implies the pro duct of its eigen
values to b e negative everywhere The pro of of the action principle can b e stated
in a few lines but it requires some intermediate results which we discuss rst
To b egin with in stead of considering variations of the metric g one can
equivalently use the variations of the inverse metric g Indeed b ecause g g
it follows that
g g g g
or
g g g g
Thus the variation of the metric can b e expressed in terms of the variation of the
inverse metric and vice versa
Next we need a rule for computing the variation of the determinant g This
is obtained from the formula
log det M Tr log M
which holds for symmetric matrices M as can easily b e checked by diagonalizing
M and using the rule that the logarithm of the pro duct equals the sum of the
logarithms of the eigenvalues From this equation we then derive
g g g g g
g
Finally we observe that the Ricci tensor R is an expression involving only the
connection Hence
R g R g R
with the variation R expressible purely terms of the variation of the connection
The total variation of the action now b ecomes
Z
p
c c
S R g R d x g g
c G G
Z
p
R
g d x
The last term is to b e interpreted in the sense that we vary R with resp ect to
the metric only via its dep endence on the connection However it turns out that
if we insert the expression for the connection this whole variation adds up
to a total derivative As the action principle applies to variations which vanish
on the b oundary this term do es not contribute to S If we then require the
variation of the action to vanish we indeed obtain the Einstein equations
c
R g R g
G
c
R g R T
G
The equation with the cosmological constant is obtained by making the replace
ment g in the action
Chapter
Geo desics
Curves and geo desics
In this chapter we discuss how basic information ab out the geometry of spaces
or space times generically manifolds can b e obtained from studying their
geo desics The geo desics of euclidean di erentiable manifolds are smo oth curves
of stationary prop er length meaning that to rst order the prop er length of an
arc do es not change under small variations of the curve More generally and more
precisely given a metric on a manifold euclidean or lorentzian the geo desics are
those smo oth curves for which the prop er interval b etween any two xed p oints
P and P through which the curve passes is b oth well de ned and stationary
The interval among the xed p oints can b e either a maximum a minimum or a
saddle p oint
A direct reason for geo desics to b e imp ortant is that time like geo desics rep
resent the allowed tra jectories of test particles in a xed background space time
But in general geo desics tell us interesting things ab out the symmetries and
top ology of a manifold which often have wider implications for physics
First we derive the fundamental equation characterizing geo desics The prop er
of an arbitrary smo oth curve C b etween the two of the arc C interval s C
PP PP
xed p oints P P is given by the integral
v
u
Z Z
u
dx dx
t
ds d s C g
PP
d d
C C
PP PP
This holds for all geo desics of euclidean manifolds and time or space like geo desics
of lorentzian manifolds is a continuous parameter on some interval of the real
line parametrizing the curve b etween the p oints P and P in any convenient way
provided the co ordinates x of p oints on the curve are monotonic di erentiable
functions of this parameter the lab el of any p oint must b e unique and the tan
gent vector x dx d must b e well de ned The parameter is then called an
a ne parameter Note that the integral do es not dep end on the particular choice
of a ne parameter it is invariant under di erentiable changes of parametriza
tion This freedom to choose for any smo oth parametrization of
the curve is often useful but requires some care in interpreting results explicitly
containing As any invariant geometrical results with physical interpretation
can not dep end on the choice of a ne parameter the b est way to pro ceed is to
remove any dep endence from equations b efore drawing physical conclusions
is stationary if the integral do es not change to The prop er interval s C
PP
rst order under arbitrary smo oth variations of the path while keeping the end
p oints of the integral xed
Z
d w x
C
PP
q
x x j Now after a partial integration where the integrand is w x jg
with vanishing b oundary terms the end p oints remain xed the variation of the
integrand can b e written
d dx g dx g dx
w x
d w d w x d d
provided w The sign on the r h s is p ositive in euclidean spaces or for
space like curves in lorentzian space times and negative for time like curves in
lorentzian space times Light like curves in lorentzian space time need sp ecial
consideration as ds is equivalent to w d
The variational principle now implies that for any smo oth lo cal varia
tion x the induced variation w must vanish if the curve x is to describ e
a geo desic For w we can divide the right hand side of eq by w once
more and use the fact that w d ds the in nitesimal prop er interval which
is measured in the lab oratory frame and indep endent of the choice of a ne pa
rameter We can also replace the partial derivative of the metric in the second
term by the connection recall that the connection was de ned in eq as
g g g
g
x x x
This expression is actually the solution of the equation
D g g
g g
D x x
stating that the metric is covariantly constant a condition known as the metric
p ostulate From the equation ab ove it follows that
g
g g
x
The vanishing of w is now seen to imply a di erential equation of the form
for the co ordinates of a p oint moving on the curve as a function of the prop er
interval parameter s
d x dx dx
ds ds ds
Note that the ab ove pro cedure eliminating the a ne parameter is equivalent
to a choice of rate of ow of the a ne parameter d equal to the prop er interval
ds measured in the co ordinate frame fx g If the signature of the manifold is
lorentzian and the curve time like this a ne parameter is the prop er time and
eq is the equation of motion of a test particle of arbitrary non vanishing
mass in a xed space time with a ne connection x Note also that the
choice of d ds implies an identity
dx dx
g
ds ds
where again the plus sign holds for euclidean or space like lorentzian curves and
the minus sign for time like curves in lorentzian space times This is a constraint
on the solutions of the geo desic equation
That this constraint is consistent with the geo desic equation follows by ob
serving that
dx dx
H s g
ds ds
is a constant of motion or more generally a constant of geo desic ow
dH d x dx dx dx
g
ds ds ds ds ds
Finally light like curves also called null curves by de nition ob ey
dx dx
g
d d
In this case the a ne parameter can not b e identi ed with prop er time or
distance As eq is to hold on all p oints of the light like curve the quantity
dx dx
g H
d d
must b e a constant of light like motion Therefore eq still holds with s
replaced by From this it follows that it is correct to interpret the solutions of
the geo desic equation
dx dx d x
d d d
satisfying the constraint H as light like geo desics
Remark Although we have shown that the geo desic equation implies the con
stancy of H for all types of geo desics the inverse statement is not true In par
ticular for non null geo desics euclidean and space like or time like lorentzian
the more general condition
d x dx dx dx
B x
d d d d
with B B an anti symmetric tensor is su cient for H to b e conserved
This happ ens for example for charged particles in an electro magnetic eld with
eld strength F
q
F x B x
m
where q m is the charge to mass ratio of the particle Similar extensions exists
for particles with spin or non ab elian charges
For light like geo desics the equation can b e further generalized to read
dx dx dx dx d x
x B
d d d d d
with an arbitrary constant It should b e noted that if then H is only
conserved by virtue of the fact that it vanishes
Summary
The results of this discussion an b e summarized as follows All geo desic curves
in euclidean or lorentzian manifolds satisfy the equation
d x dx dx
d d d
where is an a ne parameter The quantity
dx dx
g H
d d
is conserved by the geo desic ow In physical applications three cases are to b e
distinguished
For euclidean manifolds and space like geo desics of lorentzian manifolds one
can take d ds the prop er distance these geo desics are then character
ized by the condition H s
Time like geo desics of lorentzian manifolds can b e parametrized by the
prop er time d d such geo desics satisfy H
Light like null geo desics admit no such identi cation of with invariant
measures of the arc length in an inertial frame lab oratory frame null
geo desics are characterized by H
Canonical formulation
In D dimensional space time eq represents a set of D second order di er
ential equations in the a ne parameter providing a covariant description of
geo desics It is p ossible to replace this set by a set of D rst order di erential
equations by going over to the canonical formulation of the dynamics of test par
ticles in which the number of indep endent variables is doubled Introducing the
momentum variables
dx
p g x
d
the geo desic equation b ecomes
dp
p p
d
The equations and constitute a pair of rst order di erential equa
tions equivalent to the second order geo desic equation A p owerful result
is now obtained by observing that these two equations can b e derived from the
constant of motion H eq as a pair of canonical hamiltonian equations
with H in the role of hamiltonian
g p p H
With this de nition
dx dp H H
d p d x
Then any function F x p on the D dimensional phase space spanned by the
co ordinates and momenta changes along a geo desic according to
dp dx F F dF
fF H g
d d x d p
where the Poisson bracket of two functions F G on the phase space is de ned
generally as
F G F G
fF Gg fG F g
x p p x
The anti symmetry of the Poisson bracket automatically guarantees the conser
vation of the hamiltonian H by
fH H g
A similar prop erty holds for all quantities which are constant along geo desics
dF
fF H g
d
Formal prop erties of the Poisson bracket include its anti symmetry and its lin
earity in each of the arguments
fF G G g fF G g fF G g
Another imp ortant prop erty is that it satis es the Jacobi identity
fF fG K gg fG fK F gg fK fF Ggg
This prop erty implies that the Poisson bracket of two constants of geo desic ow
is again a constant of geo desic ow if fF H g and fG H g then
ffF Gg H g
As H is itself constant on geo desics the ab ove results are su cient to establish
that the constants of geo desic ow form a Lie algebra with the Poisson bracket
as the Lie bracket
The de nition of the Poisson bracket is not manifestly covariant and
therefore at rst sight it seems to destroy the build in covariance of the tensor
calculus of di erential geometry and of general relativity in particular However
note that in our de nition the connection is alway symmetric in its lower
indices therefore we can rewrite the Poisson bracket formula as
G F
fF Gg D F D G
p p
where for scalar functions F and G on the phase space we de ne the covariant
derivative
F
p D F F
p
This equation preserves manifest covariance for scalar functions F and G for
example if J x p J x p then
p D J p D J J J
The manifestly covariant form of the Poisson bracket can also b e extended to all
completely symmetric tensors T of rank n by contracting all n indices with
n
1
the momentum p to obtain the scalar
n
1
T x p p T x p
n
1
n
Inserting this scalar into the covariant Poisson bracket the bracket for a particular
n
1
tensor comp onent T is obtained by taking in the expression resulting on
the right hand side the co e cient of the n nomial in the momenta that one is
interested in For anti symmetric index structures a similar result can b e achieved
using di erential forms or more generally Grassmann algebras
Action principles
The canonical formulation of geo desic ow presented here can b e cast into the
form of a standard hamiltonian action principle Namely the canonical eqs
de ne the critical p oints of the phase space action
Z
d x p H x p S
By using the hamilton equation
H
x
p
the momentum variable can b e eliminated in favour of the velocity of geo desic
ow to give the lagrangian form of the action
Z
g x x x d S
We observe that this is not the action for the geo desics that we started from
eq In particular it is not reparametrization invariant A comparison shows
that the lagrangian in is actually the square of w used earlier eq
Clearly the critical p oints are therefore the same with the p ossible exception of
light like curves Actually the variational principle for the original action was
not even well de ned in the case of light like curves
In this section we examine somewhat more closely the relation b etween the
various action principles and in particular the role of reparametrization invari
ance It makes clear how one can derive the lagrangian and hamiltonian actions
presented ab ove within a uni ed framework
The key to the general formulation is to incorp orate reparametrization invari
ance in an action of the type This is achieved by introduction of a
new variable e a function of the a ne parameter which acts as a gauge eld
in one dimension for lo cal reparametrizations of the worldline More precisely
the variable e by de nition transforms under reparametrizations as
d
e e e
d
This is the transformation rule for the square ro ot of a metric in one dimension
on a curve for which reason the variable e is called the einbein Written in a
slightly di erent way eq states that e d is de ned to b e invariant under
reparametrizations This makes it straightforward to write a reparametrization
invariant form of the quadratic action
Z
dx dx c
d S e x g x e
e d d
The last term prop ortional to the constant c do es not dep end on the co ordinates
x However it is reparametrization invariant and its inclusion is imp ortant for
reasons given b elow
Variation of the full action S e x w r t the co ordinates x repro duces the
geo desic equation provided everywhere the di erentials d are replaced by
ed This replacement makes the geo desic equation reparametrization invariant
as ed e d On the other hand the substitution e which breaks
reparametrization invariance makes the old and new equations fully identical
Such a substitution is allowed b ecause it only serves to de ne the a ne param
eter in terms of quantities measurable by an observer in the lo cally euclidean
or lorentzian lab oratory frame like prop er length or prop er time Of course
such a choice necessarily breaks the reparametrization invariance but it pre
serves the physical content of the equations It amounts to a choice of gauge
for reparametrizations in the true sense of the expression with the result that
mo dulo a constant the action S e x reduces to the simple quadratic action
However the action S e x contains more information as one can also consider
its variation w r t the einbein e Requiring stationarity under this variation gives
a constraint on the solutions of the geo desic equation which after multiplication
by e takes the form
g x x x e c
With e comparison with the de nition of H in eq shows that this
constraint expresses the constancy of the hamiltonian along a geo desic
H c
For euclidean manifolds one can always choose units such that c but for
lorentzian space times c corresp onds to space like light like or
time like geo desics resp ectively in the latter case the constant c then represents
the sp eed of light which is unity when expressed in natural units
Finally we derive the original reparametrization invariant geo desic action
from S e x This is achieved by using eq to solve for e instead of xing a
gauge
q
e jg x x c j
Substitution of this expression for e into the action S e x then leads back mo d
ulo a p ossible sign to the original action Therefore we conclude that b oth
this action and the quadratic action follow from the same starting p oint
provided by eq
Symmetries and Killing vectors
Often space times of interest p ossess sp ecial symmetries Such symmetries are
useful in nding solutions of the geo desic equations esp ecially if they are contin
uous symmetries as these generate constants of geo desic ow This is the content
of No ether s theorem applied to the motion of test particles and its generalization
to space like curves see for example refs In this section the role
of continuous symmetries is explored in some detail
A manifold p ossesses a continuous symmetry or isometry if there are in
nitesimal co ordinate transformations leaving the metric invariant Finite trans
formations can b e obtained by reiteration of the in nitesimal transformations
and donot require separate discussion at this p oint Consider an in nitesimal
co ordinate transformation characterised by lo cal parameters x
x x x x
For the line element ds to b e invariant the metric must transform as a tensor
therefore by de nition
x x
x g x g
x x
On the other hand invariance of the metric under a co ordinate transformation
implies
g x g x
Combining the two equations and to eliminate the transformed
and inserting the in nitesimal transformation expansion to metric g
rst order in leads to the covariant condition
D D
This equation is known as the Killing equation and the vector elds de ned by
the solutions x are called Killing vectors The existence of a Killing vector is
equivalent with the existence of an in nitesimal co ordinate transformation leaving the metric invariant
Let us pause for a moment to re ect on the interpretation of this result If the
co ordinate transformation is considered to b e a passive transformation the new
co ordinates represent a relab eling of the p oints of the manifold x and x in
eq refer to the same p oint P Now consider the p oint P which in the new
co ordinate system has the same value of the co ordinates as the p oint P in the
old co ordinate system x P x P Eq states that the new metric at
the p oint P has the same functional dep endence on the new co ordinates as the
old metric on the old co ordinates at the p oint P Thus in the new co ordinate
system the neighborho o d of P lo oks identical to the neighborho o d of P in the old
co ordinates In particular it is p ossible to map the p oints in the neighborho o d of
P one to one to p oints in the neighborho o d of P in such a way that all distances
intervals are preserved As this is in fact a co ordinate invariant statement
it implies a lo cal indistinguishability of the two neighborho o ds of the manifold
as long as it is not endowed with additional structures This then is what is
implied by a continuous symmetry of the manifold generated by the solutions of
the Killing equation
Now if two neighborho o ds of the manifold are similar in the sense describ ed
ab ove then this similarity should also extend to the geo desics passing through
the p oints mapp ed on each other by the symmetry op eration Indeed as the new
metric at the p oint P has the same functional dependence on the new co ordinates
there as the old metric at P has on the old co ordinates and not just the same
value the connections and curvature comp onents there also b ecome identical after
the co ordinate transformation The same holds for the solutions of the geo desic
equations Thus the symmetry transformation maps geo desics through the p oint
P to geo desics through the p oint P and the geo desic ow in the neighborho o d
of these p oints is completely similar
The similarity of geo desic structure generated by symmetry op erations implies
the conservation of certain dynamical quantities under geo desic ow This is most
conveniently discussed in the canonical formalism of the geo desic ow presented
ab ove in sect The quantity conserved is the generator of the symmetry
transformations obtained by contraction of the Killing vector with the canonical
momentum
J x p
This is a scalar quantity hence its value is co ordinate invariant It generates the
in nitesimal co ordinate transformations through the Poisson bracket
x x fJ x g
One can similarly determine the variation of the momentum under the transfor
mation generated by J
p fJ p g
J T x p
with T This represents a co ordinate dep endent linear transformation
of the momentum
That the quantity J is conserved now follows from the observation that
the hamiltonian is invariant under the phase space transformations de ned in
eqs Indeed taking into account the canonical equations of motion
and and using the Killing equation we obtain
dJ
fJ H g
d
This result was to b e exp ected as the hamiltonian dep ends on the co ordinates
only through the metric which is invariant by de nition of an isometry
As the Killing equation is linear it follows that the Killing vectors de ne
a linear vector space any linear combination of two Killing vectors is again a
Killing vector Let the dimension of this vector space b e r then any Killing
vector can b e expanded in terms of r linearly indep endent basis vectors e x
i
i r
e x e x
i r r
Equivalently any in nitesimal symmetry transformation dep ends linearly
on r parameters to which corresp ond an equal number of conserved generators
i
J J J where J x p e x p
r r i
i
We now show that these generators de ne a Lie algebra through their Poisson
brackets The key observation is that the Poisson bracket of any two constants
of geo desic ow is itself a constant of geo desic ow This is implied by the Jacobi
identity for the Poisson bracket which can b e written as
ffF Gg H g ffF H g Gg ffG H g F g
Thus if the Poisson brackets of F and G with the hamiltonian H vanish then
the left hand side vanishes as well and fF Gg is itself a constant of geo desic ow
A second imp ortant observation is that for two constants of motion which
are linear in the momentum like the J the Poisson bracket is also linear in the
i
momentum
p fJ J g e e e e
i j
i j
j i
r
are supp osed to form a complete set the expression on the right As the fJ g
i
i
hand side can b e expanded in terms of the basis elements with the result that
k k
we have f with a certain set of co e cients f
j i ij
k
fJ J g f J
i j k
ij
An equivalent statement is that the basis vectors of the Killing space satisfy
k
e e e e f e
i j
k j i ij
Eq combined with the observations that the J form a linear vector space
i
and that the Poisson bracket is bilinear and anti symmetric in its arguments
complete the pro of that the generators of the symmetry transformations de ne a
k
Lie algebra with structure constants f
ij
Of course the linear p d e enco des the same information It can b e
interpreted in terms of the Lie derivative which is the op erator comparing the
value of a function at the p oint P with that at the p oint P which has the same
value of its co ordinates after an arbitrary transformation of the form as
the p oint P had b efore the transformation x P x P For a scalar x
for which x x this gives
L x lin x x x
where the lin Q denotes the part of the expression Q linear in For a vector
v x for which
x
v x v x
x
one nds similarly
L v x lin v x v x v x v x
For higher rank tensors the construction of the Lie derivative gives the result
1
k
1 1
k k
x T x x lin T L T
1 1 1
k k k
T x T x T x
Now comparing eqs and we see that the Lie algebra prop erty of the
Killing vectors may b e expressed as
k
e e f e L L
k j i e e
i j
ij
Finally in the phase space where by de nition p we can also write the
transformation for the momentum as
p fJ p g L p
i i e
i
Thus we have established a close relationship b etween Poisson brackets and Lie
derivatives and b etween symmetries Killing vectors and Lie algebras
Phase space symmetries and conservation
laws
In the previous section it was established that continuous symmetries of man
ifolds are generated by Killing vectors and imply conservation laws for certain
dynamical quantities the generators J along geo desics However not every con
stant of geo desic ow is necessarily connected with a Killing vector This is most
easily seen from the example of the hamiltonian itself which is conserved but not
of the form Rather the hamiltonian is quadratic in the momenta This
leads us to consider the p ossibility of conservation laws for quantities which are
higher order expressions in the momenta
Let J x p b e any function on the phase space of co ordinates x and
momenta p asso ciated with a geo desic which is non singular in the momenta
In others words we supp ose it has an expansion of the type
X
k
1
k
p J x p J x p
1
k
k
k
k
Here the co e cients J can b e taken to b e completely symmetric in all k indices
We ask under what conditions such a function can b e a constant of geo desic ow
dJ
fJ H g
d
To obtain the answer insert the expansion into this equation and compare
terms at equal order of p owers of momentum At zeroth order one nds
J
i e J is a constant Such a constant can always b e absorb ed in the de nition
of J and is of no consequence From now on it will b e dropp ed
At rst order one nds back the Killing equation
D J D J
These terms we have already discussed in detail
Finally for k we nd a generalization of the Killing equation for the
k
1
k
co e cients J x
k
D J
1
2
k +1
The parentheses denote full symmetrization over all comp onent indices enclosed
The solutions of these generalised Killing equations are called Killing tensors It
is now obvious that the hamiltonian is included b ecause the metric g is co
variantly constant and therefore automatically a solution of eq for k
Moreover this solution always exists indep endent of the particular manifold con
sidered In addition for sp eci c manifolds sp eci c metrics g other solutions
may exist as well For example in spherically symmetric spaces the square of the
angular momentum
x p x p L
is conserved and quadratic in the momenta by construction Other examples will
b e encountered later on
Just as the Killing vectors the Killing tensors generate symmetry transforma
tions not of the manifold but rather of the corresp onding phase space spanned
by the co ordinates and the momenta tangent vectors In particular
X
k
2
k
x fJ x p x g J x p J x p
2
k
k
k
We observe that the transformations generated by the higher order Killing tensors
are velocity dep endent
Finally we address the algebraic prop erties of the constants of geo desic ow
We have already remarked that the Poisson brackets of two such constants yields
another constant by the Jacobi identity In general the rank of quantities
on the left hand side are additive minus one
o n
k l k l
J J J
k l k
Now for l the rank of J is the same as that of J It follows that
i for the constants of geo desic ow linear in the momentum corresp onding to
Killing vectors k l the algebra is closed in itself it is a Lie algebra G
k k
ii for any xed value of k the brackets of a J and a J close on the J
k
however the brackets of two elements J generates an element of order k
Hence in general the algebra is not closed unless one includes elements
of arbitrary rank i e all p ossible k This implies that once higher order invari
ants app ear one may exp ect an in nite series of them Sometimes this happ ens
indeed as with the Virasoro algebra for the conformal transformation in two di
mensions However in many cases the series stops b ecause the higher order
invariants are Casimir type of op erators or they b ecome trivial in that they are
just pro ducts of commuting lower order elements commuting here is meant in
the sense of vanishing Poisson brackets
The Lie algebra G of Killing vectors is characterized by its structure constants
k k
f As the Poisson brackets of J with all elements of the Lie algebra spanned
ij
o n
k
1
k
x for the by the J are again of order k one may choose a basis e
A
Killing tensors of order k lab eled by the index A and decomp ose the right hand
side of the Poisson bracket again in terms of the corresp onding conserved charges
k
J
A
n o
k k
B
J g J J
i
iA B A
This de nes a set of higher order structure constants The Jacobi identities then
B B
imply that the matrices T g de ne a representation of the Lie algebra
i
A iA
B
B k B k B C B C
T T T f g f g g g g
i j
k A ij k A ij j C iA iC j A
A
Therefore the constants of geo desic ow
k k
1
k
J x p p e x p
1
A k A
k
span a representation space of the Lie algebra G for the representation fT g
i
The discussion of symmetries shows that group theory and Lie algebras can b e
imp ortant to ols in the analysis of geo desic motion and the structure of manifolds
Example the rigid rotor
The ab ove concepts and pro cedures can b e illustrated by the simple example of
a rigid rotor which has a physical interpretation as a mo del for the low energy
b ehaviour of diatomic molecules As such the example also serves to emphasize
the usefulness of geometric metho ds in physics outside the context of general
relativity
Consider two particles of mass m and m interacting through a central p o
tential V r dep ending only on the relative distance r jr r j The lagrangian
for this system is
m m
r r V r L
De ning as usual the total mass M m m and the reduced mass
m m M the lagrangian is separable in the center of mass co ordinates de ned
by
M R m r m r r r r
In terms of R r the expression b ecomes
M
R r V r L
As the center of mass moves like a free particle of mass M it is convenient to
work in the rest frame of the center of mass R Then we are only left with
the problem of describing the relative motion of the masses describ ed by the
three co ordinates represented by the vector r
Now supp ose that the p otential V r has a minimum for some separation
r r and rises steeply for all values of r near this minimum Then the rst
excited vibration state of the molecule has an energy well ab ove the ground
state and at low temp eratures the only degrees of freedom that play a role in
the dynamics are rotations of the molecule at xed interatomic distance r Thus
the distance is frozen out as one of the dynamical degrees of freedom and we are
e ectively left with only two co ordinates the angles describing the relative
orientation of the two masses the p otential V r V r is constant and may
b e ignored This leaves as the e ective action of the system
L sin r
ef f
This is the lagrangian of a particle of mass moving on a spherical surface of
radius r In mathematical physicists parlance the e ective low energy degrees
of freedom are sometimes called the mo dular parameters or mo duli of the
system and the sphere representing the e ective low energy con guration space
is called the mo duli space
Up to an overall scale factor r the lagrangian L represents an action of
ef f
the form in a dimensional space with euclidean signature describ ed by
the co ordinates x x and metric
g
ij
sin
As the geo desics on a sphere are great circles the great circles provide the general
solution to the equations of motion The spherical symmetry implies that the
system is invariant under dimensional rotations rotations ab out each of three
indep endent b o dy axes of the sphere and hence there are three Killing vectors
generating these rotations The corresp onding constants of motion are the
comp onents of angular momentum L L L L given explicitly by
L sin p cot cos p
L cos p cot sin p
L p
They span the Lie algebra so of the dimensional rotation group
fL L g L
i j ij k k
As the Poisson brackets of L with the hamiltonian vanish the hamiltonian can
dep end only on the Casimir constant of the rotation group i e the total angular
momentum squared L A direct computation shows that up to the constant
p otential the e ective hamiltonian is in fact prop ortional to the Casimir invariant
L
H
ef f
r
Chapter
Dynamics of space time
Classical solutions of the gravitational eld
equations
The Einstein equations describ e the gravitational elds generated by a
sp eci c distribution of material sources as sp eci ed by the energy momentum
tensor T In two or three dimensions these elds carry at b est a nite number
of physical degrees of freedom related to the sources and or the top ology In
four or higher dimensional space time the gravitational eld is a fully dynamical
system with in nitely many degrees of freedom In particular dynamical gravi
tational elds can exist in the absence of material sources in empty space time
In the geometrical interpretation of general relativity such elds represent the
intrinsic dynamics of space time itself
In eq the curvature tensor was de ned in terms of the parallel transp ort
of a vector around a closed curve A manifold is called at if after parallel
transp ort around an arbitrary closed curve in the manifold the image of any
vector coincides with the original Therefore a necessary and su cient condition
for a manifold to b e at near a given p oint is that the comp onents of the Riemann
tensor R vanish there This condition is more general than to require that
the metric takes the form
n
X
dx ds dx
i
i
with the sign dep ending on the euclidean or lorentzian signature of the manifold
Indeed in any di erentiable manifold one can always nd a lo cal co ordinate
system in the neighborho o d of a given p oint in which the line element can b e
diagonalized to this form but this is not generally the case globally Even in
cases in which it can b e done globally eq is not the most general solution
of the Einstein equations for at empty space time b ecause it is still allowed to
p erform an arbitrary lo cal change of co ordinates in this metric The atness
criterion introduced ab ove amounting to the vanishing of the Riemann curvature
is more useful as it holds in any co ordinate system and in any top ology
The existence of non trivial solutions of the Einstein equations with vanishing
cosmological constant in matter free regions of space time is p ossible b ecause
vanishing of the Ricci tensor do es not imply a vanishing of all comp onents of the
Riemann tensor A formal mathematical criterion for distinguishing solutions
of the Einstein equations in empty space from those with material sources is
through the Weyl curvature tensor de ned as the completely traceless part of
the Riemann tensor
W R g R g R g R g R
d
g g g g R
d d
Here d is the dimensionality of the manifold If we similarly de ne W as the
traceless part of the Ricci tensor
g R W W W W R
d
then we obtain a complete decomp osition of the Riemann tensor in terms of
traceless comp onents and the Riemann scalar
R W g W g W g W g W
d
g g g g R
d d
This decomp osition allows a classi cation of the solutions of the Einstein equa
tions according to which of the indep endent comp onents W W R vanish
and which don t In particular metrics for which only the Weyl tensor is non
vanishing are solutions of the source free Einstein equations R On the
other hand there also exist manifolds for which the only non zero comp onent
is the Riemann scalar R whilst the Weyl tensor and traceless Ricci tensor van
and all two dimensional surfaces x x ish the n spheres x
n
For the n spheres this follows essentially from the spherical symmetry which im
plies the absence of any prefered direction in combination with the p ermutation
symmetries b etween the comp onents of the Riemann curvature tensor
For two dimensional manifolds it is always p ossible to write the Riemann
tensor as
R g R g g g g R
For the torsion free riemannian and pseudo riemannian manifolds we consider
in higher dimensions d the number of indep endent comp onents of the
Riemann tensor R taking account of all its symmetries is given by
d d
N R
This is counted as follows the Riemann tensor is anti symmetric in each of the
index pairs and Hence R may b e considered as a square matrix
of dimension n d d Also as an n n matrix it is symmetric under
interchange of these index pairs As such it has n n indep endent elements
Finally the cyclic symmetry in the rst three lower indices of the fully covariant
comp onents of the Riemann tensor expressed by the Bianchi identity adds
d d d d algebraic constraints eliminating an equal number of
indep endent comp onents Combining these results one nds precisely the number
N R in eq ab ove
For d the number of comp onents of the traceless Ricci tensor and Riemann
scalar is
d d
N W N R
for d the Riemann curvature tensor has only one indep endent comp onent
corresp onding to the Riemann scalar Therefore for d the number of inde
p endent comp onents of the Weyl tensor is
d d d d N W
Clearly this vanishes for d and is p ositive non zero only for d Thus
we nd indeed that non trivial dynamical solutions to the source free Einstein
equations can exist only in space times of dimensions d As the minimal
case d happ ens to b e the number of dimensions of our macroscopic world
we observe that for this case the Weyl tensor and the Ricci tensor each have
algebraically indep endent comp onents Therefore half of the comp onents of the
curvature are determined by the material sources whilst the other half describ es
the intrinsic dynamics of space time
In the geometrical interpretation of the gravitational eld and its dynamics
Einstein s famous principle of the equivalence of accelerations and lo cal gravita
tional elds can b e formulated as the hypothesis that space time is a manifold in
each p oint of which one can establish a lo cally at cartesian co ordinate system
of the form Of course a fully geometric interpretation of all of physics
and also the equivalence principle breaks down in those p oints or regions where
material sources are lo cated But also solutions of the source free Einstein equa
tions can violate the equivalence principle lo cally if there are singularities for
example p oints where curvature invariants b ecome in nite and or the number of
dimensions changes Physics at these p oints is no longer describ ed at least not
completely by the Einstein equations and space time is not necessarily a mani
fold in the strict sense In some cases it can still b e a manifold with b oundaries
where sp eci c b oundary conditions hold but these b oundary conditions then
represent external data and the gravitational eld is not determined by general
relativity alone
Nevertheless singular solutions are imp ortant b ecause they are relevant for
physics as we observe it in the universe cosmological solutions of the Friedmann
Robinson Walker type with a Big Bang singularity and black hole solutions
describing stellar collapse The singularities in these cases are not necessarily
physical as the short distance physics may b e regulated by Planck scale mo d
i cations of the theory which are presently unobservable However that may
b e here we simply accept the p ossibility of singularities and discuss examples
of b oth non singular and singular geometries arising as solutions of the Einstein
equations They illustrate many interesting asp ects of gravitational physics
Plane fronted waves
To show that in four dimensions non trivial gravitational elds can indeed prop
agate through empty space time we now consider a class of solutions known in
the literature as plane fronted gravitational waves This name derives from
the prop erty that there is a co ordinate system in which these eld con gurations
i move with the sp eed of light along a straight line ii have a at planar wave
front and iii are of nite duration When discussing the physical prop erties of
these solutions it will b ecome clear that these wave like solutions actually b ehave
more like dynamical domain walls
We construct a planar wave solution with the direction of motion along the
x axis and the transverse directions de ned by the y z plane Thus we lo ok
for solutions of the Einstein equations of the form
B C
B C
g x t B C
A
f x t
g x t
The corresp onding space time line element is
1
Attempts to explain matter as a manifestation of physics in higher dimensional space
time as in Kaluza Klein mo dels or the more sophisticated string theories aim to redress this situation
ds c dt dx f x t dy g x t dz
We observe the following prop erties of these metrics
in the x t plane space time is at Minkowskian
in the y and z directions space time is generally not at but the value of the
metric dep ends only on the distance along the x axis and on time not on the val
ues of y or z themselves Hence in the co ordinate system chosen the comp onents
of these gravitational p otentials are the same everywhere in the y z plane In
particular they do not fall o to unity at spatial in nity showing that the space
time is not asymptotically Minkowskian at large space like distances Of course
they share this prop erty with plane electromagnetic waves which at a xed time
also have the same amplitude everywhere in the transverse directions
the metric has no inverse if f or g then the connections are singular
We must take sp ecial care when this happ ens
With the Ansatz for the metric one can compute the comp onents of the
connection using eq Using the notation x ct x y z for
the non vanishing comp onents have b een assembled in the following list
f
t
f f
t
cf c
g
t
g g
t
cg c
f
x
f f
x
f
g
x
g g
x
g
The next step in solving the Einstein equations is to compute the comp onents of
the Riemann curvature tensor using eq The only indep endent covariant
g not identically zero are the following ones comp onents R R
f f R g g R
tt tt
c c
R f f R g g
xx xx
R f f R g g
tx tx
c c
R f g f g f g
x x t t
c
Before discussing the general solution of the Einstein equations for this class of
metrics we rst use the ab ove results to answer the question which of the metrics
singled out by the Ansatz describ e at Minkowski space If we require all
comp onents of the curvature tensor to vanish indep endently the most
general solution for f x t and g x t is
f x t a px r ct
g x t b q x l ct
with the additional constraint
pq r l
Thus metrics of the form and corresp onding line elements describ e
a at space time if f and g are linear functions of x and t with the additional
constraint b etween their co e cients
The comp onents of the Ricci tensor R now follow by taking a covariant trace
as in eqs The non vanishing ones are
g f
tt tt
R
c f g
f g
tx tx
R R
c f g
f g
xx xx
R
f g
f
R f f f f g f g
xx tt x x t t
c g c
g
f g g f g R g g
x x tt t t xx
c f c
In the absence of material sources Einstein s equations require the Ricci tensor
to vanish Assuming f and g themselves not to vanish identically some manipu
lation of these equations then leads to the following ve necessary and su cient conditions
f f
xx tt
c
g g
tt xx
c
f g f g
x x t t
c
g f f g
xx xx
g f f g
tx tx
The rst two equations are free wave equations in two dimensional x t space
They imply that f and g can b e decomp osed into left and right moving waves
f x t f x ct f x ct
g x t g x ct g x ct
where f u g u and f v g v are functions only of the single lightcone
variable u x ct and v x ct resp ectively Each of these comp onents
then satis es the wave equation by itself and at this stage they can b e taken to
b e completely indep endent The only ambiguity here is that a p ossible constant
term can b e divided in an arbitrary way b etween the left and right moving
comp onents of the solution
If we now substitute this decomp osition of f and g in the third equation
and assume that the rst derivatives of the comp onents with resp ect to
u or v i e f and g are all non zero then the only solutions of the Einstein
equations corresp ond to at Minkowski space This can b e seen by dividing the
equation by g g to obtain
f f
g g
where on the right hand side of the last equality is a nite non zero constant
This last equality holds b ecause the two previous expressions are functions of
di erent variables hence they can b e identically equal only if they are constant
It then follows that
g f g f
Inserting these results into the last two equations addition and subtraction
gives the equivalent conditions
g f g g f g
The solution g g implies f f This leads to linear dep endence
of f and g on x and t they are of the form and corresp ond to at
Minkowski space
On the other hand in case one of the second derivatives g is non zero one of
the co e cients g f must vanish note that they cannot vanish simultaneously
for non zero f and g Hence eq gives either
f g and g f
or
f g and g f
But combining this with the results we nd again that all second deriva
tives must vanish and that the geometry is that of at space time
We conclude that in order to get non trivial solutions at least one of the rst
must vanish The last three equations then imply or g derivatives f
that after absorbing any constants terms in the remaining comp onents the only
solutions are
f x t f x ct g x t g x ct
or
f x t f x ct g x t g x ct
with the additional condition
g f
f g
Although f and g represent waves traveling either to the right or to the left at the
velocity of light the non linear character of the Einstein equations exempli ed by
the additional condition do not allow us to make arbitrary sup erp ositions
of these solutions Interestingly viewed as a two dimensional theory in x t
space f and g represent chiral b osons Wherever necessary in the following we
chose to work with right moving elds f x t f u and g x t g u for
de niteness
An example of a simple solution of eq is
f x t cos k x ct cos k u
g x t cosh k x ct cosh k u
Note that at u n k the comp onent f has a zero whilst the solution for
g x t grows inde nitely with time or distance However this do es neither imply that the geometry is singular nor that the gravitational elds b ecome arbitrarily
strong the apparent problems turn out to b e co ordinate artifacts This p oint is
discussed in detail in the next section
Let us consider a wave which reaches x at time t and extends to
x L i e the solution holds for L u If jk Lj the metric
comp onents f and g are strictly p ositive For u space time is at we can
choose a co ordinate system in which f g Note that the wave solution
assumes these same values at u whilst the rst derivatives match as
well As a result the solutions are smo othly connected and satisfy the Einstein
equations even at the b oundary u
For u L space time is again at but now we have to use the general
solution
f x t a p x ct
x ct L
g x t b q x ct
The co e cients a b p q in this domain of x t values are to b e determined
again by requiring the metric to b e continuous and di erentiable at the b oundary
u L b etween the di erent domains of the solution This condition is su cient
to guarantee that the Einstein equations are well de ned and satis ed everywhere
Imp osing these requirements we nd
cos k L a pL k sin k L p
cosh k L b q L k sinh k L q
For f and g in the domain x ct L we obtain the result
f x t cos k L k L x ct sin k L
g x t cosh k L k L x ct sinh k L
Conclusion we have constructed a solution of the Einstein equations in empty
space which interpolates b etween two Minkowski half spaces connected by a
nite region of non zero curvature where the gravitational elds have non zero
gradients and test particles are accelerated with resp ect to the initial inertial
frame After passage of the wave the test particles have acquired a nite but
constant velocity This shows that they have b een accelerated but the accelera
tion has ceased after t x L c
The plane fronted wave constructed here is a well de ned solution of the equation
R everywhere including the wave fronts x ct and x ct L as the
contributions of the metric comp onents g f and g g always comp ensate
y y z z
each other in eq However the Riemann curvature itself which coincides
with the Weyl curvature in this case do es not vanish everywhere Indeed we
nd from eq
R R R f f
R R R g g
R
Inserting the explicit solution for f and g for each of the three regions x ct
L x ct and x ct L one nds that the curvature is zero in the two
Minkowski half spaces b efore and after the wave as exp ected but nite and non
zero in the interior of the wave Indeed the second derivatives f and g have a
nite discontinuity on the wave front of magnitude k at x ct and k cos k L
resp k cosh k L at x ct L This nite discontinuity then also app ears in the
curvature comp onents where it must b e attributed to a nite discontinuity in
the Weyl curvature and thus can not b e attributed to a singular distribution of
sources But the Riemann Christo el connections which dep end only on the rst
derivatives of the metric are continuous Then the geo desics and the motions
of test particles in the space time describ ed by the wave are continuous and it
follows that the jump in the Weyl curvature presents no physical inconsistencies
Nature of the space time
The planar wave constructed ab ove can b e characterised as a region of non
vanishing curvature sandwiched b etween two at space times and propagating
with the sp eed of light either in the p ositive or the negative x direction such
that one of the two at space time regions grows at the exp ense of the other
In this section I establish the relation b etween the two at space times b efore
and after the wave The analysis shows that they are not related by a simple
Lorentz transformation a set of equidistant test particles carrying synchronised
ideal clo cks and thus de ning an inertial co ordinate system b efore the wave
are neither equidistant nor synchronised in any inertial co ordinate system after
the wave Therefore the two Minkowski space times on either side of the region
of non zero curvature are inequivalent in the sense that the Lorentz group of
rigid transformations leaving the inpro duct of four vectors invariant in these at
regions are embedded di erently in the group of lo cal co ordinate transformations
which constitutes the symmetry of the full theory
This situation is also encountered in scalar eld theories with broken sym
metry when there is a b oundary separating two domains with di erent vacuum
exp ectation values of the eld One can even show that in some mo dels these
two forms of sp ontaneous symmetry breaking of a scalar eld and of the grav
itational eld are connected Therefore we nd a new interpretation for
the planar gravitational waves they act as domain b oundaries separating di er
ent inequivalent at space solutions vacua of the Einstein equations These
domain b oundaries are dynamical they are not static but move at the sp eed of
light for this reason they are sometimes refered to as sho ck waves However
the planar wave solutions discussed here have nite extent whereas usually the
term sho ck wave is reserved for waves with a delta function pro le
Regions of space time enclosed by such dynamical b oundaries clearly grow
or shrink at the sp eed of light If we consider our planar wave solution as an
approximation to a more general domain wall geometry with a radius of curvature
large compared to the characteristic scale of the region of space time we are
interested in then we can use it to describ e the dynamics of gravitational domain
structure which may have interesting implications for example for the physics of
the early universe
Having sketched this general picture I now turn to describ e the sandwich
structure of the planar wave geometry more precisely A convenient pro cedure
is to introduce new co ordinates X cT X Y Z related to the original x
ct x y z by
c T ct
X x
Y f y
Z g z
where in terms of the original co ordinates x
x y f f z g g
This co ordinate transformation takes an even simpler form when expressed in
light come variables
U X cT x ct u
V X cT x ct v
Indeed the line element now b ecomes
ds c dt dx f dy g dz
dU dV K U Y Z dU d Y dZ
with
f g g
Z Y K U Y Z Y Z
f g g
Both for t x c and t x L c we have f g and therefore the line
element is manifestly at Minkowskian on b oth sides of the planar wave
However inside our particular wave solution one has
g f
k
f g
Then K k Z Y indep endent of U The co ordinate system X with
the line element is the closest one to an inertial frame at all times for an
observer at rest in the origin one can construct Indeed the p oint x y z
corresp onds to X Y Z at all times and b oth b efore and after the
wave the X de ne a global inertial frame Furthermore inside the wave the
new co ordinate system deviates from an inertial frame only for large values of
Z Y We also observe that in this co ordinate system the line element
never b ecomes singular showing the previous zero s of det g corresp onding to
zero s of f u to b e co ordinate artifacts indeed
Computing the various geometrical quantities metric connections and cur
vature comp onents in the new co ordinates pro duces the following summary of
results
The only non vanishing comp onents of the inverse metric are
UV VU VV
g g g k Y Z
YY ZZ
g g
The only non vanishing comp onents of the connection are
Y V V
k Y k Y
UU YU UY
V V Z
k Z k Z
UZ ZU UU
The only non vanishing covariant comp onents of the curvature are
R R k
UYUY UZUZ
Thus in these co ordinates the curvature comp onents are constant inside the wave
and we conclude that our particular planar wave solution is actually like a blo ck
wave at the b oundaries U U L the curvature comp onents jump from
zero to the values and back to zero whilst in b etween they are constant
This makes it easy to visualize how our particular solution can b e used to
construct general planar wave solutions with p olarization along the Y and Z
axes essentially it amounts to glueing together su ciently narrow blo ck waves
of xed curvature to obtain solutions in which the curvature comp onents vary
according to some arbitrary pro le
The result also allows one to verify once more by a one line computation
that the Ricci tensor vanishes identically for these planar wave geometries
With these results in hand our next aim is to establish the relation b etween the
at half spaces b efore and after the wave We have already observed that the
origin x y z b ecomes the origin X Y Z for all times t or
T resp ectively Next observe that the co ordinate transformation is not
a Lorentz transformation the co ordinates t x y z de ning an inertial frame
b efore the passage of the wave represent a non inertial frame after passage As
a check note that the jacobian of the transformation is
X
f g for u x ct
x
Indeed a system of synchronized equidistant clo cks at rest in the rst co ordinate
system which op erationally de nes an inertial frame b efore the wave arrives is
neither synchronized nor equidistant in the nal inertial frame Moreover in
general it cannot b e made so by a simple Lorentz transformation the clo cks have
b een accelerated with resp ect to each other by the wave and ultimately de ne
a non inertial frame at late times This is veri ed by the explicit calculation of
geo desics in the next section
From this analysis it will b e clear that in principle it is p ossible to deter
mine the passage and measure the characteristics of a planar gravitational wave
by measuring the acceleration it imparts to test masses in the lab oratory In
deed such measurements can determine the functions f g from the relation
b etween t x y z and T X Y Z Although in this form the statement holds
only for waves of the type discussed the principles for other types of waves are
the same even if the functions characterizing the relation b etween the initial and
nal Minkowski spaces are generally di erent
Scattering of test particles
In order to study the resp onse of test masses to passing gravitational waves
it is necessary to solve for the geo desics of space time in the gravitational eld
generated by the wave We are particularly interested in the worldlines of particles
initially at rest in a sp eci c inertial frame the lab oratory Observing frame gives
the scattering data from which the prop erties of the gravitational wave can then
b e inferred relatively easily
In principle it is very easy to nd geo desics they corresp ond to the world
lines of particles in free fall Consider therefore a particle at rest in the initial
Minkowski frame its four co ordinates are
x c
where the three vector xing the p osition of the particle is constant
note also that by de nition for a particle at rest in an inertial frame the co
ordinate time equals the prop er time
In the co ordinate system we use a particle with co ordinates is actually
always in free fall its co ordinates in the x frame do not change if it enters a
non trivial gravitational eld Of course in an arbitrary gravitational eld the
x frame is not an inertial frame and even if at late times space time is at
again the particle will have acquired a nite generally non vanishing four velocity
However its p osition in the nal Minkowski frame can b e found straightforwardly
by applying the co ordinate transformation
Let us rst con rm the ab ove statements by a simple mathematical argument
Then we turn to apply the result to nd the actual motion of the test particle
in the nal Minkowski frame We b egin by noting that in the x frame the
metric for the gravitational wave is always of the form and therefore the
only non vanishing comp onents of the connection are those given in eq In
always vanish particular in the x frame the comp onents
On the other hand for a particle with the worldline the four velocity
is purely time like and constant
dx
u c
d
Then the four acceleration in this frame vanishes identically as do es the gravita
tional force
u u
As a result the geo desic equation is satis ed proving that represents
the worldline of a particle at al l times
What we are particularly interested in is the p osition and velocity of the parti
cle at the moment it comes out of the gravitational wave It has already b een
remarked that the origin remains at rest and coincides with the origin
of the nal Minkowski frame Therefore at late times the origin has lab oratory
co ordinates
X c
Let us compare this with the four co ordinates of a particle away from the origin
in the y z plane For such a particle the values of the metric
comp onents at late times are
f a pc
c L
g b q c
Here a b p q have the values quoted b efore eq For one nds in the
same domain c L
c ap bq p q
As a result the Minkowski co o dinates of this particle at late prop er times are
X cT X Y Z
with
c
c T ap p bq q
c
X ap bq p q
Y a p c
Z b q c
Note that clo cks which were originally synchronized same value of t at
early prop er time are no longer so after passage of the gravitational wave T is
not equal to T at late prop er time Before we can calculate the p osition of
the test particle at synchronized co ordinate time T in the nal Minkowski
frame we rst have to correct for this gravitationally induced p osition dep endent
time shift
To determine the instantaneous prop er distance b etween the particle with
worldine X and the origin we have to ask at what prop er time the time
co ordinate T equals the time T measured by a clo ck at rest in the origin
T
with the result
c ap bq
c
p q
The p osition at this prop er time X X can now b e evaluated
q c p bq ap
X
p q
aq bp q a pc
Y
p q
aq bp p b q c
Z
p q
Finally we can also determine the velocity of the test particle
v
p q p q
c p q
and as a result the kinetic energy is
p q p q
M v M c
p q
From this analysis we draw the conclusion that not only in general test particles
are accelerated by the gravitational wave but also that the acceleration and
hence the nal velocity dep end on the p osition of the particle Therefore no
Lorentz transformation with any xed b o ost velocity v can bring all test particles
initially at rest w r t one another back to rest simultaneously after passage of
the wave This result explicitly demonstrates the inequivalence of the initial and
nal Minkowski space times
Symmetry breaking as a source of gravita
tional waves
The planar gravitational wave solutions describ ed in this chapter arise as transi
tion regions b etween at domains of space time which are related by a co ordi
nate transformation that cannot b e reduced to a linear Lorentz transformation
We have therefore interpreted them as domain b oundaries b etween regions of
space time trapp ed in inequivalent classical vacua of the gravitational eld in
the absence of a cosmological constant This is p ossible b ecause at space zero
curvature is asso ciated with a value of the gravitational eld the metric g
which in an appropriate co ordinate system gauge is constant non vanishing
and Lorentz invariant but not invariant under general co ordinate transforma
tions In this sense the general co ordinate invariance of the Einstein equations
is broken sp ontaneously by the vacuum solutions corresp onding to Minkowski
space time
A domain structure arising from the presence of degenerate but inequivalent
classical vacua is a well known consequence of sp ontaneous symmetry breaking in
scalar eld theories In this section this p oint is elab orated on and it is shown that
dynamical domain b oundaries in scalar eld theories scalar type sho ck waves
if one prefers can actually b e sources of planar gravitational waves of the kind
analysed in detail ab ove
The simplest scalar eld theory with a continuously degenerate set of classical
vacua is that of a complex scalar eld with a mexican hat type of p otential as
describ ed by the lagrangian
p
L g
g
In this lagrangian h c is a dimensionless coupling constant and is an inverse
length h c is a mass In the natural system of units in whichh c we can
take to b e dimensionless and to represent a mass In the following equations
one then can substitute c everywhere
i
This lagrangian is invariant under rigid U phase shifts e
with constant As a result for there is an in nitely degenerate set
of solutions of the equations of motion minimizing the p otential refered to as
classical vacua of the form
i
p
e
with an arbitrary real constant Clearly the solutions themselves are not in
variant under the U transformations the classical vacuum solutions break the
U symmetry sp ontaneously However b ecause of the U symmetry of the
dynamics these solutions are physically indistinguishable and the absolute value
of is unobservable On the other hand variations in the vacuum angle b etween
di erent regions of space time can have observable e ects
p
i x
Parametrizing the complex scalar eld as x x e the lagrangian
b ecomes
p
L g g
g
This shows that in the sp ontaneously broken mo de of the U symmetry the
q
obtained theory desrib es actually two real scalar elds the eld
q
by shifting by its classical vacuum value with a mass or inverse co
herence length equal to which can b ecome arbitrarily large and the strictly
q
representing the Goldstone excitations arising from massless eld
sp ontaneous symmetry breaking
Clearly at energies small compared to the scale set by the mass of the eld
the excitations of this massive eld play no role and one only has to consider
the Goldstone eld describ ed by the e ective lagrangian
c
p
L R g
g G
where we have added the Einstein Hilb ert action for the gravitational eld as
this also describ es a low energy degree of freedom Note that with the p otential
de ned in eqs there is no cosmological constant in the presence of
q
the vacuum value of the scalar eld or equivalently
The eld equations derived from this lagrangian are the Klein Gordon equa
tion for the Goldstone eld
p
cov
p g g 2
g
and the Einstein equations with a source term from the energy momentum of the
Goldstone eld
G
R
c
Remarkably this system of equations admits exact planar wave solutions similar
to the free sourceless Einstein equations Using the same Ansatz for
the metric we nd running planar gravitational waves f x ct g x ct
in combination with a scalar wave x ct running in the same direction
The continuity conditions require that this scalar wave interpolates continuously
b etween a domain in which b efore the wave and a domain where
after the wave where b oth are constants
The simplest solution of this kind is a pro le of interpolating linearly b e
tween these values
u for u
q
u k u for L u
q
for u L u k L
with similar expressions for the left moving solutions v It follows that the
slop e parameter k in is given by
s
k
L
At this p oint it is useful to recall that actually represents the phase of the
complex scalar eld Hence with a xed value of the mo dulus a monotonic
linear increase of implies a pure mono chromatic oscillation of the real and
imaginary comp onents of the scalar eld
ik u ik x ct
p
u e e
where the last equality assumes to equal its classical vacuum value and the
gauge choice It follows that the slop e parameter k actually represents
the wavenumber of these large oscillations of interpolating b etween di erent
classical vacuum values
It remains to solve the Einstein equations For the right moving wave
solution u f u g u these equations reduce to a slightly mo di ed ver
sion of eqs the only mo di cation b eing a constant source term in eq
g G f
for L u k
f g c
Outside the region L u the right hand side vanishes and we have the
trivial at space solutions f g
It is not very di cult to establish the existence of solutions to equation
In fact there are more solutions than the purely gravitational case b ecause rst
there are the planar gravitational blo ck waves of the type
f u cos u g u cosh u
with the constraint b etween the parameters
G
k
c
These waves have a purely gravitational contribution from a non vanishing Weyl
tensor But in addition there are also purely oscillatory solutions
f u cos u g u cos u
with a relation b etween the wavenumbers of the type
G
k
c
It is quite clear that these purely oscillatory waves require a non vanishing of
the Ricci tensor they can exist only in the presence of the scalar eld u
b ecause in the limit k one nds simultaneously that b oth This is
in contrast to the hyperb olic type of solution where k only implies
Therefore one might think of the rst type of solution as radiative
waves which have kept the quadrup ole character known from the propagation of
small p erturbations in the linearized version of gravitational eld theory whereas
the second type are matter induced waves with a dip ole like b ehaviour
By p erforming the co ordinate transformation we can again construct a
reference frame which is manifestly Minkowskian b efore and after the wave and
has constant curvature comp onents inside the wave This is based on the choice of
the simple solution for the scalar eld with constant slop e parameter k i e
a mono chromatic wave As in the case of the purely gravitational waves more
general solutions with smo oth b ehaviour at the wavefront can b e contructed
The main lesson to b e learnt from the results in this section is that the in
terpretation of the gravitational waves as moving domain b oundaries b etween
inequivalent classical vacua has its parallel in the case of scalar elds with bro
ken symmetry and that these scalar waves are actually accompanied by planar gravitational waves of the type constructed ab ove
Coupling to the electro magnetic eld
The coupling of planar gravitational waves to scalar elds of the Goldstone type
is the result of the existence of eld con gurations with constant slop e in do
main b oundaries for which the energy momentum tensor has constant non zero
comp onents in half of the directions of space time A similar type of energy
momentum density can b e generated by constant electric and magnetic elds
and again one nds that these are accompanied by gravitational waves
We consider the simplest case of constant electric and magnetic eld p erp en
dicular to each other for example in the y z plane
E E B B
and a metric which is non at only in the z z direction
ds c dt dx dy g dz
These elds provide a solution of the combined Einstein and Maxwell equations
G G
T F F g F R
mu
c c
and
D F
provided the electric and magnetic eld are constant and equal mo dulo c in
magnitude
E cB constant
whilst the metric comp onent g represents a planar wave
z z
G
g x ct cos k x ct k E
c
Here the sign in the argument i e left or right moving wave dep ends on the
sign in
It is straightforward to check that this is a solution of the Einstein equation
by substituting the g as ab ove and f in eq and writing out the energy
momentum tensor of the electro magnetic eld
T
E c B cB E
B C
B C
B C
B C
cB E E c B
B C
B C
B c C
B C
B C
E c B
B C
B C
A
2
g
E c B
Observe that the metric comp onent g itself b ecomes
z z
cos k x ct g g
z z
ans oscillates with frequency
q
E
G k c
c
It may seem surprising and worrying that constant electric and magnetic eld
generate gravitational waves of such large amplitudes however note that in
lab oratory units the frequency is
E
Hz
V m
For ordinary electric elds in the lab oratory this means that it takes a full lifetime
of the universe or longer to go through one complete cycle Even at the high
est elds imaginable in theory when the vacuum breaks down due to electron
p ositron pair creation
m c
e
E V m
cr it
e
e
where e is the electron charge m its mass and its Compton wave length the
e e
frequency can not get higher than ab out Hz Therefore the e ect will not b e
measurable under ordinary circumstances
Nevertheless it do es seem remarkable that constant electro magnetic elds
generate time dep endent oscillating gravitational elds This e ect as well
as the scalar generated waves discussed in the previous section has quite close
similarities to the Josephson e ect in sup erconductors where a constant p otential
generates an oscillating current In that case the current is related to the change
in the phase of the electron condensate which is mathematically analogous to
the phase of the complex scalar eld represented by the Goldstone mo de
Chapter
Black holes
Horizons
The example of gravitational sho ck waves domain walls has shown that space
time in four dimensions is a truely dynamical arena for physics even in the absence
of matter although adding matter for example in the form of scalar or elec
tromagnetic elds makes the p ossible structures even richer In this chapter
we turn to another class of very interesting structures in four dimensional space
time static or quasi static solutions of the Einstein equations which represent
gravitating extended b o dies whose static elds b ecome strong enough to cap
ture p ermanently anything that gets su ciently close to the central core As
these quasi static ob jects even capture light they are called black holes but it
has turned out that their prop erties are much more interesting and p eculiar than
this ominous but dull sounding name suggests
Black holes p ossess two characteristic features The rst which has given
them their name is the existence of a horizon a closed surface through which
ob jects can fall without ever b eing able to return The second is the presence of
a singularity a lo cus of p oints inside the horizon where the curvature b ecomes
in nite It has even b een conjectured by some authors that these two prop erties
are so closely linked that any space time singularity should always b e hidden
b ehind a horizon This conjecture b ears the name of the Cosmic Censorship
hypothesis Although as an unquali ed mathematical theorem the hypothesis
is certainly not correct it could b e true for singularities that arise in realistic
pro cesses from the gravitational collapse of macroscopic b o dies like implo ding
massive stars
In this chapter we restrict our attention to the sp ecial black hole solutions that
might b e called classical eternal black holes in the sense that at least outside
the horizon they describ e stationary gravitational elds and do not require for
the description of their b ehaviour the inclusion of their formation from collapsing
matter In fact they are solutions of the source free Einstein or coupled Einstein
Maxwell equations and are characterised for an outside observer completely in
terms of their mass their angular momentum and their electro magnetic charge
One might alternatively think of them as black holes that have b een formed in the
in nite past and settled in a stationary state in which all trace of their previous
history has b een lost e g by emitting gravitational radiation
Because of this restriction these lectures do not include a detailed account
of the pro cess of gravitational collapse as studied by astrophysicists but they
do allow one to isolate and study in detail the sp ecial asp ects of the physics of
black holes It may even b e that these very sp ecial simple solutions are relevant to
elementary particle physics near the Planck scale but that leads one to consider
the problems of quantum gravity a sub ject we will not deal with at length in
these lectures The interested reader can nd more information on many asp ects
of black hole structure and formation in the literature
The Schwarzschild solution
The simplest of all stationary black hole solutions of the source free Einstein
equations is that for the static spherically symmetric space time asymptotically
at at spatial in nity rst describ ed in the literature by Schwarzschild We
present a derivation of this solution making full use of the symmetries and their
connection to Killing vectors explained in chapter
In view of the spherical symmetry we introduce p olar co ordinates r
in three dimensional space and add a time co ordinate t measuring asymptotic
Minkowski time at r The Schwarzschild solution is obtained by requiring
that there exists at least one co ordinate system parametrized like this in which
the following two conditions hold
i the metric comp onents are t indep endent
ii the line element is invariant under the three dimensional rotation group
SO acting on three vectors r in the standard linear way
The rst requirement is of course equivalent to invariance of the metric under
time shifts t t t Thus formulated b oth conditions take the form of a
symmetry requirement
In chapter we discussed how symmetries of the metric are related to Killing
vectors Rotations of three vectors are generated by the di erential op erators L
of orbital angular momentum
L sin cot cos L cos cot sin
L
1 The solution was found indep endently at the same time in
Similarly time shifts are generated by the op erator
L
t
According to the discussion in sect the co e cients of the symmetry op era
tors L L are to b e comp onents of a set of Killing vectors L A
A A
A
D L D L
A A
These can equivalently b e characterised as a set of constants of geo desic ow as
in eq
J L p
A
A
The relation with the metric is that the Poisson brackets of these constants of
geo desic ow with the hamiltonian are to vanish
fJ H g H g p p
A
The general solution of these conditions for the hamiltonian H is t indep endent
whilst it can dep end on the angular co ordinates only through the Casimir invari
ant
p
L p
sin
It follows that the metric must b e of the form
ds h r dt g r dr k r d
where d d sin d is the angular distance on the unit sphere and
the co e cients h r g r and k r are functions of the radial co ordinate r only
Indeed the hamiltonian derived from this metric is
H p p p p
t r
h g k
sin
L
p p
t r
h g k
Note that the set of functions h g k can not b e unique as one can always
p erform a co ordinate transformation r r r to another metric of the same
general form with co e cients h g k There are several standard options
to remove this freedom to b egin with we choose one in which the spherical
2
In this chapter we employ natural units with c
symmetry is manifest and the spatial geometry is as close to that of at space as
p ossible That is we take r as the solution of
k r r g r
Then the line element b ecomes
h r dt g r dr ds h r dt g r dr r d
Clearly it is rotation invariant and the three dimensional space like part of the
line element is conformal to that of at space in p olar co ordinates with the
conformal factor g r dep ending only on the radius and not on the orientation
or the time In the literature the co ordinates chosen here to parametrize the
Schwarzschild metric are called isotropic co ordinates
Up to the freedom of radial reparametrizations the form of the line element
is completely determined by the symmetry requirements To x the radial
dep endence of the remaining co e cients h r and g r we nally substitute the
metric into the Einstein equations For the Einstein tensor G R g R
we nd that it is diagonal with comp onents
h hg
G h g g
tt
g r g
hg
h g hg G
r r
g h g r
G hg r
h g hg G hg
g h r g
sin
In the absence of sources or a cosmological constant these expressions have to van
ish As b oundary conditions at in nity where space time must b ecome asymp
totically at we require that r implies h and g Then we have
three di erential equations for two unknown functions and therefore the equa
tions must b e degenerate One can check that this is the case if the following
two relations are satis ed
h i
hg g r g r hg
Indeed in this case the various comp onents of G b ecome prop ortional
g
G G G G
r r tt
h r
r sin
The conditions are met for any g r of the form
b c g r g
g r a h r
r r g
It turns out that eqs indeed provide solutions of the Einstein equations
satisfying the correct b oundary conditions They are most easily obtained from
the rst Einstein equation for hg it is p ossible to factor out all
h dep endence and obtain an equation for g only
g g g g g
r
There is a unique solution with the required normalization for r
m
g r
r
where m is an undetermined constant of integration Eq then immediately
provides us with the solution for h
r m
h r
r m
The complete solution for the static spherically symmetric space time in isotropic
co ordinates therefore is
m r m
dr r d dt ds
r m r
Clearly this solution matches Minkowski space at r by construction and h
and g change monotonically decreasing and increasing resp ectively as a func
tion of r untill r m At that value h and the metric has a zero mo de
Then the derivation of the solution we have presented breaks down and its con
tinuation to values r m requires a careful interpretation discussed b elow
In any case it turns out that this singularity of the metric is not physical as
invariants of the curvature remain nite and well de ned there Indeed in the
following we construct co ordinate systems in which the metric is p erfectly regular
at these p oints and can b e continued to regions inside the surface r m In
the isotropic co ordinate system the apparent singularity of the metric for r is
another co ordinate artifact It disapp ears if one brings spatial in nity to a nite
distance by an appropriate co ordinate transformation Details are given in the
next section
Discussion
Mo dulo radial reparametrizations r r r the line element is the most
general static and spherically symmetric solution of the source free Einstein equa
tions matching at Minkowski space time at spatial in nity r Therefore
in the non relativistic weak eld range v r m it should repro duce
Newton s law for the gravitational eld of a spherically symmetric p oint mass of
magnitude M according to which the acceleration of a test particle at distance
r is
d r GM
r
dt r
As the spherically symmetric solutions of the Einstein equations are dis
tinguished only by a parameter m in the Newtonian limit this parameter must
b e related to the mass M thereby representing the equivalent gravitational mass
of the ob ject describ ed by the spherically symmetric metric
We rst show that our solutions repro duce Newtons law in the limit r m
v c The equation of motion for a test particle is the geo desic equation
i
For the spatial co ordinates x this b ecomes
i
d x dx
i
terms O v c
d d
h m h
i
h x
i
g r g
Note that in the limit v one has dx dt d whilst for r the factor
h g Up on the identi cation m GM we thus indeed repro duce Newton s
equation
Taking into account the dimensions of Newton s constant as follows from
and is expressed in eq and our natural units in which c we nd
that m has the dimension of a length consistent with the observation that m r
should b e dimensionless We have already noted that in isotropic co ordinates the
metric comp onent h vanishes at half this distance r r m Presently this
only signals that the isotropic system cannot b e extended b eyond this range but
the surface r r do es have a sp ecial physical signi cance it de nes the lo cation
of the horizon of a spherically symmetric black hole This characteristic value
r m of the radial co ordinate is known as the Schwarzschild radius
For the range m r the Schwarzschild solution in the form is
well de ned and it matches the asymptotic Minkowski metric for large values of
r We noted that on the Schwarzschild sphere r m the metric is singular
but as a solution of the Einstein equations it can b e continued to smaller values
of r and then it is again well de ned in the domain r m However it
turns out that this domain of r values do es not in any sense represent the physical
interior of the horizon as one might exp ect naively Rather it easy to establish
that the solution is invariant under the transformation
m
r
r
which maps the domain m to m Therefore the isotropic co ordinate
system actually represents a double cover of the region b etween the Schwarzschild
radius and the asymptotic region r Note that under this transformation
h r h r which can b e comp ensated by an accompanying time reversal
t t From these observations it follows that the apparent singularity for
r disapp ears as it simply describ es the another asymptotic region like that
for r in the original co ordinate system
The existence of such a co ordinate system presenting a double cover of the
space outside the black hole might lead one to susp ect that a test particle thrown
in radially towards the spherical Schwarzschild surface after a long enough time
would return to an asymptotic Minkowski region However this do es not happ en
in the rst place as a simple calculation to b e presented later shows in terms
of the time measured by an observer at asymptotic in nity it takes an in nite
p erio d of time for a test particle starting at any nite radial distance r m
with any nite velocity even to reach the Schwarzschild surface certainly such
an observer will never see it start up and set o towards in nity again Apart
from this leaving the Schwarzschild surface would also take an in nite amount of
asymptotic Minkowski time Clearly these e ects are due to the vanishing of h r
on the horizon which implies that an in nite amount of asymptotic Minkowski
time t has to pass during any nite p erio d of prop er time But this is only
the minor part of the argument more imp ortant is the second reason to wit
that time like geo desics do not ow from values r m to values r m
Instead as will b ecome clear they ow into a new region of space time describ ed
by complex values of r
Some calculational details are given later but the ab ove arguments at least
indicate that an observer at spatial in nity is essentially disconnected from the do
main of space time b eyond the Schwarzschild sphere Therefore the Schwarzschild
surface is called a horizon from spatial in nity one cannot lo ok b eyond it In
contrast things are radically di erent for an observer moving with the test par
ticle towards the horizon For such an observer in free fall co moving with the
test particle on a radial geo desic and starting from an arbitrary p oint at a
nite distance from the Schwarzschild surface only a nite amount of prop er time
passes until the radius r m is reached although for large radial distances
the required time grows linearly with the distance
Moreover having reached the horizon the particle and the observer can
continue falling inwards without noticing anything particular at least initially
The only remarkable e ect they could discover once they have passed b eyond
the Schwarzschild surface is the imp ossibility to turn back to spatial in nity
no matter how p owerful the engines they have at their disp osal to accelerate
and prop el them Therefore their passage to the inside of the Schwarzschild
sphere represents an irreversible event they are p ermanently trapp ed inside the
Schwarzschild sphere For this reason the Schwarzschild space time is called a
black hole nothing not even radiation can get out once it has fallen through
the horizon at least not in the domain of classical relativity
To describ e this motion of test particles i e the ow of geo desics through
the Schwarzschild sphere one must pass to a di erent co ordinate system one
b etter suited to particles in free fall rather than to the description of the space
time from the p oint of view of an asymptotic Minkowskian observer at rest at
spatial in nity We make this passage in two steps First we show that in other
co ordinate systems there is a di erent continuation of space time b eyond the
Schwarzschild surface one in which the interior geometry inside the horizon is
distinct from the outside geometry and can not b e mapp ed back isometrically to
the exterior After that we show by explicit calculation of the geo desics that this
alternative continuation can b e taken to decrib e the true physical situation for a
test particle falling through the Schwarzschild sphere
The interior of the Schwarzschild sphere
To go b eyond the region of space time covered by the isotropic co ordinates we
p erform a radial co ordinate transformation which instead of making the spatial
part of the metric conformal to at space as in makes the angular
part directly isomorphic to the two sphere of radius r
r r g r k r r
Then the metric takes the form
ds h r dt g r dr r d
The explicit form of the radial transformation gives for the co e cients
m
h r
g r r
This leads to the standard form of the Schwarzschild solution as presented in the
original pap ers
m
dt ds dr r d
m
r
r
As to lowest order the transformation do es not include a rescaling we have
for large r and r
O
r r r
Therefore in the asymptotic region the Newtonian equation of motion for a test
particle in the eld of a p oint mass M still holds in the same form after replacing
r by r and in the non relativistic long distance limit the two co ordinate systems
are indistinguishable
Note that the Schwarzschild radius r m is now lo cated at r m Again
the metric comp onents are singular there but now there is a zero mo de in the
tt comp onent and an in nity in the r r comp onent in such a way that the
determinant of the metric remains nite on the horizon On the other hand
the determinant g do es have a real zero for r This metric singularity was
not present in the isotropic co ordinate system as it did not cover this region of
space time but it turns out to represent a real physical singularity the curvature
b ecomes in nite there in a co ordinate indep endent way
A p eculiarity of the metric is that the comp onents h and g are not
p ositive de nite but change sign at the Schwarzschild radius This implies that
the role of time and radial distance are interchanged for r m r is the time
co ordinate and t the radial co ordinate At the same time the metric comp onents
have b ecome explicitly time dep endent they are functions of r and therefore
the metric is no longer static inside the horizon In return the metric is now
invariant under radial shifts t t t Hence it is the radial momentum p
t
which now commutes with the Hamiltonian the generator of prop er time
translations and is conserved
This strange result is the price one has to pay for continuing the Schwarzschild
geometry to the interior of the Schwarzschild sphere It is quite clear that this
could never heve b een achieved in the isotropic co ordinate system at least
not for real values of r Together with the app earance of the curvature singularity
atr this shows b eyond doubt that the region inside the Schwarzschild sphere
r m has no counter part in the standard isotropic co ordinate system
To gain a b etter understanding of the di erence b etween the continuation to
r m in isotropic co ordinates and r m in Schwarzschild co ordinates it is
useful to p erform yet another radial co ordinate transformation to a dimensionless
co ordinate de ned by
tanh h r
with the result that
r r g r m cosh
Then the Schwarzschild metric b ecomes
ds tanh dt m cosh d d
As cosh this expression for the metric like the isotropic co ordinate sys
tem is valid only outside the horizon r m Moreover continuing to negative
values of we reobtain the double cover of the exterior of the Schwarzschild black
hole
m
tanh h r h
r
Now the interior of the Schwarzschild sphere can also b e reparametrized in a
similar way taking account of the fact that h has b ecome negative
tan h r r m cos m cos
This parametrization clearly exists only for r m In these co ordinates
the metric takes the form
d d dt m cos ds tan
The time like nature of now is obvious It is also clear that this co ordinate
system presents an in nitely rep eated periodic covering of the interior of the
Schwarzschild sphere the fundamental domain may b e chosen to b e
corresp onding to m r but continuation of b eyond this region leads to
rep eated covering of the same set of r values
Comparing these two parametrizations of the exterior and the interior of the
Schwarzschild sphere it is now obvious that they are related by analytic contin
uation the interior of the sphere is not describ ed by negative values of
which corresp ond to r m in isotropic co ordinates but by imaginary values
of
i
which is like a Wick rotation changing the space like radial co ordinate to the
time like co ordinate In terms of the original isotropic co ordinates it can b e
describ ed as the analytic continuation of r to complex values
m
i
r e
Thus in the interior of the Schwarzschild surface the mo dulus of r is constant but
its phase changes whilst outside its phase is constant and the mo dulus changes
Note that whereas the p erio d of the fundamental domain of in describing the
interior region of the Schwarzschild sphere is the values of the isotropic radial
co ordinate r are p erio dic in with p erio d Again it seems that the isotropic
co ordinates cover the Schwarzschild solution twice We have more to say ab out
that in the following Finally we observe that as indicated in eq there are
two ways to p erform the analytic continuation from to i e from the exterior
to the interior of the Schwarzschild space time Also this fact has non trivial
signi cance for the geometry of the Schwarzschild space time
Geo desics
The analysis presented so far makes it clear that the geometry of the Schwarzschild
space time is quite intricate Indeed at the Schwarzschild sphere r m or
r m and resp ectively several branches of the space time meet
two branches of the exterior connected to an asymptotic Minkowski region and
two branches of the interior connecting the horizon with the space time singu
larity It is therefore of imp ortance to understand the ow of geo desics in the
neighborho o d of the horizon and see which branches test particles moving in
the Schwarzschild space time follow We b egin with time like geo desics repre
senting the motion of test particles of unit mass For such a particle one has in
asymptotic Minkowski space
H g p p
This relation was already argued from the de nition of prop er time in sect
eq and b elow As H is conserved the relation holds everywhere on the
particle s worldline
The solution of the geo desic equations for Schwarzschild space time is much
simpli ed by the large number of its symmetries They provide us with su ciently
many additional constants of motion to allow a complete solution of the equations
of motion of test particles in terms of their energy and angular momentum The
rst of these constants of motion is the momentum comp onent p conserved
t
b ecause the metric is t indep endent
dt dt
p h r
t
d d h r
At space like in nity h also with de ned as prop er time dt d for a
particle at rest there Therefore a particle at rest at r has equal to the
unit rest mass of the test particle moreover as is intuitively obvious and argued
more precisely b elow if it is not at rest it must have Indeed particles with
cannot reach spatial in nity and in this sense they are in b ound states
The factor gives the usual sp ecial relativistic kinematical time dilation for a
moving particle in asymptotic Minkowski space
In contrast for nite radial distance r r the factor h r describ es the
gravitational redshift the time dilation due only to the presence of a gravitational
eld It represents a universal e ect not in uenced by the state of motion of the
particle
The conservation of angular momentum gives us three conservation laws
which are however not indep endent
J sin p cot cos p
J cos p cot sin p
J p
with J J J L a constant Here the separate momentum comp o
nents are
d d
p k r sin p k r
d d
If one orients the co ordinate system such that the angular momentum is in the
x direction then constant and therefore p and p J
Then the angular co ordinates as a function of prop er time are given by
d
d k r
It remains to solve for the radial co ordinate r For this we use the conservation
of the hamiltonian setting H
dr
p g g
r
d h k
Note that consistency of this equation in the region outside the horizon requires
h k
It shows why in the region r the time dilation factor has to b e greater
or equal to unity At the same time for non zero values of angular
momentum the radial distance r can in general not b ecome arbitrary low like
in the ordinary Kepler problem there is a centrifugal barrier to b e overcome For
details we refer to the literature
Combining eq with that for the angle by eliminating we directly
obtain the orbital equation
s
dr k
d g h k
It is clear that an orbit can have a p oint of closest approach or furthest distance
only if dr d dr d The only physical solutions are p oints where relation
b ecomes an equality When the equality holds identically for all p oints of
the orbit the orbit is circular One can show that the smallest radius allowed for a
3 Note that each of these equations is still true for arbitrary radial parametrization
stable circular orbit corresp onds in Schwarzschild co ordinates tor m m
To the set of time like tra jectories we add the light like geo desics representing
the orbits of massless particles In that case the prop er time interval vanishes
identically on any geo desic that is to say H We parametrize the tra jectories
with an a ne parameter and obtain
dt
p h
t
d
allowing one to replace the a ne parameter by t through
h
dt d
One can again choose the plane of motion to b e and the conservation
of angular momentum holds in the form
r d d
r
h dt d
where is a constant The radial equation now b ecomes
d d dr
k k sin h g
dt dt dt
or using the choice ab ove
dr h k
g h
dt r
Like in the massive case if one can eliminate the a ne parameter also
in favour of instead of t
An interesting case as concerns the exploration of the horizon and the connection
b etween the interior and the exterior of the Schwarzschild sphere is that of radial
motion or resp ectively Let us rst consider the time like geo desic
for a particle starting from rest near r which implies and dr dt
Then
p
dr dt
h
d h d g h
It is easy to solve these equations in the Schwarzschild co ordinates indeed with
h m r one nds a solution valid in the whole range r b oth
outside and inside the horizon
2
3
r
m m
where is the prop er time at which the particle reaches the singularity atr
Taking this time as xed denotes the prop er time prior to at which the
particle is at radial distance r Observe that this time is nite for any nite
radial distancer only when starting from in nity it takes an in nite prop er time
to reach the singularity Nothing sp ecial happ ens at the horizon r m it is
crossed at prop er time
m
H
a prop er p erio d m b efore the singularity is reached of course in terms of the
time co ordinate r inside the horizon it takes a nite p erio d m
Solving for t as a function of prop er time we have to distinguish two cases
r m and r m Namely introducing the new variable
1
3
y
m
we obtain the di erential equation
y dt
m d y
the solution of which dep ends on whether the r h s is p ositive or negative We
nd for the two cases the solutions
1
3
1
r m arccoth
3
t t
m
1
3
m m m
r m arctanh
m
In b oth cases the left hand side b ecomes in nite at the horizon i e at prop er
time This con rms that an external observer at in nity will never actually
H
see the particle reach the horizon Similarly the distance measured from the
horizon to the singularity at r is in nite in terms of the co ordinate t
Now consider what happ ens in isotropic co ordinates In terms of a new vari
able
s
r
x
m
where we choose the p ositive branch of the square ro ot integration of the equation
for r is elementary and gives
x
m x
Clearly this is symmetric under interchange x x corresp onding to inversion
w r t the horizon r m Indeed b oth at r and r we see that
This con rms our earlier remark that a particle released from space
like in nity do es not return there Instead it will cross the horizon x at
as in the previous calculation Then continuing r to complex values
H
we nd
i
2
x e cos
m
The singularity is then reached for as exp ected In terms of the
radial co ordinate related by the pseudo Wick rotation the equation for
the radial distance in terms of prop er time outside the horizon b ecomes
cosh
m
The analytic continuation of co ordinates in the complex plane on crossing the
horizon is manifest
Next we consider incoming light like radial geo desics The equation is
h dr
dt g
Let t refer to the time at which the massless particle e g a photon passes a
xed p oint on the radius e g r in Schwarzschild co ordinates Then the solution
of the geo desic equation for t is
t t r r r m
ln
m m r m
With the absolute value as argument of the logarithm this equation is valid b oth
outside and inside the horizon The time for the photon to reach the horizon
from any p oint outside diverges as do es the distance it has to move from horizon
to the curvature singularity as measured by the co ordinate t
Extended Schwarzschild geometry
As we have already noticed that for a massive particle falling across the horizon
nothing very sp ecial happ ens it is unsatisfactory that we do not have a non
singular description of the same pro cess for massless particles To improve the
description of light like geo desics at the end of the last section it is not su
cient to p erform a radial co ordinate transformation one has to eliminate the
t co ordinate and replace it by one which do es not b ecome in nite on the horizon
Moreover to describ e light like geo desics it would b e convenient to have a metric
where the light cone structure is manifest by having the radial part of the met
ric conformal to radial Minkowski space a co ordinate system in which r t are
replaced by co ordinates u v such that the radial part of the metric b ecomes
h r dt g r dr f u v du dv
Such a co ordinate system was rst prop osed by Kruskal and Szekeres de ning
the new co ordinates such as to b e dimensionless the line element is written in
the form
ds
g u v d f u v dv du
m
with the standard choice for u v de ned in terms of the Schwarzschild co ordi
nates r t by
t
tanh if jv j juj
m
r r u
2m
u v e
m v
t
if jv j juj coth
m
In these co ordinates the horizon r m t is lo cated at u v whilst
the curvature singularityr corresp onds to the two branches of the hyperb ola
u v one in the past and one in the future of an observer at space like
in nity More generally from the rst result one concludes that the surfaces r
constant are mapp ed to hyperb ola s u v constant where the last constant is
p ositive outside and negative inside the horizon These hyperb ola s are worldlines
of particles in circular orbit i e b eing sub ject to a constant acceleration
Similarly hypersurfaces t constant corresp ond to u constant v repre
sented by straight lines in a u v diagram Note also that the singularity r
consists of a set of two space like surfaces in u v co ordinates
More explicitly we can write u v in terms of r t as
s
r t r
4m
u e cosh
m m
s
t r r
4m
v sinh e
m m
outside the horizon r m and
s
r
r t
4m
u e sinh
m m
s
r t r
4m
e cosh v
m m
within the horizon r m We note once again that these co ordinates pro duce
two solutions for each region of the space time geometry In the ab ove conventions
the dimensionless co e cient functions are given implicitly by
r
m r
2m
f u v e g u v
r m
We can also compare with the co ordinate systems and describing
explicitly the double cover of the Schwarzschild space time
1 t 1 t
2 2
cosh cosh
2 2 2 2
u e cosh v e sinh sinh sinh
m m
for the exterior region jv j juj and
1 t 1 t
2 2
cos cos
2 2 2 2
u e sinh v e cosh sin sin
m m
for the interior region jv j juj From these expressions we infer that the exterior
regions and corresp onding to r m and r m in isotropic
co ordinates and the two interior regions and as well coincide with
the regions u v and u v in Kruskal Szekeres co ordinates resp ectively
Hence the double cover of Schwarzschild space time given by the two solutions for
the Kruskal Szekeres co ordinates ab ove corresp ond exactly to the double cover
we found in isotropic and co ordinates
The Kruskal Szekeres co ordinates were introduced and are esp ecially useful
b ecause they represent the light like geo desics in a very simple way In particular
the radial light like geo desic are given by
du
du dv
dv
These corresp ond to straight lines in the u v diagram parallel to the diagonals
The sp ecial light rays u v are the asymptotes of the singularity and describ e
the horizon itself they can b e interpreted as photons p ermanently hovering on
the horizon Any other light like radial geo desic eventually hits the singularity
either in the past or in the future at nite u v although this is in the region
b efore t or after t in terms of the time of an observer at spatial in nity
As the light cones have a particularly simple representation in the u v
diagram for all regions inside and outside the horizon it follows that time like
and space like geo desics are distinguished by having their derivatives jdu dv j
for the time like case and jdu dv j for the space like ones One can check
by explicit computation that a particle starting from rest at any distance r and
falling into the black hole has a time like world line everywhere also inside the
horizon du dv can change on the world line of a particle but never from time
like to space like or vice versa Therefore all time like geo desics must cross the
horizon light like and the singularity space like so oner or later If they don t
reach or start from spatial in nity they must come from the past singularity
across the past horizon and then cross the future horizon to travel towards the
nal singularity all within a nite amount of prop er time
The Kruskal Szekeres co ordinate system also teaches us more ab out the top ol
ogy of the Schwarzschild geometry how the various regions of the space time are
connected First of all it shows that there is a past and a future singularity inside
a past and a future horizon juj jv j dep ending on v b eing p ositive or negative
In this resp ect it is helpful to observe that according to eq for r m the
sign of v is xed for all t In the literature these regions are sometimes refered to
as the black hole the interior of the horizon containting the future singularity and
the white hole containing the past singularity In a standard convention these
regions of space time are classi ed as regions I I future Schwarzschild sphere and
IV past Schwarzschild sphere resp ectively
There are also two exterior regions connected to asymptotic Minkowski space
They are distinguished similarly by u and u as forr m the sign of u is
xed for all t one cannot reach negative u starting from p ositive u or vice versa
by any time like or light like geo desic In the same standard convention these
regions are refered to as region I and region I I I resp ectively Being connected by
space like geo desics only the two sheets of the exterior of the Schwarzschild space
time are not in causal contact and cannot physically communicate However the
surfaces t constant do extend from u to u indeed the regions
are space like connected at the lo cus u v where the two branches of the
horizon intersect To understand the top ology of Schwarzschild space time we
must analyze this connection in some more detail this can b e done using the
technique of embedding surfaces as explained e g in ref
Consider a surface t constant and x a plane We wish to un
derstand how the regions u and u are connected for this plane The
Schwarzschild line element restricted to this plane is
dr
ds r d
m
r
Now we embed this two dimensional surface in a three dimensional at euclidean space
ds dx dy dz
by taking x r cos y r sin and de ning the surface z F r such that
ds F r dr r d
Comparison of eqs and then implies
z r F r
m m m
This de nes a surface of revolution obtained by taking the parab ola z mx
m in the x z plane and rotating it around the z axis It is a single surface
with lo oks like a throat of which the upp er half z corresp onds to
s
r
or u
m
whilst the lower half z describ es the branch
s
r
or u
m
The two regions are connected along the circle r m at z Of course the
spherical symmetry implies that the same top ology would b e observed for any
choice of angle any plane with u is connected to one with u in this way
Although no time like travel is p ossible classically b etween the regions on each
side of the throat the p ossibility of particles tunneling via quantum pro cesses
b etween the two regions p oses a curious and interesting question to theories of
quantum gravity
Charged black holes
The spherically symmetric solution of the source free Einstein equations can b e
extended to a solution of the coupled Einstein Maxwell equations including a
static spherically symmetric electric eld in the rest frame of the black hole
As the only solution of this type in at Minkowski space is the Coulomb eld we
exp ect a solution of the coupled equations to approach the Coulomb p otential at
space like in nity
q
A A dx dt
r p r
with p r for r This is a solution of the Maxwell equation
D F
in a spherically symmetric static space time geometry The Maxwell eld strength
app earing in this equations is
q p r p
F dx dx dt dr F
r p
In isotropic co ordinates the energy momentum tensor of this electro magnetic
eld reads
q p r p h
g F T F F F diag r r sin
r h p g
This has the same general structure as the Einstein tensor
hg h
G h g diag r r sin hg
g h g r g
which was derived under the conditions with the general solution
Now requiring the coupled Einstein Maxwell equations
G G T F
this reduces to a single condition on the radial functions h g p with the solution
m
p r g r
r r
r g r g
h r
g r mr
Here the constant is de ned by
q G
m e e
It may b e checked that this solution satis es the Maxwell equations in this grav
itational elds as well As g r can also b e written in the form
m e
g r
r r
we nd that the Schwarzschild solution is reobtained in the limit e For
the zero of the metric co e cient h is now shifted to
r
As in the Schwarzschild case this is a p oint where two asymptotically at regions
of space time meet the metric is again invariant under the transformation
r t t
r
under which h r dt h r dt Moreover also like in the Schwarzschild geometry
the sphere r actually de nes a horizon and time like geo desics are contin
ued into the interior of this horizon which is not covered by the real isotropic
co ordinate system
A Schwarzschild like co ordinate system which allows continuation into the
interior of the horizon exists It was found by Reissner and Nordstrom and
is related to the isotropic co ordinates by a radial co ordinate transformation
r r g r r m
r
Then the metric of the charged spherically symmetric black hole takes the form
ds h dt g dr r d
where the co e cients h g are again inversely related
m m e
h r
g r r r r r
The horizon is now seen to b e lo cated at r m in fact there is a second
horizon at r m provided m e As in general h and g are not p osi
tive de nite at each of these horizons the role of radial and time co ordinate is
interchanged Only for i e for m e the situation is di erent as the two
horizons coincide and no interchange of time and space like co ordinates o ccurs
This sp ecial case of Reissner Nordstrom solution is refered to in the literature as
an extremal charged black hole
The result of the second horizon for non extremal charged black holes is
that the real curvature singularity that is found to exist at r is a time
like singularity for m e This is exp ected from the Coulomb solution in at
Minkowski space which contains a time like singularity of the electric eld at
r Therefore such a black hole lo oks more like a p oint particle than the
Schwarzschild type of solution of the Einstein equations
The interior region b etween the horizons can b e obtained from the isotropic
co ordinates by complex analytic continuation of r from the outer horizon by
i
e r
The domain of the argument is chosen as Indeed for these values of r
the radial Reissner Nordstrom co ordinate is
r m cos
and takes all values b etween the two horizons r m precisely once In this
parametrization the isotropic Reissner Nordstrom metric b ecomes
sin
ds d d dt m cos
m cos
Again we observe the role of as the time like variable with t space like b etween
the horizons For the isotropic co ordinate r b ecomes negative
r
Thus we discover that the Reissner Nordstrom space time can b e extended con
sistently to negative values of the radial co ordinate In particular the curvature
singularity at r is seen to b e mapp ed to
m e
r
This shows rst of all that there are two time like branches of the singularity
and secondly that r can b ecome negative for non extremal black holes with m e
The two branches are found also up on analytic continuation of in to
imaginary values
i
This leads to a parametrization of the Reissner Nordstrom geometry for the ex
terior regions of the form
sinh
ds d d dt m cosh
m cosh
where the plus sign holds for the exterior region connected to asymptotic at
Minkowski space and the minus sign to the region connected to the curvature
singularity Because of the double valuedness of the analytic continuation
there are two copies of each of these exterior regions involved
Spinning black holes
The Schwarzschild and Reissner Nordstrom black holes are static and spheri
cally symmetric w r t an asymptotic Minkowski frame at spatial in nity There also exist extensions of these solutions of Einstein and coupled Einstein Maxwell
equations which are neither static nor spherically symmetric but represent ro
tating black holes with or without charge These solutions have an axial
symmetry around the axis of rotation and are stationary in the exterior region
the metric is time indep endent and the rate of rotation and all other observable
prop erties of the black hole are constant in time
The standard choice of co ordinates for these metrics with the axis of rotation
b eing taken as the z axis and reducing to those of Schwarzschild or Reissner and
Nordstrom for the spherically symmetric case of no rotation is that of Boyer
and Lindquist Denoting these co ordinates by t r to emphasize their
relation with the spherical co ordinate systems for the non rotating metrics the
line element for reads
r a cos
ds dr d dt a sin d
r a cos
r a sin a
d dt
r a cos r a
Here a is a constant parametrizing the deviation of this line element from the
usual diagonal form of the Schwarzschild Reissner Nordstrom metric its physical
interpretation is that of the total angular momentum p er unit of mass
J ma
As one can always choose co ordinates such that the angular momentum p oints
along the p ositive z axis the angular momentum p er unit of mass may b e taken
p ositive a For convenience of discussion this will b e assumed in the follow
ing
The quantity in the line element is short hand for
r m a e m
Note that like the metric co e cients h and g we have encountered earlier
this quantity is not p ositive de nite everywhere Also the zero s of this function
determine the lo cation of the horizons of which there are two an outer and an
inner one for m a e
p
m a e r m
H
The gravitational eld represented by the metric is accompanied by an
electro magnetic eld the vector p otential of which can b e chosen as the one
form
q r
A dt a sin d
For convenience of notation we have here introduced the function de ned as
r a cos
These elds are solutions of the coupled Einstein Maxwell equations and
with the Maxwell eld strength given by the two form
q
r a cos dr dt a sin d F
r cos sin q a
d adt r a d
and with the identi cation of the quantity e as in eq
q G
e
To determine the prop erties of the space time describ ed by the Kerr Newman line
element we need to analyse the geo desic ow Because the spherical symmetry
of the Schwarzschild and Reissner Nordstrom solutions is absent and only axial
symmetry remains there is one less indep endent constant of motion from angular
momentum only the comp onent along the axis of rotation the z axis say is
preserved Fortunately this is comp ensated by the app earance of a new constant
of motion quadratic in momenta which replaces the Casimir invariant of total
angular momentum As a result one can still completely solve the geo desic
equations in terms of rst integrals of motion We now give some details
First we write down the geo desic hamiltonian e ectively representing the
inverse metric g
r a p a p H p a sin p p p
t t
r
sin
Clearly to b e well de ned we must assume as the z axis represents
a co ordinate singularity in the Boyer Lindquist system where the angle is not
well de ned The ab ove hamiltonian is a constant of geo desic ow recall that for
time like geo desics it takes the value H whilst for light like ones H
Next there are two Killing vectors respresenting invariance of the geometry
under time shifts and rotations around the z axis
dt d
p g g
t tt t
d d
d dt
g p g
t
d d
Of course for time like geo desics the a ne parameter maybe taken to b e prop er
time These equations can b e viewed as establishing a linear relation b etween
b etween the constants of motion and the velocities dt d d d
t
G
with the dot denoting a derivative w r t the a ne parameter The determinant
of this linear transformation is
det G g g g sin
tt
t
and changes sign whenever do es Therefore the zero s of de ne the lo cus
of p oints where one of the eigenvalues of G changes sign from a time like to
a space like variable or vice versa at the same time the co e cient of p in
r
the hamiltonian changes sign con rming that these values represent the
horizons of the Kerr Newman space time It follows from this analysis that
outside the outer horizon in the region connected to space like in nity det G
Now if from eqs the angular prop er velocity d d is eliminated one
nds the relation
det G dt
g d
where the quantity with the dimensions of an angular velocity is de ned by
r a g
t
a
g r a a sin
a
r a r a r a
Observe that it is prop ortional to a with a constant of prop ortionality which
for large but nite r is p ositive Hence represents an angular velocity with
the same orientation as the angular momentum J With a canonical choice of
co ordinate system this quantity is p ositive and therefore as well
It also follows from eq that for time like geo desics outside the horizon
with and therefore dt d an inequality holds of the form
For large r near in nity one nds in fact the stronger inequality
Returning to the solution of the geo desic equations we take into account the
existence of a constant of motion K such that K K p p with K the
contravariant comp onents of a Killing tensor
D K
The explicit expression for K is
p r sin
K ap a cos p r p
t
r
sin
a cos
r a p ap
t
That K de nes a constant of geo desic ow is most easily established by checking
that its Poisson bracket with the hamiltonian H vanishes
fK H g
Using these constants of motion it is straightforward to establish the solutions
for time like geo desics in the form
dt r a r a a sin
a
d
a sin d r a
a
d
sin
dr
a r a K r
d
d
K a cos a sin
d
a sin
Note that for a particle starting from r one has These solutions are
well de ned everywhere except on the horizons and at p oints where At
the latter lo cus the curvature invariant R R b ecomes in nite and there is
a real physical singularity At the horizons b oth t and change in nitely fast
signalling in nite growth of these co ordinates w r t an observer in asymptotic
at Minkowski space
As with the in nite amount of co ordinate time t to reach the horizon in the
Schwarzschild and Reissner Nordstrom solutions this merely signi es a co ordi
nate singularity However here it is not only t but also which b ecomes in nite
To gain a b etter understanding of the reason for this rotation of geo desics rst
notice that according to the second equation even for the prop er
angular velocity d d do es not vanish in general b ecause Hence there are
no purely radial geo desics any incoming test particle picks up a rotation Now
computing the angular velocity as measured by an observer at rest at spatial
in nity from the rst two equations one nds
d g g
t tt
dt g g
t
It follows immediately that for a geo desic without orbital angular momentum
the observed angular velocity equals the sp ecial reference value
d
dt
At the horizon r r where this b ecomes
H
a d
H
dt r a
H
r r
+H
Comparison with the expression for eq shows that outside the horizon
the inequality
H
holds with equality only on the horizon As these quantities are nite the
time t can increase near the horizon at an in nite rate only if increases at a
prop ortional rate For this reason b oth t and have to grow without b ound as
r approaches r
H
At space like in nity the Killing vectors de ne time translations and
plane rotations However their nature changes if one approaches the horizon
Consider the time translation generated by p The corresp onding Killing
t
vector has contravariant and covariant comp onents given by
g
t t
t t
is the usual Kronecker delta symbol Similarly the comp onents of the Here
Killing vector for plane rotations generated by p are
g
It follows that these Killing vectors have norms given by
a sin
g
t tt
and
r a a sin
g sin
whilst their inner pro duct is
r a
g a sin
t t
From these equations several observations follow First as the norms and in
ner pro ducts of Killing vectors are co ordinate indep endent quantities we can
write the expressions for the reference angular velocity eq and for the
observed angular velocity eq in a co ordinate indep endent form
d
t t
t
dt
t
The last equality is equivalent to
d
t t
dt
Second with a and excluding the z axis where the Boyer
Lindquist co ordinates are not well de ned and vanishes and the invariant
inner pro duct are p ositive outside the horizon Also the p ositivity of det G
t
in the exterior region eq can b e expressed in the invariant form
t t
with the equality holding only on the horizon Then for we nd from
eq that d dt whilst for the angular velocity is less than this
reference value d dt As long as is time like this is all that can b e
t
established in fact with and the absolute value of the angular
t
velocity d dt can b ecome arbitrarily large
However it is easily established that the norm of changes sign on the surface
t
p
r r m m e a cos
which with the exception of the p oints where the surface cuts the z axis
lies outside the horizon The region b etween this surface and the horizon
It then follows directly is called the ergosphere and is space like there
t
t
from eqs and that in this region the angular velocity is always
p ositive
d
for
t
dt
Thus we have established that inside the ergosphere any test particle is dragged
along with the rotation of the black hole and no free falling observer can b e non
rotating there Related to this b oth and are space like Killing vectors there
t
Nevertheless outside the horizon there still exists time like combinations of these vectors even inside the ergosphere
t
which has non p ositive norm as a result of the inequality with
equality holding only on the horizon r r
H
The Kerr singularity
We continue the analysis of black hole geometry with a brief discussion of the
nature of the singularity of spinning black holes To b egin we have already
noticed that like the spherical charged black hole the spinning Kerr Newman
black holes have two horizons an innner and an outer one As the singularity
is lo cated inside the inner horizon we exp ect the singularity to have a
time like character
In order to elucidate the top ological structure of the space time near the
singularity we introduce new co ordinates in two steps First we de ne the Kerr
co ordinates V as the solution of the di erential equations
r a a
dV dt dr d d dr
With these de nitions the Kerr Newman line element b ecomes
h i
sin
dV a sin d r a d adV ds
dV a sin d dr d
Now we turn from quasi spherical co ordinates to quasi Cartesian ones known as
Kerr Schild co ordinates by de ning
x r cos a sin sin
y r sin a cos sin
z r cos
and we reparametrize the time co ordinate t t using
dt dV dr
We then nd that at xed the transformation
x x x r cos
y y y r sin
z z z
is equivalent to r r r This shows that like the Reissner Nordstrom case
the negative values of r are to b e taken as part of the physical domain of values
i
Next one introduces the co e cients k i by the relation
r xdx y dy a xdy y dx z dz
k dr dt
r a r
and scalar function u r
mr e r a
u r
a z
r
r
With these de nitions the metric can b e written in the hybrid form
ds dx u r k dr
where the rst term on the right hand side is to b e interpreted as in at space
time with x x y z t Therefore u and k parametrize the deviation from
at space time
Having introduced this new parametrization the physical singularity
equivalent to r and is now seen to b e mapp ed to the ring
x y a z
with the solution
x a sin y a cos
A consequence of this structure is that in contrast to the Schwarzschild geometry
geo desic crossing the horizon do not necessarily encounter the singularity but can
pass it by
Black holes and thermo dynamics
When a test particle falls into the ergosphere of a black hole the constant of
motion p b ecomes a momentum comp onent rather than an energy
t
This provides a means of extracting energy from a spinning black hole as rst
prop osed by Penrose if a particle enters the ergosphere with energy this
b ecomes a momentum during its stay in the ergosphere the momentum can b e
changed for example by a decay of the particle into two or more fragments in
such a way that the nal momentum of one of the fragments is larger than the
original momentum If this fragment would again escap e which is allowed as
the ergosphere is lo cated outside the horizon it leaves with a di erent energy
corresp onding to its changed momentum This energy is now larger than the
energy with which the original particle entered By this pro cess it is p ossible
to extract energy from a black hole even in classical physics
It would seem that the Penrose pro cess violates the conservation of energy
if there would b e no limit to the total amount of energy that can b e extracted
this way However it can b e established that such a violation is not p ossible and
that the pro cess is self limiting emission of energy is always accompanied by
a decrease in the angular momentum and or electric charge of the black hole
in such a way that after extraction of a nite amount of energy the black hole
b ecomes spherical After that there is no longer an ergosphere and no energy
extraction by the Penrose pro cess is p ossible
We will now show how this comes ab out for the simplest case of an electrically
neutral black hole of the Kerr type In this case the lo cation of the horizon is at
p p
r m m a m m J
H
m
Then the angular momentum of test particles moving tangentially to the horizon
is
J a
p
H
a r m
m m J
H
This quantity can b e interpreted in terms of the area of the spherical surface
formed by horizon of the black hole
Z Z
a
p
g A d d g r a
H
H
r r
H
H
Eq thus provides an expression for the area of the horizon in terms of the
mass and the angular momentum of the black hole
p
A
H
m m J
Now we observe that
i the area A increases with increasing mass
H
ii at xed area A the mass reaches a minimum for J
H
iii by absorbing infalling test particles including Penrose type pro cesses the
black hole area A never decreases
H
The rst two statements are obvious from eq The last one follows if
we consider the e ect of changes in the mass and angular momentum of the black
hole We assume that the absorption of a test particle with energy and angular
momentum changes a Kerr black hole of total mass m and angular momentum J
to another Kerr black hole of mass m m and angular momentum J J
This assumption is justi ed by the uniqueness of the Kerr solution for stationary
rotating black holes
Now under arbitrary changes m J the corresp onding change in the area
is
p
m m A J J
H
p p
m J m
m J m J
J
p
m J
H
m J
H
For an infalling test particle only sub ject to the gravitational force of the black
hole we have derived the inequality Moreover on the horizon such a par
ticle has b een shown to have the angular velocity w r t a stationary observer
H
at in nity indep endent of the value of Hence with m and J we
obtain
J A
H
p
H
m J
H
Thus we can indeed conclude that by such adiabatic pro cesses adding in nitesi
mal amounts of energy and angular momentum at a time the area of the black
hole never decreases
Casting the relation in the form
A J m
H H
where is given by
p
m J
mA
H
it b ears a remarkable resemblance to the laws of thermo dynamics which states
that changes of state describ ed by equilibrium thermo dynamics are such that for
isolated systems variations in energy U entropy S and angular momentum J are
related by
U T S J
where T is the temp erature and the total angular momentum and moreover
that in such pro cesses one always has S The imp ortant step needed in a
full identi cation of these relations is to asso ciate the area A with the entropy
H
and the quantity called the surface gravity with the temp erature
S c A T
H
c
where c is an unknown constant of prop ortionality Now if black holes have a
nite temp erature they should emit thermal radiation Using a semi classical
approximation it was shown by Hawking that quantum uctuations near
the horizon of a black hole indeed lead to the emission of particles with a thermal
sp ectrum This is true irresp ective of the existence of an ergosphere therefore it
covers the case of vanishing angular momentum as well The temp erature of the
radiation Hawking found was
T
H
This xes the unknown constant ab ove to the value
c
It seems therefore that the thermo dynamical description of black holes is more
than a mere analogy The issue of interpreting the entropy
A
H
S S
where S is an unknown additive constant in statistical terms by counting mi
crostates is hotly debated and has recently received a lot of imp etus from devel
opments in sup erstring theory
The extension of these results to the case of charged black holes is straight
forward The relation b etween horizon area A and angular velocity remains
H H
the as in but in terms of the macroscopic observables m J e asso ciated
with the black hole the area is now
p
A e
H
m m J m e
The changes in these observable quantities by absorbing p ossibly charged test
particles are then related by
A J e m
H H H
where the electric p otential at the horizon is
er
H
H
A
H
and the constant is now de ned as
p
m J m e
mA
H
Again by sending test particles into the black hole the area can never decrease
A In this way the thermo dynamical treatment is generalized to include
H
pro cesses in which the electric charge changes
We conclude this discussion with establishing the gravitational interpretation of
the constant asso ciated with the temp erature in the thermo dynamical descrip
tion This quantity is known as the surface gravity b ecause in the spherically
symmetric case it represents the outward radial acceleration necessary to keep a
test particle hovering on the horizon without actually falling into the black hole
For the case of the Schwarzschild black hole this is easily established De ne
the covariant acceleration as
D u
D
As the four velocity of a test particle has negative unit norm u it follows
that in the four dimensional sense the acceleration is orthogonal to the velocity
u
Also a particle at rest in the frame of an asymptotic inertial observer at in nity
has a velocity four vector of the form
p u
g
tt
Combining these two equations and taking into account the spherical symmetry
a particle at rest must have a purely radial acceleration given by
m
r
g
r tt
r
Remarkably we recover Newtons law It follws in particular that on the horizon
r
H
m
This quantity is actually a prop er scalar invariant as can b e seen from writing it
as
q
t
r
H
with the invariant magnitude of the four acceleration
q
g
For the case of the spinning black holes the prop er generalization of this result
gives the surface gravity as in eq
Chapter
Topological invariants and
self duality
Topology and top ological invariants
Gravity is a gauge theory and as such it has many similarities to Yang Mills the
ories In particular the nature of the gravitational vacuum is complicated by the
p ossibilities of non trivial top ology quantum tunneling and sp ontaneous symme
try breaking In this section we brie y introduce some concepts and notions of
top ology
In the geometrical context the global top ology of a manifold describ es prop er
ties characterising its connectedness like the presence of non contractable lo ops
spheres etc Clearly the way that various parts of a manifold are connected
determines whether it can b e covered by a single globally de ned co ordinate sys
tem or requires a sp eci c minimal number of co ordinate charts glued together
in a smo oth way i e such that in the overlap region of two charts the trans
formation from one co ordinate system to the other is di erentiable Generically
more than one co ordinate chart is necessary as in the case of a sphere in which
conventionally the northern and southern hemisphere are mapp ed separately on
the plane with the maps overlapping in the region of the equator Thus gen
eral manifolds are to b e considered as glued together from a number of smo othly
deformed regions of at space often in a non trivial way
This has imp ortant consequences For example if one computes the integral of
a certain function over the manifold this integral must necessarily b e decomp osed
into a sum of integrals over each co ordinate patch Even in the case in which
the manifold as a whole has no b oundary the various co ordinate patches do have
b oundaries Thus in the application of the generalized Gauss Stokes theorem for
the integral of a total divergence of a vector eld K x over a manifold M
P
M with M the pieces of the manifold which can b e covered with a single
i i
i
co ordinate chart the integral b ecomes a sum of integrals over the b oundaries of
the co ordinate patches
Z Z Z
X X
i i
K r K r K
n
M M M
i i
i i
i i
where K x are the comp onents of the vector eld in the i th co ordinate
patch and de subscript n denotes the comp onent normal to the b oundary M
i
Now if the complete manifold M has no b oundary all the b oundaries M are
i
internal b oundaries and these are integrated over twice with opp osite orientation
Therefore if the integrand is the same in each case the contributions of the
integrals over these internal b oundaries cancel exactly However in the change
i
of co ordinates from one region M to the next the value of the integrand K
i
n
can sometimes change and the cancellation may b e sp oiled Then the non trivial
top ology manifests itself in an apparent violation of the generalized Gauss Stokes
theorem integrals over a total divergence do not have to vanish in the absence of
an external b oundary Of course if K is a smo oth vector eld like the physically
observable electric eld it can not jump on the b oundary M a violation of
i
the Gauss Stokes theorem requires that the vector eld cannot b e smo oth Only
particular kinds of physically relevant vector elds can satisfy this requirement
without leading to physical or mathematical inconsistencies vector elds which
change in a controllable way with the change b etween two co ordinate patches
not leading to observable or measurable consequences Basically such vector elds
must b e gauge elds as these can change by unobservable gauge transformations
on the b oundary of co ordinate regions
Example
We illustrate these general and rather abstract arguments by a simple and well
known example of vector elds on the two sphere The standard co ordinate
system for a two sphere of radius r is the p olar co ordinate system
In these co ordinates the line element on the sphere reads
d sin d ds r
From this expression it can b e seen that the metric is singular it has a zero mo de
in the direction at the north and south p oles This co ordinate
singularity is avoided by covering the upp er and lower hemispheres H corre
sp onding to and separately with a cartesian co ordinate
system x y obtained for example by pro jection on the disc x y r
x r tan cos
for H
y r tan sin
and
cos x r cot
for H
y r cot sin
The minus sign in the expression for y is necessary to have x y form a right
handed co ordinate system on the exterior of the southern hemisphere it could
b e absorb ed in a rede nition Thus the orientation of the cartesian
co ordinates is preserved in region of overlap of the two co ordinate systems which
includes the equator
x x r cos
y y r sin
but which actually can b e extended to the entire sphere with the exclusion of the
north and south p oles by continuing x y outside the disc x y r
In the cartesian co ordinates x y the line element b ecomes
r
ds dy dx
y r x
This expression is manifestly non singular except in the limit x y
corresp onding to one of the p oles in each case Observe that the north p ole is
now a regular p oint of the co ordinate system x y and the south p ole of the
system x y
Now consider a vector eld on the sphere de ned in comp onents by
N
y x A
r x y
where N is a normalization constant This vector eld can b e represented as a
one form
N
A y dx x dy
r x y
N
cos d
The last expression in terms of the p olar co ordinates shows that in the region of
overlap the vector elds A di er by an exact one form
y y
A A N d arctan N d arctan N d
x x
It follows that the two form F dA dA is unique and well de ned on the
entire sphere
F dx dx F
N r N
dx dy sin d d
y r x
The dual of the two form F is de ned to b e the zero form
N
F F
p
g r
with the tensor density representing the two dimensional p ermutation symbol
As this is a constant it is closed dF or in comp onents
F
This shows that we have constructed a non trivial solution of the free Maxwell
equations on the sphere representing a constant magnetic eld of strength B
N r This solution can b e extended trivially to a static spherically symmetric
solution of Maxwell s equations in three dimensions by taking the radial com
p onent of the vector eld to vanish A In three dimensions this solution
r
represents a magnetic monop ole of strength N lo cated in the origin
Note that we can write the gauge transformation connecting the vector elds
A in the region of overlap in terms of the elements g exp iN of a U
eld g x y in the standard form
A g A g g dg
i
For this U eld to b e well de ned in the whole region of overlap of the domains
of A the whole sphere minus the p oles it must return to the same value
after a rotation This is the case only if N is an integer
The number N has a direct top ological interpretation it represents the wind
ing number of the map from the equator of the sphere to the U group man
ifold which is the unit circle in the complex plane Therefore we see that the
interpretation of the vector eld A as a connection for a U bundle a mag
netic vector p otential on the sphere requires the winding number of the gauge
transformation the magnetic charge to b e quantized a well known result rst
obtained by Dirac
1
Note that the equator cannot b e contracted to a p oint on the sphere whilst staying com
pletely inside the region of overlap where the gauge transformation is de ned as in this con traction it must always pass one of the p oles
The example of the vector eld A on the two sphere illustrates how the
naive Stokes theorem can b e violated even though F is a lo cally exact two
form F dA in H well de ned and smo oth over the entire sphere and even
though the sphere has no b oundary the integral of F over the sphere is non zero
This is b ecause F is not an exact two form globally and the vector eld makes a
jump on the b oundary b etween the regions H
Z Z Z
p
d x F F d x g F
2 2 2
S S S
Z Z I
F F A A N
H H C
+
In the line integral the lo op C H H denotes the equator with the
minus sign in front of H deriving from the reversal of the orientation of
This induces the minus sign in front of A in the lo op integral
Finally we observe that a smo oth deformation of the sphere will not change
the value of the integral if M denotes any closed surface which encloses
the magnetic charge and is the volume b etween this surface and the two sphere
S then
Z Z Z
F F F
2
S M
Z
dF
by the second Maxwell equation in the absence of magnetic charge the Bianchi
identity inside Geometrically we can understand this result b ecause the rst
line of eq shows that in comp onents the integral is manifestly indep endent
of the metric Hence a lo cal change of the metric which is equivalent to a lo cal
deformation do es not a ect the value of the integral Such a quantity is called a
top ological invariant
Topological invariants in gravity
In gravity top ology can play a role at various levels At the macroscopic classical
level one can consider multiply connected universes and wormholes whilst at the
microscopic Planck scale space time top ology may b e sub ject to quantum uctu
ations space time foam in analogy with other eld theories like sigma mo dels
and Yang Mills theories it is exp ected that the quantum tunneling pro cesses b e
tween di erent top ologies are dominated by nite action solutions of Euclidean
gravity the gravitational instantons
One way to characterise top ologically non trivial solutions of the gravitational
eld equations is by the value of top ologically invariant integrals over certain
p olynomials of the curvature tensor In four dimensions there are essentially two
indep endent top ological invariants of this type the rst Pontrjagin number and
the Euler number To de ne these we rst introduce the dual and double dual
curvatures with comp onents
R R R R
p
g
p
Here is the p ermutation symbol a tensor density requiring a factor g to
guarantee the transformation character of the dual curvature as a tensor under
lo cal co ordinate transformations The de nition is adapted to manifolds
with euclidean signature but it can easily b e generalized to include the lorentzian
case
The rst Pontrjagin number of a compact manifold M without b oundary
involves the dual curvature and is de ned by the expression
Z
R d x R p M
M
whilst the Euler number is de ned in a similar way using the double dual of the
Riemann tensor
Z
p
M d x g R R
M
For a general pro of that b oth expressions are top ological invariants it is more
convenient to use the formulation of di erential geometry in terms of the vier
b ein and spin connection rather than the metric and the Riemann Christo el
connection
ab
ab ab a b ab
R e e R
In this formulation the traces in the de nition of the Pontrjagin and Euler num
b er are to b e taken over the tangent space i e lo cal SO indices ab and we
can write b oth top ological invariants in the form
Z
c M d x Tr R R
M
where dep ending on the top ological invariant condidered the linear op erator
acting on anti symmetric SO tensors is either the identity or the SO duality
op erator
ab ab ab abcd cd
or R R R R
resp ectively From the expression it is clear that the square ro ot of the
ab
metric has canceled inside the integral the integral dep ends on the the
top ological invariance of c M is now straightforward one proves that the value
of the integral is the same for any two spin connections which can b e smo othly
deformed into each other Let denote the original spin connection and the
deformed one The di erence is a covariant SO tensor We de ne a
path in the space of connections linking and parametrized by a real number
We show that the integral expression is indep endent of
Z
dc M dR
R d x Tr
d d
M
The last result follows b ecause the integrand is a total derivative which is smo oth
and well de ned everywhere if is to check this consider the explicit form of
the deformed curvature tensor R
ab
ab ab
R R D D
Here the covariant derivativer of is constructed in terms of the original unde
formed spin connection Its derivative with resp ect to is then
d
ab
ab
R D D
d
The last term involving the commutator of two s is precisely the one necessary
to complete the covariant derivatives of to covariant derivatives D w r t the
full deformed connection
d
ab
D D R
d
Inserting this in the integral in eq one obtains
Z
dR dc M
d x Tr R
d d
M
Z
h i
d x Tr D R
M
Z
d x Tr R
M
To obtain the last line we have used the Bianchi identity for the SO curvature
ab
D R
which guarantees that the SO covariant derivative can b e taken outside the
trace in the integrand As b oth and the curvature are covariant tensors the
invariant SO traces do not have a discontinuity across the b oundaries b etween
the di erent co ordinate patches and the integrand in eq is a globally de
ned total derivative on M If the manifold has no b oundary the integral then
vanishes thus proving that the original quantity c M is indep endent of the
connection indeed
It is of interest to note that the Pontrjagin density can actually b e written
lo cally as a total covariant divergence
p
Tr RR D J g J
p
P P
g
x is de ned by the Chern Simons form where the current J
P
p
g J Tr
P
Tr
To interpret the middle expression correctly one should regard the Christo el
and the Riemann tensor or its symbols as a set of four matrices
The trace is then taken over a dual as a set of six matrices R
contravariant upp er and a covariant lower space time index In the spin connec
tion the trace is over the lo cal SO tangent space indices This result implies
directly that the only non trivial contributions to the Pontrjagin number come
from the b oundaries of the co ordinate patches
Z
N
X
i i i i i i
Tr d x p M
K
i
i
i i
where the x and denote the co ordinates and connections in the i th co
ordinate chart K That they can come from these b oundaries indeed follows
i
b ecause the current is not covariant only the variation of its divergence is
Therefore we can construct another simple pro of of the top ological invariance of
the Pontrjagin number under a deformation of the metric or the vierb ein the
variation of the current can b e written as
p
g J Tr R Tr
P
or a similar form in terms of the spin connection Because of the contraction with
the tensor the last term do es not contribute to the divergence of the variation
p
But as the rst term is covariant it do es not pro duce a non vanishing g J
P
contribution to the integral by its b oundary terms its b oundary terms cancel
pairwise b etween adjoining patches Therefore the Pontrjagin number is invariant
under smo oth deformations of the manifold
p M
As there is no similar Chern Simons type current for the Euler number this pro of
cannot b e extended to that case
Self dual solutions of the Einstein equations
There is an interesting class of space time geometries for which the two top ological
invariants the Pontrjagin and the Euler number are identical or identical
up to a sign the anti self dual geometries satisfying
R R R
p
g
Because of the symmetry of the fully covariant Riemann tensor R under
interchange of the index pairs and the double dual R then also
equals the dual up to a p ossible sign and the two top ological invariants are
equivalent indeed
The interesting asp ect of these anti self dual geometries is that they are
automatically solutions of the Einstein equations in empty space or space time
dep ending on the signature with zero cosmological constant To see this
construct the Ricci tensor and note that it vanishes due to the Bianchi identity
R R R
p
g
Then automatically also the Riemann curvature scalar vanishes and therefore
these solutions have vanishing action at least if the space time manifold has
no b oundary Euclidean solutions of this type are identi ed as gravitational
instantons
A considerable number of such anti self dual four geometries are known but
only two represent regular compact euclidean four manifolds without b oundary
the four torus T which has a at metric but is non simply connected and the
famous K manifold which is non at but simply connected Although no explicit
expression for the K metric in terms of co ordinates is known its top ological
invariants can b e calculated using cohomology arguments and index theory which
gives K p K
All other self dual solutions of the Einstein equations violate at least one of the
other assumptions the manifold is not compact or it has a non zero b oundary The earlier pro of of the top ological invariance of the Pontrjagin and Euler number
then fails in general there can app ear additional contributions from in nity or
from the b oundaries In this class of solutions one nds the Eguchi Hanson
metric and the multi center solutions of Gibb ons and Hawking
Taub NUT
A relatively simple and interesting example of these solutions is the anti self
dual Taub NUT metric It is the simplest example of the class of Gibb ons
Hawking solutions As it also has some interesting applications outside general
relativity I have chosen to discuss it here in some detail
The Taub NUT manifold is a four dimensional Riemannian manifold involv
ing on a single scale parameter m with Euclidean signature and anti self dual
Riemann curvature dep ending on the sign of m A convenient set of co ordinates
to describ e the manifold are the four dimensional p olar co ordinates r
with ranges r in terms of which
the metric is given by the line element
m m
d cos d ds dr r d r sin d
m
r
r
The parameter m with dimension of length can take b oth p ositive or negative
values Up on factoring out m and rescaling the radial co ordinate jr mj r
the expression b ecomes
d cos d ds dr r d r sin d
m r
r
where the sign represents the sign of m As m has b een reduced to a multiplica
tive constant one can choose units of length such that m The right hand
side of eq then directly de nes the metric In the following discussion of
the Taub NUT geometry this choice of metric is made but it is straightforward
to reinstate the general m dep endence when desired
In the app endix of this chapter we have collected the explicit expressions
for the metric the comp onents of the Riemann Christo el connections and the
covariant comp onents of the Riemann curvature tensor From these expressions
one may check that the curvature is self dual for p ositive and anti self dual for
negative m
The inverse metric is enco ded in the hamiltonian
2
Extensions of the construction of top ological invariants including b oundary contributions can b e found in the literature see for example the review
p p p cos p
r
H p
r r r r r sin
r
which describ es the geo desic ow The canonical momenta app earing in this
expression are derived from
r p r p
r
r r
cos p cos p r sin p
r
r
where as b efore an upp er dot denotes a derivative w r t to the a ne parameter
First integrals of motion can b e constructed with the help of the constants of
geo desic ow as explained in chapter Indeed the Taub NUT manifold admits
a set of three indep endent Killing vectors which together with the conservation
of the hamiltonian allow complete solution of the geo desic equations It
is actually convenient to express the Killing vectors as a set of four dep endent
ones instead of three indep endent ones as this leads to simpli cations b oth in
s
computations and physical interpretations The contravariant comp onents
s of these Killing vectors can b e read o from the constants of geo desic
s s
ow J p with
cos
J sin p p cot cos p J p
sin
sin
p cot sin p J p J cos p
sin
Solving these equations we get for the angular comp onents of the four momentum
p sin J cos J
p J p J
whilst b etween the constants of geo desic ow there is the additional relation
J cos J sin cos J sin J The radial momentum can b e computed from the hamiltonian
H J p
r
r r
sin J cos J J cos J
r r
sin
Before discussing some features of the solutions for the geo desics we digress
brie y to study the in nitesimal transformations genereated by the constants
of motion this gives useful general insight into the geometry of the Taub NUT
manifold From the last two equations it is clear that J and J
generate shifts in the angles and i e mutually commuting rotations around
two orthogonal axes As J are indep endent of they commute with the
rotations in the directions generated by J However they do not commute
with J or with each other A direct computation shows that their Poisson
brackets realize the commutations relations of ordinary angular momentum
o n
ij k k i j
J i j k J J
i
As the generators J change the angle in a non trivial way it follows that they
do not just realize three dimensional rotations on the submanifold spanned by the
co ordinates Rather three dimensional rotations on the spherical surface
de ned the co ordinate directions are generated by a set of related dynamical
quantities the restricted angular momentum three vector I with comp onents
cos
I J J I J
sin
sin
I J J
sin
The comp onents of the restricted angular momentum three vector satisfy the
same Poisson bracket relations as the comp onents of the total angular
momentum J
o n
ij k k i j
I I I
but with two crucial di erences i they induce rotations only in the
directions whilst commuting with the angle ii unlike the comp onents of J
they are not conserved on geo desics This discussion of symmetries shows that it
is convenient to think of the Taub NUT manifold lo cally as a dimensional
Euclidean geometry with a three dimensional space spanned by the spherical
co ordinates r and an internal compact one dimensional space characterized
by the co ordinate but with non trivial interplay b etween the ow of geo desics in these directions
The identity shows that in this three dimensional sense J is the
pro jection of J in the radial direction hence orthogonal to the spherical surface
spanned by
r J
J r J
r
The contribution to the total angular momentum orthogonal to r can b e identit
ed as the orbital angular momentum
J L J r
in the sense that
L r r
r
Thus r L indeed However in comp onents
L J sin cos p sin p cot cos p cos p
L J sin sin p cos p cot sin p cos p
L J cos p p cos p
and therefore the orbital angular momentum is not identical with the restricted
angular momentum I In fact the Poisson brackets of L do not ob ey the angular
momentum algebra Also L is not conserved although its magnitude jLj is
as guaranteed by the conservation of J and J and the orthogonality to J
of the contributions from J and L in eq This clari es the non trivial
dynamical relation b etween motion in the and directions
Finally as J and J are conserved separately along geo desics it follows from
that the three dimensional p osition vector r moves at a xed angle w r t
the axis of total angular momentum and the geo desic is restricted to lie on the
cone traced out by rays from the origin around J at a xed op ening angle
J
arccos
jLj
Obviously this is a generalization of the motion b eing con ned to the equatorial
plane corresp onding to J in three dimensional central force
elds like in the Kepler problem Again this testi es as to the non trivial role of
the momentum p J in the determination of the orbits
Runge Lenz vector and Yano tensors
It is p ossible to obtain more information ab out the nature of geo desics by observ
i
ing that there exists a triplet of constants of motion K i which are
quadratic in the momenta and which transform among themselves as a vector
K under three dimensional rotations This conserved vector is very similar
to the Runge Lenz vector in the Kepler problem It is conserved along geo desics
and for J it satis es the identity
H J
r K J J
J
As K H J J are all constant it follows that the three dimensional p osition
vector r is constrained to lie in a plane We have already seen that for J
this vector takes values on a cone with op ening angle around the axis of
angular momentum Combining these two results it follows that the geo desics
are actually conic sections obtained by intersecting the cone with the plane
It remains to construct the vector K and prove the relation First it
should b e p ointed out that as they are quadratic functions of the momenta
i i
K K p p
their existence is implied by a triplet of symmetric second rank Killing tensors
i
as in eq K
i
D K
Such a triplet of Killing tensors can indeed b e found Instead of directly giving the
explicit expressions we construct them in terms of a set of simpler ob jects known
as Yano tensors A Yano tensor is an anti symmetric tensor f f
satisfying the Killing like equation
D f D f
As a result the covariant derivative of a Yano tensor is completely anti symmetric
D f D f D f D f D f
a
Given one or more Yano tensors f one can construct symmetric Killing tensors
simply by symmetrized multiplication
b a
a b ab
f f f f K
These symmetric tensors satisfy eq by virtue of the Yano condition
It turns out that the Taub NUT manifold admits four such Yano tensors
transforming as a singlet Y and as a triplet f under three dimensional rotations
i
Their explicit expressions are given by the two forms
Y d cos d dr r r r sin d d
dx dx d cos d dx f
j k i i ij k
r
A straightforward calculation shows that the covariant exterior derivative of
these two forms gives
r Y D Y dx dx dx r r sin dr d d
rf
i
By studying the individual terms one establishes the stronger results
D Y D Y D Y
and
D f
i
Thus b oth two forms indeed ob ey the Yano condition with the triplet of
tensors f actually b eing covariantly constant It follows immediately that the
i
triplet of symmetric tensors de ned by
K Y f f Y
i i
i
form a vector of Killing tensors For this triplet of Killing tensors eq can
b e proven First note that as a two form
r f r r sin d d r d cos d dr
Performing matrix multiplication and contracting with the momenta to get an expression of the form it follows that
r K r f Y r Y f r K p p
r r
p r p p p cos p
r
r r sin
r H J J
Using eq this gives the desired result Note that this equation
remains true also for J in which case the geo desic ow in the three
dimensional r space is a along a hyperb ola in the plane p erp endicular to
J If J is along the z axis then the vanishing of p implies that at the
same time is constant
The constants of geo desic ow constructed ab ove may b e interpreted as an
analogue of the Runge Lenz vector for the Taub NUT geometry b ecause they
can b e written in the manifest three vector form
K J J J r J r H
r
Finally we comment up on the existence of the covariantly constant two forms f
i
They have the prop erty that as anti symmetric two tensors each of them squares
to minus the identity In fact they realize the quaternion algebra
f f f
i j ij ij k k
Thus the two forms f de ne a triplet of almost complex structures and the four
i
dimensional real geometry of the Taub NUT manifold can b e recast in terms of
analytic complex or quaternionic geometry Indeed the Taub NUT manifold b e
longs to the class of complex manifolds known as hyper K ahler manifolds An
imp ortant consequence of b eing K ahleror hyper K ahleris that these manifolds
have N resp N sup ersymmetric extensions Such su
p ersymmetries give much additional insight into the prop erties of manifolds like
Taub NUT
App endix
In this app endix I have collected the comp onents of the connection and curvature
tensor of the Taub NUT manifold in the co ordinate system in which the metric
takes the form with m The non zero comp onents of the Riemann
Christo el connection are
r r
r r
r r
r r r
r
cot
r r
r r r
r
r r
r r sin r
cos cos
sin
r r
r sin r
cot
r r
r r r
and
r r r
r
sin cos cos sin
r r r
sin r cos
r r
r r
r
r
r
From these expressions one can compute the comp onents of the Riemann curva
ture tensor
R
To allow easy check of the anti self duality we give here the completely
covariant comp onents R the non vanishing ones are
cot sin
R R
r r r r
r r r
cos
R R
r r r r
r r
r r sin
R sin cos R
r r
r r
r sin r sin
R R
r r
r r
cot r r sin
R R
r r r
r cos r sin
R R
r r
All other comp onents not related by p ermutation symmetry of the indices are zero
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