7.5 F u r t h e r re 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).

7.6E xe rc i s e s

1If 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)?

2To first order in φ, compute gαβ for Eq. (7.8).

3Calculate 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.

4Verify 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

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

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 (Ui = 0, p,0 = 0, etc.) obeys the relativistic equation of hydrostatic equilibrium

(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.

6Deduce Eq. (7.25) from Eq. (7.10).

7Consider 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 dφ2), where M is a constant;

(iii)

where M and a are constants and we have introduced the shorthand notation

= r2 − 2Mr + a2, ρ2 = r2 + a2 cos2 θ ;

(iv)ds2 = −dt2 + R2(t) "(1 − kr2)−1dr2 + r2(dθ 2 + sin2 θ dφ2)#, 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 dφ2).

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 Tν μ;ν = 0 for any stress–energy tensor becomes

(b)Suppose that in these coordinates Tαβ =0 only in some bounded region of each spacelike hypersurface x0 = const. Show that Eq. (7.41) implies

'

Tν μ√ − g nν d3x

x0=const.

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

'

J = T0φ r2 sin θ dr dθ dφ (7.42)

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 Tαβ on the Cartesian basis (t, x, y, z), showing that

'

J = (xTy0 − yTx0)dx dy dz. (7.43)

This is the continuum version of the nonrelativistic expression (r × p)z for a particle’s angular momentum about the z axis.

9(a) Find the components of the Riemann tensor Rαβμν for the metric, Eq. (7.8), to first order in φ.

(b)Show that the equation of geodesic deviation, Eq. (6.87), implies (to lowest order in φ and velocities)

d2ξ i

(7.44)

dt2

(c) 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.

10 (a) Show that if a vector field ξ α satisfies Killing’s equation

(d)Show that Lorentz transformations of the fields in (b) simply produce linear combinations as in (c).

(e)If you did Exer. 7, use the results of Exer. 7(a) to find Killing vectors of metrics (ii)–(iv).