Chapter 4 Gravitation - University of Minnesota...

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 = mia (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 Principlestates that

mi = mg. (4.3)

This means that for a freely falling particle the effect of the gravitationalfield is independent on its mass

a = −∇Φ. (4.4)

This also means that by preforming local experiments it is impossible todistinguish the gravitational field from being in a uniformly accelerated ref-erence frame.

Of course all this was known long before Einstein, but what was new isthe 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

CHAPTER 4. GRAVITATION 55

detect the existence of a gravitational field.”

An immediate and celebrated prediction of the Einstein Equivalence Princi-ple is the gravitational redshift. Consider two observers at a distance z fromeach 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

∆t =z

c. (4.5)

By that time the velocity of both boxes changed by

∆v = a∆t = az

c(4.6)

and due to non-relativistic Doppler effect the wavelength will be shifted by

∆λ

λ0=

∆v

c=

az

c2. (4.7)

According to the Einstein Equivalence Principle this effect should be indistin-guishable when the “acceleration is due to gravity, then the photon emittedfrom the ground should be shifted by

∆λ

λ=

agz

c2. (4.8)

and in terms of Newtonian potential

∆λ

λ= −

1

c

!

∇Φdt =

= −1

c2

!

∂zΦdz =

= −∆Φ

c2. (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 respectto which one can measure acceleration and, thus, the “acceleration due togravity” is ill defined. (As we shall see even the gravitational mass of extendedobjects is also ill defined.) Therefore, to satisfy the principle we shall give upon the idea of extending inertial frames of Special Relativity throughout theentire space-time or otherwise the far away objects would accelerate in such


frames. Of course we can still keep the notion of the local inertial frameswhere by local we mean a sufficiently small region of space-time volume.Thus it seems necessary (and as we will see also sufficient) to incorporatethe ideas of differential geometry into the laws of gravitation. Evidently, thelocal inertial frames are nothing but local normal coordinates for whichthe Christoffel symbols vanish locally and all of the laws of Spatial Relativityare maintained; and the impossibility to define acceleration of the far awayobjects is due to the fact that the parallel transport of vectors depends onthe path for a curved space-time with non-vanishing Riemann tensor.

In the normal coordinates a freely falling observer moves without ‘accel-eration’

d2xµ

dλ2= 0. (4.10)

This equation can be made covariant (or independent on coordinates) withhelp of Christoffel symbols

d2xµ

dλ2+ Γµ

ρσ

dxρ

dλ

dxσ

dλ= 0. (4.11)

which is just the geodesics equation that tells that the freely falling particlemove along geodesics.

We shall now check that (4.11) reduces to the Newton’s law of gravity(4.2) in the limit of:

• small velocitiesdxi

dτ≪

dx0

dτ(4.12)

where τ is proper time and thus from (4.11)

d2xµ

dτ 2+ Γµ

00

dx0

dτ

dx0

dτ≈ 0. (4.13)

• slowly changing metric∂0gµν ≪ ∂igρσ (4.14)

and thus

Γµ00 =

1

2gµλ (∂0gλ0 + ∂0g0λ − ∂λg00) ≈ −

1

2gµλ∂λg00. (4.15)

• weak gravitational field

gµν = ηµν + hµν (4.16)


where the perturbation|hµν | ≪ 1 (4.17)

and thus from (4.16) and (4.14)

Γµ00 ≈ −

1

2ηµi∂ih00. (4.18)

From (4.13) and (4.18)

d2xµ

dτ 2≈

1

2ηµi∂ih00

"

dx0

dτ

#2

(4.19)

whose µ = 0 componentd2x0

dτ 2≈ 0 (4.20)

ordx0

dτ≈ const. (4.21)

and µ = i component"

dτ

dx0

#2 d2xi

dτ 2≈

1

2∂ih00 (4.22)

or by setting t = x0,d2xi

dt2≈

1

2∂ih00. (4.23)

Note that (4.23) is the same as (4.4) with

h00 = −2Φ (4.24)

org00 = − (1 + 2Φ) . (4.25)

Similarly one can adopt all of the laws of physics to respect the EinsteinEquivalence Principle. This usually involves starting with normal coordinateswhere the laws of physics are known and rewriting them in a covariant form itin terms of legitimate tensors (e.g. partial derivative are replaced by covariantetc.).


4.2 Einstein Equations

As was previously stated we want to derive a dynamical equation for themetric tensor which generalizes the Poisson equation

∇2Φ = 4πGρ (4.26)

where ∇2 = δij∂i∂j and ρ is the mass density. Since

T00 = ρ (4.27)

we combine (4.24) and (4.26) into

∇2h00 = −8πGT00. (4.28)

Of course, the next step should be to make this equations tensorial.The first obvious choice

∇λ∇λgµν ∝ Tµν (4.29)

is not going to work just because the left hand side is identically zero since∇λgµν = 0. The second choice

Rµν ∝ Tµν (4.30)

is a lot more reasonable, but does not work either just because the energyand momentum conservations imply

∇µTµν = 0 (4.31)

but∇µRµν ̸= 0 (4.32)

for general space-times. An important exception is a vacuum Einstein equa-tion for vanishing energy momentum tensor

Rµν = Tµν = 0. (4.33)

There is however a geometric object the Einstein tensor

Gµν = Rµν −1

2R gµν (4.34)

with the right properties on arbitrary spaces. It is a symmetric (0, 2) tensorbuilt of the second (and first and zeroth) derivatives of the metric and

∇µGµν = 0. (4.35)


Thus we can conjecture that

Gµν = Rµν −1

2R gµν = 8πGTµν (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 actuallycontains 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 numberof dynamical equations in (4.36) when the constraints due to conservation ofthe energy and momentum (4.31) or the Bianchi identity (4.35) is taken intoaccount.

The Einstein equation can also be derived using variational principle forthe Hilbert action

SH =

!

d4x√−g R. (4.37)

To obtain the Einstein equations with lower indices we will vary with respectto the inverse metric. Since

R = gµνRµν (4.38)

the variation of the acton will have three contributions:

δSH =

!

d4x$√

−g gµνδRµν +√−g Rµνδg

µν +Rδ√−g%

. (4.39)

But the first term is given by

δRρµλν = δ

$

∂λΓρµν − ∂νΓ

ρµλ + Γρ

λσΓσµν − Γρ

νσΓσµλ

%

=

= ∂λ$

δΓρµν

%

− ∂ν$

δΓρµλ

%

+ δΓρλσΓ

σµν + Γρ

λσδΓσµν − δΓρ

νσΓσµλ − Γρ

νσδΓσµλ =

=&

∂λ$

δΓρµν

%

+ ΓρλσδΓ

σµν − Γσ

λµδΓρσν − Γσ

λνδΓρµσ

'

−&

∂ν$

δΓρµλ

%

+ ΓρνσδΓ

σµλ − Γσ

νµδΓρσλ − Γσ

νλδΓρµσ

'

=

= ∇λ

$

δΓρµν

%

−∇ν

$

δΓρµλ

%

(4.40)

Note that unlike Γρµν its variation δΓρ

µν is a tensor as can be seen from the


transformation law (3.6). Thus,!

d4x$√

−g gµνδRµν

%

=

!

d4x√−g$

gµν∇λ

$

δΓλµν

%

− gµν∇ν

$

δΓλµλ

%%

=

=

!

d4x√−g$

∇λ

$

gµνδΓλµν

%

−∇λ

$

gµλδΓνµν

%%

=

=

!

d4x√−g∇λ

$

gµνδΓλµν − gµλδΓν

µν

%

= 0 (4.41)

is a boundary term which is set to zero by making the variations vanish atthe boundary of space-time.

Now the third term in (4.39) can be simplified using the following usefulidentity

Tr(log (M)) = log (det (M)) (4.42)

or

Tr$

M−1δM%

=δ (det (M))

det (M)(4.43)

Applying it to inverse metric

Mµν = gµν (4.44)

gives us

Tr$

gµνδgνλ%

=δ (g−1)

g−1

gµνδgµν = −g−1δg. (4.45)

δ√−g = −

1

2

δg√−g

= −1

2

(−g) gµνδgµν√−g

= −1

2

√−ggµνδg

µν (4.46)

and by substituting in (4.39) we get

δSH =

!

d4x√−g

"

Rµν −1

2Rgµν

#

δgµν . (4.47)

Therefore for the action to vanish for arbitrary variations of the metric

1√−g

δS

δgµν= 0 (4.48)

the vacuum Einstein equations must be satisfied

Gµν = Rµν −1

2Rgµν = 0. (4.49)


4.3 Energy-Momentum Tensor

To obtain a non-vacuum equation on should also include matter Lagrangian,

S =1

16πG

!

d4x√−g R. +

!

d4x√−g LM , (4.50)

and the variational principle implies

δS

δgµν=

1

16πG

!

d4x√−g

"

Rµν −1

2Rgµν

#

+

!

d4x√−g

"

1√−g

δ (√−gLM)

δgµν

#

= 0.

For this to agree with Einstein equation

Rµν −1

2R gµν = 8πGTµν

we define the energy momentum tensor as

Tµν = −21√−g

δLM

δgµν. (4.51)

For example, action for a scalar field

Sφ =

!

dx4√−g

"

−1

2gµν∇µφ∇νφ− V (φ)

#

(4.52)

can be varied

δSφ =

!

dx4

(

−√−g

1

2∇µφ∇νφ (δg

µν) +

"

−1

2gµν∇µφ∇νφ− V (φ)

#

δ$√

−g%

)

=

=

!

dx4

(

−√−g

1

2∇µφ∇νφ+

"

−1

2gλρ∇λφ∇ρφ− V (φ)

#"

−1

2

√−ggµν

#)

δgµν =

=

!

dx4√−g

(

−1

2∇µφ∇νφ+

1

4gµνg

λρ∇λφ∇ρφ+1

2V (φ)gµν

)

δgµν (4.53)

which leads to the following energy momentum tensor

Tµν = −21√−g

δS

δgµν= ∇µφ∇νφ−

1

2gµνg

λρ∇λφ∇ρφ− V (φ)gµν . (4.54)

Note that the non-vacuum Einstein equation can be solved for an arbi-trary manifold. Just specify the metric, calculate the Einstein tensor anddemand 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


whether the corresponding energy momentum tensor is realistic. Thus, it isuseful to have some restrictions on Tµν .

Weak Energy Condition:

Tµνtµtν ≥ 0

for all time-like vectors tµ .Null Energy Condition:

Tµν lµlν ≥ 0

for all light-like vectors lµ .Dominant Energy Condition:

Tµνtµtν ≥ 0

for all time-like tµ and time-like or light-like T µνtµ .Strong Energy Condition:

Tµνtµtν −

1

2T λ

λtσtσ ≥ 0

for all time-like vectors tµ. This condition is used to prove famous singularitytheorems, but is also violated in quantum systems, for example, subject toCasimir force.

For a perfect fluid

Tµν = (ρ+ p)UµUν + 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 fourvelocities Uµ. Then the expansion, rotation and shear of the ball can bedefined as

θ = ∇λUλ. (4.55)

ωµν = ∇[νUµ] (4.56)


σµν = ∇(νUµ) −1

3∇λU

λ (gµν + UµUν) . (4.57)

and the famous Raychaudhuri equation describes the time evolution of theexpansion parameter,

dθ

dτ= −

1

3θ2 − σµνσµν + ωµνωµν − RµνU

µUν . (4.58)

If the ball of particles is at rest U = (1, 0, 0, 0) with respect to a locallyinertial system gµν = ηµν and has vanishing expansion θ = 0, rotation ω = 0and shear σ = 0, then the Raychaudhuri and Einstein equations give

dθ

dτ= −R00 = −8πG

"

T00 −1

2T g00

#

= −4πG (T00 + T11 + T22 + T33)

(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

dτ= −4πG (ρ+ px + py + pz) . (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 gravityis attractive.

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