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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
(2Rμλ − δμλR);μ = 0. |
(6.97) |
These are the twice-contracted Bianchi identities, often simply also called the Bianchi identities. If we define the symmetric tensor
Gαβ ≡ Rαβ − 21 gαβ R = Gβα , |
(6.98) |
then we see that Eq. (6.97) is equivalent to
Gαβ ;β = 0. |
(6.99) |
The tensor Gαβ is constructed only from the Riemann tensor and the metric, and is automatically 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
Gαβ = 8π Tαβ
(where Tαβ is the stress-energy tensor). The Bianchi identities then imply
Tαβ ;β ≡ 0,
which is the equation of local conservation of energy and momentum. But this is looking a bit far ahead.]
6.7 C u r va t u re i n p e r s p e c t i v e
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 material 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
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Curved manifolds |
gαβ = ηαβ ,
gαβ,γ = 0 βαγ = 0.
(3) The element of proper volume is
|g|1/2d4x,
where g is the determinant of the matrix of components gαβ .
(4)The covariant derivative is simply the ordinary derivative in locally inertial coordinates. Because of curvature ( α 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 identically 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 gαβ 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 F u r t h e r re a d i n g
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).
6.9E xe rc i s e s
1Decide 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 pi and qi;
(b)the interior of a circle of unit radius in two-dimensional Euclidean space;
(c)the set of permutations of n objects;
(d)the subset of Euclidean space of two dimensions (coordinates x and y) which is a solution to xy (x2 + y2 − 1) = 0.
2Of 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?
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6.9 Exercises |
3 It is well known that for any symmetric matrix A (with real entries), there exists a matrix H for which the matrix HT AH is a diagonal matrix whose entries are the eigenvalues of A.
(a)Show that there is a matrix R such that RT HT AHR is the same matrix as HT AH 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 NT RT HT AHRN is a diagonal matrix whose entries on the diagonal are −1, 0, or +1.
(c)Show that if A has an inverse, none of the diagonal elements in (b) is zero.
(d)Show from (a)–(c) that there exists a transformation matrix which produces
Eq. (6.2).
4Prove the following results used in the proof of the local flatness theorem in § 6.2:
(a)The number of independent values of ∂2xα /∂xγ ∂xμ |0 is 40.
(b)The corresponding number for ∂3xα /∂xλ ∂xμ ∂xν |0 is 80.
(c)The corresponding number for gαβ,γ μ |0 is 100.
5(a) Prove that μαβ = μβα 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.
6Prove 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).
(c)Generalize Eq. (6.39) (by induction or otherwise) to arbitrary n × n matrices.
8Fill in the missing algebra leading to Eqs. (6.40) and (6.42).
9Show that Eq. (6.42) leads to Eq. (5.56). Derive the divergence formula for the metric in Eq. (6.19).
10A ‘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
α = α = α + α μ =
( V) β V ;β V ,β μβ V 0?
(a)A necessary condition, called the integrability condition for this equation, follows from the commuting of partial derivatives. Show that Vα ,νβ = Vα ,βν implies
α μβ,ν − α μν,β Vμ = α μβ μσ ν − α μν μσβ Vσ .
(b) By relabeling indices, work this into the form
α μβ,ν − α μν,β + σ σ ν σ μβ − α σβ σ μν Vμ = 0.
This turns out to be sufficient, as well.
12 Prove that Eq. (6.52) defines a new affine parameter.
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(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) from Eq. (6.58).
(b) Fill in the algebra needed to justify Eq. (6.61). 17 (a) Prove that Eq. (6.5) implies gαβ ,μ(P) = 0.
(b) Use this to establish Eq. (6.64).
(c) Fill in the steps needed to establish Eq. (6.68). 18 (a) Derive Eqs. (6.69) and (6.70) from Eq. (6.68).
(b) Show that Eq. (6.69) reduces the number of independent components of Rαβμν 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 Rαβμν.)
(c) Show that Eq. (6.70) imposes only one further relation independent of Eq. (6.69) on the components, reducing the total of independent ones to 20.
19 Prove that Rα 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
Vα ;β = Vα ,β + α μβ Vμ and Vα;β = Vα,β − μαβ Vμ?
22Fill in the algebra necessary to establish Eqs. (6.84), (6.85), and (6.86).
23Prove 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.)
24Establish Eq. (6.89) from Eq. (6.88).
25(a) Prove that the Ricci tensor is the only independent contraction of Rα βμν : all others are multiples of it.
(b)Show that the Ricci tensor is symmetric.
26Use Exer. 17(a) to prove Eq. (6.94).
27Fill 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 (gθ θ = r2, gφφ = r2 sin2 θ , gθ φ = 0) in the usual spherical coordinates.
(c)Find the components gαβ for the sphere.
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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 Rθ φθ φ and obtain all other components in terms of it.)
30Calculate 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!)
31Show that covariant differentiation obeys the usual product rule, e.g. (Vαβ Wβγ );μ = Vαβ ;μ Wβγ + Vαβ Wβγ ;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 guv = gww = gpp = 1, all |
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vanishing. |
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(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 coordinate transformation to the usual coordinates (t, x, y, z). (You may find it a useful
hint to calculate eν · eν and eu · eu .)
33A ‘three-sphere’ is the three-dimensional surface in four-dimensional Euclidean space (coordinates x, y, z, w), given by the equation x2 + y2 + z2 + w2 = r2, 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 coordinates gχ χ = r2, gθ θ = r2 sin2 χ , gφφ = r2 sin2 χ sin2 θ , all other components vanishing. (Use the same method as in Exer. 28.)
34Establish the following identities for a general metric tensor in a general coordinate system. You may find Eqs. (6.39) and (6.40) useful.
(a) μμν = 21 (ln |g|),ν |
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(b) gμν α μν = −(gαβ |
√ − g),β /√ − g; |
(c) for an antisymmetric tensor Fμν , Fμν ;ν = (√ − g Fμν ),ν /√ − g;
(d)gαβ gβμ,ν = −gαβ ,ν gβμ (hint: what is gαβ gβμ?);
(e)gμν ,α = − μβα gβν − ν βα gμβ (hint: use Eq. (6.31)).
35 Compute 20 independent components of Rαβμν for a manifold with line element ds2 = −e2 dt2 + e2 dr2 + r2(dθ 2 + sin2 θ dφ2), where and are arbitrary func-
tions of the coordinate r alone. (First, identify the coordinates and the components gαβ ; then compute gαβ and the Christoffel symbols. Then decide on the indices of the 20 components of Rαβμν 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
ds2 = −(1 + 2φ) dt2 + (1 − 2φ)(dx2 + dy2 + dz2),
where |φ(t, x, y, z)| 1 everywhere. At any point P with coordinates (t0, x0, y0, z0), 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?
