Any small body, for example a planet, that falls freely in the relativistic source’s gravitational field but stays far away from it will follow the geodesics of the metric, Eq. (8.50), with φ given by Eq. (8.59). In Ch. 7 we saw that these geodesics obey Kepler’s laws for the gravitational field of a body of mass M. We therefore define this constant M to be the total mass of the relativistic source.

Notice that this definition is not an integral over the source: we do not add up the masses of its constituent particles. Instead, we simply measure its mass – ‘weigh it’ – by the orbits it produces in test bodies far away: this is how astronomers determine the masses of the Earth, Sun, and planets. This definition enables us to write Eq. (8.50) in its form far from any stationary source:

It is important to understand that, because we are not doing an integral over the source, the source of the far-field constant M could be quite different from our expectations from Newtonian theory, a black hole, for example. See also the discussion of the active gravitational mass in Exer. 20, § 8.6.

The assumption that the source was stationary was necessary to reduce the wave equation, Eq. (8.42), to Laplace’s equation, Eq. (8.51). A source which changes with time can emit gravitational waves, and these, as we shall see in the next chapter, travel out from it at the speed of light and do not obey the inequalities Eq. (8.56), so they cannot be regarded as Newtonian fields. Nevertheless, there are situations in which the definition of the mass we have just given may be used with confidence: the waves may be very weak,

¯00

so that the stationary part of h dominates the wave part; or the source may have been stationary in the distant past, so that we can choose r large enough that any waves have not yet had time to reach that large an r. The definition of the mass of a time-dependent source is discussed in greater detail in more-advanced texts, such as Misner et al. (1973) or Wald (1985).

8.5 F u r t h e r re a d i n g

There are a wide variety of ways to ‘derive’ (really, to justify) Einstein’s field equations, and a selection of them may be found in the texts listed below. The weak-field or linearized equations are useful for many investigations where the full equations are too difficult to solve. We shall use them frequently in subsequent chapters, and most texts discuss them. Our extraction of the Newtonian limit is very heuristic, but there are more rigorous approaches that reveal the geometric nature of Newton’s equations (Misner et al. 1973, Cartan 1923) and the asymptotic nature of the limit (Damour 1987, Futamase and Schutz 1983). Post-Newtonian theory is described in review articles by Blanchet (2006) and Futamase and Itoh (2007). Einstein’s own road to the field equations was much less direct than the one we took. As scholars have studied his notebooks, many previous assumptions about his thinking have proved wrong. See the monumental set edited by Renn (2007).

It seems appropriate here to list a sampling of widely available textbooks on GR. They differ in the background and sophistication they assume of the reader. Some excellent texts expect little background – Hartle (2003), Rindler (2006); some might be classed as first-year graduate texts – Carroll (2003), Glendenning (2007), Gron and Hervik (2007), Hobson, et al. (2006), parts of Misner et al. (1973), Møller (1972), Stephani (2004), Weinberg (1972), and Woodhouse (2007); and some that make heavy demands of the student – Hawking and Ellis (1973), Landau and Lifshitz (1962), much of Misner et al. (1973), Synge (1960), and Wald (1984). The material in the present text ought, in most cases, to be sufficient preparation for supplementary reading in even the most advanced texts.

Solving problems is an essential ingredient of learning a theory, and the book of problems by Lightman et al. (1975), though rather old, is still an excellent supplement to those in the present book. An introduction with limited mathematics is Schutz (2003).

8.6E xe rc i s e s

1Show that Eq. (8.2) is a solution of Eq. (8.1) by the following method. Assume the point particle to be at the origin, r = 0, and to produce a spherically symmetric field.

Then use Gauss’ law on a sphere of radius r to conclude

dφ dr

Deduce Eq. (8.2) from this. (Consider the behavior at infinity.)

2 (a) Derive the following useful conversion factors from the SI values of G and c:

G/c2 = 7.425 × 10−28m kg−1 = 1, c5/G = 3.629 × 1052J s−1 = 1.

(b)Derive the values in geometrized units of the constants in Table 8.1 from their given values in SI units.

(c)Express the following quantities in geometrized units:

(i)a density (typical of neutron stars) ρ = 1017 kg m−3;

(ii)a pressure (also typical of neutron stars) p = 1033 kg s−2 m−1;

(iii)the acceleration of gravity on Earth’s surface g = 9.80 m s−2;

(iv)the luminosity of a supernova L = 1041 J s−1.

(d)Three dimensioned constants in nature are regarded as fundamental: c, G, and .

With c = G = 1, has units m2, so 1/2 defines a fundamental unit of length, called the Planck length. From Table 8.1, we calculate 1/2 = 1.616 × 10−35 m.

Since this number involves the fundamental constants of relativity, gravitation, and quantum theory, many physicists feel that this length will play an important role in quantum gravity. Express this length in terms of the SI values of c, G, and . Similarly, use the conversion factors to calculate the Planck mass and Planck time, fundamental numbers formed from c, G, and that have the units of mass and time respectively. Compare these fundamental numbers with characteristic masses, lengths, and timescales that are known from elementary particle theory.

3(a) Calculate in geometrized units:

(i)the Newtonian potential φ of the Sun at the Sun’s surface, radius 6.960 × 108 m;

(ii)the Newtonian potential φ of the Sun at the radius of Earth’s orbit, r = 1 AU = 1.496 × 1011 m;

(iii)the Newtonian potential φ of Earth at its surface, radius = 6.371 × 106 m;

(iv)the velocity of Earth in its orbit around the Sun.

(b)You should have found that your answer to (ii) was larger than to (iii). Why, then, do we on Earth feel Earth’s gravitational pull much more than the Sun’s?

(c)Show that a circular orbit around a body of mass M has an orbital velocity, in

Newtonian theory, of v2 = −φ, where φ is the Newtonian potential.

4 (a) Let A be an n × n matrix whose entries are all very small, |Aij| 1/n, and let I be the unit matrix. Show that

(I + A)−1 = I − A + A2 − A3 + A4 − + . . .

by proving that (i) the series on the right-hand side converges absolutely for each of the n2 entries, and (ii) (I + A) times the right-hand side equals I.

(b) Use (a) to establish Eq. (8.21) from Eq. (8.20).

5(a) Show that if hαβ = ξα,β + ξβ,α , then Eq. (8.25) vanishes.

(b)Argue from this that Eq. (8.25) is gauge invariant.

(c)Relate this to Exer. 10, § 7.6.

8Derive Eq. (8.32) in the following manner:

(a)show that Rα βμν = ηασ Rσβμν + 0(h2αβ );

(b)from this calculate Rαβ to first order in hμν ;

(c)show that gαβ R = ηαβ ημν Rμν + 0(h2αβ );

(d)from this conclude that

10 Use the Lorentz gauge condition, Eq. (8.33), to simplify Gαβ to Eq. (8.41).

11When we write Maxwell’s equations in special-relativistic form, we identify the scalar

potential φ and three-vector potential Ai (signs defined by Ei = −φ,i − Ai,0) as components of a one-form A0 = −φ, Ai (one-form) = Ai (three-vector). A gauge transformation is the replacement φ → φ − ∂f /∂t, Ai → Ai + f,i. This leaves the electric and magnetic fields unchanged. The Lorentz gauge is a gauge in which ∂φ/∂t + iAi = 0. Write both the gauge transformation and the Lorentz gauge condition in four-tensor

notation. Draw the analogy with similar equations in linearized gravity.

12Prove Eq. (8.34).

13The inequalities |T00| |T0i| |Tij| for a Newtonian system are illustrated in Exers. 2(c). Devise physical arguments to justify them in general.

14From Eq. (8.46) and the inequalities among the components hαβ , derive Eqs. (8.47)– (8.50).

15We have argued that we should use convenient coordinates to solve the weakfield problem (or any other!), but that any physical results should be expressible in coordinate-free language. From this point of view our demonstration of the Newtonian limit is as yet incomplete, since in Ch. 7 we merely showed that the metric Eq. (7.8), led to Newton’s law

dp/dt = −m φ

But. surely this is a coordinate-dependent equation, involving coordinate time and position. It is certainly not a valid four-dimensional tensor equation. Fill in this gap in our reasoning by showing that we can make physical measurements to verify that the relativistic predictions match the Newtonian ones. (For example, what is the relation between the proper time one orbit takes and its proper circumference?)

16Re-do the derivation of the Newtonian limit by replacing 8π in Eq. (8.10) by k and fol-

lowing through the changes this makes in subsequent equations. Verify that we recover Eq. (8.50) only if k = 8π .

17(a) A small planet orbits a static neutron star in a circular orbit whose proper circumference is 6 × 1011 m. The orbital period takes 200 days of the planet’s proper time.

Estimate the mass M of the star.

(b)Five satellites are placed into circular orbits around a static black hole. The proper circumferences and proper periods of their orbits are given in the table below. Use the method of (a) to estimate the hole’s mass. Explain the trend of the results you get for the satellites.

18Consider the field equations with cosmological constant, Eq. (8.7), with arbitrary and k = 8π .

(a)Find the Newtonian limit and show that we recover the motion of the planets only if | | is very small. Given that the radius of Pluto’s orbit is 5.9 × 1012 m, set an upper bound on | | from solar-system measurements.

(b)By bringing over to the right-hand side of Eq. (8.7), we can regard − gμν /8π

as the stress-energy tensor of ‘empty space’. Given that the observed mass of the region of the universe near our Galaxy would have a density of about 10−27 kg m3

if it were uniformly distributed, do you think that a value of | | near the limit you established in (a) could have observable consequences for cosmology? Conversely, if is comparable to the mass density of the universe, do we need to include it in the equations when we discuss the solar system?

19In this exercise we shall compute the first correction to the Newtonian solution caused by a source that rotates. In Newtonian gravity, the angular momentum of the source does not affect the field: two sources with the same ρ(xi) but different angular momenta have the same field. Not so in relativity, since all components of Tμν generate the field. This is our first example of a post-Newtonian effect, an effect that introduces an aspect of general relativity that is not present in Newtonian gravity.

(a) Suppose a spherical body of uniform density ρ and radius R rotates rigidly about the x3 axis with constant angular velocity . Write down the components T0 ν in a Lorentz frame at rest with respect to the center of mass of the body, assuming ρ, , and R are independent of time. For each component, work to the lowest nonvanishing order in R.

(b)The general solution to the equation 2f = g, which vanishes at infinity, is the generalization of Eq. (8.2),

which reduces to Eq. (8.2) when g is nonzero in a very small region. Use this to

only outside the body, and only to the lowest nonvanishing order in r−1, where r

¯0j

is the distance from the body’s center. Express the result for h in terms of the body’s angular momentum. Find the metric tensor within this approximation, and transform it to spherical coordinates.

(c)Because the metric is independent of t and the azimuthal angle φ, particles orbiting this body will have p0 and pφ constant along their trajectories (see § 7.4). Consider a particle of nonzero rest-mass in a circular orbit of radius r in the equatorial plane. To lowest order in , calculate the difference between its orbital period in the positive sense (i.e., rotating in the sense of the central body’s rotation) and in the

negative sense. (Define the period to be the coordinate time taken for one orbit of

φ = 2π .)

(d)From this devise an experiment to measure the angular momentum J of the

central body. We take the central body to be the Sun (M = 2 × 1030 kg, R = 7 × 108 m, = 3 × 10−6 s−1) and the orbiting particle Earth (r = 1.5 × 1011 m). What would be the difference in the year between positive and negative orbits?

20 This exercises introduces the concept of the active gravitational mass. After deriving the weak-field Einstein equations in Eq. (8.42), we immediately specialized to the low-velocity Newtonian limit. Here we go a few steps further without making the assumption that velocities are small or pressures weak compared to densities.

(a) Perform a trace-reverse operation on Eq. (8.42) to get

(b)If the system is isolated and stationary, then its gravitational field far away will be dominated by h00, as argued leading up to Eq. (8.50). If the system has weak internal gravity but strong pressure, show that h00 = −2 where satisfies a Newtonian-like Poisson equation

-.

For a perfect fluid, this source term is just ρ + 3p, which is called the active gravitational mass in general relativity. If the system is Newtonian, then p ρ and we have the usual Newtonian limit. This is another example of a post-Newtonian effect.