Schutz B. A first course in general relativity

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163 6.6 Bianchi identities: Ricci and Einstein tensors

Now, using 

βγ

= 0atA,wehave

∇

dλ



dλ

+ 

β0



+ 0

dλ

+ 

β0,0

(6.85)

at A. (We have also used ξ

= 0atA, which is the condition that curves begin parallel.)

So we get

∇





β0,0

− 

00,β



= R

00β

= R

μνβ

, (6.86)

where the second equality follows from Eq. (6.63). The ﬁnal expression is frame invariant,

and A was an arbitrary point, so we have, in any basis,

∇

= R

μνβ

. (6.87)

Geodesics in ﬂat space maintain their separation; those in curved spaces don’t. This is

called the equation of geodesic deviation and shows mathematically that the tidal forces of

a gravitational ﬁeld (which cause trajectories of neighboring particles to diverge) can be

represented by curvature of a spacetime in which particles follow geodesics.

6.6 Bianchi identities: Ricci and Einstein tensors

Let us return to Eq. (6.63) for the Riemann tensor’s components. If we differentiate it

with respect to x

(just the partial derivative) and evaluate the result in locally inertial

coordinates, we ﬁnd

αβμν,λ



αν,βμλ

− g

αμ,βνλ

+ g

βμ,ανλ

− g

βν,αμλ



. (6.88)

From this equation, the symmetry g

αβ

= g

βα

and the fact that partial derivatives commute,

we can show that

αβμν,λ

+ R

αβλμ,ν

+ R

αβνλ,μ

= 0. (6.89)

Since in our coordinates 

αβ

= 0 at this point, this equation is equivalent to

αβμν;λ

+ R

αβλμ;ν

+ R

αβνλ;μ

= 0. (6.90)

But this is a tensor equation, valid in any system. It is called the Bianchi identities, and will

be very important for our work.

164 Curved manifolds

The Ricci tensor

Before pursuing the consequences of the Bianchi identities, we shall need to deﬁne the

Ricci tensor R

αβ

:= R

σμβ

= R

βα

. (6.91)

It is the contraction of R

ανβ

on the ﬁrst and third indices. Other contractions would in

principle also be possible: on the ﬁrst and second, the ﬁrst and fourth, etc. But because

αβμν

is antisymmetric on α and β and on μ and ν, all these contractions either vanish

identically or reduce to ±R

αβ

. Therefore the Ricci tensor is essentially the only contraction

of the Riemann tensor. Note that Eq. (6.69)impliesitisasymmetric tensor (Exer. 25, § 6.9).

Similarly, the Ricci scalar is deﬁned as

R := g

μν

= g

μν

αβ

αμβν

. (6.92)

The Einstein tensor

Let us apply the Ricci contraction to the Bianchi identities, Eq. (6.90):

αμ

αβμν;λ

+ R

αβλμ;ν

+ R

αβνλ;μ

= 0

βν;λ

+ (−R

βλ;ν

) + R

βνλ;μ

= 0. (6.93)

To derive this result we need two facts. First, by Eq. (6.31)wehave

αβ;μ

= 0.

Since g

αμ

is a function only of g

αβ

it follows that

αβ

;μ

= 0. (6.94)

Therefore, g

αμ

and g

βν

can be taken in and out of covariant derivatives at will: index-

raising and -lowering commutes with covariant differentiation. The second fact is that

αμ

αβλμ;ν

=−g

αμ

αβμλ;ν

=−R

βλ;ν

, (6.95)

accounting for the second term in Eq. (6.93). Eq. (6.93) is called the contracted Bianchi

identities. A more useful equation is obtained by contracting again on the indices β and ν:

βν

βν;λ

− R

βλ;ν

+ R

βνλ;μ

= 0

;λ

− R

λ;μ

+ (−R

λ;μ

) = 0. (6.96)

165 6.7 Curvature in perspective

Again the antisymmetry of R has been used to get the correct sign in the last term. Note

that since R is a scalar, R

;λ

≡ R

,λ

in all coordinates. Now, Eq. (6.96) can be written in

the form

(2R

− δ

;μ

= 0. (6.97)

These are the twice-contracted Bianchi identities, often simply also called the Bianchi

identities. If we deﬁne the symmetric tensor

αβ

≡ R

αβ

−

αβ

R = G

βα

, (6.98)

then we see that Eq. (6.97) is equivalent to

αβ

;β

= 0. (6.99)

The tensor G

αβ

is constructed only from the Riemann tensor and the metric, and is auto-

matically divergence free as an identity. It is called the Einstein tensor, since its importance

for gravity was ﬁrst understood by Einstein. [In fact we shall see that the Einstein ﬁeld

equations for GR are

αβ

= 8π T

αβ

(where T

αβ

is the stress-energy tensor). The Bianchi identities then imply

αβ

;β

≡ 0,

which is the equation of local conservation of energy and momentum. But this is looking

a bit far ahead.]

6.7 Curvature in perspective

The mathematical machinery for dealing with curvature is formidable. There are many

important equations in this chapter, but few of them need to be memorized. It is far more

important to understand their derivation and particularly their geometrical interpretation.

This interpretation is something we will build up over the next few chapters, but the mate-

rial already in hand should give the student some idea of what the mathematics means. Let

us review the important features of curved spaces.

(1) We work on Riemannian manifolds, which are smooth spaces with a metric deﬁned

on them.

(2) The metric has signature +2, and there always exists a coordinate system in which, at

a single point, we can have

166 Curved manifolds

αβ

= η

αβ

αβ,γ

= 0 ⇒ 

βγ

= 0.

(3) The element of proper volume is

|g|

1/2

where g is the determinant of the matrix of components g

αβ

(4) The covariant derivative is simply the ordinary derivative in locally inertial coordi-

nates. Because of curvature (

βγ,σ

= 0) these derivatives do not commute.

(5) The deﬁnition of parallel-transport is that the covariant derivative along the curve is

zero. A geodesic parallel-transports its own tangent vector. Its afﬁne parameter can be

taken to be the proper distance itself.

(6) The Riemann tensor is the characterization of the curvature. Only if it vanishes identi-

cally is the manifold ﬂat. It has 20 independent components (in four dimensions), and

satisﬁes the Bianchi identities, which are differential equations. The Riemann tensor in

a general coordinate system depends on g

αβ

and its ﬁrst and second partial derivatives.

The Ricci tensor, Ricci scalar, and Einstein tensor are contractions of the Riemann

tensor. In particular, the Einstein tensor is symmetric and of second rank, so it has ten

independent components. They satisfy the four differential identities, Eq. (6.99).

6.8 Further reading

The theory of differentiable manifolds is introduced in a large number of books. The

following are suitable for exploring the subject further with a view toward its physical

applications, particularly outside of relativity: Abraham and Marsden (1978), Bishop and

Goldberg (1981), Hermann (1968), Isham (1999), Lovelock and Rund (1990), and Schutz

(1980b). Standard mathematical reference works include Kobayashi and Nomizu (1963,

1969), Schouten (1990), and Spivak (1979).

6.9 Exercises

1 Decide if the following sets are manifolds and say why. If there are exceptional points

at which the sets are not manifolds, give them:

(a) phase space of Hamiltonian mechanics, the space of the canonical coordinates and

momenta p

and q

;

(b) the interior of a circle of unit radius in two-dimensional Euclidean space;

(d) the subset of Euclidean space of two dimensions (coordinates x and y) which is a

solution to xy (x

+ y

− 1) = 0.

2 Of the manifolds in Exer. 1, on which is it customary to use a metric, and what is that

metric? On which would a metric not normally be deﬁned, and why?

167 6.9 Exercises

3 It is well known that for any symmetric matrix A (with real entries), there exists

amatrixH for which the matrix H

AH is a diagonal matrix whose entries are the

eigenvalues of A.

(a) Show that there is a matrix R such that R

AHR is the same matrix as H

except with the eigenvalues rearranged in ascending order along the main diagonal

from top to bottom.

(b) Show that there exists a third matrix N such that N

AHRN is a diagonal

matrix whose entries on the diagonal are −1, 0, or +1.

(d) Show from (a)–(c) that there exists a transformation matrix  which produces

Eq. (6.2).

4 Prove the following results used in the proof of the local ﬂatness theorem in § 6.2:

(a) The number of independent values of ∂

/∂x

γ 

∂x

μ

is 40.

(b) The corresponding number for ∂

/∂x

λ

∂x



∂x



is 80.

αβ,γ



is 100.

5 (a) Prove that 

αβ

= 

βα

in any coordinate system in a curved Riemannian space.

(b) Use this to prove that Eq. (6.32) can be derived in the same manner as in ﬂat space.

6 Prove that the ﬁrst term in Eq. (6.37) vanishes.

7 (a) Give the deﬁnition of the determinant of a matrix A in terms of cofactors of

elements.

(b) Differentiate the determinant of an arbitrary 2 ×2 matrix and show that it satisﬁes

Eq. (6.39).

8 Fill in the missing algebra leading to Eqs. (6.40) and (6.42).

9 Show that Eq. (6.42) leads to Eq. (5.56). Derive the divergence formula for the metric

in Eq. (6.19).

10 A ‘straight line’ on a sphere is a great circle, and it is well known that the sum of the

interior angles of any triangle on a sphere whose sides are arcs of great circles exceeds

180

◦

. Show that the amount by which a vector is rotated by parallel transport around

such a triangle (as in Fig. 6.3) equals the excess of the sum of the angles over 180

◦

11 In this exercise we will determine the condition that a vector ﬁeld



V can be considered

to be globally parallel on a manifold. More precisely, what guarantees that we can ﬁnd

a vector ﬁeld



V satisfying the equation

(∇



= V

;β

= V

,β

+ 

μβ

= 0?

(a) A necessary condition, called the integrability condition for this equation, follows

from the commuting of partial derivatives. Show that V

,νβ

= V

,βν

implies





μβ ,ν

− 

μν,β







μβ



σν

− 

μν



σβ



(b) By relabeling indices, work this into the form





μβ ,ν

− 

μν,β

+ 

σν



μβ

− 

σβ



μν



= 0.

This turns out to be sufﬁcient, as well.

12 Prove that Eq. (6.52) deﬁnes a new afﬁne parameter.

168 Curved manifolds

13 (a) Show that if



A and



B are parallel-transported along a curve, then g(



B) =



A ·



B is

constant on the curve.

(b) Conclude from this that if a geodesic is spacelike (or timelike or null) somewhere,

it is spacelike (or timelike or null) everywhere.

14 The proper distance along a curve whose tangent is



V is given by Eq. (6.8). Show that

if the curve is a geodesic, then proper length is an afﬁne parameter. (Use the result of

Exer. 13.)

15 Use Exers. 13 and 14 to prove that the proper length of a geodesic between two points is

unchanged to ﬁrst order by small changes in the curve that do not change its endpoints.

16 (a) Derive Eqs. (6.59) and (6.60)fromEq.(6.58).

(b) Fill in the algebra needed to justify Eq. (6.61).

17 (a) Prove that Eq. (6.5) implies g

αβ

,μ

(P) = 0.

(b) Use this to establish Eq. (6.64).

18 (a) Derive Eqs. (6.69) and (6.70)fromEq.(6.68).

(b) Show that Eq. (6.69) reduces the number of independent components of R

αβμν

from 4 ×4 ×4 ×4 = 256 to 6 × 7/2 = 21. (Hint: treat pairs of indices. Calculate

how many independent choices of pairs there are for the ﬁrst and the second pairs

on R

αβμν.

)

on the components, reducing the total of independent ones to 20.

19 Prove that R

βμν

= 0 for polar coordinates in the Euclidean plane. Use Eq. (5.45)or

equivalent results.

20 Fill in the algebra necessary to establish Eq. (6.73).

21 Consider the sentences following Eq. (6.78). Why does the argument in parentheses

not apply to the signs in

;β

= V

,β

+ 

μβ

and V

α;β

= V

α,β

− 

αβ

22 Fill in the algebra necessary to establish Eqs. (6.84), (6.85), and (6.86).

23 Prove Eq. (6.88). (Be careful: one cannot simply differentiate Eq. (6.67) since it is valid

only at P, not in the neighborhood of P.)

24 Establish Eq. (6.89)fromEq.(6.88).

25 (a) Prove that the Ricci tensor is the only independent contraction of R

βμν

: all others

are multiples of it.

(b) Show that the Ricci tensor is symmetric.

26 Use Exer. 17(a) to prove Eq. (6.94).

27 Fill in the algebra necessary to establish Eqs. (6.95), (6.97), and (6.99).

28 (a) Derive Eq. (6.19) by using the usual coordinate transformation from Cartesian to

spherical polars.

(b) Deduce from Eq. (6.19) that the metric of the surface of a sphere of radius r

has components (g

θθ

= r

, g

φφ

= r

sin

θ, g

θφ

= 0) in the usual spherical coor-

dinates.

αβ

for the sphere.

169 6.9 Exercises

29 In polar coordinates, calculate the Riemann curvature tensor of the sphere of unit

radius, whose metric is given in Exer. 28. (Note that in two dimensions there is only

one independent component, by the same arguments as in Exer. 18(b). So calculate

θφθφ

and obtain all other components in terms of it.)

30 Calculate the Riemann curvature tensor of the cylinder. (Since the cylinder is ﬂat, this

should vanish. Use whatever coordinates you like, and make sure you write down the

metric properly!)

31 Show that covariant differentiation obeys the usual product rule, e.g. (V

αβ

βγ

)

;μ

αβ

;μ

βγ

+ V

αβ

βγ;μ.

(Hint: use a locally inertial frame.)

32 A four-dimensional manifold has coordinates (u, v, w, p) in which the met-

ric has components g

= g

= 1, all other independent components

vanishing.

(a) Show that the manifold is ﬂat and the signature is +2.

(b) The result in (a) implies the manifold must be Minkowski spacetime. Find a coor-

dinate transformation to the usual coordinates (t, x, y, z). (You may ﬁnd it a useful

hint to calculate e

·e

and e

·e

33 A ‘three-sphere’ is the three-dimensional surface in four-dimensional Euclidean space

(coordinates x, y, z, w), given by the equation x

+ y

+ z

+ w

= r

, where r is the

radius of the sphere.

(a) Deﬁne new coordinates (r, θ, φ, χ ) by the equations w = r cos χ, z = r sin χ cos θ ,

x = r sin χ sin θ cos φ, y = r sin χ sin θ sin φ. Show that (θ , φ, χ) are coordinates

for the sphere. These generalize the familiar polar coordinates.

(b) Show that the metric of the three-sphere of radius r has components in these coor-

dinates g

χχ

= r

, g

θθ

= r

sin

χ, g

φφ

= r

sin

χ sin

θ, all other components

vanishing. (Use the same method as in Exer. 28.)

34 Establish the following identities for a general metric tensor in a general coordinate

system. You may ﬁnd Eqs. (6.39) and (6.40)useful.

(a) 

μν

(ln |g|)

,ν

;

(b) g

μν



μν

=−(g

αβ

√

− g)

,β

√

− g;

μν

, F

μν

;ν

= (

√

− gF

μν

)

,ν

√

− g;

(d) g

αβ

βμ,ν

=−g

αβ

,ν

βμ

(hint: what is g

αβ

βμ

?);

(e) g

μν

,α

=−

βα

βν

− 

βα

μβ

(hint: use Eq. (6.31)).

35 Compute 20 independent components of R

αβμν

for a manifold with line element

=−e

2

+ e

2

+ r

(dθ

+ sin

θ dφ

), where  and  are arbitrary func-

tions of the coordinate r alone. (First, identify the coordinates and the components g

αβ

;

then compute g

αβ

and the Christoffel symbols. Then decide on the indices of the 20

components of R

αβμν

you wish to calculate, and compute them. Remember that you

can deduce the remaining 236 components from those 20.)

36 A four-dimensional manifold has coordinates (t, x, y, z) and line element

=−(1 +2φ)dt

+ (1 − 2φ)(dx

+ dy

+ dz

where |φ(t, x, y, z)|1 everywhere. At any point P with coordinates (t

, x

, y

, z

ﬁnd a coordinate transformation to a locally inertial coordinate system, to ﬁrst order in

φ. At what rate does such a frame accelerate with respect to the original coordinates,

again to ﬁrst order in φ?

170 Curved manifolds

37 (a) ‘Proper volume’ of a two-dimensional manifold is usually called ‘proper area’.

Using the metric in Exer. 28, integrate Eq. (6.18) to ﬁnd the proper area of a sphere

of radius r.

(b) Do the analogous calculation for the three-sphere of Exer. 33.

38 Integrate Eq. (6.8) to ﬁnd the length of a circle of constant coordinate θ on a sphere of

radius r.

39 (a) For any two vector ﬁelds



U and



V, their Lie bracket is deﬁned to be the vector ﬁeld

[



V] with components

[



= U

∇

− V

∇

. (6.100)

Show that

[



V] =−[



U],

[



= U

∂ V

/∂x

− V

∂U

/∂x

This is one tensor ﬁeld in which partial derivatives need not be accompanied by

Christoffel symbols!

(b) Show that [



V] is a derivative operator on



V along



U, i.e. show that for any

scalar f ,

[



U, f



V] = f [



V] +



U ·∇f ). (6.101)

This is sometimes called the Lie derivative with respect to



U and is denoted by

[



V]:= £



U ·∇f := £



f . (6.102)

Then Eq. (6.101) would be written in the more conventional form of the Leibnitz

rule for the derivative operator £



V) = f£



V +



V£



f . (6.103)

The result of (a) shows that this derivative operator may be deﬁned without a con-

nection or metric, and is therefore very fundamental. See Schutz (1980b) for an

introduction.

knowledge that, for any vector ﬁeld



V, ˜ω(



V) is a scalar like f above, and from the

deﬁnition that £



˜ω is a one-form ﬁeld:



[ ˜ω(



V)] = (£



˜ω)(



V) +˜ω(£



V).

This is the analog of Eq. (6.103).

Physics in a curved spacetime

7.1 The transition from differential geometry

to gravity

The essence of a physical theory expressed in mathematical form is the identiﬁcation of

the mathematical concepts with certain physically measurable quantities. This must be our

ﬁrst concern when we look at the relation of the concepts of geometry we have developed

to the effects of gravity in the physical world. We have already discussed this to some

extent. In particular, we have assumed that spacetime is a differentiable manifold, and we

have shown that there do not exist global inertial frames in the presence of nonuniform

gravitational ﬁelds. Behind these statements are the two identiﬁcations:

(I) Spacetime (the set of all events) is a four-dimensional manifold with a metric.

(

II) The metric is measurable by rods and clocks. The distance along a rod between two

nearby points is |dx ·dx|

1/2

and the time measured by a clock that experiences two

events closely separated in time is |−dx ·dx|

1/2

So there do not generally exist coordinates in which dx · dx =−(dx

)

+ (dx

)

(dx

)

+ (dx

)

everywhere. On the other hand, we have also argued that such frames do

exist locally. This clearly suggests a curved manifold, in which coordinates can be found

which make the dot product at a particular point look like it does in a Minkowski spacetime.

Therefore we make a further requirement:

(III) The metric of spacetime can be put in the Lorentz form η

αβ

at any particular event

by an appropriate choice of coordinates.

Having chosen this way of representing spacetime, we must do two more things to get

a complete theory. First, we must specify how physical objects (particles, electric ﬁelds,

ﬂuids) behave in a curved spacetime and, second, we need to say how the curvature is

generated or determined by the objects in the spacetime.

Let us consider Newtonian gravity as an example of a physical theory. For Newton,

spacetime consisted of three-dimensional Euclidean space, repeated endlessly in time.

(Mathematically, this is called R

× R.) There was no metric on spacetime as a whole

172 Physics in a curved spacetime

manifold, but the Euclidean space had its usual metric and time was measured by a

universal clock. Observers with different velocities were all equally valid: this form of rela-

tivity was built into Galilean mechanics. Therefore there was no universal standard of rest,

and different observers would have different deﬁnitions of whether two events occurring at

different times happened at the same location. But all observers would agree on simultane-

ity, on whether two events happened in the same time-slice or not. Thus the ‘separation in

time’ between two events meant the time elapsed between the two Euclidean slices con-

taining the two events. This was independent of the spatial locations of the events, so in

Newtonian gravity there was a universal notion of time: all observers, regardless of posi-

tion or motion, would agree on the elapsed time between two given events. Similarly, the

‘separation in space’ between two events meant the Euclidean distance between them. If

the events were simultaneous, occurring in the same Euclidean time-slice, then this was

simple to compute using the metric of that slice, and all observers would agree on it. If the

events happened at different times, each observer would take the location of the events in

their respective space slices and compute the Euclidean distance between them. The loca-

tions would differ for different observers, but again the distance between them would be

the same for all observers.

However, in Newtonian theory there was no way to combine the time and distance

measures: there was no invariant measure of the length of a general curve that changed

position and time as it went along. Without an invariant way of converting times to dis-

tances, this was not possible. What Einstein brought to relativity was the invariance of the

speed of light, which then permits a uniﬁcation of time and space measures. Einstein’s

four-dimensional spacetime has a much simpler structure than Newton’s!

Now, within this model of spacetime, Newton gave a law for the behavior of objects

that experienced gravitational forces: F = ma, where F =−m∇φ for a given gravitational

ﬁeld φ. And he also gave a law determining how φ is generated: ∇

 = 4πGρ. These

two laws are the ones we must now ﬁnd analogs for in our relativistic point of view on

spacetime. The second one will be dealt with in the next chapter. In this chapter, we ask

only how a given metric affects bodies in spacetime.

We have already discussed this for the simple case of particle motion. Since we know

that the ‘acceleration’ of a particle in a gravitational ﬁeld is independent of its mass, we

can go to a freely falling frame in which nearby particles have no acceleration. This is

what we have identiﬁed as a locally inertial frame. Since freely falling particles have no

acceleration in that frame, they follow straight lines, at least locally. But straight lines in a

local inertial frame are, of course, the deﬁnition of geodesics in the full curved manifold.

So we have our ﬁrst postulate on the way particles are affected by the metric:

(IV) Weak Equivalence Principle: Freely falling particles move on timelike geodesics

of the spacetime.

It is more common to deﬁne the WEP without reference to a curved spacetime, but just to say that all parti-

cles fall at the same rate in a gravitational ﬁeld, independent of their mass and composition. But the Einstein

Equivalence Principle (Postulate



) is normally taken to imply that gravity can be represented by spacetime

curvature, so we shall simply start with the assumption that we have a curved spacetime.