ñòð. 7 |

X X

where defines a Lorentz transformation. Thus, local Cartesian freely falling

(non-rotating) frames at an event P are related to one another by boosts, spatial

152 The equivalence principle and spacetime curvature

rotations or combinations of the two. For any one of these coordinate systems, the

Ë†

timelike basis vector e0 P is simply the normalised 4-velocity vector u P of the

origin of that frame at the event P, and the three mutually orthogonal spacelike

vectors ei P i = 1 2 3 define the orientation of the spatial axes of the frame.

For points near to P, the metric in a local inertial coordinate system X (whose

origin is at P) is given by

= +2 X +Â·Â·Â·

1

g g PX

The sizes of the second derivatives g P thus determine the region over

which the approximation g â‰ˆ remains valid. We shall see the significance

of these second derivatives shortly.

7.5 Observers in a curved spacetime

We discussed the subject of observers in Minkowski spacetime in Chapter 5, but

let us now consider the subject in its full generality, in a curved spacetime. An

observer will trace out some general (timelike) worldline x through spacetime,

as expressed in some arbitrary coordinate system, where is the observerâ€™s proper

time. An idealisation of his local laboratory is a frame of four orthonormal vectors

Ë† (or tetrad) satisfying

e

Ë† Ë†

Â·e =

e

which are carried with him along his worldline (these vectors may, in general,

be totally unrelated to the basis vectors e of the coordinate system that we are

using to label points in spacetime, although we can always express one set of

vectors in terms of the other). In particular, at any point along his worldline the

Ë† Ë†

coincides with the normalised 4-velocity u = u /c of

timelike vector e0

Ë†

the observer. Similarly, the evolution of the spacelike vectors ei along the

worldline reflect the different ways in which his local laboratory may be spinning

or tumbling. Quantities measured in this laboratory correspond to projections of

the relevant physical 4-vectors and 4-tensors onto this orthonormal frame.

As shown in Chapter 5, if the observer has a 4-acceleration a = du/d but

is not rotating, the tetrad basis vectors are Fermiâ€“Walker-transported along the

observerâ€™s worldline:

e

dË† 1

Ë† Ë†

=2 uÂ·e aâˆ’ aÂ·e u (7.5)

d c

This expression holds equally well in a curved spacetime. An important special

case is that of a non-rotating, freely falling observer, i.e one who is moving only

153

7.6 Weak gravitational fields and the Newtonian limit

Ë†

under the influence of gravity. The vectors e then define what is called a

freely falling frame (FFF). Free from any external forces, the observerâ€™s worldline

Ë†

traces out a geodesic in the curved spacetime. Thus the timelike vector e0 changes

with proper time along the worldline according to

e

dË† 0

=0

d

Ë†

In other words, e0 is parallel-transported along the worldline, and the observerâ€™s

4-acceleration a is zero. In this case we see from (7.5) that Fermiâ€“Walker transport

Ë†

reduces to parallel transport. Thus the spacelike frame vectors ei (i = 1 2 3) are

also parallel-transported along the geodesic, so that

e

dË† i

=0

d

Ë† Ë†

=e Â·e

Hence, in an arbitrary coordinate system x , the components e

of any frame vector evolve as follows:

Ë† Ë†

De de

Ë†

= + u =0

e

D d

This equation is extremely useful for determining what a freely falling observer

would measure at a given event in spacetime. It is also clear that the frame vectors

Ë†

e at any event P along the observerâ€™s worldline are the basis vectors of a local

Cartesian inertial coordinate system at P.

7.6 Weak gravitational fields and the Newtonian limit

It is clear that, by construction, our description of gravity in terms of spacetime

curvature reduces to special relativity in local inertial frames. It is important to

check, however, that such a description also reduces to Newtonian gravity in the

appropriate limits.

In the absence of gravity, spacetime has a Minkowski geometry. Therefore a

weak gravitational field corresponds to a region of spacetime that is only â€˜slightlyâ€™

curved. In other words, in such a region there exist coordinates x in which the

metric takes the form

= +h where h 1 (7.6)

g

Note that it is important to say â€˜there exist coordinatesâ€™ since (7.6) does not hold

for all coordinates; as we saw in Chapter 5, one can find coordinates even in

Minkowski space in which g is not close to the simple form . Let us assume

that in the coordinate system (7.6) the metric is stationary, which means that all

154 The equivalence principle and spacetime curvature

the derivatives 0 g are zero. An example of such a coordinate system might be

a fixed Cartesian frame at some point on the surface of the (non-rotating) Earth.

The worldline of a particle freely falling under gravity is given in general by

the geodesic equation

d2 x dx dx

+ =0

d2 dd

We shall assume, however, that the particle is slow-moving, so that the compo-

c i = 1 2 3 , where t is defined by

nents of its 3-velocity satisfy dxi /dt

0 â‰¡ ct. This is equivalent to demanding that, for i = 1 2 3,

x

dx0

dxi

d d

Thus we can ignore the 3-velocity terms in the geodesic equation to obtain

2

d2 x dt

+ =0

2

(7.7)

00 c

d2 d

Now, recalling the expression (3.21) giving the connection in terms of the metric

and using the form (7.6) for g , we find that the connection coefficients 00

are given by

= 2g + âˆ’ g00 = âˆ’ 2 g g00 = âˆ’ 2

1 1 1

0 g0 0 g0 h00

00

where the last equality is valid to first order in h . Since we have assumed that

the metric is stationary, we have

=0 = 1 ij

0 i

and j h00

00 00 2

where the Latin index runs over i = 1 2 3. Inserting these coefficients into (7.7)

gives

2

d2 t d2 x dt

=0 = âˆ’ 2 c2

1

and h00

d2 d2 d

The first equation implies that dt/d = constant, and so we can combine the two

equations to yield the following equation of motion for the particle:

d2 x

= âˆ’ 2 c2 h00

1

dt2

If we compare this equation with the usual Newtonian equation of motion for

a particle in a gravitational field (7.2), we see that the two are identical if we

make the indentification h00 = 2 /c2 . Hence for a slowly moving particle our

155

7.7 Electromagnetism in a curved spacetime

description of gravity as spacetime curvature tends to the Newtonian theory if the

metric is such that, in the limit of a weak gravitational field,

2

g00 = 1 + (7.8)

c2

How big is the correction to the Minkowski metric? Some values of /c2 for

various systems are as follows:

âŽ§

âŽªâˆ’10âˆ’9 at the surface of the Earth

GM âŽ¨

= âˆ’ 2 = âˆ’10âˆ’6 at the surface of the Sun

âŽª

c2 âŽ©

cr

âˆ’10âˆ’4 at the surface of a white dwarf star

Thus, we see that even at the surface of a dense object like a white dwarf, the

size of /c2 is much smaller than unity and hence the weak-field limit will be

an excellent approximation.

From (7.8), the observant reader will have noticed that the description of gravity

in terms of spacetime curvature has another immediate consequence, namely that

the time coordinate t does not, in general, measure proper time. If we consider a

clock at rest at some point in our coordinate system (i.e. dxi /dt = 0), the proper

time interval d between two â€˜clicksâ€™ of the clock is given by

=g dx dx = g00 c2 dt2

c2 d 2

from which we find that

1/2

2

d = 1+ 2 dt

c

This gives the interval of proper d corresponding to an interval dt of coordinate

time for a stationary observer near a massive object, in a region where the

gravitational potential is . Since is negative, this proper time interval is shorter

than the corresponding interval for a stationary observer at a large distance from

the object, where â†’ 0 and so d = dt. Thus, as a bonus, our analysis has also

yielded the formula for time dilation in a weak gravitational field.

7.7 Electromagnetism in a curved spacetime

Before going on to discuss the mathematics of curvature in detail, let us look

back at our development of electromagnetism in Chapter 6. It is clear that our

derivation of the electromagnetic field equations in arbitrary coordinates did not

156 The equivalence principle and spacetime curvature

depend on the intrinsic geometry of the manifold on which the electromagnetic

field tensor F and the 4-current j are defined. In other words, one can arrive at

these equations without assuming the spacetime to have a Minkowski geometry.

Thus, in the presence of gravitating matter, spacetime becomes curved but the

field equations of electromagnetism in an arbitrary coordinate system are still

given by

=

F 0j

(7.9)

+ + =0

F F F

The effects of gravitation are automatically included in these field equations

through the covariant derivatives, which depend on the metric g describing

the spacetime geometry. Moreover, if we construct a local Cartesian coordinate

system about some point P in the manifold then (as discussed above) these

coordinates correspond to a local inertial frame in the neighbourhood of P. In these

coordinates, the equations of electromagnetism then take their familiar special

relativistic forms.

An electromagnetic field tensor F defined on a curved spacetime gives rise (as

in Minkowski space) to a 4-force f = qF Â· u, which acts on a particle of charge

q with 4-velocity u. Thus the equation of motion of a charged particle moving

under the influence of an electromagnetic field in a curved spacetime has the

same form as that in Minkowski spacetime, i.e.

du

= qF Â· u

m0

d

where m0 is the rest mass of the particle. In this case, however, because of

the curvature of spacetime the particle is moving under the influence of both

electromagnetic forces and gravity. In some arbitrary coordinate system, the

particleâ€™s worldline is again given by

d2 x dx dx q dx

+ = F

d2 dd m0 d

Obviously, in the absence of an electromagnetic field (or for an uncharged parti-

cle), the right-hand side is zero and we recover the equation of a geodesic.

We must remember, however, that the energy and momentum of the electro-

magnetic field will itself induce a curvature of spacetime, so the metric in this

case is determined not only by the matter distribution but also by the radiation.

157

7.8 Intrinsic curvature of a manifold

7.8 Intrinsic curvature of a manifold

Since the notion of curvature is central to general relativity, we must now inves-

tigate how to quantify the intrinsic curvature of a manifold at any given point P.3

A manifold (or region of a manifold) is flat if there exist coordinates X such

that, throughout the region, the line element can be written

ds2 = dX 1 2 + dX 2 2 + Â· Â· Â· + 2

dX N (7.10)

1 2 N

where a = Â±1 (in other words â€˜flatâ€™ is a shorthand for pseudo-Euclidean). If,

however, points in the manifold are labelled with some arbitrary coordinate system

xa then in general the line element ds2 will not be of the above form. Thus, if for

some manifold the line element is given by

ds2 = gab x dxa dxb

how can we tell whether the intrinsic geometry of the manifold in some region is

flat or curved in some way?

Consider, for example, the following line element for a three-dimensional space:

ds2 = dr 2 + r 2 d + r 2 sin2 d

2 2

Of course, we recognise this as the line element of ordinary three-dimensional

Euclidean space written in spherical polar coordinates. In other words, the trans-

formation

x = r sin cos y = r sin sin z = r cos

will turn the above line element into the form

ds2 = dx2 + dy2 + dz2 (7.11)

But what about other line elements? For example, recall from Chapter 2 the

three-dimensional space described by the line element (2.21):

a2

ds = 2 dr 2 + r 2 d 2 + r 2 sin2 d 2

2

a âˆ’ r2

How can we tell whether this metric, or a more complicated metric, corresponds

to flat space but merely looks complicated because of a weird choice of coordi-

nates? It would be immensely tedious to try to discover whether there exists a

coordinate transformation that reduces a metric to the form (7.11). We therefore

need some means of telling whether a manifold is flat directly from the metric

gab , independently of the coordinate system being used.

3

Since the material presented here is applicable to any N -dimensional pseudo-Riemannian manifold, we will

use indices a b etc. that have a range 1 to N , rather than etc., with a range 0 to 3. Of course, the final

application to general relativity will govern the scope of our results.

158 The equivalence principle and spacetime curvature

The physical significance of this to general relativity is as follows. If, throughout

some region of a four-dimensional spacetime, we can reduce the line element

ds2 = g dx dx

to Minkowski form then there can be no gravitational field in this region. The

equivalence of a general line element to that of Minkowski spacetime therefore

guarantees that the gravitational field will vanish. The solution to our mathematical

problem of finding a coordinate-independent way of defining the curvature of

spacetime will lead us to the field equations of gravity.

7.9 The curvature tensor

We can find a solution to the problem of measuring the curvature of a manifold at

any point by considering changing the order of covariant differentiation. Covariant

differentiation is clearly a generalisation of partial differentiation. There is one

important respect in which it differs, however: it matters in which order covariant

differentiation is performed, and changing the order (in general) changes the result.

Since for a scalar field the covariant derivative is simply the partial deriva-

tive, the order of differentiation does not matter. However, let us consider some

arbitrary vector field defined on a manifold, with covariant components va . The

covariant derivative of the va is given by

= b va âˆ’

d

b va ab vd

A second covariant differentiation then yields

= âˆ’ ac b ve âˆ’

e e

c b va b va bc e va

c

= c b va âˆ’ vd âˆ’

d d

ab c vd

c ab

âˆ’ b ve âˆ’ âˆ’ e va âˆ’

e d e d

eb vd ae vd

ac bc

which follows since b va is itself a rank-2 tensor. Swapping the indices b and c

to obtain a corresponding expression for b c va and then subtracting gives

c b va âˆ’ = Rd abc vd (7.12)

b c va

where

Rd abc â‰¡ ac âˆ’ c ab + eb âˆ’

d d e d e d

(7.13)

b ac ab ec

To determine directly whether the N 4 quantities Rd abc transform as the compo-

nents of a tensor under a coordinate transformation would be an arduous algebraic

task. Fortunately the quotient theorem (Section 4.11) provides a much shorter

route. The left-hand side of (7.12) is a tensor, for arbitrary vectors va , so the

159

7.10 Properties of the curvature tensor

contraction of Rd abc with vd is also a tensor. Since Rd abc does not depend on va ,

we conclude from the quotient theorem that the Rd abc are indeed the components

of some rank-4 tensor R. This tensor is called the curvature tensor (or Riemann

tensor), and equation (7.13) shows that it is defined in terms of the metric tensor

gab and its first and second derivatives.

We must now establish how the tensor (7.13) is related to the curvature of the

manifold. In a flat region of a manifold, we may choose coordinates such that

the line element takes the form (7.10) throughout the region. In these coordinates

a

bc and its derivatives are zero, and hence

Rd abc = 0

at every point in the region. This is a tensor relation, however, and so it must hold

in any coordinate system. Conversely, if Rd abc = 0 at every point in some region

of a manifold, then it may be shown that it is possible to introduce a coordinate

system in which the line element takes the form (7.10), and hence this region

is flat.4 Thus the vanishing of the curvature tensor is a necessary and sufficient

condition for a region of a manifold to be flat.

7.10 Properties of the curvature tensor

The curvature tensor (7.13) possesses a number of symmetries and satisfies certain

identities, which we now discuss. The symmetries of the curvature tensor are most

easily derived in terms of its covariant components

Rabcd = gae Re bcd

For completeness, we note that in an arbitrary coordinate system an explicit form

for these components is found, after considerable algebra, to be

Rabcd = d a gbc âˆ’ d b gac + c b gad âˆ’ c a gbd âˆ’ g ef eac fbd âˆ’ ead fbc

1

2

One could use this expression straightforwardly to derive the symmetry properties

of the curvature tensor, but we take the opportunity here to illustrate a general

mathematical device that is often useful in reducing the algebraic burden of tensor

manipulations.

Let us choose some arbitrary point P in the manifold and construct a geodesic

coordinate system about this point (see Section 3.11), in which the connec-

tion vanishes, a bc P = 0, although in general its derivatives will not. In this

4

For a proof of this result, see (for example) P. A. M. Dirac, General Theory of Relativity, Princeton Landmarks

in Physics Series, Princeton University Press, 1996.

160 The equivalence principle and spacetime curvature

coordinate system, one may easily show directly from (7.13) that the covariant

components of the curvature tensor at P are given by

= d a gbc âˆ’ d b gac + c b gad âˆ’ c a gbd P

1

Rabcd P 2

From this expression one may immediately establish the following symmetry

properties at P:

Rabcd = âˆ’Rbacd (7.14)

Rabcd = âˆ’Rabdc (7.15)

Rabcd = Rcdab (7.16)

The first two properties show that the curvature tensor is antisymmetric with

respect to swapping the order of either the first two indices or the second two

indices. The third property shows that it is symmetric with respect to swapping

the first pair of indices with the second pair of indices. Moreover, we may also

easily deduce the cyclic identity

Rabcd + Racdb + Radbc = 0 (7.17)

which on using (7.15) may be written more succinctly as Ra bcd = 0. Although

the results (7.14â€“7.17) have been derived in a special coordinate system, each

condition is a tensor relation and so if it is valid in one coordinate system then it is

valid in all. Moreover, since the point P is arbitrary, the results hold everywhere.

Although first appearances might suggest that the curvature tensor has N 4

components, the conditions (7.14â€“7.17) reduce the number of independent compo-

nents to N 2 N 2 âˆ’ 1 /12. Recall from Section 2.11 that this is also the number of

degrees of freedom among the second derivatives d c gab . This is not surprising

since, at any point P in a manifold, we can perform a transformation to local

Cartesian coordinates in which gab P = and c gab P = 0. Thus, a general

metric at any point P is characterised by the N 2 N 2 âˆ’ 1 /12 second derivatives

that cannot be made to vanish there.

For manifolds of different dimensions we have the following results:

No. of dimensions 2 3 4

No. of independent components of Rabcd 1 6 20

You can see from this table that in four dimensions the number of independent

components is reduced from a possible 256 to 20. You will also see that in one

dimension the curvature tensor is always equal to zero: R1111 = 0. How can this

161

7.11 The Ricci tensor and curvature scalar

be? Can a line not be curved? Think about this â€“ the curvature measures the

â€˜innerâ€™ properties of the space. When we say that a line is curved we refer to

a particular embedding in a higher-dimensional space, but this does not tell us

about the inner properties of the space. In one dimension, it is evident that we

can always find a coordinate transformation that will reduce an arbitrary metric to

the form (7.10). As a two-dimensional example, in Appendix 7A we calculate the

single independent component of the curvature tensor for the surface of a sphere.

The Gaussian curvature K of a two-dimensional surface is given by

R1212

K=

g

where g = det gab is the determinant of the metric tensor.

The curvature tensor also satisfies a differential identity, which may be derived

as follows. Let us once again adopt a geodesic coordinate system about some

arbitrary point P. In this coordinate system, differentiating and then evaluating

the result at P gives

= = e c abd âˆ’ e d abc P

e Rabcd P e Rabcd P

Cyclically permuting c, d and e to obtain two further analogous relations and

adding, one finds that at P

e Rabcd + c Rabde + =0 (7.18)

d Rabec

This is, however, a tensor relation and thus holds in all coordinate systems;

moreover, since P is arbitrary the relationship holds everywhere. This result is

known as the Bianchi identity and, using the antisymmetry relation (7.14), it may

be written more succinctly as

=0

e Rab cd

7.11 The Ricci tensor and curvature scalar

It follows from the symmetry properties (7.14â€“7.16) of the curvature tensor that

it possesses only two independent contractions. We may find these by contracting

either on the first two indices or on the first and last indices respectively. From

(7.14), raising the index a and then contracting on the first two indices gives

Ra acd = 0

Contracting on the first and last indices, however, gives in general a non-zero

result and this leads to a new tensor, the Ricci tensor. It is traditional to use the

162 The equivalence principle and spacetime curvature

same kernel letter for the Ricci tensor as for the curvature tensor, so we denote

its components by

Rab â‰¡ Rc abc

By raising the index a in the cyclic identity (7.17) and contracting with d, one

may easily show that the Ricci tensor is symmetric. Thus we have Ra b = Ra b and

we can denote both by Rb .

a

A further contraction gives the curvature scalar (or Ricci scalar)

R â‰¡ g ab Rab = Ra

a

where again the same kernel letter is used. This is a scalar quantity defined at

each point of the manifold.

The covariant derivatives of the Ricci tensor and the curvature scalar obey a

particularly important relation, which will be central to our development of the

field equations of general relativity. Raising a in the Bianchi identity (7.18) and

contracting with d gives

e Rbc + c R bae + =0

a a

a R bec

which, on using the antisymmetry property (7.16) in the second term, gives

e Rbc âˆ’ c Rbe + =0

a

a R bec

If we now raise b and contract with e, we find

b Rc âˆ’ cR + =0

b ab

(7.19)

a R bc

Using the antisymmetry properties (7.14, 7.15) we may write the third term as

= = =

ab ba a b

a R bc a R cb a Rc b Rc

so the first and last terms in (7.19) are identical and we obtain

b Rc âˆ’ = 2Rb âˆ’ =0

b b

2 cR cR

b c

Finally, raising the index c, we obtain the important result

Rbc âˆ’ 2 g bc R = 0

1

b

The term in parentheses is called the Einstein tensor

Gab â‰¡ Rab âˆ’ 2 g ab R

1

163

7.12 Curvature and parallel transport

It is clearly symmetric and thus possesses only one independent divergence a Gab ,

which vanishes (by construction). As we will see, it is this tensor that describes

the curvature of spacetime in the field equations of general relativity.

7.12 Curvature and parallel transport

In Chapter 3, we remarked that parallel transport in a curved manifold was path

dependent. We now have a more formal description of curvature. If a region

of manifold is flat then the curvature tensor vanishes throughout the region;

otherwise, it is curved. Thus there must be some link between the curvature tensor

and parallel transport.

Let us consider the parallel transport of a vector v around a closed curve

in a manifold. We can define an arbitrary surface bounding the curve and

break this surface up into a lot of small areas each bounded by closed curves

a

N , as indicated in Figure 7.2. The change in the components v on being

parallel-transported around the closed curve is then

va = va N

N

where va N is the change in va around the small closed curve N . This follows

because the changes in va around any of the interior closed curves cancel,

leaving just the contributions around the outer edges that bound the curve .

Let us now calculate va N around the small closed curve N defined by the

parametric equations xa u . The equation for parallel transport is given by (3.41):

dva c

b dx

=âˆ’ a

bc v

du du

Define an arbitrary surface

bounding the curve â€“ break

this area up into lots of little

closed curves N

Figure 7.2 An arbitrary surface bounding a closed curve .

164 The equivalence principle and spacetime curvature

Thus, if va is parallel-transported along the small closed curve from some

N

initial point P then at some other point along this curve we have

c

u

b dx

v u= âˆ’

a a a

(7.20)

vP bc v du

du

uP

However, since the closed curve is small we can expand the factors in the integrand

about P to first order in xa âˆ’ xP :

a

u= + xd u âˆ’ xP + Â· Â· Â·

a a a d

bc bc P d bc P

va u = vP âˆ’ xc u âˆ’ xP + Â· Â· Â·

a a b c

bc P vP

Substituting these expressions into (7.20) and retaining terms only up to first order

in xa âˆ’ xP , we obtain

a

dxc

u

vu= âˆ’

a a a b

vP bc P vP du

du

uP

dxc

u

âˆ’ bc âˆ’ âˆ’ xP

a a e b d d

bd P vP x du

d ec

du

uP

If we integrate the coordinate differentials around a closed loop we have dxc = 0,

and so we find that

u

v =âˆ’ bc âˆ’

a a a e b

xd dxc

bd P vP

d ec

uP

We may obtain an analogous result by interchanging the dummy indices c and d.

Now using the result

d xc xd = xc dxd + xd dxc = 0

we find that

va = âˆ’ 2 bd âˆ’ d bc + bd âˆ’

1 b

a a a e a e xc dxd

bc P vP

c ec ed

On using the expression (7.13), we finally obtain

va = âˆ’ 2 Ra bcd

1 b

xc dxd (7.21)

P vP

Equation (7.21) establishes the link between the curvature tensor at a point

P and parallel transport around a small loop close to P. It tells us that the

components va will remain unchanged after parallel transportation around a small

closed loop near P if and only if the curvature tensor vanishes at P. So, returning

165

7.13 Curvature and geodesic deviation

A

C

A

B D

B C

Figure 7.3 Parallel transport around a closed curve on the surface of a sphere

and the surface of a cylinder.

va N , the vector components va will not change on

to our construction of

parallel transportation around the entire closed curve if the curvature tensor

Ra bcd vanishes over the entire area bounding the curve.

As an example, consider the parallel transportation of a vector around the

closed triangle ABC on the surface of a sphere (see Figure 7.3). As shown in

Appendix 7A, the curvature tensor is nowhere zero, and it is evident that the

vector changes direction after parallel transportation around the triangle. However,

as also mentioned in Appendix 7A, the curvature tensor vanishes everywhere

on the surface of a cylinder and hence the components of a vector will remain

unchanged if the vector is parallel-transported around any closed curve (see

Figure 7.3).

7.13 Curvature and geodesic deviation

Another important consequence of curvature is that two nearby geodesics that

are initially parallel either converge or diverge, depending on the local curvature.

This is embodied in the equation of geodesic deviation, which we now derive.

Consider two neighbouring geodesics, given by xa u and Â¯ given by xa u , Â¯

a u be the small â€˜vectorâ€™ connecting

where u is an affine parameter, and let

points on the two geodesics with the same parameter value (see Figure 7.4), i.e.

xa u = xa u +

Â¯ a

u

In particular, let us suppose that for some arbitrary value of u the vector a u

connects the point P on to the point Q on Â¯ .

Once again our derivation is simplified considerably by constructing local

geodesic coordinates about the point P, in which the connection coefficients

166 The equivalence principle and spacetime curvature

x a (u)

x a(u)

Î¾ a (u)

Figure 7.4 Two neighbouring geodesics.

vanish at P but their derivatives are in general non-zero there. In this coordinate

system, since and Â¯ are geodesics we have

d 2 xa

=0 (7.22)

du2 P

Â¯ Â¯Â¯

d 2 xa d xb d xc

+ =0

a

(7.23)

bc

du2 du du Q

a,

at the points P and Q respectively. However, to first order in

Q= P+ =

a a a d a d

bc bc d bc P d bc P

Thus, subtracting (7.22) from (7.23) gives, to first order, at P

Â¨a + Ë™Ë™ =0

a

xb xc d

d bc

where the dots denote d/du. However, in our geodesic coordinates the second-

order intrinsic derivative of a at P is given by

D2 a d Ë™a

+ a bc b xc = Â¨ a + d a bc b xc xd

= Ë™Ë™

2

Du du

where we have used the fact that a bc P = 0; we note that nevertheless the

derivatives of a bc at P may not vanish. Thus, combining the last two equations

and relabelling dummy indices, we find that at P

D2 a

+ b a cd âˆ’ d a bc + b xc xd = 0

Ë™Ë™

2

Du

We may now identify the terms in parentheses on the left-hand side as components

Ra cbd of the Riemann tensor when expressed in local geodesic coordinates about

P. Thus we may write the above result as

D2 a

+ Ra cbd x x =0

Ë™Ë™

bcd

(7.24)

2

Du

167

7.14 Tidal forces in a curved spacetime

A

B C

Figure 7.5 Converging geodesics on the surface of a sphere.

which is clearly a tensor relation and is hence valid in any coordinate system.

Moreover, since P is an arbitrary point on , this relation is valid everywhere

along the curve. The result (7.24) is the equation of geodesic deviation.

The geometric meaning of (7.24) is straightforward. In a flat region of a mani-

fold, Ra bcd = 0 and we may adopts Cartesian coordinates throughout. In this case,

D/Du = d/du and the equation of geodesic deviation reduces to d2 a /du2 = 0,

which implies that a u = Aa u + Ba where Aa and Ba are constants. So in a

flat region the separation vector a u connecting the two geodesics (which are

simple straight lines in this case) in general increases linearly with u. In the

special case where the two lines are initially parallel then they will remain so

and hence never intersect. In a curved region of a manifold, however Ra bcd = 0

and so neighbouring geodesics either converge or diverge. For example, the two

neighbouring geodesics AB and AC on the surface of a sphere (see Figure 7.5)

converge as we approach the point A at the pole because the surface is positively

curved. Equation (7.24) allows us to compute the rates of convergence or diver-

gence of neighbouring geodesics for Riemannian spaces of arbitrary complexity.

All one needs to do is to compute the curvature tensor (7.13) at each point using

the metric.

7.14 Tidal forces in a curved spacetime

Now that we have derived the equation of geodesic variation (7.24), we can

give a more quantitative account of the gravitational tidal forces mentioned

in our discussion of the equivalence principle in Section 7.2. Let us begin by

working in Newtonian gravity and consider an initially spherical distribution of

non-interacting particles freely falling towards the Earth (see Figure 7.6). Each

particle moves on a straight line through the centre of the Earth, but those nearer

the Earth fall faster because the gravitational attraction is stronger. Thus the

sphere no longer remains a sphere but is distorted into an ellipsoid of the same

168 The equivalence principle and spacetime curvature

sphere

of particles

ellipsoid

of particles

Figure 7.6 Tidal force on a collection of non-interacting particles.

volume: gravity has produced a tidal force in the sphere of particles that results

in an elongation of the distribution in the direction of motion and a compression

of the distribution in the transverse directions. Indeed, it is straightforward to

show that, for two nearby particles with trajectories xi t and xi t i = 1 2 3

Â¯

respectively in Cartesian coordinates, that the components of the separation vector

i = xi âˆ’ xi evolve as

Â¯

d2 i 2

=âˆ’ j

2 i xj

dt x

where is the Newtonian gravitational potential (see Exercise 7.21).

A similar tidal effect occurs in general relativity and can be understood in

terms of the curvature of the spacetime. In particular, we can gain some idea

of the general-relativistic tidal forces by considering the equation of geodesic

deviation (7.24). Consider any pair of our non-interacting particles. Each one

is in free fall and so they must move along the timelike geodesics x and

Â¯ respectively, where is the proper time experienced by the first particle

x

(say). If we define a small separation vector between the two particle worldlines

=x Â¯ âˆ’x

by , then (7.24) shows that it evolves according to the

equation

D2

=S (7.25)

2

D

where we have defined the tidal stress tensor

â‰¡R (7.26)

S uu

in which u â‰¡ du /d is the 4-velocity of the first particle. Note that in defining

S we have made use of the fact that the curvature tensor is antisymmetric in

its last two indices. The result (7.25) is a fully covariant tensor equation and

therefore holds in any coordinate system.

169

7.14 Tidal forces in a curved spacetime

To understand the physical consequences of the geodesic deviation effect, it is

helpful to consider how some observer will view the relative spatial acceleration

of the two particles. Suppose that our observer is sitting on the first particle,

the worldline x of which passes through some event P. In order to calculate

the relative spatial acceleration measured by our observer, we may erect a set of

Ë†

orthonormal basis vectors e at P that define the instantaneous rest frame (IRF)

of the first particle (and the observer) at this event. The timelike basis vector is

Ë† Ë†

given simply by e0 = u, where u is the 4-velocity at P of the first particle, and

Ë†

we may choose the spacelike basis vectors ei in any way, provided that the full

set satisfies

Ë†Ë†

e Â·e =

Ë† Ë†

In this way, the duals of these basis vectors, which are given by e = e,

also form an orthonormal set. The general situation is illustrated schematically in

Figure 7.7.

The components of the separation vector with respect to our new frame are

Ë†

Ë† Ë†

â‰¡e Â· = e

these components give the temporal and spatial separations of the events P and

Ë† =

Q on the two particle worldlines, as measured by our observer. Since the e

0 1 2 3 are the basis vectors of an inertial Cartesian coordinate system at P,

the intrinsic derivative in this coordinate system is simply equal to the ordinary

x Âµ (Ï„)

x Âµ (Ï„)

Q

Î¾(Ï„)

Ãª0

Ãª3

P Î¶(Ï„)

Ãª2

Ãª1

Figure 7.7 Schematic illustration of the basis vectors of the instantaneous rest

frame at P. A general connecting vector and the orthogonal connecting vector

are also shown.

170 The equivalence principle and spacetime curvature

derivative. Moreover, with respect to the IRF, the 4-velocity of the first particle

is simply u Ë† = c 0 . Thus from (7.25) we have

Ë†

d2 Ë†

= c 2 R Ë† 00 Ë† (7.27)

Ë†Ë†

2

d

where the components of the curvature tensor in the Cartesian inertial frame at P

may be written as

RË† Ë† Ë† Ë† â‰¡ R Ë† Ë† Ë† Ë† (7.28)

e e e e

Ë†

Equation (7.27) in fact holds for any orthonormal freely falling frame e .

Clearly, the general separation vector is inappropriate for our discussion of

the evolution of the spatial separation seen by our observer at P, since typically

will have some temporal component in the observerâ€™s frame. Thus, we must

work instead with the orthogonal connecting vector shown in Figure 7.7,

Ë†

Ë†

which has a zero component in the e0 -direction, i.e. 0 = 0. Since (7.27) is valid

for any small connecting vector it must also hold for the orthogonal connecting

Ë†

= 0 for all .

vector , but we must remember that 0

A useful alternative interpretation of (7.25) or (7.27) is that it gives the force

per unit mass required to keep two particles moving along parallel curves; this

force must be supplied by some mechanical means. For example, the worldline

of the centre of mass of a rigid body in free fall is a timelike geodesic, but this

is not true of the other parts of the object, which are constrained to move along

curves parallel to the centre of mass rather than along neighbouring geodesics.

The necessary forces must be supplied by internal stresses in the object. The

physical magnitude of the stresses is most easily found by solving the eigenvalue

problem

v=v (7.29)

S

where S is given by (7.26). One of the eigenvalues is always zero (for v = u ),

and the remaining three eigenvalues give the principal stresses in the object.

Appendix 7A: The surface of a sphere

The metric5 of the surface of a sphere in spherical polar coordinates is

ds2 = a2 d + a2 sin2 d

2 2

5

Note that this term is often applied, as here, to the line element itself.

171

Appendix 7A: The surface of a sphere

To get used to handling problems involving curved spaces you should calculate

the components of the affine connection, starting from this metric. The definition

of the affine connection is

= 2 g ad b gdc + c gbd âˆ’ d gbc

1

a

bc

as given in (3.21), and in two dimensions there are six independent connection

coefficients,

1 1 1 2 2 2

11 12 22 11 12 22

These coefficients are given by (exercise):

= 2 g 11 1 g11 + 1 g11 âˆ’ 1 g11 =0

1 1

11

= 2 g 11 2 g11 + 2 g21 âˆ’ 1 g12 =0

1 1

12

= 2 g 11 2 g21 + 2 g21 âˆ’ 1 g22 = âˆ’ 2 g 11 1 g22

1 1 1

22

= 2 g 22 2 g12 + 2 g12 âˆ’ 2 g11 =0

2 1

11

= 2 g 22 1 g22 + 2 g12 âˆ’ 2 g11 = 2 g 22 1 g22

2 1 1

12

= 2 g 22 2 g22 + 2 g22 âˆ’ 2 g22 =0

2 1

22

So, the only two non-zero coefficients are

1

=âˆ’ 2a2 sin cos = âˆ’ sin cos

1

22 2

2a

1 cos

= 2a2 sin cos =

2

12

2a2 sin2 sin

The curvature tensor is

Rabcd = d a gbc âˆ’ d b gac + c b gad âˆ’ c a gbd âˆ’ gef bd âˆ’

1 e f e f

ac ad bc

2

and in two dimensions the symmetry properties of this tensor mean that there is

only one independent component. We can take this to be R1212 , so fortunately we

only have to calculate this single component:

R1212 = 2 1 g21 âˆ’ 2 g11 + 1 2 g12 âˆ’ 1 g22 âˆ’ gef 22 âˆ’

2 2

1 e f e f

11 12 21

2

= âˆ’2 1 g22 âˆ’ g11 22 âˆ’ âˆ’ g22 22 âˆ’

2 1 1 1 1 2 2 2 2

1

11 12 12 11 21 21

= a2 sin2

Thus the Gaussian curvature K of a spherical surface is given by

R1212 a2 sin2 1

K= = =

a4 sin2 a2

g

172 The equivalence principle and spacetime curvature

Instead of a spherical surface, we could instead consider the surface of a

cylinder of radius a. The metric of the surface in cylindrical polar coordinates is

ds2 = a2 d + dz2

2

and it is obvious that this two-dimensional space is spatially flat because we can

transform the metric into the form

ds2 = dx2 + dz2

by the coordinate transformation x = a . It therefore follows that the curvature

of a cylindrical surface vanishes.

Exercises

7.1 From Poissonâ€™s equation 2 = 4 G show that the gravitational potential outside

a spherical object of mass M at a radial distance r from its centre is given by

r = âˆ’GM/r. What is the form of r inside a uniform spherical body?

7.2 A charged object held stationary in a laboratory on the surface of the Earth does not

emit electromagnetic radiation. If the object is then dropped so that it is in free fall, it

will begin to radiate. Reconcile these observations with the principle of equivalence.

Hint: Consider the spatial extent of the electric field of the charge.

7.3 If X is a local Cartesian coordinate system at some event P, show that so too is the

coordinate system X = X , where defines a Lorentz transformation.

7.4 If two vectors v and w are Fermiâ€“Walker-transported along some observerâ€™s world-

line, show that their scalar product v Â· w is preserved at all points along the line.

7.5 Photons of frequency E are emitted from the surface of the Sun and observed by an

astronaut with fixed spatial coordinates at a large distance away. Obtain an expression

for the frequency O of the photons as measured by the astronaut. Hence estimate

the observed redshift of the photon.

7.6 An experimenter A drops a pebble of rest mass m in a uniform gravitational field g.

At a distance h below A, experimenter B converts the pebble (with no energy loss)

into a photon of frequency B . The photon passes by A, who observes it to have

frequency A . Use simple physical arguments to show that to a first approximation

gh

= 1+

B

c2

A

Use this result to argue that for two stationary observers A and B in a weak gravi-

tational field with potential , the ratio of the rates at which their laboratory clocks

run is 1 + /c2 , where is the potential difference between A and B.

7.7 A satellite is in circular polar orbit of radius r around the Earth (radius R, mass M).

A standard clock C on the satellite is compared with an identical clock C0 at the

173

Exercises

south pole on Earth. Show that the ratio of the rate of the orbiting clock to that of

the clock on Earth is approximately

GM 3GM

1+ âˆ’

Rc2 2rc2

Note that the orbiting clock is faster only if r > 2 R, i.e. if r âˆ’ R > 3184 km.

3

7.8 Consider the limit of a weak gravitational field in a coordinate system in which

g= + h , with h 1, and 0 g = 0. Keeping only terms that are first

order in v/c, show that the equation of motion for a slowly moving test particle

takes the form

d 2 xi

â‰ˆ âˆ’ 2 c2 j h00 + c j h0k âˆ’ k h0j

1 ij ik

vj

2

dt

Give a physical interpretation of the second term on the right-hand side.

7.9 Show that in a two-dimensional Riemannian manifold all the components of Rabcd

are equal either to zero or to Â±R1212 .

7.10 Show that the line element ds2 = y2 dx2 + x2 dy2 represents the Euclidean plane, but

the line element ds2 = y dx2 + x dy2 represents a curved two-dimensional manifold.

7.11 For a two-dimensional manifold with line element ds2 = dr 2 + f 2 r d 2 , show that

the Gaussian curvature is given by K = âˆ’f /f , where a prime denotes d/dr.

7.12 By calculating the components of the curvature tensor Rd abc in each case, show that

the line element

a2

ds = 2 dr 2 + r 2 d + r 2 sin2 d

2 2 2

a âˆ’r 2

represents a curved three-dimensional manifold. Show that the manifold is flat in

the limit a â†’ 0.

7.13 A spacetime has the metric

ds2 = c2 dt2 âˆ’ a2 t dx2 + dy2 + dz2

Show that the only non-zero connection coefficients are are

= = = aa

Ë™ = = = a/a

Ë™

0 0 0 1 2 3

and

11 22 33 10 20 30

Deduce that particles may be at rest in such a spacetime and that for such particles

the coordinate t is their proper time. Show further that the non-zero components of

the Ricci tensor are

R00 = 3a/a

Â¨ R11 = R22 = R33 = âˆ’aa âˆ’ 2a2

Â¨ Ë™

and

Hence show that the 00-component of the Einstein tensor is G00 = âˆ’3a2 /a2 .

Ë™

7.14 Show that the covariant components of the curvature tensor are given by

Rabcd = âˆ’ + âˆ’ âˆ’ g ef âˆ’

1

d a gbc d b gac c b gad c a gbd eac fbd ead fbc

2

and hence verify its symmetries. Show further that, for an N -dimensional manifold,

the number of independent components is N 2 N 2 âˆ’ 1 /12.

174 The equivalence principle and spacetime curvature

7.15 Show that for any two-dimensional manifold the covariant curvature tensor has the form

Rabcd = K gac gbd âˆ’ gad gbc

where the scalar K may be a function of the coordinates. Why does this result not

generalise to arbitrary manifolds of higher dimension?

7.16 If va are the contravariant components of a vector and T ab are the contravariant

components of a rank-2 tensor, prove the results

âˆ’ = âˆ’Ra dbc vd

a a

bv cv

c b

âˆ’ = âˆ’Ra ecd T eb âˆ’ Rb ecd T ae

ab ab

cT dT

d c

Can you guess the corresponding result for the mixed components T ab c of a rank-3

tensor?

7.17 Show that any Killing vector va , as defined in Exercise 4.11, satisfies the relations

= Ra bcd vd

a

bv

c

=0

va aR

7.18 Calculate explicit forms for the Ricci tensor Rab and the Ricci scalar R in terms of

the metric, the connection and its partial derivatives.

7.19 Prove that the Ricci tensor Rab is symmetric.

7.20 A conformal transformation, such as that in Exercise 2.7, is not a change of

coordinates but an actual change in the geometry of a manifold such that the metric

tensor transforms as

gab x =

Ëœ 2

x gab x

where x is some non-vanishing scalar function of position. Show that, under

such a transformation, the metric connection transforms as

1

= + + âˆ’ gbc g ad

a a a a

bc bc cb bc d

Hence show that the curvature tensor, the Ricci tensor and the Ricci scalar transform

respectively as

e f

Ra bcd = Ra bcd âˆ’ 2 âˆ’ gb c

f

ae e

g af

b

cd d

e f

+2 2 âˆ’ 2gb c g af + gb c

f

ae e a

g ef

b

cd d d 2

e f

Rbc = Rbc + N âˆ’ 2 + gbc g ef

f e

b c

e f

âˆ’ 2 N âˆ’2 âˆ’ N âˆ’ 3 gbc g ef

f e

b c 2

R e f e f

R= + 2 N âˆ’ 1 g ef + N âˆ’ 1 N âˆ’ 4 g ef

2 3 4

where N is the dimension of the manifold.

175

Exercises

7.21 Show that parallel transportation of a vector around the closed triangle ABC on the

surface of a sphere, as shown in Figure 7.3, results in a vector that is orthogonal to

its original direction.

7.22 On the surface of a sphere, show that, along the geodesic = constant, the geodesic

deviation vector i satisfies

2

D2 D2 d

=0 =âˆ’

Ds2 Ds2 ds

Choose a geodesic = 0 with path length s = measured from = 0, and a

neighbouring geodesic = 0 + 0 , also with s = , and define i as the vector

between s = on one and s = on the other. Show that i =0 for all .

= 0 when = 0 then

Show in addition that if

= l2 sin2

where l2 is a constant, and that the two geodesics pass through = .

7.23 In Newtonian gravity, consider two nearby particles with trajectories xi t and

xi t i = 1 2 3 respectively in Cartesian coordinates. Show that the components

Â¯

of the separation vector i = xi âˆ’ xi evolve as

Â¯

d2 i 2

=âˆ’ j

dt2 xi xj

where is the Newtonian gravitational potential.

7.24 In the weak-field, Newtonian, limit of general relativity, we may choose coordinates

such that g = +h , where h 1, and we assume that all particle velocities

are small compared with c. By considering the equation of geodesic deviation,

show that the general-relativistic tidal force reduces to the Newtonian limit given

in Exercise 7.23.

8

The gravitational field equations

Let us now follow Einsteinâ€™s suggestion that gravity is a manifestation of space-

time curvature induced by the presence of matter. We must therefore obtain a set

of equations that describe quantitatively how the curvature of spacetime at any

event is related to the matter distribution at that event. These will be the gravi-

tational field equations, or Einstein equations, in the same way that the Maxwell

equations are the field equations of electromagnetism.

Maxwellâ€™s equations relate the electromagnetic field F at any event to its

source, the 4-current density j at that event. Similarly, Einsteinâ€™s equations relate

spacetime curvature to its source, the energyâ€“momentum of matter. As we shall

see, the analogy goes further. In any given coordinate system, Maxwellâ€™s equa-

tions are second-order partial differential equations for the components F of

the electromagnetic field tensor (or equivalently for the components A of the

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