Chapter 4
Gravitation
Problem Set #4: 4.1, 4.3, 4.5 (Due Monday Nov. 18th)
4.1
Equivalence Principle
The Newton’s second law states that
f = mi a
(4.1)
where mi is the inertial mass. The Newton’s law of gravity states that
f = −mg ∇Φ
(4.2)
where mg is the gravitational mass. The Weak Equivalence Principle
states that
mi = mg .
(4.3)
This means that for a freely falling particle the effect of the gravitational
field is independent on its mass
a = −∇Φ.
(4.4)
This also means that by preforming local experiments it is impossible to
distinguish the gravitational field from being in a uniformly accelerated reference frame.
Of course all this was known long before Einstein, but what was new is
the Special Theory of Relativity which can be included in the
Einstein Equivalence Principle: “In a small enough region of space-time,
the laws of physics reduce to those of special relativity; it is impossible to
54
55
CHAPTER 4. GRAVITATION
detect the existence of a gravitational field.”
An immediate and celebrated prediction of the Einstein Equivalence Principle is the gravitational redshift. Consider two observers at a distance z from
each other moving with acceleration a so that the distance remains fixed.
One observer emits a photon and it reaches the other observer after a time
z
∆t = .
c
(4.5)
By that time the velocity of both boxes changed by
∆v = a∆t = a
z
c
(4.6)
and due to non-relativistic Doppler effect the wavelength will be shifted by
∆λ
∆v
az
=
= 2.
λ0
c
c
(4.7)
According to the Einstein Equivalence Principle this effect should be indistinguishable when the “acceleration is due to gravity, then the photon emitted
from the ground should be shifted by
∆λ
ag z
= 2 .
λ
c
(4.8)
and in terms of Newtonian potential
!
∆λ
1
= −
∇Φdt =
λ
c
!
1
= − 2 ∂z Φdz =
c
∆Φ
= − 2 .
c
(4.9)
This was checked experimentally by Pound and Rebka in 1960.
Einstein Equivalence Principle also suggests that the gravitational field
acts universally on massive and as we have seen on massless particles.
This means that there are no gravitationally neutral objects with respect
to which one can measure acceleration and, thus, the “acceleration due to
gravity” is ill defined. (As we shall see even the gravitational mass of extended
objects is also ill defined.) Therefore, to satisfy the principle we shall give up
on the idea of extending inertial frames of Special Relativity throughout the
entire space-time or otherwise the far away objects would accelerate in such
56
CHAPTER 4. GRAVITATION
frames. Of course we can still keep the notion of the local inertial frames
where by local we mean a sufficiently small region of space-time volume.
Thus it seems necessary (and as we will see also sufficient) to incorporate
the ideas of differential geometry into the laws of gravitation. Evidently, the
local inertial frames are nothing but local normal coordinates for which
the Christoffel symbols vanish locally and all of the laws of Spatial Relativity
are maintained; and the impossibility to define acceleration of the far away
objects is due to the fact that the parallel transport of vectors depends on
the path for a curved space-time with non-vanishing Riemann tensor.
In the normal coordinates a freely falling observer moves without ‘acceleration’
d2 xµ
= 0.
(4.10)
dλ2
This equation can be made covariant (or independent on coordinates) with
help of Christoffel symbols
ρ
σ
d2 xµ
µ dx dx
+
Γ
= 0.
ρσ
dλ2
dλ dλ
(4.11)
which is just the geodesics equation that tells that the freely falling particle
move along geodesics.
We shall now check that (4.11) reduces to the Newton’s law of gravity
(4.2) in the limit of:
• small velocities
dxi
dx0
≪
dτ
dτ
where τ is proper time and thus from (4.11)
0
0
d2 xµ
µ dx dx
+
Γ
≈ 0.
00
dτ 2
dτ dτ
• slowly changing metric
∂0 gµν ≪ ∂i gρσ
(4.12)
(4.13)
(4.14)
and thus
1
1
Γµ00 = g µλ (∂0 gλ0 + ∂0 g0λ − ∂λ g00 ) ≈ − g µλ ∂λ g00 .
2
2
(4.15)
• weak gravitational field
gµν = ηµν + hµν
(4.16)
57
CHAPTER 4. GRAVITATION
where the perturbation
(4.17)
|hµν | ≪ 1
and thus from (4.16) and (4.14)
1
Γµ00 ≈ − η µi ∂i h00 .
2
(4.18)
From (4.13) and (4.18)
1
d2 xµ
≈ η µi ∂i h00
2
dτ
2
whose µ = 0 component
or
or by setting t = x0 ,
dx0
dτ
#2
(4.19)
d2 x0
≈0
dτ 2
(4.20)
dx0
≈ const.
dτ
(4.21)
#2
(4.22)
and µ = i component
"
"
dτ
dx0
d2 xi
1
≈ ∂i h00
2
dτ
2
1
d2 xi
≈ ∂i h00 .
dt2
2
Note that (4.23) is the same as (4.4) with
(4.23)
h00 = −2Φ
(4.24)
g00 = − (1 + 2Φ) .
(4.25)
or
Similarly one can adopt all of the laws of physics to respect the Einstein
Equivalence Principle. This usually involves starting with normal coordinates
where the laws of physics are known and rewriting them in a covariant form it
in terms of legitimate tensors (e.g. partial derivative are replaced by covariant
etc.).
58
CHAPTER 4. GRAVITATION
4.2
Einstein Equations
As was previously stated we want to derive a dynamical equation for the
metric tensor which generalizes the Poisson equation
∇2 Φ = 4πGρ
(4.26)
where ∇2 = δ ij ∂i ∂j and ρ is the mass density. Since
T00 = ρ
(4.27)
∇2 h00 = −8πGT00 .
(4.28)
we combine (4.24) and (4.26) into
Of course, the next step should be to make this equations tensorial.
The first obvious choice
∇λ ∇λ gµν ∝ Tµν
(4.29)
is not going to work just because the left hand side is identically zero since
∇λ gµν = 0. The second choice
Rµν ∝ Tµν
(4.30)
is a lot more reasonable, but does not work either just because the energy
and momentum conservations imply
∇µ Tµν = 0
(4.31)
∇µ Rµν ̸= 0
(4.32)
Rµν = Tµν = 0.
(4.33)
but
for general space-times. An important exception is a vacuum Einstein equation for vanishing energy momentum tensor
There is however a geometric object the Einstein tensor
1
Gµν = Rµν − R gµν
2
(4.34)
with the right properties on arbitrary spaces. It is a symmetric (0, 2) tensor
built of the second (and first and zeroth) derivatives of the metric and
∇µ Gµν = 0.
(4.35)
CHAPTER 4. GRAVITATION
59
Thus we can conjecture that
1
Gµν = Rµν − R gµν = 8πGTµν
2
(4.36)
where the proportionality constant was determined from (4.28).
This is the Einstein equation which is a rather complicated second-order
non-linear equation with very few explicit solutions. The equation actually
contains 10 coupled differential equation since both sides are symmetric (0, 2)
tensors which is exactly the number of unknown parameters in a metric gµν .
But because of the general covariance 4 of the 10 equations are unphysical
′
(represented by four function xµ (xµ )) and we are left with only 6 physical
(or coordinate independent) degrees of freedom. This agrees with the number
of dynamical equations in (4.36) when the constraints due to conservation of
the energy and momentum (4.31) or the Bianchi identity (4.35) is taken into
account.
The Einstein equation can also be derived using variational principle for
the Hilbert action
!
√
SH = d4 x −g R.
(4.37)
To obtain the Einstein equations with lower indices we will vary with respect
to the inverse metric. Since
R = g µν Rµν
the variation of the acton will have three contributions:
!
$√
√
√ %
δSH = d4 x −g g µν δRµν + −g Rµν δg µν + Rδ −g .
(4.38)
(4.39)
But the first term is given by
$
%
δRρµλν = δ ∂λ Γρµν − ∂ν Γρµλ + Γρλσ Γσµν − Γρνσ Γσµλ =
$
%
$
%
= ∂λ δΓρµν − ∂ν δΓρµλ + δΓρλσ Γσµν + Γρλσ δΓσµν − δΓρνσ Γσµλ − Γρνσ δΓσµλ =
& $
'
%
= ∂λ δΓρµν + Γρλσ δΓσµν − Γσλµ δΓρσν − Γσλν δΓρµσ
& $
%
'
− ∂ν δΓρµλ + Γρνσ δΓσµλ − Γσνµ δΓρσλ − Γσνλ δΓρµσ =
$
%
%
$
= ∇λ δΓρµν − ∇ν δΓρµλ
(4.40)
Note that unlike Γρµν its variation δΓρµν is a tensor as can be seen from the
60
CHAPTER 4. GRAVITATION
transformation law (3.6). Thus,
!
!
$√
%
$
%
$
%%
√ $
d4 x −g g µν δRµν =
d4 x −g g µν ∇λ δΓλµν − g µν ∇ν δΓλµλ =
!
%
$
%%
√ $ $
=
d4 x −g ∇λ g µν δΓλµν − ∇λ g µλ δΓνµν =
!
$
%
√
=
d4 x −g∇λ g µν δΓλµν − g µλ δΓνµν = 0 (4.41)
is a boundary term which is set to zero by making the variations vanish at
the boundary of space-time.
Now the third term in (4.39) can be simplified using the following useful
identity
Tr(log (M)) = log (det (M))
(4.42)
or
$
% δ (det (M))
Tr M −1 δM =
det (M)
Applying it to inverse metric
M µν = g µν
(4.43)
(4.44)
gives us
$
%
δ (g −1)
Tr gµν δg νλ =
g −1
µν
gµν δg
= −g −1 δg.
√
1 (−g) gµν δg µν
1√
1 δg
√
δ −g = − √
=−
=−
−ggµν δg µν
2 −g
2
−g
2
and by substituting in (4.39) we get
#
"
!
√
1
4
δSH = d x −g Rµν − Rgµν δg µν .
2
(4.45)
(4.46)
(4.47)
Therefore for the action to vanish for arbitrary variations of the metric
1 δS
√
=0
−g δg µν
(4.48)
the vacuum Einstein equations must be satisfied
1
Gµν = Rµν − Rgµν = 0.
2
(4.49)
CHAPTER 4. GRAVITATION
4.3
61
Energy-Momentum Tensor
To obtain a non-vacuum equation on should also include matter Lagrangian,
!
!
√
√
1
4
d x −g R. + d4 x −g LM ,
S=
(4.50)
16πG
and the variational principle implies
# !
#
"
"
√
!
√
√
δS
1
1
1 δ ( −gLM )
4
4
=
d x −g Rµν − Rgµν + d x −g √
= 0.
δg µν
16πG
2
−g
δg µν
For this to agree with Einstein equation
1
Rµν − R gµν = 8πGTµν
2
we define the energy momentum tensor as
1 δLM
Tµν = −2 √
.
−g δg µν
For example, action for a scalar field
#
"
!
√
1 µν
4
Sφ = dx −g − g ∇µ φ∇ν φ − V (φ)
2
(4.51)
(4.52)
can be varied
(
"
#
)
!
$√ %
√ 1
1 µν
4
µν
δSφ =
dx − −g ∇µ φ∇ν φ (δg ) + − g ∇µ φ∇ν φ − V (φ) δ −g =
2
2
#)
(
"
#"
!
√
1
1
1√
−ggµν
δg µν =
=
dx4 − −g ∇µ φ∇ν φ + − g λρ ∇λ φ∇ρ φ − V (φ)
−
2
2
2
)
(
!
1
1
1
4√
λρ
=
dx −g − ∇µ φ∇ν φ + gµν g ∇λ φ∇ρ φ + V (φ)gµν δg µν
(4.53)
2
4
2
which leads to the following energy momentum tensor
1 δS
1
Tµν = −2 √
= ∇µ φ∇ν φ − gµν g λρ ∇λ φ∇ρ φ − V (φ)gµν .
µν
−g δg
2
(4.54)
Note that the non-vacuum Einstein equation can be solved for an arbitrary manifold. Just specify the metric, calculate the Einstein tensor and
demand that it is proportional to the energy momentum tensor. Of course,
this does not tell us what should be the field content of the theory and
62
CHAPTER 4. GRAVITATION
whether the corresponding energy momentum tensor is realistic. Thus, it is
useful to have some restrictions on Tµν .
Weak Energy Condition:
Tµν tµ tν ≥ 0
for all time-like vectors tµ .
Null Energy Condition:
Tµν lµ lν ≥ 0
for all light-like vectors lµ .
Dominant Energy Condition:
Tµν tµ tν ≥ 0
for all time-like tµ and time-like or light-like T µν tµ .
Strong Energy Condition:
1
Tµν tµ tν − T λλ tσ tσ ≥ 0
2
for all time-like vectors tµ . This condition is used to prove famous singularity
theorems, but is also violated in quantum systems, for example, subject to
Casimir force.
For a perfect fluid
Tµν = (ρ + p)Uµ Uν + pgµν
where ρ is the energy density, p is pressure and Uµ is the four velocity.
• weak energy conditions implies ρ ≥ 0 and ρ + p ≥ 0,
• null energy condition implies ρ + p ≥ 0,
• dominant energy condition implies ρ ≥ |p|, and
• strong energy condition implies ρ + 3p ≥ 0 and ρ + p ≥ 0.
Consider time evolution of a small ball of small particles moving with four
velocities U µ . Then the expansion, rotation and shear of the ball can be
defined as
θ = ∇λ U λ .
(4.55)
ωµν = ∇[ν Uµ]
(4.56)
CHAPTER 4. GRAVITATION
63
1
(4.57)
σµν = ∇(ν Uµ) − ∇λ U λ (gµν + Uµ Uν ) .
3
and the famous Raychaudhuri equation describes the time evolution of the
expansion parameter,
dθ
1
= − θ2 − σ µν σµν + ω µν ωµν − Rµν U µ U ν .
dτ
3
(4.58)
If the ball of particles is at rest U = (1, 0, 0, 0) with respect to a locally
inertial system gµν = ηµν and has vanishing expansion θ = 0, rotation ω = 0
and shear σ = 0, then the Raychaudhuri and Einstein equations give
"
#
1
dθ
= −R00 = −8πG T00 − T g00 = −4πG (T00 + T11 + T22 + T33 )
dτ
2
(4.59)
where we used the fact that
R = −8πGT.
(4.60)
If we denote T00 = ρ, T11 = px , T22 = py , T33 = pz then,
dθ
= −4πG (ρ + px + py + pz ) .
dτ
(4.61)
Note that the gravity would tend to decrease the size of the ball if ρ+ 3p ≥ 0.
Thus, for a perfect fluid the strong energy condition implies that the gravity
is attractive.