Special Relativity in a Nutshell

Cosmology Special Relativity in a Nutshell

Basic Postulates: Einstein 1905

All observers moving uniformly at constant velocity along straight lines detect identical physical laws

Only changes in the velocity, i.e., accelerations are detectable!

Who is sitting in a box without windows by no means is able to ﬁnd out whether he is at rest or moving at constant velocity.

Light rays propagate in all frames moving relative to each other at constant velocity in all directions with the same velocity

c = 29907920458 m s−1

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 137 Cosmology

The speed of light c is limiting velocity. No physical body can move at speed bigger than c. It is not possible to transport information with speed larger than c (signal velocity 3 ≤ c). has to be considered as a convention for the synchronization of clocks, which move relative to each other at constant velocity!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 138 Cosmology Lorentz Transformations How did Einstein ﬁnd these postulates?

Classical mechanics: basic laws of classical mechanics are Galilei invariant g = TGR

~x 0 = R~x Rotations R: t0 = t

~x 0 = ~x + ~a Translations T: t0 = t + b

~x 0 = ~x − ~3 t Galilei Transformations G: t0 = t

G implements the relative motion; Time t is absolute and not transformed besides shifts, i.e. time intervals are invariant.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 139 Cosmology

Classical Electrodynamics: basic equations of classical Electrodynamics (Maxwell) are Poincare´ invariant Einstein 1905 (Larmor 1897, Lorentz 1899, 1904, Poincare´ 1905)

P = TRL is the symmetry of Electrodynamics: R, T together with

Special Lorentz transformations L: ~x 0 = ~x + ~n ~n · ~3 (γ − 1) − γ~3 t t0 γ t − ~3·~x = c2 with (3 = |~3 |, γ = Lorentz factor)

~3 1 ~n = ; γ = q 3 2 − 3 1 c2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 140 Cosmology

New form of velocity transformation. Not t is invariant but limiting velocity c. Time lost absolute meaning, time is relative! L is the prescription which allows us to translate space–time coordinate information between systems having constant relative velocity.

Note that in Electrodynamics, Maxwell’s uniﬁcation of electric and magnetic phenomena, the speed of light shows up as a constant c = √ 1 0 µ0 which is the speed of propagation of electromagnetic waves. 0 permittivity of free space (electric constant), µ0 permeability of free space (magnetic constant), both are universal constants.

Lorentz invariance of Electrodynamics is true under the natural assumption that c is a universal constant. Einstein with his covariant reformulation of Electrodynamics, did not attempt to modify Maxwell’s theory, so the universality of the speed of light had to be incorporated in the reformulation of the laws of motion (relativistic kinematics). In fact the universality and ﬁniteness of the speed of light is a key element of special relativity.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 141 Cosmology

Einstein’s covariant reformulation of classical Electrodynamics was triggered by the observation that moving a conducting coil in a magnetic ﬁeld is described by the Lorentz force law, while moving the magnet relative to the coil at rest is described by the law of induction. Except for the constant relative motion the physic is the same. Therefore physical laws have to be formulated in a manner which does not depend on constant relative motion.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 142 Cosmology Consequences

1) Physical laws are invariant with respect to Lorentz transformations (generally: Poincare´ group)

x0 = γ (x − 3t) y0 = y z0 = z t0 γ t − 3x = c2 (special L–transformation in x–direction)

+ Rotations + Translations ≡ Poincare´ group

Time not absolute anymore, transforms non–trivially Space-Time

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 143 Cosmology

Classical limiting case: 3 c formally c → ∞ L–transformation → Galilei–transformation Indeed: c → ∞ ; β → 0 ; γ → 1 x0 = x − vt y0 = y z0 = z t0 = t ⇐

Note replaces postulate of universal time. In the limiting case c → ∞ one gets this back, ultra relativistic limit: 3 → c £ β → 1 0 3 γ → ∞ t → ∞ if not t = c, which is a light ray.

2) Two Lorentz transformations executed one after the other is again a Lorentz transformation 31+32 3 = 3 3 1+ 1 2 c2 relativistic formula of addition of velocities.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 144 Cosmology

light ﬂash c′ = c = c v c′ = c = c + v 60 6 − 6 80

60+80=140?

Additivity of velocities does not hold for high velocities:

3 100 1 1 Error at a speed of 100 km/h: c ' 300 000 60·60 ' 10 000 000 In place of cars light ﬂashes: independent of movement of train always c and not c + 3train

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 145 Cosmology

Limit velocity: 32, 32 < c → 3 < c where 3(31, 32) is monotonically increasing in 31 and 32 and for 31 → c £ 3 → c

Exercise: show that two Lorentz transformations along the x-axis, L1(31) and L2(32), executed one after the other yield a Lorentz transformation along the x-axis L(3) = L2(32) L1(31). Calculate 3. Of course this can be generalized to arbitrary Lorentz transformations. The simple scalar velocity addition is then replaced by an appropriate vector relation between ~3, ~31 and ~32.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 146 Cosmology

Exercise: comment the following paradox: the locomotive driver knows his train will not ﬁt into the tunnel the track man is sure it will

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 147 Cosmology

3) Lorentz contraction

A measuring rod of length `0 along the x–direction at rest in a IS A appears to be length contracted in a moving IS B (Moving measuring rods appear shortened in the direction they are moving) t0 ⇒ t 3 x = 0 : = c2 2 p £ x0 γ x − 3 x x − β2 = c2 = 1 hence

p 2 `b = `0 1 − β (< `0)

Transversal to the direction of the movement there is no contraction. This, however, does not mean that we see a moving sphere as an ellipsoid -≫. Illustration: a chain of moving dice -≫

Other: -≫; Download: theory -≫ (Weiskopf, Kraus, Ruder)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 148 Cosmology

4) Time dilatation

Consider a clock with oscillation time T0 at rest at a point ~x = 0 in the IS A. Observed from the moving IS B the clock appears to oscillate slower: x = 0 : ⇒ t0 = t γ hence

1 Tb = T0 √ (> T0) 1−β2

All this is experimentally established, muons in Cosmic Rays, muons in a Muon storage ring etc.

Exercise: In a muon storage ring muons move at the magic energy E = 3.1 GeV. Compare their velocity with the one of the light. Calculate their lifetime in the Laboratory frame.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 149 Cosmology

Note that the time dilatation is a real phenomenon. The muons circulating in the storage ring indeed decay sooner or later depending on their speed. Muons at rest would decay on place, while highly relativistic muons circulate a number of times in the ring before they decay. This phenomenon is also well known from cosmic muons, for example. Note that unstable particles are clocks, with an intrinsic lifetime in their rest frame. In the rest frame the lifetime is shortest.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 150 Cosmology Space-Time: Minkowski Geometry

Event-space: deﬁned by relationship Event ⇔ Point P = P(x0, x1, x2, x3) in event-space x0 = c t “time” measured as distance which a light signal in vacuum would cover in time t x1 = x x2 = y x ⇔ (x0, x1, x2, x3) x3 = z

Coordinates: xµ ; µ = 0, 1, 2, 3 so called four-vector

This is the geometrization of the events (t, ~x)

Motion of a point-particle: continuous series of events =ˆ continuous curve in the event-space £ World-line of the point-particle

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 151 Cosmology

x0 = ct

world line

path

The path of the particle in space is the projection of the world line onto the conﬁguration space.

The world line of a photon (light quanta, light wave) passing at t = 0 at ~x = 0 at t > 0 is located at

~x = ~n c t ; |~n| = 1 , ~n direction/unit/vector

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 152 Cosmology

Therefore the world line of a photon in any case is a line on the envelope of a cone: the light cone ~x2 − (x0)2 = 0

x0 = ct world line photon

45◦ x2 world line x1 particle m> 0

Particle of mass m > 0 ⇒ 3 < c

£ world line at each point strictly inside light cone.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 153 Cosmology Measuring Space and Time Minkowski distance measurement ≡ Minkowski metric

Theorem: the indeﬁnite quadratic form

s2 (x0)2 − ~x2 is invariant under Lorentz transformations:

deﬁnes invariant distance of two events

2 2 2 2 sAB = (xA − xB) (x0A − x0B) − ~xA − ~xB

Minkowski distance is independent £ of the choice of the inertial frame

Classiﬁcation: s2 = constant for a plane x2 = x3 = 0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 154 Cosmology

Events A and B relative to each other lie:

2 sAB > 0 time like two–shell hyperboloid 2 sAB = 0 light like double cone 2 sAB < 0 space like one–shell hyperboloid

absolute future time-like 0 x 2 =0 s s2 = c2 T 2 s2 = ℓ2 T − space-like ℓ light-like x1 absolute distant

absolute past Minkowski Space-Time

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 155 Cosmology Interpretation

2 sAB > 0: the events A and B lie time-like to each other. The straight world line between A and B corresponds to the motion of a massive particle at velocity 3 < c. In the rest frame of the particle the events A and B happen at the same place ~xA = ~xB = ~x at different times tA and tB. In other words, there exist an IS in which

sAB = c |tA − tB| ; ~xA = ~xB 2 sAB < 0: the events A and B lie space-like to each other. There exist an IS in which A and B happen at equal time, but at different places

sAB = |~xA = ~xB| ; tA = tB 2 sAB = 0: the events A and B lie light-like to each other. There exists a light signal which connects A and B:

|~xA − ~xB| = c |tA − tB| ; sAB = 0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 156 Cosmology Time measurement with a clock in arbitrary motion x0 B

B B q R R 2 ~3 ~3 t T T t − ~3 = ( ) AB = d = d 1 c2 A A A x1 q p 2 with s c2 t2 − ~x 2 c t − ~3 we have d = d d = d 1 c2

B 1 R Proper time TAB = c ds independent of IS! A

Ì ds is the Minkowski line element

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 157 Cosmology

RB Ì ds is the Minkowski arc length A The proper time is a maximum for straight line uniform motion x0

RB RB ds < ds = max . Γ2 Γ1 Γ1 Γ2 A A

A x1

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 158 Cosmology

Twin Paradox

Note that time has no longer an absolute meaning, different observers can draw different conclusions concerning non-invariant quantities. The main point in the twin paradox is that one observer stays at rest in the supposed IS, while the other leaves the position and later comes back. The latter observer cannot stay all the time in an IS and in fact acceleration change the ticking of watches as we will see later. If later the twins again sit together in the same IS, they have a different history and there is no reason why this should not distinguish them.

2 P3 P3 µ ν Metric: quadratic form ds = µ=0 ν=0 gµνdx dx

Minkowski metric (indeﬁnite):

 1 0 0 0     0 −1 0 0  g :=   µν  0 0 −1 0     0 0 0 −1 

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 159 Cosmology

To be compared with common 3-d Euclidean space

2 P3 P3 i k Metric: quadratic form d` = i=1 k=1 eikdx dx Euclidean metric (positive deﬁnite)= unit matrix  1 0 0    e :=  0 1 0  ik    0 0 1 

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 160 Cosmology Relativistic Kinematics

World line of a point particle: xµ = xµ(λ), λ a parameter (parametrization of world line). The natural parameter is the arc length λ = s, chosen such that x0 ≷ 0 for s ≷ 0, with reference point xµ(0) = 0, ~x (0)

x0 xµ(λ)

λ3

λ2 x2 λ1

µ dxµ We then deﬁne: four-velocity u ds

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 161 Cosmology

   !2 2 µ ν 02 2 02  d~x  ds = gµνdx dx = dx − d~x = dx 1 −  dx0  d~x = ~3, ~3 Newton’s velocity d~x dx0 c dt q £ s x0 − ~3 2 d = d 1 c2 0 dx0 1 u = = r ds 2 1−~3 u0 c2 i dxi dx0 dxi 3i 1 u = ds = ds 0 = c r dx ~3 2 uµ 1− 1 c2 u1

µ ~3 u = γ 1, c

µ ν 2 is dimensionless of modulus 1 since uµuνg = g dx dx = ds = 1 µν µν ds ds ds2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 162 Cosmology Momentum and Energy

Again the requirements are

O relativistic covariance (how to unambiguously transfer information from one IS to another)

O need quantity which is conserved for all IS’s

O for 3 c: must yield classical form

The recipe is not difﬁcult to guess: by analogy µ ~3 velocity ~3 « Four-velocity u = γ 1, c momentum ~p = m~3 « Four-momentum pµ mc uµ

By deﬁnition pµ is a contravariant four-vector.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 163 Cosmology

µ P µ Consequence: if P = pi = 0 in the rest system (assumed to be an IS) then i Pµ = 0 in any IS.

Meaning of m, p0 and ~p :

Let pµ = (p0, ~p ) be the four momentum of a free particle,

1. relativistic momentum:

!! m~3 1 32 34 ~p ' m~3 ( O = q 1 + 2 + 4 3 2 |{z} 2 c c − ~ classical form 1 c2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 164 Cosmology

2. relativistic energy:

!! m c 1 32 34 p0 ' m c O = q 1 + 2 + 4 − ~3 2 2 c c 1 c2

1 0 2 3 2 ··· E = c p = |{z}m c + m~ + rest energy |{z}2 classical kinetic energy

Rest energy new term!

rest energy → contribution to the potential energy

“Mass as frozen energy”

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 165 Cosmology

3. In rest system E = m c2 inner energy of the particle, m rest mass, E = m c2 is a relativistic invariant.

The inertial mass is: m(3) = m γ(3)

~p = m(3) · ~3 ; E = m(3) · c2

represent the relativistic generalizations of the non-relativistic versions of momentum and energy!

What is new?

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 166 Cosmology

Example 1.: Decay of a neutral Kaon into two identical neutral pions

K → π0π0

2 Experimentally: mK = 3.7 mπ0 with mπc = 134.97 MeV In the rest system of the K :

P = mKc,~0 = p1 + p2 E E = 1 + 2,~0 ; ~p = −~p c c 1 2 with

m~3 0 m c ~p = q ; p = q − ~3 2 − ~3 2 1 c2 1 c2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 167 Cosmology indeed

p p0 2 − ~p 2 = m2c2 ⇒ p0 = + m2c2 + ~p 2

where we note that the energy must be positive (spectral condition).

2 2 2 ~p1 = −~p2 £ ~p1 = ~p2 = ~p 0 0 0 E1 = E2 ⇔ p1 = p2 = p

Thus ! E 0 p = (m c, 0) = 2 π , 0 K c

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 168 Cosmology i.e. q 2 2 2 2Eπ0 = mK c = 2 c mπc + ~p m 2 K c2 = m2c2 + ~p 2 2c π m 2 ~p 2 = K c2 − m2 c2 2c π r m 2 |~p | = c K − m2 2 π

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 169 Cosmology

This very important result illustrates a novel phenomenon in a relativistic theory: the difference of the squared rest masses 2 mK − (2 mπ) is transmuted to kinetic energy!

There exist no conservation law for rest masses!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 170 Cosmology

Example 2.: Decay of the neutral π0 into two photons π0 → γγ In the rest frame of the π0: P = pπ = mπ c,~0 E E = 1 + 2, ~p + ~p c c 1 2

~p1 + ~p2 = 0 £ ~p2 = −~p1 ; |~p1| = |~p2| = |~p | q 0 2 2 2 0 p = mγ c + ~p ; mγ = 0 i.e. p = |~p | .

Thus

E1 = E2 = |~p | · c m c m c = 2 |~p | ⇒ |~p | = π . π γ 2

As photons have no rest mass, we have the result: π0 decays into pure energy of motion !!!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 171 Cosmology The Mass Shell

We ﬁrst note that 2 2 µ ν 0 2 2 2 µ ν 2 2 p = gµν p p = p − ~p = m c u u gµν = m c > 0 ! independent of the IS, and the rest mass is independent of the inertial reference frame. pµ is in any case time-like (m > 0)

q ~p 2 1 ~p 4 E = c p0 = c m2 c2 + ~p 2 ' m c2 + − + ··· 2m 8 m3c2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 172 Cosmology

all physical states p0 E > 0 2 =0 p

pµ Tachyons m =0 ⊲⊳ p0 = ~p ~p | | unphysical| |

E < 0 unphysical

Physical Spectral Condition: all physical states must have p0 > 0 and p2 ≤ 0 Massive particles: p2 > 0 ; m > 0 ; p0 > 0 Massless particles: p2 = 0 ; m = 0 ; p0 > 0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 173 Cosmology for massless particles

E2 p2 = g pµ pν = − ~p 2 = 0 ./ E = c |~p | µν c

As E = γ m c2 and ~p = γ m~3 ⇒| ~3 | = c. Thus all massless particles propagate with velocity c!

G each massless state of spin s > 0 has exactly two possible helicities

G the two helicity states do not mix [two independent degrees of freedom]

G for massive particles (m > 0), the number of states is 2s + 14

4 i.e. for s ≥ 1 massless states have a lower number of independent degrees of freedom than massive one’s

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 174 Cosmology Waves and Doppler Shift Photons in the particle picture carry energy E and momentum ~p, which at the same time are electromagnetic waves angular frequency ω and wave-vector ~k in the wave picture:

2π ω = 2πν , |~k| = λ E h ν = , λ = h |~p | E λ · ν ./ = c . |~p |

As a wave we have

i (~k~x−ω t) E~(~x, t) = E~0 e + h.c.

µ 0 0 and we may write ωt − ~k~x = kx = kµx = k x − ~k~x

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 175 Cosmology µ ω ~ k = c , k wave-vector

The particle–wave dualism is characterized by pµ or kµ

h pµ = ~ kµ ; ~ = ; p2 = k2 = 0 2π both light-like contravariant vectors.

Relativistic Doppler Shift

We consider the frame dependence of the frequency of an electromagnetic wave. It is given by a Lorentz transformation

µ0 µ ν k = Λ ν k .

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 176 Cosmology

Consider a wave of wave-vector

~k = |~k | (cos θ, − sin θ, 0)

~ 2π with |k | = λ . We now perform a special Lorentz transformation in x-direction:

0 3 k0 = γ k0 − γ k1 c ! ω0 ω 3 2π·ν = γ − cos θ c c c λ ·ν 3 ν0 = ν γ 1 − cos θ . c

Take Λ to be the Lorentz transformation to the system in which the source is at 0 rest: ν = ν0.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 177 Cosmology

We thus get r 2 ν 1−~3 0 c2 ν = 3 1−c cos θ

t =0 ~v t = T θ x γ ~k

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 178 Cosmology

Exercise 2.: Calculate energy and frequency of the γ’s in the decay of a π0 at rest, as well as the decay in ﬂight for 3π0 = 0.9 c.

Exercise 3.: Show that the relativistic action principle of a free particle reads

Z2 S = −m c ds

q 2 i.e. the action is proportional to the Minkowski arc length. L ~x, ~x˙ −m c2 − ~3 , ( ) = 1 c2 is the relativistic Lagrangian of a free particle. Formulate the principle of least action in geometrical form.

Exercise 4: Maxwell’s uniﬁcation of electric and magnetic phenomena resulted in the prediction of electromagnetic waves, propagating at the speed of light and in fact light represents electromagnetic waves. The covariant formulation of Maxwell’s equations reads:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 179 Cosmology

µν ν ˜µν ∂µ F = −e jem ; ∂µ F = 0 , where Fµν = ∂µAν − ∂νAµ is the anti-symmetric electromagnetic field strength µ tensor, with Aµ is the electromagnetic four-potential (classical photon field), jem is ˜µν 1 µνρσ the electromagnetic current and F = 2 Fρσ is the dual pseudotensor. Show µ that in vacuum, where jem = 0, the photon potential Aµ, in the Lorentz gauge ν ∂νA = 0, satisfies the wave-equation 2Aµ = 0. Contemplate about the role of this fact, the role of light rays in Einstein’s special relativity theory and the meaning of Lorentz invariance in this context.

Remark about the gauge condition needed: note that the field Aµ has unphysical degrees of freedom as it has four components while a physical photon only has two physical degrees of freedom. One thus requires some subsidiary conditions to eliminate the unphysical degrees of freedom, i.e. we must fix a gauge. In general ν the longitudinal component ∂νA (x) φ(x) is an independent scalar field, which however has nothing to do with the physical photon. The Lorentz condition

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 180 Cosmology

ν ∂νA = 0 eliminates this scalar component. In general the unphysical degrees of freedom ore sorted out by the requirement of gauge invariance. Indeed, Maxwell’s equations are invariant under photon ﬁeld transformation Aµ(x) → Aµ(x) − ∂µα(x), where α(x) is an arbitrary scalar function.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 181 Cosmology Appendix: technicalities on Minkowski space and Lorentz transformations

The known simple form of Maxwell’s equations of classical electrodynamics holds for a restricted class of coordinate frames only, the so called inertial frames. The space–time transformations which leave the Maxwell equations invariant form the Lorentz group. Accordingly, Lorentz transformations are transformations between different inertial frames. The invariance group of Maxwell’s equations in this way singles out a particular space-time structure, the Minkowski space. Lorentz invariance is a basic principle which applies for the other fundamental interactions. We brieﬂy sketch the elements which we will need for a discussion of a relativistic theory.

A space-time event is described by a point (contravariant vector)

xµ = x0, x1, x2, x3 = x0, ~x ; x0 = t (= time) in units where c = 1

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 182 Cosmology

in Minkowski space with metric

 1 0 0 0     0 −1 0 0  g = gµν =   . µν  0 0 −1 0     0 0 0 −1 

The metric deﬁnes a scalar product

0 0 µ ν xy = x y − ~x · ~y = gµνx y

invariant under Lorentz transformations (L-invariance) which include

1. rotations

2. special Lorentz transformations (boosts)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 183 Cosmology

As usual we adopt the summation convention: repeated indices are summed over unless stated otherwise. For Lorentz indices µ, ··· = 0, 1, 2, 3 summation only makes sense (i.e. respects L-invariance) between upper (contravariant) and lower (covariant) indices and is called contraction. Thus µ ν P P µ ν µ P µ xy = gµν x y ≡ µ ν gµν x y = x yµ ≡ µ x yµ

The set of linear transformations

µ µ0 µ ν µ x → x = Λ νx + a

which leave invariant the distance

2 µ µ ν ν (x − y) = gµν(x − y )(x − y )

between two events x and y form the Poincare´ group P. P includes the Lorentz transformations and the translations. We denote the group elements by (Λ, a). To two transformations (Λ1, a1) and (Λ2, a2), applied successively, there corresponds

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 184 Cosmology a transformation (Λ2Λ1, Λ2a1 + a2) (multiplication law of the group). The Lorentz transformations (Λ, 0) by themselves form the Lorentz group. L-invariance of the scalar products implies the invariance condition

µ ρ Λ νΛ σgµρ = gνσ

µ for the metric. This condition on the matrices Λ ν actually fully determines them 0 and implies det Λ = ±1 and | Λ 0 |≥ 1. Transformations with determinant Λ = +1 are called proper (+). Such transformations do not change the orientation of 0 frames (no space-reﬂections). Transformations with the property Λ 0 ≥ 1 are called orthochronous (↑), since they exclude time inversions (no time-reversal).

0 Λ i (i = 1, 2, 3) are the special Lorentz transformations (=boosts=velocity i transformations) and Λ k (i, k = 1, 2, 3) are the rotations. The Lorentz transformations form a group of real 4 × 4 matrices of determinant unity (pseudo-orthogonal transformations).

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 185 Cosmology

Special relativity requires physical laws to be invariant under proper ↑ ↑ orthochronous Poincare´ transformations P+. Thus P+ exhibits the general transformation law between inertial frames.

We denote by ! ∂ ∂ ∂µ = = , ∇~ ∂µ ∂0 µ 0 the derivative with respect to x = (x , ~x). ∂µ transforms as a covariant vector i.e. ν 0 it has the same transformation property as xµ = gµνx = (x , −~x). The invariant D’Alembert operator (four-dimensional Laplace operator) is given by

2 µ µν ∂ 2 = ∂µ∂ = g ∂µ∂ν = − 4. ∂x02

A contravariant tensor T µ1µ2···µn of rank n is an object which has the same µ1 µ2 µn transformation property as the products of n contravariant vectors x1 x2 ··· xn . Covariant or mixed tensors are deﬁned correspondingly.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 186 Cosmology

The Kronecker symbol

 1 0 0 0     0 1 0 0  δµ = gµρ g =   ν ρν  0 0 1 0     0 0 0 1 

is a 2nd rank mixed tensor. With its help, contracting the invariance condition of the metric with gσλ, we may write

µ λ λ Λ νΛµ = δν

which shows that µ −1 µ Λ ν = (Λ )ν is the transpose of the inverse of Λ. Covariant vectors transform like

0 ν −1 ν xµ = Λµ xν = xν(Λ ) µ

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 187 Cosmology

2 µ and the L-invariance of x = xµx follows immediately.

Keep in mind: Ì xµ, dxµ are prototype contravariant four-vectors 0µ µ ν transformation law: x = Λ ν x etc. ν ∂ Ì xµ = gµν x , ∂µ ≡ ∂xµ are prototype covariant vectors 0 ν transformation law xµ = Λµ xν etc. Thus 0 0 0µ 0 µ ρ ν ρ ν ν x y = x yµ = Λ νΛµ x xρ = δν x xρ = x xν = xy or, with help of the metric 0 0 0µ 0ν µ ν ρ σ ρ σ x y = gµνx y = gµνΛ ρΛ σ x y = gρσ x y = xy .

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 188 Cosmology

Finally we will need the totally antisymmetric pseudo-tensor5

 +1 (µνρσ) even permutation of (0123)  εµνρσ =  −1 (µνρσ) odd permutation of (0123)   0 otherwise .

With the help of this tensor the determinant of any 4x4 matrix A is given by

µνρσ α β γ δ αβγδ ε A µA νA ρA σ = det A ε . 5One easily checks that it transforms as a rank 4 tensor and that it is numerically invariant (identically the same in any inertial frame). Useful relations are

µνρσ ε εµνρσ = −24

εµνρσε 0 − δσ µνρσ = 6 σ0 ρ ρ εµνρσε 0 0 = −2δ δσ + 2δ δσ µνρ σ ρ0 σ0 σ0 ρ0 ρ ρ ρ ρ ρ ρ εµνρσε 0 0 0 = −δν δ δσ + δν δ δσ + δν δ δσ − δν δ δσ − δν δ δσ + δν δ δσ µν ρ σ ν0 ρ0 σ0 ν0 σ0 ρ0 ρ0 ν0 σ0 ρ0 σ0 ν0 σ0 ν0 ρ0 σ0 ρ0 ν0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 189 Cosmology

The 4–dimensional volume element is invariant under proper Lorentz transformations which satisfy detΛ = 1:

d4x → detΛ d4x = d4x

A 3–dimensional hyper-surface element is deﬁned by a covariant vector

ν ρ σ dS µ = εµνρσ dx dx dx and by partial integration we obtain Z Z 4 d x ∂µ f (x) = dS µ f (x) V Σ where Σ = ∂V is the boundary of V. The Gauss law takes the form Z Z 4 µ µ d x ∂µ f (x) = dS µ f (x) . V Σ

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 190 Cosmology

For an inﬁnite volume

Z Z Z +∞ Z +∞ Z +∞ Z +∞ ... → d4x ... = dx0 dx1 dx2 dx3 ... V −∞ −∞ −∞ −∞ the surface terms vanish if the function falls off sufﬁciently fast in all directions. In this case a component–wise partial integration yields Z Z 4 4 d x g(x) ∂µ f (x) = − d x (∂µ g(x)) f (x) and the integral of a divergence is vanishing Z 4 d x ∂µ f (x) = 0 .

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 191 Cosmology Postulates of Special Relativity

(1) There exist reference frames in which laws of Nature have a particularly simple form: the inertial frames (IS)

(2) The relative motion of two IS’s is straight line uniform motion

(3) Natural laws have identical form in all IS’s (if written in covariant form)

(4) In each IS Euclidean geometry is valid: the distance of two events which are simultaneous in that frame if q 2 d = ~x1 − ~x2 ; E1 :(~x1, t) , E2 :(~x2, t)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 192 Cosmology

(5) The velocity of light in vacuum is the same in all directions in all IS’s:

c = 29907920458 m s−1

c plays the role of the maximal signal velocity.

Consequence: one has to ﬁnd an IS then all others are known: Important Addition: Empirical fact is that a frame of reference in which the ﬁxed stars in average rest is an IS! Ernst Mach) → Connection to Cosmology!

Note that the key point of relativity is not relativity but invariance! The invariance of physics with respect to Poincare` transformations, in particular special Lorentz transformations (velocity transformations). The proper (det Λ = +1 i.e. no 0 space-time reﬂections), orthochronous (Λ 0 ≥ 1, i.e. no time-reﬂections) Poincare` + group P↑ is a symmetry group for the class of inertial systems.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 193 Cosmology General Relativity

1. Acceleration in Special Relativity

In an inertial system (IS) a body which is not subject to any forces is freely falling. Its world line is xµ = uµ · s where uµ is the four-velocity, u2 = 1, and s the proper time.

Now we transform this to a non–IS, i.e. to curved coordinates.

Coordinates in the IS: xµ; arbitrary curved coordinates: x0µ = f µ(x0, x1, x2, x3). The change in coordinates x0µ under a change of the original coordinates xµ is

∂x0µ ∂xα we assume to be non–singular (non-degenerate) and the differentials change according to

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 194 Cosmology

µ ∂xµ 0α dx = ∂x0α dx

◦ 2 µ ν 0 0µ 0ν ds =gµν dx dx = gµν(x ) dx dx

Thus in the non–IS we see the metric ◦ 0 g ∂xρ ∂xσ gµν(x ) = ρσ ∂x0µ ∂x0ν

◦ 0 Note that gµν(x ) can be arbitrarily complicated, while gµν is simple, the Minkowski metric:

 1 0 0 0    ◦  0 -1 0 0  g =   µν  0 0 -1 0     0 0 0 -1 

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 195 Cosmology

In case the system x0µ is an IS xµ → x0µ is a Lorentz transformation and thus ◦ 0 gµν(x ) =gµν. In all other cases the deviation ◦ gµν− gµν describes acceleration.

Note that in the new non–IS with world-line

dx0µ x0µ = x0µ(λ) ;x ˙0µ = , dλ

the distance between events A and B is

B B Z Z q 0 0µ 0ν sAB = ds = gµν(x )x ˙ x˙ dλ . A A

sAB independent of arbitrary coordinate changes and independent of the parametrization.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 196 Cosmology

2. Gravity

Gravitation is the force which lets fall the apple from the tree and which lets circle the moon around the earth. Newton was the ﬁrst who realized that the same force is responsible for both phenomena. Gravitation rules the motion of celestial bodies: planets, the Sun, double stars, galaxies, clusters of galaxies etc. and the universe as a whole.

Gravitation acts universally on all bodies in the same manner. In contrast, for example, to the electromagnetic force, which only acts on charged bodies. In addition there is no neutralization, there is no screening of gravitation, no repulsion only attraction.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 197 Cosmology

Newton’s classical form

∆φ(x) = 4πGN µ(x) m ~x¨ = −∇ φ(x) is not compatible with the special relativity principle. Naive relativistic generalizations do not take into account of the true nature of gravity. Einstein’s conclusion: the principle of relativity must be modiﬁed!

Key: The equivalence principle!

3. The Equivalence Principle

Gravitation is a very weak universally attractive interaction. It manifests itself essentially only in the presence of large masses and it cannot be screened by any means.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 198 Cosmology

F~ According to Newton’s theory of gravity ~r M m ~ Force: F γ g g m F = G r2 ; g additive − F mg Mg Acceleration: a = = γG mi mi r2

Experimental fact: mg ≡ mi for an appropriate choice of the units (deﬁnition of γG). This has been veriﬁed with very high accuracy!

The consequence: there exist a gravitational potential φ = φ(~x, t), such that ~x¨ = −∇ φ(~x, t) describes the motion of arbitrary bodies in a gravitational ﬁeld.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 199 Cosmology

This means that bodies fall completely independent of any properties at the same speed. Therefore freely falling bodies are gravity free. In the framework of Newton’s theory this fact remains accidental.

Einstein: mg = mi has fundamental meaning!

The following “Gedankenexperiment” illustrates the point: an observer in a sufﬁciently small box

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 200 Cosmology

a = g −

free body free body g g inertial force gravity force

engine gravity

Left: box in gravity free space accelerated (I). Right: box in gravitational ﬁeld non-accelerated (II).

If the box is sufﬁciently small, such that the gravitational ﬁeld ~g=constant within the box we can say:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 201 Cosmology

There exists no experiment by means of which the observer in the box can tell in which situation (I) or (II) he is. Therefore the equivalence principle must hold:

To each event in an accelerated reference frame (RF) there corresponds the same event in a non-accelerated RF, which is subject to a gravitational ﬁeld. In other words: a no–IS is equivalent to a gravitational ﬁeld.

Consequences:

O also accelerations have no absolute meaning (as the velocities within the framework of special relativity already).

O inertial forces and gravity are uniﬁed (they are locally indistinguishable) and dis- tinguishing inertial and gravitational mass is not possible as a matter of principle

O freely falling boxes (IS’s) are force–free; in them all bodies move uniformly on straight lines, i.e. freely falling systems represent local inertial systems (IS’s)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 202 Cosmology

The equivalence principle may by formulated as a generalized relativity principle:

For freely falling observers special relativity holds (existence of local Lorentz systems)

Fact: all gravitational fields observed in nature are inhomogeneous at large. They result from mass concentrations like stars etc. and vanish at infinity. It is at least natural to assign such behavior to true gravitational fields.

Consequently:

O in the presence of gravitational ﬁelds there only exist local ,i.e. in a limited region of space–time, freely falling Lorentz frames. In contrast, if gravitational ﬁelds are absent, a global Lorentz system would exist.

O in the presence of gravitational ﬁelds the relationship between different IS’s in general is very complicated.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 203 Cosmology

O the difference between inertial fields and “true” gravitational fields is a matter of the behavior of the fields at infinity.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 204 Cosmology

The Role of the Metric gµν(x) No gravitational ﬁelds: there exists a global IS (whole Minkowski space), such that

◦ 2 µ ν 4 ds =gµν dx dx ∀x ∈ M (space − time) in all space i.e. space-time is ﬂat In a non–IS one obtains

2 µ ν ds = gµν(x) dx dx where gµν(x), the space-time metric, describes the geometry of the curvilinear coordinate system. As a non–IS is equivalent to a gravitational field the “metric tensor” gµν(x) = gνµ(x) (10 independent components) obviously must describe the local gravitational field also if the field is a real gravitational field.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 205 Cosmology

Real gravitational ﬁelds: according to the equivalence principle they must be described by the metric

2 µ ν ds = gµν(x) dx dx

For a true gravitational ﬁeld the is no choice of a RF for which the ﬁeld can be transformed away globally (i.e. in all of space).

Since gµν has 10 independent components with 4 independent transformations one cannot achieve that ◦ gµν(x) =gµν on the entire space.

Consequently: in the presence of gravity space-time is curved!

=⇒ Riemann Geometry!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 206 Cosmology

The metric gµν(x) in any case is determined only modulo arbitrary coordinate transformations (four arbitrary non-singular differentiable functions).

Illustration: accelerated rocket and the behavior of light as a consequence of the equivalence principle.

An accelerated box acts like a gravitational ﬁeld =⇒ light rays must be bent in a gravitational ﬁeld!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 207 Cosmology

light ray

Indeed: solar eclipse 1919 starlight is bent by the gravitational ﬁeld of the sun!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 208 Cosmology Curved Spaces Familiar examples: A) Positively curved space

Example: Neighborhood of the Sun

path of a A α light ray γ

α + β + γ > 180◦ C

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 209 Cosmology A) Negatively curved space

Example: Carousel (merry-go-round)

path of a light ray when carrousel at rest

B path of a light ray A’ as seen in B’ B rotating path of a system β light ray A as seen in A α rotating system

α + β + γ < 180◦ C

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 210 Cosmology

Result: locally, i.e., in sufﬁciently small regions

accelerated systems clocks go slower are equivalent to « scales get shorter systems subject to gravity

Shortest distance: straight line ⇔ geodesic line

The shortest distance between two points or the path of a light signal do not coincide any longer with the straight line between the points.

“Straight Line” must be replaces by “Geodesic Line”, the path a freely falling body takes between the two points.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 211 Cosmology

Notions like straightest, directest, shortest path may have different meaning now. For example: the cheapest way to get from New York to Paris is not taking the straight line between the two cities but the great circle:

geodesic on the sphere cut through earth center

NewYork

not the easiest way to get from New York to Paris Paris

Einstein: geometrical model

Instead of saying that light rays or shortest distances are bent, one may say space is curved!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 212 Cosmology The New View of the World On a bumpy square some people play with bowls. One observes that some spots are avoided others preferred.

An observer on place says: the bowls are rolling according to the curvature of the terrain (Einstein’s point of view).

A distant observer, to whom the square looks plain, would conclude that certain (unexplained) forces act on the bawls (Newton’s point of view).

Newton matter ⇐===⇒ acts on matter ⇐ ⇒ ⇓ ⇑ Einstein curvature « curvature here over there

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 213 Cosmology x “Space tells matter how to move” y “Matter tells space how to curve”

⋆ ⋆

seeming light path true light path Sun apparent direction true direction

Earth

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 214 Cosmology Non-Euclidean Geometry

Gauss 1817, Lobachevsky 1826, Bolyai 1832, Riemann 1854 -≫

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 215 Cosmology

« Positively curved, ﬁnite, unbounded (no boundary), prototype sphere curvature type: k = 1 circles (radius r): A < π r2, C < 2 π r triangles: α + β + γ > 180◦

« Not curved, inﬁnite, Minkowski space = pseudo Euclidean is also ﬂat! prototype plain curvature type: k = 0 circles (radius r): A = π r2, C = 2 π r triangles: α + β + γ = 180◦

« Negatively curved, inﬁnite, prototype saddle curvature type: k = −1 circles (radius r): A > π r2, C > 2 π r triangles: α + β + γ < 180◦

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 216 Cosmology Example: 4D surface of a sphere

We consider a sphere of radius R in 5D ﬂat space. The corresponding 4D surface is the curved space we want to look at. The 4D surface appears “embedded” in 5D ﬂat space.

R 4D

We denote the coordinates on the 4D sphere by xi (i = 1, 2, 3, 4), w is the 5th coordinate in 5D space and R the radius of the sphere:

2 2 2 2 2 2 x1 + x2 + x3 + x4 + w = R = constant

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 217 Cosmology

The metric follows from an inﬁnitesimal variation of the coordinates

xi → xi + dxi expanded to linear order:

2 x1 dx1 + 2 x2 dx2 + 2 x3 dx3 + 2 x4 dx4 + 2 w dw = 0 , which we may write

µ xµ dx x dxµ + w dw = 0 or dw = . µ w

Thus, with |x|2 + w2 = R2 ⇒ w2 = r2 − |x|2

µ ν xµxνdx dx (dw)2 = R2 − |x|2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 218 Cosmology

The metric of 5D Euclidean space is

2 µ ν 2 ds = δµνdx dx + (dw) and for the 4D spherical surface we obtain by eliminating dw s2 δ xµxν xµ xν d = µν + R2−|x|2 d d and thus we ﬁnd g x δ xµxν µν( ) = µν + R2−|x|2 for the metric of the surface of the sphere. It delivers the distance measurement on the sphere.

How can we know whether we life in a curved space? For example, suppose space-time is a 4–dimensional surface of the sphere: if we proceed in a given direction we come back to the starting point.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 219 Cosmology

Locally curvature is characterized by vector parallel transport.

Examples:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 220 Cosmology

ν General: R µ,ρσ Riemann’s curvature tensor

Parallel transport of a vector Aµ along a closed square determined by the shifts (∆u, ∆3)) in the parameters which parametrize the world sheet: xµ = xµ(u, 3). The mismatch ∆Aν of the parallel transported vector Aµ is a local measure of the curvature, and is proportional to the vector and to the area of the square. The curvature tensor is the corresponding tensor coefﬁcient.

A2 2 ∆Aν ν µ ∂xρ ∂xσ 3 ∆u∆3 = R µ,ρσ A ∂u ∂3 A3 ∆u, ∆3 → 0 ∆A A A 1 A4 0 ∆v→ 1 0 ∆u

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 221 Cosmology Einstein’s Equations

Newton: K γ m1 m2 m a m a = r2 = 2 2 = 1 1 (mass) (mass) force ∝ 1 2 = (inertial mass) × acceleration (distance)2 where mass=gravitational mass = inertial mass

1 Einstein: Rµν − 2gµν R = κ Tµν

! ! geometric curvature matter distribution ∝ of space in space

Riemann: metric gµν → Rµν,ρσ curvature tensor

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 222 Cosmology

ρσ µν gρσ Rµν = Rµν ; gµν R = R

1 Rµν Ricci tensor, R scalar curvature, Gµν = Rµν − 2gµν R Einstein tensor The Einstein’s equation is a partial differential equation of 2nd order for the metric gµν for given Tµν (sources)

Note that the coordinate volume element d4x is not the geometrical volume element, which must be invariant under coordinate transformations. The latter is obtained by including the root of the modulus of the determinant of the metric:

4 p dVinvariant = d x − det g .

Exercise: show that dVinvariant is invariant under general coordinate transformations.

The geometrical volume element is an important ingredient in constructing the

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 223 Cosmology invariant action for gravity:

c4 R 4 √ SG = d x −g R 16π GN where g = det gµν is the determinant and R is the curvature scalar.

Application to the Sun (stars):

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 224 Cosmology Schwarzschild Solution Prediction of GR for spherically symmetric mass distributions:

A spherically symmetric system in a framework of general coordinate invariance means: there exists a coordinate system in which the solution looks spherically symmetric

θ ~r

y ds2 = ea c2 dt2 − eb dr2 − r2 dΩ2 ϕ

x where dΩ2 = dθ2 + sin2 θdϕ2

The last term is ﬁxed by rotational invariance and the asymptotic ﬂatness condition, i.e., for r → ∞ the last term dominates and and must be Euclidean.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 225 Cosmology

Einstein’s equation in external space, where Tµν = 0 yields

r rr0 2 φ ea = eb = 1 − 0 → 1 + r c2 where

G M φ = − N r is Newton’s potential. Thus r0 is ﬁxed by the total mass to

r 2 GN M 0 = c2 the Schwarzschild radius.

Typical numbers for r0 follow:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 226 Cosmology

33 Sun: M ' 2 × 10 gr 2 r0 ' (2 GN M ) /c ' 3 km 5 R ' 7 × 10 km

Earth: M ' 6 × 1027 gr ' 2 ' r0 ♁ 2 GN M /c 0.9 cm R ' 6.4 × 103 km ♁ ♁ ♁ Planet closest to the Sun: distance to Sun r ' 0.5 × 108 km

Mercury: M ' 3 × 1026 gr ' 2 ' r0 ' 2 GN M /c 0.45 mm 3 R '' 2.4 × 10 'km ' −8 Note that r0 /r ∼ 6 × 10 1 which is Newton’s regime −7 similarly, m /M ' 1.5 × 10 1 which tells us that mercury is a test particle. ' c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 227 Cosmology

2 Black Hole (BH): RBH < r0 BH = (2 GN MBH) /c → very massive object (see below)

A black hole is obtained if the size of the “star” is smaller than the Schwarzschild radius. The latter is a coordinate singularity and for a far away observer photons emitted get infinite red shift at r0, which means photons emitted by the BH (inside r0) get trapped by the strong gravitational field and can never reach a distant observer. r0 in this case is a horizon. A traveler approaching the BH reaches the horizon in a finite proper time and does not see a singularity in his local coordinate frame,i.e, the singularity is not physical, still communication with a very distant observer (i.e. r → ∞) breaks down.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 228 Cosmology Geometrical Interpretation

Spatial metric

1 d`2 = dr2 + r2 dθ2 + sin2 θdϕ2 r0 1 − r

r0 d` is the frame independent proper length. We denote β = 1 − r ≥ 1.

For r r0, β → 1: Euclidean and we have the proper normalization.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 229 Cosmology

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 230 Cosmology

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 231 Cosmology

Radial behavior: θ, ϕ = constant

r0 ∆s1 ∆s2

r1 r2

For equal ∆r

1 ∆s1 = q ∆r 1 − r0 r1 1 ∆s2 = q ∆r 1 − r0 r2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 232 Cosmology

Thus proper length of rods shorten the more the closer to r0:

q r 1− 0 ∆s1 r2 = q r ∆s2 1− 0 r1

This compares to the Lorentz contraction between moving IS’s in Special Relativity.

Circumference of circle:

∆C = 2π∆r p ∆s = β ∆r

This yields:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 233 Cosmology

∆C r0 ∆s ' 2 π 1 − 2R

−5 and for the Sun with r0/R ' 0.43 × 10 we get

R −9 ∆C|∆s=1 m ' 2 π 1 − 0.7 × 10 m ∆s

Radial behavior, time dependent

1 ds2 = c2 dt2 − β dr2 β

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 234 Cosmology

Clock in Schwarzschild ﬁeld at rest: dr = 0

s c T c √1 t d = d = β d

£ clock go slower!

Compares to time dilatation in Special relativity.

This also directly implies a red shift. On observer far away observes a red shifted object: T x0/c T √1 x0/c ∆ ∞ = ∆ ; ∆ R = β ∆

−6 This implies e.g. ∆ν ' φ − φ ' 2.12 × 10 (φ Newton potential). ♁

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 235 Cosmology General covariance and Riemannian geometry

In presents of gravitational ﬁelds physical laws must be written in a general covariant way, which does not depend of the particular Coordinate System (CS)

The notion of CS has a fundamentally different meaning than in special relativity, where a class of ISs exist for which physical laws are particularly simple. The equivalence principle implies that the principle of special relativity can only hold locally, for freely falling bodies/observers.

Of course, also in ﬂat space, where global ISs exist, we may formulate physical laws in arbitrary coordinates. This happens by substitution by general covariant expressions:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 236 Cosmology

IS Non-IS ◦ ◦ Metric: gµν → gµν(x) ,gµν µ µ ◦ ◦ µ µ Γ ρσ≡ 0, R ν,ρσ≡ 0 Γ ρσ , 0, R ν,ρσ ≡ 0 Derivatives: ∂µ → Dµ → Tensors: Poincare´ tensor generally covariant√ tensor Tensor densities: 1 → −g In special RT different ISs are related by linear transformations (pseudo orthogonal) and physical laws are linearly equivalent. In general RT CSs are related by non-linear transformations and one cannot claim that all CSs are physically equivalent. The concrete form of motions depends on the CS.

Most importantly, in the presence of gravitational ﬁelds the metric takes a dynamics role and is promoted to a physical quantity. General covariance is the key tool to separate internal geometrical properties of space-time from coordinate effects (inertial effects).

The formal consequence of the above discussion is that in the presence of

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 237 Cosmology gravitational ﬁelds space-time is promoted to a Riemannian space:

The event space is assumed to be a topological Hausdorff space (distinct points have disjoint neighborhoods).

For freely falling bodies/observers Minkowskian geometry holds (existence of local maps), i.e., there exist local CSs and we can perform coordinate transformations.

The event space can be covered by countable many local CSs (existence of an atlas of maps) and is assumed to be connected (one universe).

There exists a covariant 2nd rank tensor ﬁeld, the space-time metric gµν(x) of signature (+ − −−) and 2 µ ν ds = gµν(x) dx d x is the fundamental metric form. The metric gµν(x) provides the scale lengths, the clock times and exhibits the gravitational ﬁeld.

Up to singular points continuous differentiability is assumed.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 238 Cosmology

Formally, the metric gµν(x) comprises the entire structure of the Riemannian space

In particular the metric determines the geometrical arc length, geodesic curves (minimum or maximum length), parallel transport of vectors and tensors (Christoffel symbols or the afﬁne connection) as well as the curvature (curvature tensor) of a Riemannian manifold. More details are given in the following and in an Appendix below.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 239 Cosmology GR Formal Tools

Space-time equipped with metric, events:

4 x ∈ M ; gµν ; g = det gµν < 0 ; signature + − − −

∂ Tetrad (natural “4-bein”): eµ ∂xµ basis of tangent–space at point x.

Relationship between neighboring tetrads of inﬁnitesimally close points:

α ν deµ = Γ µν eα dx

Christoffel symbols: = afﬁne connection (gauge potential in gauge theories)

ρ 1 ρσ n o Γ µν g ∂µ gνσ + ∂ν gµσ − ∂σ gµν 2

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 240 Cosmology

Covariant derivative: Dµ ϕ(x) = ∂µ ϕ(x);Dµ gρσ(x) = 0

ν ν ν ρ Dµ A (x) = ∂µ A (x) + Γ ρµ A (x) ρ Dµ Bν(x) = ∂µ Bν(x) − Γ νµ Bρ(x) etc.

Parallelism: let xµ = xµ(λ) be a path C ν ν ν µ A is parallel along C ./ DA = Dµ A x˙ dλ = 0

ν A (P0) ν A (Pε) || P0 ν ν − ν Aν (P ) DA A||(Pε) A (P0) Pε ε =lim dλ ε→0 ε

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 241 Cosmology

Curvature Tensor: DµDν − DνDµ ϕ(x) = 0

ρ ρ σ DµDν − DνDµ A (x) = R σ,νµ A (x) σ DµDν − DνDµ Bρ(x) = −R ρ,νµ Bσ(x) etc.

Area A: xµ = xµ(u, 3) Surface element: ∂xρ ∂xσ ∂xσ ∂xρ dΣρσ ∆u ∆3 − ∆u ∆3 ∂u ∂3 ∂u ∂3 Shift of vector in parallel transport along inﬁnitesimal closed path: ν ν ν ∆A = A||(C) − A

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 242 Cosmology

2 ν 1 ν µ ρσ A1 ∆A = R A dΣ 2 µ,ρσ 3 ν ρ σ ∆A A0 ∆A ν µ ∂x ∂x lim = R µ,ρσ A A3 1 ∆u∆3→0 ∆u∆3 ∂u ∂3 A4 ∆u ∆րv 0

measure for the curvature of space!

ρ Ricci Tensor: Rµν(x) = R µ,νρ(x) µν Scalar Curvature: R(x) = g Rµν(x) 1 Einstein Tensor: Gµν(x) Rµν(x) − 2 gµν(x)R(x) µ Bianchi Identity: Dµ G ν ≡ 0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 243 Cosmology

The Riemann tensor can be written in terms of the Christoffel symbols:

ρ ρ ρ ρ λ ρ λ R σ,µν = ∂µ Γ νσ − ∂ν Γ µσ + Γ µλ Γ νσ − Γ νλ Γ µσ as it may be worked out from the deﬁnition by parallel transport around inﬁnitesimal “parallelograms”.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 244 Cosmology

Symmetry properties: gµν(x) = gνµ(x)

ν 1 Γµ,ρσ gµν Γ = ∂ρ gσµ + ∂σ gρµ − ∂µ gρσ ρσ 2 Γµ,ρσ = Γµ,σρ ; Γµ,ρσ + Γσ,ρµ = ∂ρ gµσ α Rµν,ρσ gµα R ν,ρσ

Rµν,ρσ = −Rµν,σρ ; Rµν,ρσ = −Rνµ,ρσ

Rµν,ρσ = Rρσ,µν X Rµ[ν,ρσ] = 0 ; [...]: cycl. def Dλ Rµν,ρσ = Rµν,ρσ;λ : Rµν,[ρσ;λ] = 0 Bianchi − Identity

Rµν = Rνµ ; Gµν = Gνµ

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 245 Cosmology

Geometrical arc length:

ZB 2 µ ν sAB = ds ; ds = gµν(x) dx dx A fundamental metric form.

B µ µ R p µ ν x = x (λ) sAB = gµνx˙ x˙ dλ A A RB Geodesic lines: ds extremal A 2 µ ρ σ d x + Γµ dx dx ds2 ρσ ds ds

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 246 Cosmology

◦ ρ Theorem 1: In ﬂat space (∃ CS, such that gµν =gµν globally) ⇔ R σ,µν ≡ 0

0 ∂xρ ∂xσ Note: g (x ) = g null 0 0 is system of differential equations, coordinates are µν ρσ ∂x µ ∂x ν ρ globally determinable, but is not necessarily Minkowskian, except iff R σ,µν = 0 (integrability condition). On the one hand the space corresponding to a homogeneous gravitational ﬁeld is ﬂat.

However, a global homogeneous gravitational ﬁeld is unphysical! On the other hand ρ R σ,µν , 0 (space curved) is a measure for existing gravitational ﬁelds

2 µ Theorem 2: In ﬂat space (∃ CS, such that Γρ = 0 globally) ⇔ d x = 0 µν ds2 µ µ µ i.e. x = u s + x0 £ geodesics are straight lines.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 247 Cosmology Einstein Equation In case of Maxwell’s equations one has to construct a vector valued function of the gauge four-potential Aν, at most of second order in the derivatives, such that

µ ν µ f (∂α∂βA ) = µ0 j (x) where jµ(x) is the Hermitian conserved covariant electromagnetic four-current: µ ∂µ j = 0. This ﬁxes

µ ν µ ν µ µν f (∂α∂βA ) = ∂ ∂νA − 2 A (x) = ∂ν F .

In gravity, matter is described by the energy-momentum tensor Tµν in place of the current. Explicit examples will be discussed later. Here we assume Tµν to exist and to have the properties

O it is covariantly conserved: Dµ Tµν = 0,

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 248 Cosmology

O it is symmetric: Tµν = Tνµ,

O it is Hermitian.

Up to a constant, which is ﬁxed by the Newtonian limit, the r.h.s. of the ﬁeld equation we are looking for is

8πG N T c2 µν while the l.h.s. should be a function of gµν and derivatives of it up to second order,

8πG f µν(g , ∂ g , ∂ ∂ g ) = N T αβ ρ αβ σ ρ αβ c2 µν

By the properties of Tµν we require

µν µν νµ Dµ f = 0 ; f = f .

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 249 Cosmology

Here one may use a Theorem of Cartan: If we require f µν to be linear in the second derivatives, then f µν must have the form

fµν = a Rµν + b gµν R + c gµν , which on the one hand says that ﬁrst derivatives of gµν give no covariant tensor, because Dρgµν ≡ 0. On the other hand second derivatives give a tensor which has to be related to the curvature tensor.

We then have

µ µ D fµν = a D Rµν + b DνR ,

µ 1 which together with the Bianchi identity D Rµν = 2 DνR and the condition that Tµν being conserved yields

a Dµ f = + b D R = 0 , µν 2 ν

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 250 Cosmology and thus a b = − . 2

The constant multiplying Tµν on the r.h.s. is chosen such that we may set a = 1 (classical limit) and we have the result

1 Rµν − 2 gµν R + c gµν = κ Tµν

8πG κ = N = 1.86 × 10−27 cm/gr c2

◦ Einstein (Mach) concept: empty space Tµν ≡ 0 ./ ∃ Coordinate System: gµν =gµν requires c = 0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 251 Cosmology c cosmological constant. Iff c = 0 empty space ↔ ﬂat space equivalent matter-geometry duality matter ↔ curved geometry

Einstein’s equation: 1 Rµν − 2 gµν R = κ Tµν

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 252 Cosmology Experimental Tests of GR see also: -≫ and references there

Newton limit

GR yields all results of Newton’s gravity theory in weak gravitational ﬁelds and for slowly moving bodies. This is true for smallest details of planetary motion etc.

Mercury perihelion shift

According to Newton planetary orbits are exact ellipses (discarding usually tiny perturbations by third party bodies). Einstein predicts that ellipses in fact are to be replaced by rosette trajectories. The effect is largest for the planet closest to the sun, mercury.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 253 Cosmology

The shift predicted for the rotation of the perihelion (the closest point of the trajectory to the sun), L the angular momentum

3 G2 M δϕ ' π N 2 c4 L2

00 Per 100 years predicted: 42.98 (arc seconds) 00 observed: (42.98 ± 0.043)

Experiment Test: Theory = 1.000 ± 0.001

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 254 Cosmology

Observations are optical and radar, accuracy 0.1%

The perihelion shift has also been observed for Venus: predicted 8.6 seconds of arc per century, and 8.4 seconds has been observed.

In 1975, binary pulsar PSR 1913+16 was discovered. The two massive sources, in very tight orbit, display a perihelion advance of 4.23 degrees per year!

Light deﬂection by gravitation

Light passing at distance R from the Sun gets deﬂected by

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 255 Cosmology

δ ∼ 4 GN M c2 R

1 2 which is twice to value one get in Newton’s theory δNewton = 2 δ using m = E/c as an effective photon mass, E the photon energy.

P”

δ ϕ Q R observer ⊙ Sun P P’ star

Recent measurements Very Long Baseline Interferometry (VLBI), observation of radio sources and quasars. 00 Light passing Sun predicted: 1.75 (arc seconds) 00 observed: (2.2 ± 1.5) (1919) 00 Radio waves observed: (1.73 ± 0.05) (1972)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 256 Cosmology

Experiment Test: Theory = 1.000 ± 0.001

Gravitational red shift

2 Photon is heavy: mγ,eﬀ = Eγ/c , light from the sun requires energy to leave gravitational ﬁeld of the sun. The corresponding red shift is

z = δλ ' M ∼ 10−6 λ R which is too small in comparison with other perturbations.

For a back hole: z = ∞ photons cannot escape any longer!

Well known: laboratory experiments (Mossbauer¨ effect), earth gravitational effect

Pound & Rebka 1960:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 257 Cosmology

δλ m z ' h ∼ −16 h = 2 10 (in meters) λ R♁ ♁

∆ν −16 For h = 22.6 m ; ν = 2.5 × 10 ± 10%

Today: accuracy 1%, Voyager 1 in the ﬁeld of Saturn

Radar echo delay (Shapiro)

The idea is to compare radar echo measurement of close to the sun planets and satellites:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 258 Cosmology

P Planet

R1 a) beam passes close to sum b) beam passes very far from sum R 0 Q Signal delay:

Sun R2 4 R1 R2 δt ∼ 4 m log 2 + 1 R0

Earth Viking mission: two mirrors on Mars, two mirrors in orbit

Predictions conﬁrmed at 0.1% accuracy!

Cassini probe: more recently gives a 0.002% test of general relativity

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 259 Cosmology Gravitational Lenses Unlike optical lenses gravitational lenses have a focal line in place of a focal point, which in the ideal case of exactly aligned optical axes gives raise to the so called Einstein ring (quasar 1938). Theory see -≫.

The picture shows an Einstein cross, a quasar (2237) behind a galaxy (in the center) is seen four times due to collimation of light be the galaxy -≫. In fact

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 260 Cosmology gravitational lenses are defocusing rather than focusing the light from a source! For Einstein-Cross calculations see: -≫

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 261 Cosmology

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 262 Cosmology

NASA Hubble Space Telescope image of the rich galaxy cluster, Abell 2218, is a spectacular example of gravitational lensing. The arc-like pattern spread across the picture like a spider web is an illusion caused by the gravitational ﬁeld of the cluster -≫ -≫.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 263 Cosmology Evidence for the Existence of Black Holes Black Hole in center of our Galaxy established A. Eckart & R. Genzel, Nature 383 (1996) 415. See: -≫ -≫

6 Velocity ﬁeld reveals black hole of mass 2.45 ± 0.4 × 10 M , New key tool: Infrared/Submillimeter Astronomy

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 264 Cosmology

In normal optical astronomy center of Milky Way hidden by stars and dust. In infrared motion of stars in galactic center observable: ﬂux of a faint source increased by a factor of 5-6 and fainted again after about 30 min (ﬂares)

Proﬁle of proper motion reveals unambiguously the existence of a black hole of a 6 few 10 M

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 265 Cosmology

6 More recently (2002) 4.31 ± 0.38 × 10 M has been determined for the Sagittarius A complex in the galactic center.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 266 Cosmology

Flares:

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 267 Cosmology

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 268 Cosmology

Look at animations here: -≫

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 269 Cosmology Black Holes and Galactic Centers There are strong indications that many (most?) galaxies host a black hole:

9 Center: Black Hole ∼ 1.2 × 10 M ?

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 270 Cosmology

Credit: D. Richstone (U. Mich) et al., HST, NASA

Do all galaxies have black holes at their centers? Although not even a single galaxy has yet been proven to have a central black hole, the list of candidates is growing. Recent results from the Hubble Space Telescope now indicate that most - and possibly even all - large galaxies may harbor a black hole. In all the galaxies studied, star speeds continue to increase closer the very center. This in itself

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 271 Cosmology indicates a center millions of times more massive than our Sun is needed to contain the stars. This mass when combined with the limiting size make the case for the black holes.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 272 Cosmology Geodesic effect and Lense-Thirring effect

NASA’s Gravity Probe B by means of extremely precise gyroscopes for the ﬁrst time has been able to evidence two effects predicted by Einstein’s GRT in the gravitational ﬁeld of the earth:

G The ﬁrst is the geodetic effect, which is the warping of spacetime around a gravitational body, such as a planet.

G The second effect of gravity is frame dragging (Lense-Thirring effect), which is the amount that a spinning object pulls the fabric of spacetime along with it.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 273 Cosmology

An artist’s concept of Gravity Probe B orbiting Earth, which is warping spacetime.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 274 Cosmology Gravitational Waves GR predicts gravitational waves, they are extremely hard to detect, usually background effects are much bigger than the expected signal.

A new generation of gravitational wave antennas based on Laser Interferometry are presently under construction or are taking data: LIGO (photo), VIRGO, GEO600, LISA and TAMA 300. They serve to detect violent processes in space like gamma ray burst. So far no signal established.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 275 Cosmology

Gravitational waves were indirectly conﬁrmed to exist when observations were made of the binary pulsar PSR 1913+16, for which the Nobel Prize was awarded to Hulse and Taylor in 1993 -≫.

Explained by energy loss due to gravitational wave emission!

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 276 Cosmology Perspectives G Solar system weak ﬁelds at 10−3 precision

G Binary Pulsars strong ﬁelds at 10−3 accuracy

G Equivalence Principle: STEP (Satellite Test of the Equivalence Principle) experiment by NASA+ESA 10−17 (at present 10−12)

G Constancy of gravitational constant GN:

1 ∆GN < 10−12/year GN ∆t

Monitoring the position of the Moon with lasers, Viking radar, binary pulsar,...

G planned: NASA space-gyroscope, studying weak ﬁeld effect at 10−5

G Gravitational wave searches;laser interferometry (Michelson experiment with Fabry Perot´ interferometer): LIGO, VIRGO, GEO600, LISA, TAMA300

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 277 Cosmology

Expect to observe violent events in galaxies (gamma-ray bursts etc)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 278 Cosmology

For more science ﬁction see -≫

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 279 Cosmology Tests beyond GR Above we have discussed predictions of GR and their confrontation with experiments. There is another approach, so to say the inverse problem: which attempts to determine the metric by measurements of as many observables as possible. Experiments are guided here by the general approach assuming an arbitrary “Metric Theory of Gravity” where the metric is parametrized according to the “parametrized post-Newtonian (PPN) formalism”:

GR PPN gµν = gµν + δgµν ,

PPN where the perturbation δgµν in leading order depends on 3 physical parameters α,¯ β¯ and γ¯. Given that GR describes observations successfully the PPN parameters are small perturbations. For a recent review see -≫.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 280 Cosmology Appendix: geodesic lines and covariant derivatives A) Geodesic Curves For simplicity we assume a Riemannian manifold with positive deﬁnite metric here. The basic features carry over to what we need in GR.

Curve Length

µ Let x (λ), λ0 ≤ λ ≤ λ1 be a parametrized curve between points P0 and P1. Then

Z λ1 p µ ν dλ gµνx˙ x˙ λ0 i) is invariant under coordinate transformations, ii) and invariant under reparametrization, and deﬁnes the length of the curve from P0 to P1.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 281 Cosmology

– i) follows from

µ ν 0 0ρ 0σ gµν(x)x ˙ x˙ = gρσ(x )x ˙ x˙ which is easily veriﬁed.

– ii) follows considering a change of the parameter λ = λ(λ0) which we take as a 0 0 monotonic and smooth function with λ0 = λ(λ0) and λ1 = λ(λ1). Then

λ λ 1 !1/2 1 !1/2 Z dxµ dxν Z dxµ dλ0 dxν dλ0 dλ g = dλ g µν dλ dλ µν dλ0 dλ dλ0 dλ λ0 λ0 0 λ λ 1 !1/2 1 !1/2 Z dxµ dxν dλ0 Z dxµ dxν = dλ g = dλ0 g µν dλ0 dλ0 dλ µν dλ0 dλ0 λ 0 0 λ0

q.e.d. K

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 282 Cosmology

Arc length from P0 to a variable point P is given by

Z λ µ 0 p µ ν µ dx p µ ν s(λ) = dλ gµνx˙ x˙ ;x ˙ = 0 , or equivalently˙ s = gµνx˙ x˙ , λ0 dλ respectively, 2 µ ν 2 µ ν (s˙) = gµνx˙ x˙ ⇔ ds = gµν dx dx , which is the fundamental metric from of the Riemannian manifold M. Since s(λ) is a monotonically increasing function of λ the arc length s may be utilized as a curve-parameter, where

!2 ds dxµ dxν = 1 = g , ds µν ds ds

µ dxµ which means that the corresponding four-velocity u ds is a vector of modulus unity.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 283 Cosmology

Distance

For positive definite metric spaces on defines: Def: Distance d(P0, P1) between two points P0 and P1 ≡≡ infimum of the lengths of piecewise smooth curves between P0 and P1. For Minkowski type metric spaces (as relevant for special and general relativity) the corresponding generalized distance notions have been discussed earlier.

Geodesic Lines

Determining the shortest distance between two points on a manifold in general is a difﬁcult problem. A much simpler problem is to ﬁnd a system of differential equations for the more general class of curves of stationary length which includes the shortest ones.

Formulation of the problem µ Let x (λ); λ0 ≤ λ ≤ λ1 describe a smooth curve K0 between two ﬁxed points P0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 284 Cosmology and P1. Each possible neighboring curve K1 may be represented in the form

µ µ µ K1 : z1(λ) = x (λ) + ε y (λ) with ε = 1 ,

µ µ µ with the help of a smooth function y (λ); λ0 ≤ λ ≤ λ1 with y (λ0) = y (λ1) = 0.

P1 P1

K1 Kε P0 P0

With

µ µ µ Kε : zε(λ) = x (λ) + ε y (λ)

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 285 Cosmology we obtain a family Kε (−1 ≤ ε ≤ +1) of neighboring curves of K0 of length L(ε). Def: K0 is called curve of stationary length or geodesic line, if

dL (0) = 0 dε

µ µ µ for any choice of the function y (λ) which satisﬁes y (λ0) = y (λ1) = 0. Formally, we may consider K0 as an extremal curve of the variational problem:

Z λ1 p µ ν δL = δ dλ gµνx˙ x˙ = 0 . λ0

Analytic implementation Given

Z λ1 n µ µ ν ν o1/2 L(ε) = dλ gµν(x + ε y) (x˙ (λ) + ε y˙ (λ))(x˙ (λ) + ε y˙ (λ)) λ0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 286 Cosmology we obtain

Z λ1 ( ) dL 1 ∂gµν α µ ν µ ν µ ν (0) = dλ y x˙ x˙ + gµν y˙ x˙ + gµν x˙ y˙ . p ρ σ α dε λ0 2 gρσ x˙ x˙ ∂x

A partial integration yields

¨¨*0 λ1 ¨ λ λ1 Z µ µ ¨¨ 1 Z  µ  g x g x ¨ g x µν ˙ µν ˙ ¨¨ d  µν ˙  ν ¨ ν   ν dλ y˙ = ¨ y − dλ   y p p ¨  p  ρ σ ¨ ρ σ dλ  ρ σ 2 gρσ x˙ x˙ 2 ¨¨gρσ x˙ x˙ 2 gρσ x˙ x˙ ¨ λ λ0 ¨ 0 λ0 where the ﬁrst term vanishes because of the boundary condition y(λ0) = y(λ1) = 0. Therefore,

Z λ   µ ν dL 1  1 ∂gµν d gαµ x˙ + gαν x˙  (0) = dλ yα  x˙µ x˙ν −   .  p ρ σ α  p ρ σ  dε λ0 2 gρσ x˙ x˙ ∂x dλ 2 gρσ x˙ x˙ 

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 287 Cosmology

dL µ Necessary and sufﬁcient condition that dε (0) = 0 for any choice of y (λ) satisfying y(λ0) = y(λ1) = 0 is that {··· } = 0 in the integrand. If we choose the arc length s as ρ σ a parameter on K0 we have gρσ x˙ x˙ = 1 and the geodesic line condition reads

( ) ∂gµν d d x˙µx˙ν − g x˙ν − g x˙µ = 0 ∂xα ds αν ds αµ or (eventually renaming summed over indices)

∂gµν ∂g ∂gαµ x˙µx˙ν − αν x˙µx˙ν − x˙µx˙ν − 2 g x¨ν = 0 . ∂xα ∂xµ ∂xν αν

Thus

! 1 ∂g ∂gαµ ∂gµν g x¨ρ = − αν + − x˙µx˙ν αρ 2 ∂xµ ∂xν ∂xα

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 288 Cosmology and contracting with gαβ we arrive at ! 1 ∂g ∂gαµ ∂gµν x¨β = − gβα αν + − x˙µx˙ν . 2 ∂xµ ∂xν ∂xα

This gives rise to the deﬁnition of the Christoffel symbols: Def: Christoffel symbols

! 1 ∂gνα ∂gµα ∂gµν β βα Γα,µν + − ; Γ µν g Γα,µν . 2 ∂xµ ∂xν ∂xα

Symmetry of gµν implies the symmetry relations

∂g Γ = Γ ; Γβ = Γβ ; Γ + Γ = να . α,µν α,νµ µν νµ α,µν ν,αµ ∂xµ

As a result the system of differential equations of geodesic lines takes the form

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 289 Cosmology

2 µ ρ σ d x = −Γµ dx dx ds2 ρσ ds ds This system is of the form ( y˙p = F(x1, ··· , xn, y1, ··· yn) ; p = 1, ··· , n x˙p = yp and in the theory of differential equations is proven to have a unique solution xµ(λ) to given initial values xµ(0) and x˙µ(0) for λ ∈ (−ε, ε). As a consequence we have

Theorem 1. Through each point of a Riemannian manifold there exists exactly one geodesic line in a given direction.

Furthermore, it follows that to each point P of a Riemannian manifold there exists a neighborhood U (P) such that for any point Q ∈ U (P) there is exactly one geodesic line between P and Q within the given neighborhood. For the positive metric case this is also the shortest connection and hence the distance between P and Q.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 290 Cosmology

Another important fact is that the Christoffel symbols do not represent a tensor! While the metric is a second rank tensor, the ordinary derivatives of the metric which deﬁne the Γρ,µν are not tensors, as one easily checks be direct calculation: with

µ ν 0 0 gµ ν = Xµ0 Xν0 gµν and

µ µ µ ∂ µ ∂ ∂x ∂ ∂x µ X 0 0 0 X 0 = 0 0 = 0 0 = X 0 0 µ α ∂xα µ ∂xα ∂xµ ∂xµ ∂xα α µ we obtain

0 0 ∂gµ ν µ ν µ ν µ ν ∂gµν α = X X 0 + X X 0 0 g + X X 0 X 0 . ∂xα0 µ0 α0 ν µ0 ν α µν µ0 ν ∂xα α

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 291 Cosmology

For the Christoffel symbol the three terms combine to

β0 β0 ρ ν α β0 β Γ ν0α0 = Xρ Xν0 α0 + Xν0 Xα0 Xβ Γ να which more explicitly reads

0 0 0 β ∂xν ∂xα ∂xβ β ∂2xρ ∂xβ Γ = 0 0 Γ + 0 0 ν0α0 ∂xν ∂xα ∂xβ να ∂xν ∂xα ∂xρ everything at the same point. The Γρ,µν’s thus have inhomogeneous transformation properties and they play the same role as gauge potential in local gauge theories. Like the gauge potentials, the Christoffel symbols are the basic ingredients, called afﬁne connections, needed for the construction covariant derivatives.

The fact that the Γ’s are not tensors actually has an important consequence. Locally they can be transformed away: in each point P of a manifold there exist a µ coordinate system in which Γ ρσ(P) = 0; this particular CS is called geodetic coordinate system (GCS) deﬁned by

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 292 Cosmology

∂gµν ∂xλ (P) = 0 .

This is a system where geodesic lines are chosen as coordinate axes. GCSs are often a useful tool for proofs of tensor relations, like covariant derivatives of sums, products, tensor contractions etc.

Note that in contrast to the non-covariant Christoffel symbols, if a tensor vanishes in one system it vanishes in any other system, because of the homogeneous transformation law tensors obey. B) Covariant derivatives In order to construct a covariant derivative, we observe that the disturbing second derivative of the coordinates may be expressed in terms of the Christoffel

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 293 Cosmology symbols:

2 ρ ρ ν α ∂ x ∂x β0 ∂x ∂x ρ ρ ρ β0 ν α ρ = Γ − Γ or X = X Γ − X 0 X 0 Γ . ∂xν0∂xα0 ∂xβ0 ν0α0 ∂xν0 ∂xα0 να ν0α0 α0 ν0α0 ν α να

µ··· A general mixed tensor ﬁeld S α··· has the transformation law

µ0··· µ0 α µ··· S α0··· = Xµ ··· Xα0 ··· S α··· .

Using the abbreviations/relations

µ ∂ µ ∂2xµ µ µ X = 0 X = 0 0 ⇒ X = X ν0 α0 ∂xα ν0 ∂xα ∂xν ν0 α0 α0 ν0 µ ∂ µ ∂2xµ ∂ µ α0 µ µ α0 X = X = 0 = 0 X X ⇒ X = X X ν0 α ∂xα ν0 ∂xα∂xν ∂xα ν0 α ν0 α ν0 α0 α we try to structure the expression we obtain by looking at

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 294 Cosmology

the derivative of the transformed tensor:

µ0··· ∂S α0··· µ0 α ρ··· µ0 ρ µ··· = X ··· X 0 ··· S + ··· + X ··· X ··· S + ··· ∂xν0 ρ ν0 α α··· µ α0 ν0 ρ··· µ··· µ0 α ∂S α··· +X ······ X 0 ··· µ α ∂xν0 µ0 µ ρ0 σ0 µ0 ν α ρ··· = Xµ Γ ρν − Xρ Xν Γ ρ0σ0 Xν0 ··· Xα0 ··· S α··· + ··· µ0 ρ ρ0 α ν ρ µ··· +Xµ ··· Xρ0 Γ α0ν0 − Xα0 Xν0 Γ αν ··· S ρ··· + ··· µ··· µ0 α ν ∂S α··· +X ··· X 0 ··· X 0 µ α ν ∂xν0

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 295 Cosmology

0 µ ··· ( µ··· ) ∂S α0··· µ0 α ν ∂S α··· µ ρ··· ρ µ··· = X ··· X 0 ··· X 0 + Γ S + · · · − Γ S − · · · ∂xν0 µ α ν ∂xν ρν α··· αν ρ··· ρ0 σ0 ν µ0 α ρ··· −Xρ Xν Xν0 Γ ρ0σ0 ··· Xα0 ··· S α··· − · · · | {z0 } δσ ν0 0 0 +Xµ ··· Xρ Γρ ··· S µ··· + ··· µ ρ0 α0ν0 ρ··· ( µ··· ) µ0 α ν ∂S α··· µ ρ··· ρ µ··· = X ··· X 0 ··· X 0 + Γ S + · · · − Γ S − · · · µ α ν ∂xν ρν α··· αν ρ··· µ0 ρ0··· −Γ ρ0ν0 S α0··· − · · · ρ0 µ0··· +Γ α0ν0 S ρ0··· + ···

Note that in the bracket on the ﬁrst line only unprimed objects have been included. If we move the primed extra terms to the l.h.s of the equation, we indeed obtain an

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 296 Cosmology object witch transforms as a tensor:

 0   µ ···  ∂S 0 0 0··· 0 0···   α ··· µ S ρ · · · − ρ S µ − · · ·   ν0 + Γ ρ0ν0 α0··· + Γ α0ν0 ρ0···   ∂x  ( µ··· ) µ0 α ν ∂S α··· µ ρ··· ρ µ··· = X ··· X 0 ··· X 0 + Γ S + · · · − Γ S − · · · µ α ν ∂xν ρν α··· αν ρ···

We may formulate this results as a theorem:

µν··· Theorem 2. If S αβ··· is a tensor ﬁeld of type (p, q), then

µν··· def µν··· µ ρν··· ν µρ··· ρ µν··· ρ µν··· Dλ S αβ··· = ∂λ S αβ··· + Γ ρλS αβ··· + Γ ρλS αβ··· + · · · − Γ αλS ρβ··· − Γ βλS αρ··· − · · · is a tensor field of type (p, q+1), which defines the covariant derivative of the tensor µν··· field S αβ···.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 297 Cosmology

µν··· µν··· Often one uses the notation Dλ S αβ··· = S αβ··· ;λ .

µ Exercise: prove the following fundamental properties: 1) Dλ gµν = 0, 2) Dλ δ ν = 0, 3) Dλ ϕ(x) = ∂λ ϕ(x), the gradient of ϕ for scalar ﬁelds ϕ(x).

The covariant derivative allows us to define a coordinate independent covariant derivative of a tensor field along a curve: for a given curve xµ(λ) we define

D def S µν··· = D S µν··· x˙α . dλ αρ··· α αρ···

This in turn is basic for deﬁning parallelism: a contravariant vector ﬁeld Aµ(x) (as an example) is parallel along the given curve if its covariant derivative vanishes along the curve:

! D ∂Aµ Aµ = (D Aµ) x˙α = + Γµ Aβ x˙α = 0 . dλ α ∂xα βα

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 298 Cosmology

Furthermore, the curvature tensor now may be obtained from the commutator of two covariant derivatives:

µ µ µ α [D]ρσ A ≡ DρDσ − DσDρ A = Rα,ρσ A .

µ Exercise: work out Rα,ρσ in terms of the Christoffel symbols by calculating the covariant commutator.

Exercise: show that 1)

1 ∂gρσ 1 ∂g ∂ p Γρ = gρσ = = ln |g| µρ 2 ∂xµ 2g ∂xµ ∂xµ with g = det gµν.

c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 299 Cosmology

2) p ∂ |g| gρσ 1 σλ ρ p + g Γ σλ = 0 . |g| ∂xσ

Exercise: using the above results to verify the following relations:

Gradient : (Grad ϕ)µ Dµ ϕ = ∂µ ϕ , Rotation : (Rot A)µν DµAν − DνAµ = ∂µAν − ∂νAµ µ 1 p µ Divergence : (Div A) DµA = √ ∂µ |g| A |g| 1 p µν Laplacian : ∆ ϕ Div Grad ϕ = √ ∂µ |g| g ∂νϕ |g|

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c 2009, F. Jegerlehner ≪R Lect. 4 R≫ 300