The Einstein field equations

be ten differential equations, one for each independent component of Eq. (8.3), in place of the single one, Eq. (8.1). (Recall that T is symmetric, so it has only ten independent components, not 16.)

By analogy with Eq. (8.1), we should look for a second-order differential operator O that

satisfies these conditions, if μ and are constants. To determine μ we use a property of Tαβ , which we have not yet used, namely that the Einstein equivalence principle demands local conservation of energy and momentum (Eq. (7.6))

which again must be true for any metric tensor. Since gαβ ;μ = 0, we now find, from Eq. (8.4)

is to be an identity for arbitrary gαβ . So we are led by this chain of argument to the equation

with undetermined constants and k. In index-free form, this is

These are called the field equations of GR, or Einstein’s field equations. We shall see below that we can determine the constant k by demanding that Newton’s gravitational field equation comes out right, but that remains arbitrary.

But first let us summarize where we have got to. We have been led to Eq. (8.7) by asking for equations that (i) resemble but generalize Eq. (8.1), (ii) introduce no preferred coordinate system, and (iii) guarantee local conservation of energy–momentum for any metric tensor. Eq. (8.7) is not the only equation which satisfies (i)–(iii). Many alternatives

have been proposed, beginning even before Einstein arrived at equations like Eq. (8.7). In recent years, when technology has made it possible to test Einstein’s equations fairly precisely, even in the weak gravity of the solar system, many new alternative theories have been proposed. Some have even been designed to agree with Einstein’s predictions at the precision of foreseeable solar-system experiments, differing only for much stronger fields. GR’s competitors are, however, invariably more complicated than Einstein’s equations themselves, and on simply aesthetic grounds are unlikely to attract much attention from physicists unless Einstein’s equations are eventually found to conflict with some experiment. A number of the competing theories and the increasingly accurate experimental tests which have been used to eliminate them since the 1960s are discussed in Misner et al. (1973), Will (1993), and Will (2006). (We will study two classical tests in Ch. 11.) Einstein’s equations have stood up extremely well to these tests, so we will not discuss any alternative theories in this book. In this we are in the good company of the Nobel-Prize-winning astrophysicist S. Chandrasekhar (1980):

The element of controversy and doubt, that have continued to shroud the general theory of relativity to this day, derives precisely from this fact, namely that in the formulation of his theory Einstein incorporates aesthetic criteria; and every critic feels that he is entitled to his own differing aesthetic and philosophic criteria. Let me simpy say that I do not share these doubts; and I shall leave it at that.

Although Einstein’s theory is essentially unchallenged at the moment, there are still reasons for expecting that it is not the last word, and therefore for continuing to probe it experimentally. Einstein’s theory is, of course, not a quantum theory, and strong theoretical efforts are currently being made to formulate a consistent quantum theory of gravity. We expect that, at some level of experimental precision, there will be measurable quantum corrections to the theory, which might for example come in the form of extra fields coupled to the metric. The source of such a field might violate the Einstein equivalence principle. The field itself might carry an additional form of gravitational waves. In principle, any of the predictions of general relativity might be violated in some such theory. Precision experiments on gravitation could some day provide the essential clue needed to guide the theoretical development of a quantum theory of gravity. However, interesting as they might be, such considerations are outside the scope of this introduction. For the purposes of this book, we will not consider alternative theories any further.

Geometrized units

We have not determined the value of the constant k in Eq. (8.7), which plays the same role as 4π G in Eq. (8.1). Before discussing it below we will establish a more convenient set of units, namely those in which G = 1. Just as in SR where we found it convenient to choose units in which the fundamental constant c was set to unity, so in studies of gravity it is more natural to work in units where G has the value unity. A convenient conversion factor from SI units to these geometrized units (where c = G = 1) is