Schutz B. A first course in general relativity

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173 7.1 The transition from differential geometry to gravity

By ‘freely falling’ we mean particles unaffected by other forces, such as electric ﬁelds,

etc. All other known forces in physics are distinguished from gravity by the fact that there

are particles unaffected by them. So the Weak Equivalence Principle (Postulate IV) is a

powerful statement, capable of experimental test. And it has been tested, and continues

to be tested, to high accuracy. Experiments typically compare the rate of fall of objects

that are composed of different materials; current experimental limits bound the fractional

differences in acceleration to a few parts in 10

(Will 2006). The WEP is therefore one of

the most precisely tested laws in all of physics. There are even proposals to test it up to the

level of 10

−18

using satellite-borne experiments.

But the WEP refers only to particles. How are, say, ﬂuids affected by a nonﬂat metric?

We need a generalization of (

IV):

(IV



) Einstein Equivalence Principle: Any local physical experiment not involving

gravity will have the same result if performed in a freely falling inertial frame

as if it were performed in the ﬂat spacetime of special relativity.

In this case ‘local’ means that the experiment does not involve ﬁelds, such as electric ﬁelds,

that may extend over large regions and therefore extend outside the domain of validity of

the local inertial frame. All of local physics is the same in a freely falling inertial frame as it

is in special relativity. Gravity introduces nothing new locally. All the effects of gravity are

felt over extended regions of spacetime. This, too, has been tested rigorously (Will 2006).

This may seem strange to someone used to blaming gravity for making it hard to climb

stairs or mountains, or even to get out of bed! But these local effects of gravity are, in

Einstein’s point of view, really the effects of our being pushed around by the Earth and

objects on it. Our ‘weight’ is caused by the solid Earth exerting forces on us that prevent us

from falling freely on a geodesic (weightlessly, through the ﬂoor). This is a very reasonable

point of view. Consider astronauts orbiting the Earth. At an altitude of some 300 km, they

are hardly any further from the center of the Earth than we are, so the strength of the

Newtonian gravitational force on them is almost the same as on us. But they are weightless,

as long as their orbit prevents them encountering the solid Earth. Once we acknowledge

that spacetime has natural curves, the geodesics, and that when we fall on them we are in

free fall and feel no gravity, then we can dispose of the Newtonian concept of a gravitational

force altogether. We are only following the natural spacetime curve.

The true measure of gravity on the Earth are its tides. These are nonlocal effects, because

they arise from the difference of the Moon’s Newtonian gravitational acceleration across

the Earth, or in other words from the geodesic deviation near the Earth. If the Earth were

permanently cloudy, an Earthling would not know about the Moon from its overall grav-

itational acceleration, since the Earth falls freely: we don’t feel the Moon locally. But

Earthlings could in principle discover the Moon even without seeing it, by observing and

understanding the tides. Tidal forces are the only measurable aspect of gravity.

Mathematically, what the Einstein Equivalence Principle means is, roughly speaking,

that if we have a local law of physics that is expressed in tensor notation in SR, then its

mathematical form should be the same in a locally inertial frame of a curved spacetime.

174 Physics in a curved spacetime

This principle is often called the ‘comma-goes-to-semicolon rule’, because if a law

contains derivatives in its special-relativistic form (‘commas’), then it has these same

derivatives in the local inertial frame. To convert the law into an expression valid in

any coordinate frame, we simply make the derivatives covariant (‘semicolons’). It is an

extremely simple way to generalize the physical laws. In particular, it forbids ‘curvature

coupling’: it is conceivable that the correct form of, say, thermodynamics in a curved space-

time would involve the Riemann tensor somehow, which would vanish in SR. Postulate

(



) would not allow any Riemann-tensor terms in the equations.

As an example of how (



) translates into mathematics, we discuss ﬂuid dynamics,

which will be our main interest in this course. The law of conservation of particles in SR

is expressed as

(nU

)

,α

= 0, (7.1)

where n is the density of particles in the momentarily comoving reference frame (MCRF),

and where U

is the four-velocity of a ﬂuid element. In a curved spacetime, at any event,

we can ﬁnd a locally inertial frame comoving momentarily with the ﬂuid element at that

event, and deﬁne n in exactly the same way. Similarly we can deﬁne



U to be the time basis

vector of that frame, just as in SR. Then, according to the Einstein equivalence principle

(see Ch. 5), the law of conservation of particles in the locally inertial frame is exactly

Eq. (7.1). But because the Christoffel symbols are zero at the given event because it is the

origin of the locally inertial frame, this is equivalent to

(nU

)

;α

= 0. (7.2)

This form of the law is valid in all frames and so allows us to compute the conservation law

in any frame and be sure that it is the one implied by the Einstein equivalence principle.

We have therefore generalized the law of particle conservation to a curved spacetime. We

will follow this method for other laws of physics as we need them.

Is this just a game with tensors, or is there physical content in what we have done? Is

it possible that in a curved spacetime the conservation law would actually be something

other than Eq. (7.2)? The answer is yes: consider postulating the equation

(nU

)

;α

= qR

, (7.3)

where R is the Ricci scalar, deﬁned in Eq. (6.92) as the double trace of the Riemann ten-

sor, and where q is a constant. This would also reduce to Eq. (7.1) in SR, since in a ﬂat

spacetime the Riemann tensor vanishes. But in curved spacetime, this equation predicts

something very different: curvature would either create or destroy particles, according to

the sign of the constant q. Thus, both of the previous equations are consistent with the

laws of physics in SR. The Einstein equivalence principle asserts that we should general-

ize Eq. (7.1) in the simplest possible manner, that is to Eq. (7.2). It is of course a matter

for experiment, or astronomical observation, to decide whether Eq. (7.2)orEq.(7.3)is

correct. In this book we shall simply make the assumption that is nearly universally made,

that the Einstein equivalence principle is correct. There is no observational evidence to the

contrary.

175 7.2 Physics in slightly curved spacetimes

Similarly, the law of conservation of entropy in SR is

,α

= 0. (7.4)

Since there are no Christoffel symbols in the covariant derivative of a scalar like S,thislaw

is unchanged in a curved spacetime. Finally, conservation of four-momentum is

μν

,ν

= 0. (7.5)

The generalization is

μν

;ν

= 0, (7.6)

with the deﬁnition

μν

= (ρ + p)U

+ pg

μν

, (7.7)

exactly as before. (Notice that g

μν

is the tensor whose components in the local inertial

frame equal the ﬂat-space metric tensor η

μν

7.2 Physics in slightly curved spacetimes

To see the implications of (IV



) for the motion of a particle or ﬂuid, we must know the

metric on the manifold. Since we have not yet studied the way a metric is generated, we will

at this stage have to be content with assuming a form for the metric which we shall derive

later. We will see later that for weak gravitational ﬁelds (where, in Newtonian language,

the gravitational potential energy of a particle is much less than its rest-mass energy) the

ordinary Newtonian potential φ completely determines the metric, which has the form

=−(1 +2φ)dt

+ (1 − 2φ)(dx

+ dy

+ dz

). (7.8)

(The sign of φ is chosen negative, so that, far from a source of mass M,wehaveφ =

−GM/r.) Now, the condition above that the ﬁeld be weak means that |mφ|m, so that

|φ|1. The metric, Eq. (7.8), is really only correct to ﬁrst order in φ, so we shall work to

this order from now on.

Let us compute the motion of a freely falling particle. We denote its four-momentum

by p. For all except massless particles, this is m



U, where



U = dx/dτ .Now,by(

IV), the

particle’s path is a geodesic, and we know that proper time is an afﬁne parameter on such

a path. Therefore



U must satisfy the geodesic equation,

∇



U = 0. (7.9)

For convenience later, however, we note that any constant times proper time is an afﬁne

parameter, in particular τ/m. Then dx/d(τ/m) is also a vector satisfying the geodesic equa-

tion. This vector is just mdx/dτ =p. So we can also write the equation of motion of the

particle as

∇

p

p = 0. (7.10)

176 Physics in a curved spacetime

This equation can also be used for photons, which have a well-deﬁned p but no



U since

m = 0.

If the particle has a nonrelativistic velocity in the coordinates of Eq. (7.8), we can ﬁnd an

approximate form for Eq. (7.10). First let us consider the zero component of the equation,

noting that the ordinary derivative along p is m times the ordinary derivative along



U,orin

other words m d/dτ :

dτ

+ 

αβ

= 0. (7.11)

Because the particle has a nonrelativistic velocity we have p

 p

,soEq.(7.11)is

approximately

dτ

+ 

)

= 0. (7.12)

We need to compute 



0α

α0,0

+ g

α0,0

− g

00,α

). (7.13)

Now because [g

αβ

] is diagonal, [g

αβ

] is also diagonal and its elements are the reciprocals

of those of [g

αβ

]. Therefore g

0α

is nonzero only when α = 0, so Eq. (7.13) becomes



00,0

−(1 + 2φ)

(−2φ)

= φ

+ 0(φ

). (7.14)

To lowest order in the velocity of the particle and in φ, we can replace (p

)

in the second

term of Eq. (7.12)bym

, obtaining

dτ

=−m

∂φ

∂τ

. (7.15)

Since p

is the energy of the particle in this frame, this means the energy is conserved unless

the gravitational ﬁeld depends on time. This result is true also in Newtonian theory. Here,

however, we must note that p

is the energy of the particle with respect to this frame only.

The spatial components of the geodesic equation give the counterpart of the Newtonian

F = ma.Theyare

,α

+ 

αβ

= 0, (7.16)

or, to lowest order in the velocity,

dτ

+ 

)

= 0. (7.17)

Again we have neglected p

compared to p

in the  summation. Consistent with this we

can again put (p

)

= m

to a ﬁrst approximation and get

dτ

=−m

. (7.18)

We calculate the Christoffel symbol:



iα

α0,0

+ g

α0,0

− g

00,α

). (7.19)

177 7.3 Curved intuition

Now, since [g

αβ

] is diagonal, we can write

iα

= (1 −2φ)

−1

iα

(7.20)

and get



(1 − 2φ)

−1

(2g

j0,0

− g

00j

), (7.21)

where we have changed α to j because δ

is zero. Now we notice that g

≡ 0 and so we get



=−

00,j

+ 0(φ

) (7.22)

=−

(−2φ)

. (7.23)

With this the equation of motion, Eq. (7.17), becomes

/dτ =−mφ

. (7.24)

This is the usual equation in Newtonian theory, since the force of a gravitational ﬁeld

is −m∇φ. This demonstrates that general relativity predicts the Keplerian motion of the

planets, at least so long as the higher-order effects neglected here are too small to measure.

We shall see that this is true for most planets, but not for Mercury.

Both the energy-conservation equation and the equation of motion were derived

as approximations based on two things: the metric was nearly the Minkowski metric

(|φ| 1), and the particle’s velocity was nonrelativistic (p

 p

). These two limits are

just the circumstances under which Newtonian gravity is veriﬁed, so it is reassuring –

indeed, essential – that we have recovered the Newtonian equations. However, there is no

magic here. It almost had to work, given that we know that particles fall on straight lines

in freely falling frames.

We can do the same sort of calculation to verify that the Newtonian equations hold for

other systems in the appropriate limit. For instance, the student has an opportunity to do this

for the perfect ﬂuid in Exer. 5, § 7.6. Note that the condition that the ﬂuid be nonrelativistic

means not only that its velocity is small but also that the random velocities of its particles

be nonrelativistic, which means p  ρ.

This correspondence of our relativistic point of view with the older, Newtonian theory

in the appropriate limit is very important. Any new theory must make the same predictions

as the old theory in the regime in which the old theory was known to be correct. The

equivalence principle plus the form of the metric, Eq. (7.8), does this.

7.3 Curved intuition

Although in the appropriate limit our curved-spacetime picture of gravity predicts the same

things as Newtonian theory predicts, it is very different from Newton’s theory in concept.

We must therefore work gradually toward an understanding of its new point of view.

178 Physics in a curved spacetime

The ﬁrst difference is the absence of a preferred frame. In Newtonian physics and in

SR, inertial frames are preferred. Since ‘velocity’ cannot be measured locally but ‘acceler-

ation’ can be, both theories single out special classes of coordinate systems for spacetime

in which particles which have no physical acceleration (i.e. d



U/dτ = 0) also have no coor-

dinate acceleration (d

/dt

= 0). In our new picture, there is no coordinate system which

is inertial everywhere, i.e. in which d

/dt

= 0 for every particle for which d



U/dτ = 0.

Therefore we have to allow all coordinates on an equal footing. By using the Christoffel

symbols we correct coordinate-dependent quantities like d

/dt

to obtain coordinate-

independent quantities like d



U/dτ . Therefore, we need not, and in fact we should not,

develop coordinate-dependent ways of thinking.

A second difference concerns energy and momentum. In Newtonian physics, SR, and

our geometrical gravity theory, each particle has a deﬁnite energy and momentum, whose

values depend on the frame they are evaluated in. In the latter two theories, energy and

momentum are components of a single four-vector p. In SR, the total four-momentum

of a system is the sum of the four-momenta of all the particles,



p

(i)

. But in a curved

spacetime, we cannot add up vectors that are deﬁned at different points, because we do not

know how: two vectors can only be said to be parallel if they are compared at the same

point, and the value of a vector at a point to which it has been parallel-transported depends

on the curve along which it was moved. So there is no invariant way of adding up all the ps,

and so if a system has deﬁnable four-momentum, it is not just the simple thing it was in SR.

It turns out that for any system whose spatial extent is bounded (i.e. an isolated system),

a total energy and momentum can be deﬁned, in a manner which we will discuss later. One

way to see that the total mass energy of a system should not be the sum of the energies

of the particles is that this neglects what in Newtonian language is called its gravitational

self-energy, a negative quantity which is the work we gain by assembling the system from

isolated particles at inﬁnity. This energy, if it is to be included, cannot be assigned to any

particular particle but resides in the geometry itself. The notion of gravitational potential

energy, however, is itself not well deﬁned in the new picture: it must in some sense repre-

sent the difference between the sum of the energies of the particles and the total mass of

the system, but since the sum of the energies of the particles is not well deﬁned, neither

is the gravitational potential energy. Only the total energy–momentum of a system is, in

general, deﬁnable, in addition to the four-momentum of individual particles.

7.4 Conserved quantities

The previous discussion of energy may make us wonder what we can say about conserved

quantities associated with a particle or system. For a particle, we must realize that gravity,

in the old viewpoint, is a ‘force’, so that a particle’s kinetic energy and momentum need

not be conserved under its action. In our new viewpoint, then, we cannot expect to ﬁnd

a coordinate system in which the components of p are constants along the trajectory of a

particle. There is one notable exception to this, and it is important enough to look at in

detail.

179 7.4 Conserved quantities

The geodesic equation can be written for the ‘lowered’ components of p as follows

β;α

= 0, (7.25)

β,α

− 

βα

= 0,

dτ

= 

βα

. (7.26)

Now, the right-hand side turns out to be simple



αβ

γν

νβ,α

+ g

να,β

− g

αβ,ν

νβ,α

+ g

να,β

− g

αβ,ν

γν

νβ,α

+ g

να,β

− g

αβ,ν

. (7.27)

The product p

is symmetric on ν and α, while the ﬁrst and third terms inside paren-

theses are, together, antisymmetric on ν and α. Therefore they cancel, leaving only the

middle term



βα

να,β

. (7.28)

The geodesic equation can thus, in complete generality, be written

dτ

να,β

. (7.29)

We therefore have the following important result: if all the components g

αν

are independent

of x

for some ﬁxed index β, then p

is a constant along any particle’s trajectory.

For instance, suppose we have a stationary (i.e. time-independent) gravitational ﬁeld.

Then a coordinate system can be found in which the metric components are time indepen-

dent, and in that system p

is conserved. Therefore p

(or, really, −p

) is usually called the

‘energy’ of the particle, without the qualiﬁcation ‘in this frame’. Notice that coordinates

can also be found in which the same metric has time-dependent components: any time-

dependent coordinate transformation from the ‘nice’ system will do this. In fact, most

freely falling locally inertial systems are like this, since a freely falling particle sees a grav-

itational ﬁeld that varies with its position, and therefore with time in its coordinate system.

The frame in which the metric components are stationary is special, and is the usual ‘lab-

oratory frame’ on Earth. Therefore p

in this frame is related to the usual energy deﬁned

in the lab, and includes the particle’s gravitational potential energy, as we shall now show.

Consider the equation

p ·p =−m

= g

αβ

=−(1 +2φ)(p

)

+ (1 − 2φ)[(p

)

+ (p

)

+ (p

)

], (7.30)

180 Physics in a curved spacetime

where we have used the metric, Eq. (7.8). This can be solved to give

)

= [m

+ (1 − 2φ)(p

)](1 + 2φ)

−1

, (7.31)

where, for shorthand, we denote by p

the sum (p

)

+ (p

)

+ (p

)

. Keeping within the

approximation |φ|1, |p|m, we can simplify this to

)

≈ m

(1 − 2φ + p

)

≈ m(1 −φ +p

/2m

). (7.32)

Now we lower the index and get

= g

0α

= g

=−(1 +2φ)p

, (7.33)

− p

≈ m(1 +φ + p

/2m

) = m +mφ + p

/2m. (7.34)

The ﬁrst term is the rest mass of the particle. The second and third are the Newtonian

pieces of its energy: gravitational potential energy and kinetic energy. This means that

the constancy of p

along a particle’s trajectory generalizes the Newtonian concept of a

conserved energy.

Notice that a general gravitational ﬁeld will not be stationary in any frame,

so no

conserved energy can be deﬁned.

In a similar manner, if a metric is axially symmetric, then coordinates can be found in

which g

αβ

is independent of the angle ψ around the axis. Then p

will be conserved. This

is the particle’s angular momentum. In the nonrelativistic limit we have

= g

ψψ

≈ g

ψψ

m dψ/dt ≈ mg

ψψ

, (7.35)

where  is the angular velocity of the particle. Now, for a nearly ﬂat metric we have

ψψ

=e

·e

≈ r

(7.36)

in cylindrical coordinates (r, ψ, z) so that the conserved quantity is

≈ mr

. (7.37)

This is the usual Newtonian deﬁnition of angular momentum.

So much for conservation laws of particle motion. Similar considerations apply to ﬂuids,

since they are just large collections of particles. But the situation with regard to the total

mass and momentum of a self-gravitating system is more complicated. It turns out that an

isolated system’s mass and momentum are conserved, but we must postpone any discussion

of this until we see how they are deﬁned.

It is easy to see that there is generally no coordinate system which makes a given metric time independent. The

metric has ten independent components (same as a 4 × 4 symmetric matrix), while a change of coordinates

enables us to introduce only four degrees of freedom to change the components (these are the four functions

¯α

)). It is a special metric indeed if all ten components can be made time independent this way.

181 7.6 Exercises

7.5 Further reading

The question of how curvature and physics ﬁt together is discussed in more detail by

Geroch (1978). Conserved quantities are discussed in detail in any of the advanced texts.

The material in this chapter is preparation for the theory of quantum ﬁelds in a ﬁxed curved

spacetime. See Birrell and Davies (1984) and Wald (1994). This in turn leads to one of the

most active areas of gravitation research today, the quantization of general relativity. While

we will not treat this area in this book, readers in work that approaches this subject from

the starting point of classical general relativity (as contrasted with approaching it from the

starting point of string theory) may wish to look at Rovelli (2004) Bojowald (2005), and

Thiemann (2007).

7.6 E xe rc is e s

1 If Eq. (7.3) were the correct generalization of Eq. (7.1) to a curved spacetime, how

would you interpret it? What would happen to the number of particles in a comoving

volume of the ﬂuid, as time evolves? In principle, can we distinguish experimentally

between Eqs. (7.2) and (7.3)?

2 To ﬁrst order in φ, compute g

αβ

for Eq. (7.8).

3 Calculate all the Christoffel symbols for the metric given by Eq. (7.8), to ﬁrst order in

φ. Assume φ is a general function of t, x, y, and z.

4 Verify that the results, Eqs. (7.15) and (7.24), depended only on g

: the form of g

doesn’t affect them, as long as it is 1 +0(φ).

5 (a) For a perfect ﬂuid, verify that the spatial components of Eq. (7.6) in the Newtonian

limit reduce to

+ (υ ·∇)υ + ∇p/ρ + ∇φ = 0 (7.38)

for the metric, Eq. (7.8). This is known as Euler’s equation for nonrelativistic ﬂuid

ﬂow in a gravitational ﬁeld. You will need to use Eq. (7.2) to get this result.

(b) Examine the time-component of Eq. (7.6) under the same assumptions, and

interpret each term.

obeys the equation of hydrostatic equilibrium

∇p +ρ∇φ = 0. (7.39)

A metric tensor is said to be static if there exist coordinates in which e

is timelike,

= 0, and g

αβ,0

= 0. Deduce from Eq. (7.6) that a static ﬂuid (U

= 0, p

= 0,

etc.) obeys the relativistic equation of hydrostatic equilibrium

+ (ρ +p)



ln(−g

)



= 0. (7.40)

182 Physics in a curved spacetime

(d) This suggests that, at least for static situations, there is a close relation between g

and −exp(2φ), where φ is the Newtonian potential for a similar physical situation.

Show that Eq. (7.8) and Exer. 4 are consistent with this.

6 Deduce Eq. (7.25)fromEq.(7.10).

7 Consider the following four different metrics, as given by their line elements:

(i) ds

=−dt

+ dx

+ dy

+ dz

;

(ii) ds

=−(1 −2M/r)dt

+ (1 − 2M/r)

−1

+ r

(dθ

+ sin

θ dφ

), where

M is a constant;

(iii)

=−

 − a

sin

− 2a

2Mr sin

dt dφ

+ a

)

− a

 sin

sin

θ dφ



+ ρ

dθ

where M and a are constants and we have introduced the shorthand notation

 = r

− 2Mr + a

, ρ

= r

+ a

cos

θ;

(iv) ds

=−dt

+ R

(t)

(1 − kr

)

−1

+ r

(dθ

+ sin

θ dφ

)

, where k is a

constant and R(t) is an arbitrary function of t alone.

The ﬁrst one should be familiar by now. We shall encounter the other three

in later chapters. Their names are, respectively, the Schwarzschild, Kerr, and

Robertson–Walker metrics.

(a) For each metric ﬁnd as many conserved components ρ

of a freely falling particle’s

four momentum as possible.

(b) Use the result of Exer. 28, § 6.9 to put (i) in the form



)ds

=−dt

+ dr

+ r

(dθ

+ sin

θ dφ

From this, argue that (ii) and (iv) are spherically symmetric. Does this increase the

number of conserved components p



) and (ii)–(iv), a geodesic that begins with θ = π/2

and p

= 0 – i.e. one which begins tangent to the equatorial plane – always has

θ = π/2 and p

= 0. For cases (i



), (ii), and (iii), use the equation p ·p =−m

to solve for p

in terms of m, other conserved quantities, and known functions of

position.

(d) For (iv), spherical symmetry implies that if a geodesic begins with p

= p

= 0,

these remain zero. Use this to show from Eq. (7.29) that when k = 0, p

is a

conserved quantity.

8 Suppose that in some coordinate system the components of the metric g

αβ

are

independent of some coordinate x

(a) Show that the conservation law T

μ;ν

= 0forany stress–energy tensor becomes

√

− g

(

√

− gT

)

,ν

= 0. (7.41)

(b) Suppose that in these coordinates T

αβ

= 0 only in some bounded region of each

spacelike hypersurface x

= const. Show that Eq. (7.41) implies