175		7.2 Physics in slightly curved spacetimes
		Similarly, the law of conservation of entropy in SR is
		Uα S,α = 0.	(7.4)
	Since there are no Christoffel symbols in the covariant derivative of a scalar like S, this law
	is unchanged in a curved spacetime. Finally, conservation of four-momentum is
		Tμν ,ν = 0.	(7.5)
	The generalization is
		Tμν ;ν = 0,	(7.6)
	with the definition
		Tμν = (ρ + p)UμUν + pgμν ,	(7.7)

exactly as before. (Notice that gμν is the tensor whose components in the local inertial frame equal the flat-space metric tensor ημν .)

7.2 P h y s i c s i n s l i g h t l y c u r v e d s p a ce t i m e s

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

ds2 = −(1 + 2φ) dt2 + (1 − 2φ) (dx2 + dy2 + dz2).

(7.8)

(The sign of φ is chosen negative, so that, far from a source of mass M, we have φ = −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 mU, where U	dx/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 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 dx/d(τ /m) is also a vector satisfying the geodesic equation. This vector is just mdx/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, or in

other words m d/dτ :

m

d

0

+

0

α

β

= 0.

(7.11)

p

αβ p

p

dτ

Because the particle has a nonrelativistic velocity we have p0 p1 , so Eq. (7.11) is approximately

	d
m		p0 + 000(p0)2 = 0.	(7.12)
	dτ
We need to compute 000:
000 = 21 g0α (gα0,0 + gα0,0 − g00,α ).			(7.13)

Now because [gαβ ] is diagonal, [gαβ ] is also diagonal and its elements are the reciprocals of those of [gαβ ]. Therefore g0α is nonzero only when α = 0, so Eq. (7.13) becomes

000 = 21 g00g00,0 =	1				1		(−2φ),0
	2	−		(1	+	2φ)
= φ,0 + 0(φ2).						(7.14)

To lowest order in the velocity of the particle and in φ, we can replace (p0)2 in the second term of Eq. (7.12) by m2, obtaining

	d		∂φ
		p0	= −m		.	(7.15)
	dτ			∂τ

Since p0 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 p0 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. They are

pα pi,α + iαβ pα pβ = 0,			(7.16)
or, to lowest order in the velocity,
dpi		+ i00(p0)2 = 0.	(7.17)
m
	dτ

Again we have neglected pi compared to p0 in the summation. Consistent with this we can again put (p0)2 = m2 to a first approximation and get

dpi		= −m i00.	(7.18)
	dτ
We calculate the Christoffel symbol:
i00 = 21 giα (gα0,0 + gα0,0 − g00,α ).			(7.19)

177			7.3 Curved intuition
		Now, since [gαβ ] is diagonal, we can write
			giα = (1 − 2φ)−1δiα	(7.20)
		and get
		i	00 = 21 (1 − 2φ)−1δij(2gj0,0 − g00j),	(7.21)
		where we have changed α to j because δi0 is zero. Now we notice that gj0 ≡ 0 and so we get
			i00 = − 21 g00,jδij + 0(φ2)	(7.22)
			= − 21 (−2φ),jδij.	(7.23)
		With this the equation of motion, Eq. (7.17), becomes
			dpi/dτ = −mφ,jδij.	(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 (p0 pi). 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 C u r v e d i n t u i t i o n

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. dU/dτ = 0) also have no coor-
	dinate acceleration (d2xi/dt2 = 0). In our new picture, there is no coordinate system which
	is inertial everywhere, i.e. in which d2xi/dt2 = 0 for every particle for which dU/dτ = 0.
	Therefore we have to allow all coordinates on an equal footing. By using the Christoffel
	symbols we correct coordinate-dependent quantities like d2xi/dt2 to obtain coordinate-
	independent quantities like dU/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,			i 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 represent 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 Co n s e r v e d q u a n t i t i e s

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

pα pβ;α = 0,

(7.25)

or

pα pβ,α − γ βα pα pγ = 0,

or

m

dpβ

= γ βα pα pγ .

(7.26)

dτ

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

γ

αβ p

α

pγ

=

1

γ ν

(gνβ,α + gνα,β − gαβ,ν )p

α

pγ

g

2

=

1

(gνβ,α + gνα,β

− gαβ,ν )gγ ν pγ pα

2

=

1

(gνβ,α + gνα,β

− gαβ,ν )pν pα .

(7.27)

2

The product pν pα is symmetric on ν and α, while the first and third terms inside parentheses are, together, antisymmetric on ν and α. Therefore they cancel, leaving only the middle term

γ βα pα pγ =					1	gνα,β pν pα .	(7.28)
					2
The geodesic equation can thus, in complete generality, be written
	dpβ	1		gνα,β pν pα .
m		=					(7.29)
	dτ		2

We therefore have the following important result: if all the components gαν are independent of xβ for some fixed index β, then pβ 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 independent, and in that system p0 is conserved. Therefore p0 (or, really, −p0) 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 timedependent 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 gravitational 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 ‘laboratory frame’ on Earth. Therefore p0 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 = −m2 = gαβ pα pβ
= −(1 + 2φ)(p0)2 + (1 − 2φ)[(px)2 + (py)2 + (pz)2],	(7.30)

180			Physics in a curved spacetime
	where we have used the metric, Eq. (7.8). This can be solved to give
			(p0)2 = [m2 + (1 − 2φ)(p2)](1 + 2φ)−1,	(7.31)
		where, for shorthand, we denote by p2 the sum (px)2 + (py)2 + (pz)2. Keeping within the
	approximation |φ|		1, |p| m, we can simplify this to
			(p0)2 ≈ m2(1 − 2φ + p2/m2)
	or
			p0 ≈ m(1 − φ + p2/2m2).	(7.32)
	Now we lower the index and get
			p0 = g0α pα = g00p0 = −(1 + 2φ)p0,	(7.33)
			− p0 ≈ m(1 + φ + p2/2m2) = m + mφ + p2/2m.	(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 p0 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,2 so no conserved energy can be defined.

In a similar manner, if a metric is axially symmetric, then coordinates can be found in which gαβ is independent of the angle ψ around the axis. Then pψ will be conserved. This is the particle’s angular momentum. In the nonrelativistic limit we have

pψ = gψ ψ pψ ≈ gψ ψ m dψ/dt ≈ mgψ ψ ,

(7.35)

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

gψ ψ = eψ · eψ ≈ r2	(7.36)
in cylindrical coordinates (r, ψ , z) so that the conserved quantity is
pψ ≈ mr2 .	(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.

2It 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 xα¯ (xμ)). It is a special metric indeed if all ten components can be made time independent this way.