!
168
Curved manifolds
! and B
! B
! ·B
! are parallel-transported along a curve, then g(A,
!) = A
! is
13 (a) Show that if A
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) 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
V α ;β = V α ,β + $ α µβ V µ
and
Vα;β = Vα,β − $ µ αβ Vµ ?
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) 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.
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 (gθθ = r2 , gφφ = r2 sin2 θ, gθφ = 0) in the usual spherical coordinates.
(c) Find the components gαβ 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
Rθφθφ 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 Wβγ );µ =
V αβ ;µ Wβγ + V αβ Wβγ ;µ. (Hint: use a locally inertial frame.)
32 A four-dimensional manifold has coordinates (u, v, w, p) in which the metric has components guv = gww = gpp = 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 coordinate transformation to the usual coordinates (t, x, y, z). (You may find it a useful
hint to calculate !eν · !eν and !eu · !eu .)
33 A ‘three-sphere’ is the three-dimensional surface in four-dimensional Euclidean space
(coordinates x, y, z, w), given by the equation x2 + y2 + z2 + w 2 = 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 gχχ = r2 , gθθ = r2 sin2 χ , gφφ = r2 sin2 χ sin2 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) ( µ µν = 12 (ln |g|),ν ;
√
√
(b) gµν ( α µν = −(gαβ − g),β / − g;
√
√
(c) for an antisymmetric tensor F µν , F µν ;ν = ( − g F µν ),ν / − g;
(d) gαβ gβµ,ν = −gαβ ,ν gβµ (hint: what is gαβ gβµ ?);
(e) gµν ,α = −( µ βα 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 dφ 2 ), where ) and * are arbitrary functions 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?
!
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.
! and V,
! their Lie bracket is defined to be the vector field
39 (a) For any two vector fields U
! V]
! with components
[U,
! V]
! α = U β ∇β V α − V β ∇β U α .
[U,
(6.100)
Show that
! V]
! = −[V,
! U],
!
[U,
! V]
! α = U β ∂ V α /∂xβ − V β ∂U α /∂xβ .
[U,
This is one tensor field in which partial derivatives need not be accompanied by
Christoffel symbols!
! V]
! is a derivative operator on V! along U,
! i.e. show that for any
(b) Show that [U,
scalar f ,
! f V]
! = f [U,
! V]
! + V(
! U
! · ∇f ).
[U,
(6.101)
! and is denoted by
This is sometimes called the Lie derivative with respect to U
! V]
! := £ ! V,
!
[U,
U
! · ∇f := £ ! f .
U
U
(6.102)
Then Eq. (6.101) would be written in the more conventional form of the Leibnitz
rule for the derivative operator £U! :
! !f.
! = f £ ! V! + V£
£U! (f V)
U
U
(6.103)
The result of (a) shows that this derivative operator may be defined without a connection or metric, and is therefore very fundamental. See Schutz (1980b) for an
introduction.
(c) Calculate the components of the Lie derivative of a one-form field ω̃ from the
! ω̃(V)
! is a scalar like f above, and from the
knowledge that, for any vector field V,
definition that £U! ω̃ is a one-form field:
!
! = (£ ! ω̃)(V)
! + ω̃(£ ! V).
£U! [ω̃(V)]
U
U
This is the analog of Eq. (6.103).
!
181
7.6 Exercises
7.5 F u r t h e r r e a d i n g
The question of how curvature and physics fit together is discussed in more detail by
Geroch (1978). Conserved quantities are discussed in detail in any of the advanced texts.
The material in this chapter is preparation for the theory of quantum fields in a fixed curved
spacetime. See Birrell and Davies (1984) and Wald (1994). This in turn leads to one of the
most active areas of gravitation research today, the quantization of general relativity. While
we will not treat this area in this book, readers in work that approaches this subject from
the starting point of classical general relativity (as contrasted with approaching it from the
starting point of string theory) may wish to look at Rovelli (2004) Bojowald (2005), and
Thiemann (2007).
7.6 E x e r c i s e s
1 If Eq. (7.3) were the correct generalization of Eq. (7.1) to a curved spacetime, how
would you interpret it? What would happen to the number of particles in a comoving
volume of the fluid, as time evolves? In principle, can we distinguish experimentally
between Eqs. (7.2) and (7.3)?
2 To first order in φ, compute gαβ for Eq. (7.8).
3 Calculate all the Christoffel symbols for the metric given by Eq. (7.8), to first order in
φ. Assume φ is a general function of t, x, y, and z.
4 Verify that the results, Eqs. (7.15) and (7.24), depended only on g00 : the form of gxx
doesn’t affect them, as long as it is 1 + 0(φ).
5 (a) For a perfect fluid, verify that the spatial components of Eq. (7.6) in the Newtonian
limit reduce to
υ ,t + (υ · ∇)υ + ∇p/ρ + ∇φ = 0
(7.38)
for the metric, Eq. (7.8). This is known as Euler’s equation for nonrelativistic fluid
flow in a gravitational field. You will need to use Eq. (7.2) to get this result.
(b) Examine the time-component of Eq. (7.6) under the same assumptions, and
interpret each term.
(c) Eq. (7.38) implies that a static fluid (ν = 0) in a static Newtonian gravitational field
obeys the equation of hydrostatic equilibrium
∇p + ρ∇φ = 0.
(7.39)
A metric tensor is said to be static if there exist coordinates in which !e0 is timelike,
gi0 = 0, and gαβ,0 = 0. Deduce from Eq. (7.6) that a static fluid (U i = 0, p,0 = 0,
etc.) obeys the relativistic equation of hydrostatic equilibrium
!
"
(7.40)
p,i + (ρ + p) 12 ln(−g00 ) = 0.
,i
!
182
Physics in a curved spacetime
(d) This suggests that, at least for static situations, there is a close relation between g00
and − exp(2φ), where φ is the Newtonian potential for a similar physical situation.
Show that Eq. (7.8) and Exer. 4 are consistent with this.
6 Deduce Eq. (7.25) from Eq. (7.10).
7 Consider the following four different metrics, as given by their line elements:
(i) ds2 = −dt2 + dx2 + dy2 + dz2 ;
(ii) ds2 = −(1 − 2M/r) dt2 + (1 − 2M/r)−1 dr2 + r2 (dθ 2 + sin2 θ dφ 2 ), where
M is a constant;
(iii)
# − a2 sin2 θ 2
2Mr sin2 θ
dt
−
2a
dt dφ
ρ2
ρ2
(r2 + a2 )2 − a2 # sin2 θ
ρ2
+
sin2 θ dφ 2 + dr2 + ρ 2 dθ 2 ,
2
#
ρ
ds2 = −
where M and a are constants and we have introduced the shorthand notation
# = r2 − 2Mr + a2!, ρ 2 = r2 + a2 cos2 θ;
"
(iv) ds2 = −dt2 + R2 (t) (1 − kr2 )−1 dr2 + r2 (dθ 2 + sin2 θ dφ 2 ) , where k is a
constant and R(t) is an arbitrary function of t alone.
The first one should be familiar by now. We shall encounter the other three
in later chapters. Their names are, respectively, the Schwarzschild, Kerr, and
Robertson–Walker metrics.
(a) For each metric find as many conserved components ρα of a freely falling particle’s
four momentum as possible.
(b) Use the result of Exer. 28, § 6.9 to put (i) in the form
(i" ) ds2 = −dt2 + dr2 + r2 (dθ 2 + sin2 θ dφ 2 ).
From this, argue that (ii) and (iv) are spherically symmetric. Does this increase the
number of conserved components pα ?
(c) It can be shown that for (i" ) and (ii)–(iv), a geodesic that begins with θ = π/2
and pθ = 0 – i.e. one which begins tangent to the equatorial plane – always has
θ = π/2 and pθ = 0. For cases (i" ), (ii), and (iii), use the equation p# · p# = −m2
to solve for pr in terms of m, other conserved quantities, and known functions of
position.
(d) For (iv), spherical symmetry implies that if a geodesic begins with pθ = pφ = 0,
these remain zero. Use this to show from Eq. (7.29) that when k = 0, pr is a
conserved quantity.
8 Suppose that in some coordinate system the components of the metric gαβ are
independent of some coordinate xµ .
(a) Show that the conservation law T ν µ;ν = 0 for any stress–energy tensor becomes
√
1 √
( − gT ν µ ),ν = 0.
−g
(7.41)
(b) Suppose that in these coordinates T αβ % = 0 only in some bounded region of each
spacelike hypersurface x0 = const. Show that Eq. (7.41) implies
!
183
7.6 Exercises
!
x0 =const.
Tνµ
√
− g nν d3 x
is independent of x0 , if nν is the unit normal to the hypersurface. This is the
generalization to continua of the conservation law stated after Eq. (7.29).
(c) Consider flat Minkowski space in a global inertial frame with spherical polar
coordinates (t, r, θ , φ). Show from (b) that
!
T 0 φ r2 sin θ dr dθ dφ
(7.42)
J=
t=const.
is independent of t. This is the total angular momentum of the system.
(d) Express the integral in (c) in terms of the components of T αβ on the Cartesian basis
(t, x, y, z), showing that
!
J = (xT y0 − yT x0 )dx dy dz.
(7.43)
9 (a)
(b)
(c)
10 (a)
This is the continuum version of the nonrelativistic expression (r × p)z for a
particle’s angular momentum about the z axis.
Find the components of the Riemann tensor Rαβµν for the metric, Eq. (7.8), to first
order in φ.
Show that the equation of geodesic deviation, Eq. (6.87), implies (to lowest order
in φ and velocities)
d2 ξ i
= −φ,ij ξ j .
(7.44)
dt2
Interpret this equation when the geodesics are world lines of freely falling particles
which begin from rest at nearby points in a Newtonian gravitational field.
Show that if a vector field ξ α satisfies Killing’s equation
∇α ξβ + ∇β ξα = 0,
(b)
(c)
(d)
(e)
(7.45)
then along a geodesic, pα ξα = const. This is a coordinate-invariant way of characterizing the conservation law we deduced from Eq. (7.29). We only have to know
whether a metric admits Killing fields.
Find ten Killing fields of Minkowski spacetime.
Show that if ξ% and η% are Killing fields, then so is α ξ% + β η% for constant α and β.
Show that Lorentz transformations of the fields in (b) simply produce linear
combinations as in (c).
If you did Exer. 7, use the results of Exer. 7(a) to find Killing vectors of metrics
(ii)–(iv).