187

8.2

Einstein’s equations

Table 8.1 Comparison of SI and geometrized values of fundamental constants

Constant

SI value

Geometrized value

c

2.998 ×

108 ms−1

1

G

6.674 ×

10−11 m3 kg−1 s−2

1

× 10−70 m2

1.055 ×

10−34 kg m2 s−1

2.612

me

9.109 ×

10−31 kg

6.764

× 10−58 m

mp

1.673 ×

10−27 kg

1.242

× 10−54 m

M

1.988

×

1030 kg

1.476

103 m

M+

5.972

1024 kg

4.434

× 10−3 m

L

,

3.84

×

1.06

×

×

1026 kg m2 s−3

×

10−26

+

Notes: The symbols me and mp stand respectively for the rest masses of the electron and proton;

M

+

and M

,

denote, respectively, the masses of the Sun and Earth; and L

+

is the Sun’s lumi-

nosity (the SI unit is equivalent to joules per second). Values are rounded to at most four figures

even when known more accurately. Data from Yao (2006).

1 = G/c2 = 7.425 × 10−28m kg−1.

(8.9)

We shall use this to eliminate kg as a unit, measuring mass in meters. We list in Table 8.1 the values of certain useful constants in SI and geometrized units. Exer. 2, § 8.6, should help the student to become accustomed to these units.

An illustration of the fundamental nature of geometrized units in gravitational problems is provided by the uncertainties in the two values given for M,. Earth’s mass is measured by examining satellite orbits and using Kepler’s laws. This measures the Newtonian potential, which involves the product GM,, c2 times the geometrized value of the mass. This number is known to ten significant figures, from laser tracking of satellites orbiting the Earth. Moreover, the speed of light c now has a defined value, so there is no uncertainty in it. Thus, the geometrized value of M, is known to ten significant figures. The value of G, however, is measured in laboratory experiments, where the weakness of gravity introduces large uncertainty. The conversion factor G/c2 is uncertain by two parts in 105, so that is also the accuracy of the SI value of M,. Similarly, the Sun’s geometrized mass is known to nine figures by precise radar tracking of the planets. Again, its mass in kilograms is far more uncertain.

8.2 E i n s t e i n ’ s e q u a t i o n s

In component notation, Einstein’s equations, Eq. (8.7), take the following form if we specialize to = 0 (a simplification at present, but one we will drop later), and if we take k = 8π ,

Gαβ = 8π Tαβ .

(8.10)

188			The Einstein field equations
	The constant		is called the cosmological constant, and was originally not present

in Einstein’s equations; he inserted it many years later in order to obtain static cosmological solutions – solutions for the large-scale behavior of the universe – that he felt at the time were desirable. Observations of the expansion of the universe subsequently made him reject the term and regret he had ever invented it. However, recent astronomical observations strongly suggest that it is small but not zero. We shall return to the discussion of in Ch. (12), but for the moment we shall set= 0. The justification for doing this, and the possible danger of it, are discussed in Exer. 18, § 8.6.

The value k = 8π is obtained by demanding that Einstein’s equations predict the correct behavior of planets in the solar system. This is the Newtonian limit, in which we must demand that the predictions of GR agree with those of Newton’s theory when the latter are well tested by observation. We saw in the last chapter that the Newtonian motions are produced when the metric has the form Eq. (7.8). One of our tasks in this chapter is to show that Einstein’s equations, Eq. (8.10), do indeed have Eq. (7.8) as a solution when we assume that gravity is weak (see Exer. 3, § 8.6). We could, of course, keep k arbitrary until then, adjusting its value to whatever is required to obtain the solution, Eq. (7.8). It is more convenient, however, for our subsequent use of the equations of this chapter if we simply set k to 8π at the outset and verify at the appropriate time that this value is correct.

Eq. (8.10) should be regarded as a system of ten coupled differential equations (not 16, since Tαβ and Gαβ are symmetric). They are to be solved for the ten components gαβ when the source Tαβ is given. The equations are nonlinear, but they have a wellposed initial-value structure – that is, they determine future values of gαβ from given initial data. However, one point must be made: since {gαβ } are the components of a tensor in some coordinate system, a change in coordinates induces a change in them. In particular, there are four coordinates, so there are four arbitrary functional degrees of freedom among the ten gαβ . It should be impossible, therefore, to determine all ten gαβ from any initial data, since the coordinates to the future of the initial moment can be changed arbitrarily. In

fact, Einstein’s equations have exactly this property: the Bianchi identities
Gαβ ;β = 0	(8.11)

mean that there are four differential identities (one for each value of α above) among the ten Gαβ . These ten, then, are not independent, and the ten Einstein equations are really only six independent differential equations for the six functions among the ten gαβ that characterize the geometry independently of the coordinates.

These considerations are of key importance if we want to solve Einstein’s equations to watch systems evolve in time from some initial state. In this book we will do this in a limited way for weak gravitational waves in Ch. (9). Because of the complexity of Einstein’s equations, dynamical situations are usually studied numerically. The field of numerical relativity has evolved a well-defined approach to the problem of separating the coordinate freedom in gαβ from the true geometric and dynamical freedom. This is described in more advanced texts, for instance Misner et al. (1973), or Hawking and Ellis (1973), see also Choquet-Bruhat and York (1980) or the more recent review by Cook (2000). It will suffice