4. General Relativity and Gravitation
4.1. The Principle of Equivalence
4.2. Gravitational Forces
4.3. The Field Equations of General Relativity
4.4. The Gravitational Field of a Spherical Body
4.5. Black and White Holes
4.1. The Principle of Equivalence
Under a coordinate transformation xμ → xμ ,
' ' ' 'g g Tg Λ g Λ 1T Λ ΛIn general,
Every real symmetric matrix can be diagonalized by an orthogonal transformation:
1, ,TD ddiag g g O g O g
1T O O
gj are the (real) eigenvalues of g.
Consider Λ = O D, where D = diag(D1 , …, Dd ).
Λ1 exists & real → Dj 0 & real j.
2 21 1, , d ddiag D g D g T T
Dg OD Og O OD DD g D→
2 1j
j
DgChoosing → 1, 1, , 1, 1, 1, , 1diag g
canonical form of the metric tensor
Spacetime is locally flat (Minkowskian): 1, 1, 1, 1
set
diag g at 1 point.
4.2. Gravitational ForcesLagrangian:
Special relativity:
Principle of covariance → all EOMs covariant under Λ that leaves η unchanged( Poincare transformations )
General relativity:
Principle of covariance → all EOMs covariant under all Λ
→ L is a scalar ( contraction of tensors )
Principle of equivalence → L is Minkowskian in any local inertial frame.
→ L contains only contractions involving gμν and gμν, σ.
Free Particles
Minkowski → General: → g :
1
2
dx dxL m g x
d d
1
2
dx dxS m d g x
d d
; 0g → This is also the only choice that is both covariant and linear in g.
,
1
2
Lmg x x
x
1
2
Lmg x x
x
1
2m g x x g
mg x
d Lm g x g x
d x
,m g x x g x
,
10
2
dg x g x x
d
Euler-Lagrange equation:
,
10
2
dg x g x x
d
dg x g x g x
d
,g x g x x
, ,
10
2g x g x x g x x
, , ,
10
2g x g g g x x
, , ,
10
2x g g g g x x
0x x x Geodesic equation
4.2.3. Gravity
g h Let ( hμν small )
, , ,
1
2h h h h
2, , ,
1
2h h h O h
2cd h dx dx
Non-relativistic motion:
dv c
dt
x
0d cdt dx
d d d
x→
0idx dx→
22 000 00cd h dx 2001 h cdt
00
1
1
dtt
d h
00
11
2h
0x x x 0 0x 200 0j jx ct →
jj d dt d x
xd d dt
2
22
j jd x d xt t
dt dt
22
2
jd xt
dt
22
002
jjd x
cdt
→
The only non-vanishing components of g μν,σ are g00 , j = h00 , j .
00 0,0 0,0 00,
1
2j j h h h
00,
1
2j h ,
00
1
2jh
001
2j h
x
001
2jk
k
h
x
0 00 0j j 00
0 ,0 00, 0,0
1
2 j j jh h h 00,
1
2 jh 001
2 j
h
x
22 00
2
1
2
jjk
k
hd xc
dt x
→
Setting 200
1
2V c h
2
2
dm m V
dt
xgivesNewtonian gravity
4.3. The Field Equations of General Relativity
Electrodynamics: Particles Fields InteractionL L L L
Gravitational field : LParticles + LInteraction 1
2
dx dxm g x
d d
Task is to find LFields → invariant infinitesimal spacetime volume dV
For a Minkowski spacetime 3dV dt d x 41d x
c
' 'x x x 4 1 4'd x J d x 'detJ
→ = Jacobian
d 4 x is a scalar density of weight 1
Only choice is41
dV g d xc ( g = det | g | is a scalar density of weight +2 )
Check: In a Minkowski spacetime det 1g
41dV g d x
c
By definition 4 4d x x y f x f y f = scalar function
4 4d x x y
4 x y
→ is a scalar
is a scalar density of weight +1
41x y
g
is a scalar.
41c dV x y f x f y
g
Lagrangian Densities
4matter gravS d x x x L L
4
1
1
2
N
matter n n n n n n n nn
x m d x x g x x x
L
1 1
2grav x g x R xc
L
R R g R
= Ricci curvature
κ = coupling constant Λ = cosmological constant
Einstein introduced Λ to allow for a static solution, even though the vacuum solution would no longer be Minkowskian.
At present: Λ = 0 within experimental precision.
Recent theories: Λ 0 immediately after the Big Bang.
Field Equations
Euler-Lagrange equations for the metric tensor field degrees of freedom gμν are called Einstein’s field equations:
1
2R R g T
4
1
N
n n n nn
cT x d m x x x x
g x
1
2G R R g
G g T G g T
stress tensor
Einstein curvature tensor
0 G T→
Ex.4.2
Another Form of the Field Equation
T g T T 4g g
1
2g R R g g T
14
2R R T
4R T
1
2R T g T
1
2R T Tg g
gνμ field eqs →
Field eqs:
Newtonian Limit
Newtonian theory : 2 4V G GMV x
x M x x
κ is determined by the principle of correspondence.
g h 00 00h h x 0 00 0 00, 00
1
2k
j j j jkh
, ,R
R R 0R
00 0 0R R 0 0jjR 00, 0 ,0 00 0 0
j j j jj j j j
→ non-vanishing components of R must have at least two “0” indices.
→
000, 00 0j j
j j 200 00 00
1
2h h h
ρ is stationary → ,dx
cd
0
2
00
,0,0,01
cT diag
h
00 200 001T g T g T h c
00
001
TTg
h
2
001
c
h
To lowest order in h, 2 200
1 1
2 2h c
2 2 21
2V c c
4
8 G
c
→ 0
4.4. The Gravitational Field of a Spherical Body
The Schwarzschild Solution (1916 ):
1. ρ is spherically symmetric; so is g.
2. ρ is bounded so that g ~ η at large distances.
3. g is static (t-independent) in any coordinate system in which ρ is stationary.
, , ,x ct r 2 2 2 2 2 2 2 2sinds A r c dt B r dr r d d
1A B 2. →
Note: ( r, θ, φ) are spatial coordinates only when r → .
An extra C(r) factor in the “angular” term can be absorbed by C r r r
→
2 2 2, , , sing x diag A B r r
2 2
2
0 intervals
c d time like
for null
dl space like
Exterior Solutions
0 0T 1
02
G R Rg →
2 0G R R R
0G R (2nd order partial differential equations for gμν )→
Schwarzschild solution [see Chapter 14, D’Inverno ]:
1
2 2 2 2 2 2 2 21 1 sinS Sr rds c dt dr r d d
r r
2
2S
GMr
c = Schwarzschild radius
Singularity at r = rs will be related to the possibility of black holes.
, 0.886S Earthr cm , 2.95S Sunr km
4.4.2. Time Near a Massive Body
Coordinate t = time measured by a stationary Minkowskian (r→) observer.
To this observer, two events at (ct1 , x1) and (ct2 , x2) are simultaneous if t1 = t2.
For another stationary observer at finite r > rS , time duration experienced = proper time interval d with dx = 0
d A dt 1 Sr dtr
→ two events simultaneous to one stationary observer (Δτ1 = 0 ) are simultaneous to all stationary observers (Δt = Δτ2 = 0 ) .
The finite duration Δτ of the same events (fixed dt 0) differs for stationary observers at different r.
If something happens at spatial point (r1 ,θ1 ,φ1) for duration 1 11
1 Sr tr
another stationary observer at (robs ,θobs ,φobs) will find 1obst t
11 Sobs
obs
rt
r 1
1
1
1
S
obs
S
rrrr
2
1
12
21
21
obsVc
Vc
For the observation of emision of light
obs emis
emis obs
1
1
S
emis
S
obs
rrrr
Verified to an accuracy of 103 by Pound and Rebka in 1960 for the emission of rays at a height of 22m above ground using the Mossbauer effect.
For measurements done on the sun and star light, Earth’s gravity can be ignored.
2
2
21
21
emisobs
emisobs
Vc
Vc
2 2
1 11 1obs
emis obsemis
V Vc c
2
11 emis obsV V
c
obs emis
emis
v v v
v v
2
1emis obsV V
c
For starlights observed on earth, emis obsV V 0 →gravitational red shift
Originally, observed red shifts ~ validation of the theory of general relativity.
Now: ~ validation of the principle of equivalence.
→ Allows for other gravitational theories, such as the Brans-Dicke theory.
4.4.3. Distances Near a Massive Body
2 2 2 2 2 2 21sin
1 S
ds dl dr r d dr
r
0dt →
Radial distance between 2 points with the same and coordinates is defined as
2
1
r
r
d lr d r
d r
2
1
1
1
r
r S
d rr
r
2 2 1 1r f r r f r
2 1 Sr r r 1 ln 1 1S S S
S
r r r rf r
r r r r
where
Only exterior solution known
→ radial distance of a point from the origin is not defined.
Consider circular path described by the equations r = a and θ = π/2.
Its length, or circumference, is L dl2
0
2a d a
( same as 3 )
Its radius is not defined.Closest distance between 2 concentric circles r = a1 and r = a2 is
2 2 1 1a f a a f a not 2 1a a
A “circle” of a well defined radius a about a point would appear lopsided when plotted using the spherical coordinates.
Since lim 1r
f r
2 1r r r 1 2,r r
for
The lowest order of corrections valid for 1 2, , Sr r r r are
1 ln 2S
S
r rf r
r r
2 12 1
2 1
1 ln 2 1 ln 2S S
S S
r r r rr r r
r r r r
2
2 11
1ln
2 S
rr r r
r
radial distance
difference in circumference
2
2 1 2 1 1
1 11 ln
2 2 2Sr r r
r r r r r
difference in circumference
radial distance 2
2 1 1
12 1 ln
2Sr r
r r r
4.4.4. Particle Trajectories Near a Massive Body
Einstein field equations are non-linear → principle of superposition is invalid → perturbation theory inapplicable → even the 2-body problem is in general intractable
One tractable class of problems:Motion of a “test” particle ( geodesics of g )
For time-like geodesics in the Schwarzschild spacetime,
1 0Sd rt
d r
2 2 2sin cos 0d
r rd
2 2sin 0d
rd
1 12 2 2 2 2 21 1
1 1 1 sin 02 2
S S Sr d r d rr c t r r r
r dr r dr r
1 22 2 2 2 2 2
2 2
1 11 1 sin 0
2 2S S S Sr r r r
r c t r r rr r r r
Setting m = 0 makes S = 0.
Hence, for massless particles, we switch to another affine parameter
1d d
m so that
1
2
dx dxS d g
d d
Null geodesic eqs are obtained from the geodesics by replacing τ with λ.
Notable phenomena:
• Bending of light by the sun.
• Precession of Mercury.
See Chap 15, D’Inverno.
In practice, the r eq is usually replaced by
1
0
1
time like
g x x for null geodesics
space like
4.5. Black and White Holes
R = radius of the mass distribution.
If R > rS then singularity at r = rS is fictitious.
Problem of interest:
R < rS and R < r < rS
Radial Motion: Solution for r
Free particle with purely radial motion ( dθ = dφ = 0 ):
12 2 2 2 21 1S Sr r
c d c dt drr r
22 2 2 21 1S Sr r
r c t cr r
2 1
2 2 2 21 1S Sr rc t r c
r r
→ →
EOM for r:
1 12
2
11 1 0
2S S Sr r r
r cr r r
→2
2
1
2Srr c
r
2
GM
r
21
2
d rr
d r
2 2 2 1 1
e Se
r r r cr r
→
1 22 2 2 2 2 2
2 2
1 11 1 sin 0
2 2S S S Sr r r r
r c t r r rr r r r
→
Newton’s law
ForS
ee
rr c
r we have
22 Sr c
rr
→S
rdr cd
r
3/ 2 3/ 20
2
3 S
r r cr
0 0r r
2 / 3
3/ 20
3
2 Sr r r c →
Outgoing Incoming
Singularity at r = rS not felt
2 2 2 1 1e S
e
r r r cr r
Radial Motion: Solution for t 2
2 Sr cr
r
2 12 2 2 21 1S Sr r
c t r cr r
Putting into
gives
22 2 2 21 1S S Sr r r
c c t cr r r
221 1Sr t
r
1
1 S
trr
→ →
dt d rt
d r d
dtr
d r Srdt
cd r r
1
1
S
S
rdtc
rd r rr
→
→ Outgoing Incoming
for r > rS
Incoming Outgoing
for r < rS
3/ 2
SS
dt rc r
d r r r
3/ 2
SS
dt rc r
d r r r →
3/ 21
SS
rc t dr
r rr
3/ 2 3/ 2 11 22 2 tanh
3S SSS
rct r r r r
rr
3/ 2 3/ 2
11 2
2 ln3
1
SS S
S
S
rr
r r r rr r
r
3/ 2 3/ 2
11 2
2 ln3
1
SS S
S
S
rr
ct r r r rr r
r
3/ 2 3/ 20
2
3S
S
r rc r
For an incoming particle in the region r > rS
r → rS as t → To a Minkowskian observer, the particle takes forever to reach r = rS , the singularity in coord system ( ct, r, θ,φ)
To an observer travelling with the particle, the time τ it takes to fall from r0 to rS is finite:
Null Geodesics
The null geodesics (light paths) are given by ds= 0.
For radial ( d θ = d φ = 0 ) null geodesics, 1
2 2 20 1 1S Sr rc dt dr
r r
→dr r
cdt ct
Sr r
r
Note:
dtt
d
drr
d are not defined individually on the null geodesics.&
S
rct d r
r r
lnS S Sr r r r r const
lnS Sr r r r Outgoing Incoming
For r < rS , r becomes time-like & t space-like.
t = const is a time-like line → forward light cones must point towards the origin.
→ Increasing time: dr > 0, increasing radial distance: c dt > 0.
lnS Sct r r r r
To a Minkowskian observer, incoming light takes forever to reach r = rS .
Eddington-Finkelstein Coordinates
Eddington- Finkelstein coordinates: null radial geodesics are straight lines.Incoming null geodesics: lnS Sct r r r r
Set lnS Sct ct r r r r for r > rS (straight line)
→ SS
d rcd t cdt r
r r
1
1S Sr rcdt d r
r r
Line element:
2 2 2 2 2 2 2 21 2 1 sinS S Sr r rc d d t d t dr d r r d d
r r r
regular for all r 0
Region I: rS < r <
Region II: 0 < r < rS
Assuming the line element to be valid for all r is called an analytic extension of from region I into region II as t →
Advanced time parameter : v ct r lnS Sct r r r r
2 2 2 2 2 2 21 2 sinSrc d dv dvdr r d dr
Line element:
Incoming null geodesics: v const
2 2 2 2 2 2 2 21 2 1 sinS S Sr r rc d d t d t dr d r r d d
r r r
becomes
lnS Sct r r r r becomes
For outgoing particles with time-reversed coordinate
* lnS Sct ct r r r
Retarded time parameter : * lnS Sw ct r ct r r r r
Line element:
2 2 2 2 2 2 21 2 sinSrc d dw dwdr r d dr
Analytical extension from region I into region II* (0 < r < rS ).
Outgoing null geodesics: lnS Sct r r r r
v const becomes
w const becomes
Forward light cones in region II point to the right because we are dealing with a time-reversed solution.
Black Holes
Eddington-Finkelstein coordinates are not time-symmetric
Incoming (outgoing) particles, time is measured by t or v ( t* or w).
I: future light cones point upward
II: future light cones point left
→ no light can go from II to I
II = black hole
Spherical surface at rS = event horizon
To a Minkowskian observer , light emitted by ingoing particles are redshifted.
Possible way to form black holes: collapse of stars or cluster of stars.All information are lost except for M, Q, and L. Rotating black hole ~ Kerr solution.
Black holes can be detected by the high energy radiation ( X and rays) emitted by matter drawn to it from nearby stars or nabulae.
E.g., gigantic black hole at the center of our galaxy.
Estimated minimum mass density of a black hole of total mass M:
343 S
M
r 3
2
4 23
M
GMc
6
2 2
3
32
c
G M
26
2 2
3
32
Mc
G M M
2
16 310M
g cmM
331.99 10M g
For M < 10 M , ρ is too large so the star collapses only into a neutron star.
Extension Regions
II
I
II
I
Direction of extension
Direction of particle motionis denoted by
For extension I → II, no light ray can stay in II (white hole).Extension II (II) → I (I) shows that I and I are identical. However, I(I) is distinct from region I → no overlap or extension between them.
The collection of these 4 regions is called the maximal extension of the Schwarzschild solution [see Chapter 17, D’Inverno].