here simply to note that there are really only six equations for six quantities among the gαβ , and that Einstein’s equations permit complete freedom in choosing the coordinate system.

8.3 E i n s t e i n ’ s e q u a t i o n s f o r w e a k g ra v i t a t i o n a l fi e l d s

Nearly Lorentz coordinate systems

Since the absence of gravity leaves spacetime flat, a weak gravitational field is one in which spacetime is ‘nearly’ flat. This is defined as a manifold on which coordinates exist in which the metric has components

where

everywhere in spacetime. Such coordinates are called nearly Lorentz coordinates. It is important to say ‘there exist coordinates’ rather than ‘for all coordinates’, since we can find coordinates even in Minkowski space in which gαβ is not close to the simple diagonal (−1, +1, +1, +1) form of ηαβ . On the other hand, if one coordinate system exists in which Eqs. (8.12) and (8.13) are true, then there are many such coordinate systems. Two fundamental types of coordinate transformations that take one nearly Lorentz coordinate system into another will be discussed below: background Lorentz transformations and gauge transformations.

But why should we specialize to nearly Lorentz coordinates at all? Haven’t we just said that Einstein’s equations allow complete coordinate freedom, so shouldn’t their physical predictions be the same in any coordinates? Of course the answer is yes, the physical predictions will be the same. On the other hand, the amount of work we would have to do to arrive at the physical predictions could be enormous in a poorly chosen coordinate system. (For example, try to solve Newton’s equation of motion for a particle free of all forces in spherical polar coordinates, or try to solve Poisson’s equation in a coordinate system in which it does not separate!) Perhaps even more serious is the possibility that in a crazy coordinate system we may not have sufficient creativity and insight into the physics to know what calculations to make in order to arrive at interesting physical predictions. Therefore it is extremely important that the first step in the solution of any problem in GR must be an attempt to construct coordinates that will make the calculation simplest. Precisely because Einstein’s equations have complete coordinate freedom, we should use this freedom intelligently. The construction of helpful coordinate systems is an art, and it is often rather difficult. In the present problem, however, it should be clear that ηαβ is the

Gauge transformations

There is another very important kind of coordinate change that leaves Eqs. (8.12) and (8.13) unchanged: a very small change in coordinates of the form

xα = xα + ξ α (xβ ),

generated by a ‘vector’ ξ α , where the components are functions of position. If we demand

If all |ξ α ,β | are small, then the new hαβ is still small, and we are still in an acceptable coordinate system. This change is called a gauge transformation, which is a term used because of strong analogies between Eq. (8.24) and gauge transformations of electromagnetism. This analogy is explored in Exer. 11, § 8.6. The coordinate freedom of Einstein’s equations means that we are free to choose an arbitrary (small) ‘vector’ ξ α in Eq. (8.24). We will use this freedom below to simplify our equations enormously.

A word about the role of indices such as α and β in Eqs. (8.21) and (8.22) may be helpful here, as beginning students are often uncertain on this point. A prime or bar on an index is an indication that it refers to a particular coordinate system, e.g. that gα β is a component of g in the {xν } coordinates. But the index still takes the same values (0, 1, 2, 3). On the right-hand side of Eq. (8.22) there are no primes because all quantities are defined in the unprimed system. Thus, if α = β = 0, we read Eq. (8.22) as: ‘The 0–0 component of g in the primed coordinate system is a function whose value at any point is the value of the 0–0 component of η plus the value of the 0–0 “component” of hαβ in the unprimed coordinates at that point minus twice the derivative of the function ξ0 – defined by Eq. (8.23) – with respect to the unprimed coordinate x0 there.’ Eq. (8.22) may look strange because – unlike, say, Eq. (8.15) – its indices do not ‘match up’. But that is acceptable, since Eq. (8.22) is not what we have called a valid tensor equation. It expresses the relation between components of a tensor in two specific coordinates; it is not intended to be a general coordinate-invariant expression.