Massachusetts Institute of Technology

Department of Physics

Physics Spring

Introduction to Tensor Calculus for

General Relativity

c

Edmund Bertschinger

Introduction

There are three essential ideas underlying general relativity GR The rst is that space

time may b e describ ed as a curved fourdimensional mathematical structure called a

pseudoRiemannian manifold In brief time and space together comprise a curved four

dimensional nonEuclidean geometry Consequently the practitioner of GR must b e

familiar with the fundamental geometrical prop erties of curved spacetime In particu

lar the laws of physics must b e expressed in a form that is valid indep endently of any

co ordinate system used to lab el p oints in spacetime

The second essential idea underlying GR is that at every spacetime p oint there exist

lo cally inertial reference frames corresp onding to lo cally at co ordinates carried by freely

falling observers in which the physics of GR is lo cally indistinguishable from that of

sp ecial relativity This is Einsteins famous strong equivalence principle and it makes

general relativity an extension of sp ecial relativity to a curved spacetime The third key

idea is that mass as well as mass and momentum ux curves spacetime in a manner

describ ed by the tensor eld equations of Einstein

These three ideas are exemplied by contrasting GR with Newtonian gravity In the

Newtonian view gravity is a force accelerating particles through Euclidean space while

time is absolute From the viewp oint of GR as a theory of curved spacetime there is no

gravitational force Rather in the absence of electromagnetic and other forces particles

ed by mass follow the straightest p ossible paths geo desics through a spacetime curv

Freely falling particles dene lo cally inertial reference frames Time and space are not

absolute but are combined into the fourdimensional manifold called spacetime

In sp ecial relativity there exist global inertial frames This is no longer true in the

presence of gravity However there are local inertial frames in GR such that within a

suitably small spacetime volume around an event just how small is discussed eg in

MTW Chapter one may cho ose co ordinates corresp onding to a nearlyat spacetime

Thus the local prop erties of sp ecial relativity carry over to GR The mathematics of

vectors and tensors applies in GR much as it do es in SR with the restriction that vectors

and tensors are dened indep endently at each spacetime event or within a suciently

small neighborho o d so that the spacetime is sensibly at

Working with GR particularly with the Einstein eld equations requires some un

derstanding of dierential geometry In these notes we will develop the essential math

ematics needed to describ e physics in curved spacetime Many physicists receive their

introduction to this mathematics in the excellent book of Weinb erg Weinberg

minimizes the geometrical content of the equations by representing tensors using com

p onent notation We b elieve that it is equally easy to work with a more geometrical

description with the additional b enet that geometrical notation makes it easier to dis

tinguish physical results that are true in any co ordinate system eg those expressible

using vectors from those that are dep endent on the co ordinates Because the geometry

of spacetime is so intimately related to physics we b elieve that it is b etter to highlight

the geometry from the outset In fact using a geometrical approach allows us to develop

the essential dierential geometry as an extension of vector calculus Our treatment

is closer to that Wald and closer still to Misner Thorne and Wheeler

MTW These b o oks are rather advanced For the newcomer to general relativity we

utz Our notation and presentation is patterned largely warmly recommend Sch

after Schutz It expands on MTW Chapters and The student wishing addi

tional practice problems in GR should consult Lightman et al A slightly more

advanced mathematical treatment is provided in the excellent notes of Carroll

These notes assume familiarity with sp ecial relativity We will adopt units in which

the sp eed of light c Greek indices etc which take the range f g

will b e used to represent comp onents of tensors The Einstein summation convention

is assumed rep eated upp er and lower indices are to b e summed over their ranges

B A B A B A B A B Fourvectors will b e represented with eg A

an arrow over the symb ol eg A while oneforms will b e represented using a tilde

Spacetime p oints will b e denoted in b oldface type eg x refers to a p oint eg B

Our metric has signature the at spacetime Minkowski metric with co ordinates x

comp onents are diag

Vectors and oneforms

The essential mathematics of general relativity is dierential geometry the branch of

mathematics dealing with smo othly curved surfaces dierentiable manifolds The

physicist do es not need to master all of the subtleties of dierential geometry in order

to use general relativity For those readers who want a deep er exp osure to dierential

geometry see the introductory texts of Lovelo ck and Rund Bishop and Goldb erg

or Schutz It is sucient to develop the needed dierential geometry as a

straightforward extension of linear algebra and vector calculus However it is imp ortant

to keep in mind the geometrical interpretation of physical quantities For this reason

we will not shy from using abstract concepts like p oints curves and vectors and we will

distinguish b etween a vector A and its comp onents A Unlike some other authors eg

Weinberg we will introduce geometrical ob jects in a co ordinatefree manner only

later introducing co ordinates for the purp ose of simplifying calculations This approach

requires that we distinguish vectors from the related ob jects called oneforms Once

the dierences and similarities b etween vectors oneforms and tensors are clear we will

adopt a unied notation that makes computations easy

Vectors

We b egin with vectors A vector is a quantity with a magnitude and a direction This

primitive concept familiar from undergraduate physics and mathematics applies equally

in general relativity An example of a vector is dx the dierence vector b etween two

Vectors form a linear algebra ie a vector innitesimally close p oints of spacetime

space If A is a vector and a is a real numb er scalar then aA is a vector with the

same direction or the opp osite direction if a whose length is multiplied by jaj If

A and B are vectors then so is A B These results are as valid for vectors in a curved

fourdimensional spacetime as they are for vectors in threedimensional Euclidean space

Note that we have intro duced vectors without mentioning co ordinates or co ordinate

transformations Scalars and vectors are invariant under co ordinate transformations

vector comp onents are not The whole p oint of writing the laws of physics eg F ma

using scalars and vectors is that these laws do not dep end on the co ordinate system

imp osed by the physicist

We denote a spacetime p oint using a b oldface symbol x This notation is not meant

to imply co ordinates Note that x refers to a p oint not a vector In a curved spacetime

x is not useful the concept of a radius vector x p ointing from some origin to each p oint

b ecause vectors dened at two dierent p oints cannot b e added straightforwardly as

they can in Euclidean space For example consider a sphere embedded in ordinary

threedimensional Euclidean space ie a twosphere A vector p ointing east at one

p oint on the equator is seen to p oint radially outward at another p oint on the equator

�

The radially outward direction is undened on the whose longitude is greater by

sphere

Technically we are discussing tangent vectors that lie in the tangent space of the

manifold at each p oint For example a sphere may b e emb edded in a threedimensional

Euclidean space into which may b e placed a plane tangent to the sphere at a p oint A two

dimensional vector space exists at the p oint of tangency However such an emb edding

is not required to dene the tangent space of a manifold Wald As long as the

space is smo oth as assumed in the formal denition of a manifold the dierence vector

dx between two innitesimally close p oints may b e dened The set of all dx denes

the tangent space at x By assigning a tangent vector to every spacetime p oint we

can recover the usual concept of a vector eld However without additional preparation

one cannot compare vectors at dierent spacetime p oints b ecause they lie in dierent

tangent spaces In later notes we intro duce will parallel transp ort as a means of making

this comparison Until then we consider only tangent vectors at x To emphasize the

status of a tangent vector we will o ccasionally use a subscript notation A

X

Oneforms and dual vector space

Next we intro duce oneforms A oneform is dened as a linear scalar function of a vector

That is a oneform takes a vector as input and outputs a scalar For the oneform P

V is also called the scalar pro duct and may b e denoted using angle brackets P

V i P P V h

The oneform is a linear function meaning that for all scalars a and b and vectors V and

W the oneform P satises the following relations

P aV bW hP aV bW i ahP V i bhP W i aP V bP W

Just as we may consider any function f as a mathematical entity indep endently of

indep endently of any particular any particular argument we may consider the oneform P

e may also asso ciate a oneform with each spacetime p oint resulting in a vector V W

oneform eld P P Now the distinction b etween a p oint a vector is crucial P is

X X

V is a scalar dened implicitly at p oint x The scalar a oneform at p oint x while P

i V pro duct notation with subscripts makes this more clear hP

X X

Oneforms ob ey their own linear algebra distinct from that of vectors Given any two

scalars a and b and oneforms P and Q we may dene the oneform aP bQ by

V i bhQ V i aP V bQV ahP aP bQV haP bQ V i

Comparing equations and we see that vectors and oneforms are linear op erators

on each other pro ducing scalars It is often helpful to consider a vector as b eing a linear

scalar function of a oneform Thus we may write hP V i P V V P The set of

all oneforms is a vector space distinct from but complementary to the linear vector

space of vectors The vector space of oneforms is called the dual vector or cotangent

space to distinguish it from the linear space of vectors tangent space

Although oneforms may app ear to b e highly abstract the concept of dual vector

spaces is familiar to any student of quantum mechanics who has seen the Dirac braket

notation Recall that the fundamental ob ject in quantum mechanics is the state vector

represented by a ket j i in a linear vector space Hilb ert space A distinct Hilb ert

space is given by the set of bra vectors hj Bra vectors and ket vectors are linear scalar

functions of each other The scalar pro duct hj i maps a bra vector and a ket vector to a

scalar called a probability amplitude The distinction b etween bras and kets is necessary

b ecause probability amplitudes are complex numb ers As we will see the distinction

between vectors and oneforms is necessary b ecause spacetime is curved

Tensors

Having dened vectors and oneforms we can now dene tensors A tensor of rank m n

also called a m n tensor is dened to b e a scalar function of m oneforms and n vectors

that is linear in all of its arguments It follows at once that scalars are tensors of rank

vectors are tensors of rank and oneforms are tensors of rank We

Q one of rank by P Q A etc P may denote a tensor of rank by

T T

Our notation will not distinguish a tensor from a tensor although a

T T

notational distinction could b e made by placing m arrows and n tildes over the symbol

or by appropriate use of dummy indices Wald

The scalar pro duct is a tensor of rank which we will denote and call the

I

identity tensor

P V hP V i P V V P

I

the identity b ecause when applied to a xed oneform P We call and any vector V it

I

y tensor was dened as a mapping from a oneform V Although the identit returns P

and a vector to a scalar we see that it may equally b e interpreted as a mapping from a

P oneform to the same oneform P where the dot indicates that an argument

I

may b e considered a vector is needed to give a scalar A similar argument shows that

I

the identity op erator on the space of vectors V V V

I

A tensor of rank m n is linear in all its arguments For example for m n

we have a straightforward extension of equation

q S aP b Q cR dS ac P R ad P S bc Q R bd

T T T T T

Tensors of a given rank form a linear algebra meaning that a linear combination of

tensors of rank m n is also a tensor of rank m n dened by straightforward extension

of equation Two tensors of the same rank are equal if and only if they return the

same scalar when applied to all p ossible input vectors and oneforms Tensors of dierent

rank cannot b e added or compared so it is imp ortant to keep track of the rank of each

tensor Just as in the case of scalars vectors and oneforms tensor elds are dened

T

X

by asso ciating a tensor with each spacetime p oint

There are three ways to change the rank of a tensor The rst called the tensor or

outer pro duct combines two tensors of ranks m n and m n to form a tensor

of rank m m n n by simply combining the argument lists of the two tensors

and thereby expanding the dimensionality of the tensor space For example the tensor

pro duct of two vectors A and B gives a rank tensor

A B P Q AP B Q

T T

We use the symbol to denote the tensor pro duct later we will drop this symbol for

notational convenience when it is clear from the context that a tensor pro duct is implied

Note that the tensor pro duct is noncommutative A B B A unless B cA for

some scalar c b ecause AP B Q AQ B P for all P and Q We use the symbol

to denote the tensor pro duct of any two tensors eg P P A B is a tensor

T

of rank The second way to change the rank of a tensor is by contraction which

reduces the rank of a m n tensor to m n The third way is the gradient We

ts later will discuss contraction and gradien

Metric tensor

The scalar pro duct eq requires a vector and a oneform Is it p ossible to obtain a

scalar from two vectors or two oneforms From the denition of tensors the answer is

clearly yes Any tensor of rank will give a scalar from two vectors and any tensor

of rank combines two oneforms to give a scalar However there is a particular

;

called the metric tensor and a related tensor eld g called tensor eld g

X

X

the inverse metric tensor for which sp ecial distinction is reserved The metric tensor is

a symmetric bilinear scalar function of two vectors That is given vectors V and W g

returns a scalar called the dot pro duct

g V W V W W V gW V

;

wo oneforms P and Q which we also call the dot Similarly g returns a scalar from t

pro duct

; ;

g P Q P Q P Q g P Q

;

Although a dot is used in b oth cases it should b e clear from the context whether g or g

is implied We reserve the dot pro duct notation for the metric and inverse metric tensors

just as we reserve the angle brackets scalar pro duct notation for the identity tensor eq

;

Later in eq we will see what distinguishes g from other tensors and g

from other symmetric tensors

One of the most imp ortant prop erties of the metric is that it allows us to convert

vectors to oneforms If we forget to include W in equation we get a quantity denoted

that b ehaves like a oneform V

V g V g V

where we have inserted a dot to remind ourselves that a vector must b e inserted to give

a scalar Recall that a oneform is a scalar function of a vector We use the same letter

to indicate the relation of V and V

Thus the metric g is a mapping from the space of vectors to the space of oneforms

; ;

g V V By denition the inverse metric g is the inverse mapping g V V

V The inverse always exists for nonsingular spacetimes Thus if V is dened for any

by equation the inverse metric tensor is dened by

; ;

V g V g V

Equations and give us several equivalent ways to obtain scalars from vectors

V and W and their asso ciated oneforms V and W

;

V W hV W i hW V i V W V W V W W V gV W g

I I

Basis vectors and oneforms

It is p ossible to formulate the mathematics of general relativity entirely using the abstract

formalism of vectors forms and tensors However while the geometrical co ordinatefree

interpretation of quantities should always b e kept in mind the abstract approach often is

not the most practical way to p erform calculations To simplify calculations it is helpful

to introduce a set of linearly indep endent basis vector and oneform elds spanning

our vector and dual vector spaces In the same way practical calculations in quantum

mechanics often start by expanding the ket vector in a set of basis kets eg energy

eigenstates By denition the dimensionality of spacetime four equals the number of

linearly indep endent basis vectors and oneforms

We denote our set of basis vector elds by fe g where lab els the basis vector

X

eg and x lab els the spacetime p oint Any four linearly indep endent basis

vectors at each spacetime p oint will work we do not not imp ose orthonormality or any

other conditions in general nor have we implied any relation to co ordinates although

y expand any vector eld A as a linear combination later we will Given a basis we ma

of basis vectors

A A e A e A e A e A e

X X X X X X

X X X X

X

Note our placement of subscripts and sup erscripts chosen for consistency with the Ein

stein summation convention which requires pairing one subscript with one sup erscript

are called the comp onents of the vector often the contravariant The co ecients A

comp onents Note well that the co ecients A dep end on the basis vectors but A do es

not

Similarly we may cho ose a basis of oneform elds in which to expand oneforms

Although any set of four linearly indep endent oneforms will suce for each like A

X

spacetime p oint we prefer to choose a sp ecial oneform basis called the dual basis and

denoted fe g Note that the placement of subscripts and sup erscripts is signicant

X

we never use a subscript to lab el a basis oneform while we never use a sup erscript

is not related to e through the metric eq to lab el a basis vector Therefore e

ge Rather the dual basis oneforms are dened by imp osing the following e

requirements at each spacetime p oint

he e i

X

X

where is the Kronecker delta if and otherwise with the

same values for each spacetime p oint We must always distinguish subscripts from

sup erscripts the Kronecker delta always has one of each Equation is a system of

and it four linear equations at each spacetime p oint for each of the four quantities e

has a unique solution The reader may show that any nontrivial transformation of the

dual basis oneforms will violate eq Now we may expand any oneform eld P in

X

the basis of oneforms

P e P

X X

X

The comp onent P of the oneform P is often called the covariant comp onent to distin

ariant comp onent P of the vector P In fact b ecause we have guish it from the contrav

consistently treated vectors and oneforms as distinct we should not think of these as

b eing distinct comp onents of the same entity at all

There is a simple way to get the comp onents of vectors and oneforms using the fact

that vectors are scalar functions of oneforms and vice versa One simply evaluates the

vector using the appropriate basis oneform

Ae he Ai he A e i he e iA A A

and conversely for a oneform

e i he e iP P P P e hP e i hP e

We have suppressed the spacetime p oint x for clarity but it is always implied

Tensor algebra

We can use the same ideas to expand tensors as pro ducts of comp onents and basis

tensors First we note that a basis for a tensor of rank m n is provided by the tensor

pro duct of m vectors and n oneforms For example a tensor like the metric tensor

can b e decomp osed into basis tensors e e The components of a tensor of rank m n

lab eled with m sup erscripts and n subscripts are obtained by evaluating the tensor using

m basis oneforms and n basis vectors For example the comp onents of the metric

tensor the inverse metric tensor and the identity tensor are

;

e e he e i g e e e e g g g e e e e

I

The last equation follows from eqs and The tensors are given by summing over

the tensor pro duct of basis vectors and oneforms

;

g g e e g g e e e e

I

The reader should check that equation follows from equations and the duality

condition equation

Basis vectors and oneforms allow us to represent any tensor equations using com

p onents For example the dot pro duct b etween two vectors or two oneforms and the

scalar pro duct b etween a oneform and a vector may b e written using comp onents as

A hP Ai P A P Q g P P B g A A

The reader should prove these imp ortant results

If two tensors of the same rank are equal in one basis ie if all of their comp onents

are equal then they are equal in any basis While this mathematical result is obvious

from the basisfree meaning of a tensor it will have imp ortant physical implications in

GR arising from the Equivalence Principle

As we discussed ab ove the metric and inverse metric tensors allow us to transform

vectors into oneforms and vice versa If we evaluate the comp onents of V and the

dened by equations and we get oneform V

;

e V g V g e V V g V V g

Because these two equations must hold for any vector V we conclude that the matrix

is the inverse of the matrix dened by g dened by g

g g

We also see that the metric and its inverse are used to lower and raise indices on comp o

nents Thus given two vectors V and W we may evaluate the dot pro duct any of four

equivalent ways cf eq

V W g V W V W V W g V W

In fact the metric and its inverse may b e used to transform tensors of rank m n

into tensors of any rank m k n k where k m m n Consider for

example a tensor with comp onents

T

T e e e

T

If we fail to plug in the oneform e the result is the vector T e A oneform must b e

This vector may then b e inserted into the metric inserted to return the quantity T

tensor to give the comp onents of a tensor

ge T T e g T

verse metric to raise the third index say giving us the comp onent We could now use the in

of a tensor distinct from equation

;

T g e T e g T g g T

mn

In fact there are dierent tensor spaces with ranks summing to m n The metric

or inverse metric tensor allow all of these tensors to b e transformed into each other

Returning to equation we see why we must distinguish vectors with comp onents

from oneforms with comp onents V The scalar pro duct of two vectors requires V

the metric tensor while that of two oneforms requires the inverse metric tensor In

g The only case in which the distinction is unnecessary is in at general g

Lorentz spacetime with orthonormal Cartesian basis vectors in which g case

However gravity curves is everywhere the diagonal matrix with entries

spacetime Besides we may wish to use curvilinear co ordinates even in at spacetime

g everywhere As a result it is imp ossible to dene a co ordinate system for which g

We must therefore distinguish vectors from oneforms and we must b e careful ab out the

placement of subscripts and sup erscripts on comp onents

At this stage it is useful to introduce a classication of vectors and oneforms drawn

from sp ecial relativity with its Minkowski metric Recall that a vector A A e

is called spacelike timelike or null according to whether A A A A is p ositive

negative or zero resp ectively In a Euclidean space with p ositive denite metric A A

is never negative However in the Lorentzian spacetime geometry of sp ecial relativity

A In time enters the metric with opp osite sign so that it is p ossible to have A

dx d of a massive particle where d is prop er time particular the fourvelocity u

is a timelike vector This is seen most simply by p erforming a Lorentz b o ost to the

t x y z

rest frame of the particle in which case u u u u and u u

Now u u is a Lorentz scalar so that u u in any Lorentz frame Often this is

where p mu is the fourmomentum for a particle of mass m written p p m

For a massless particle eg a photon the prop er time vanishes but the fourmomentum

is still welldened with p p the momentum vector is null We adopt the same

notation in general relativity replacing the Minkowski metric comp onents with the

actual metric g and evaluating the dot pro duct using A A gA A g A A The

;

same classication scheme extends to oneforms using g a oneform P is spacelike

;

P g P P g P P is p ositive negative timelike or null according to whether P

or zero resp ectively Finally a vector is called a unit vector if A A and similarly

for a oneform The fourvelocity of a massive particle is a timelike unit vector

Now that we have intro duced basis vectors and oneforms we can dene the contrac

tion of a tensor Contraction pairs two arguments of a rank m n tensor one vector and

one oneform The arguments are replaced by basis vectors and oneforms and summed

over For example consider the tensor which may b e contracted on its second

R

vector argument to give a tensor also denoted but distinguished by its shorter

R

argument list

X

e A B A e B e A e B

R R R

In later notes we will dene the Riemann curvature tensor of rank its contraction

dened by equation is called the Ricci tensor Although the contracted tensor would

ectors app ear to dep end on the choice of basis b ecause its denition involves the basis v

and oneforms the reader may show that it is actually invariant under a change of basis

and is therefore a tensor as long as we use dual oneform and vector bases satisfying

equation Equation b ecomes somewhat clearer if we express it entirely using

tensor comp onents

R R

Summation over is implied Contraction may b e p erformed on any pair of covariant

and contravariant indices dierent tensors result