General relativity is a theory which describes space, time and the gravitational field in terms of a Lorentzian metric . A complete understanding of the gravitational field requires an understanding of the matter sources which generate it. In the Einstein equations, , the left hand side depends only on and is a feature of the geometry alone. On the other hand the right hand side, the energy-momentum tensor, depends not only on the metric but also on some matter fields. The right hand side of Einstein’s equations seems to have suffered from bad press coverage from an early stage. Einstein himself is often quoted as having said that the left hand side of his equations is made of marble while the right hand side is made of wood. I do not have a source for this quote – if anyone reading this does I would be grateful to hear about it. In this post I want to suggest treating that humble right hand side with more respect. If I lived in a palace made of marble with beautiful wooden furniture then I might be more impressed by the marble than by the wood. I would nevertheless do my best to prevent little boys from carving their initials into the furniture with penknives or the cat (much as I love cats) from using it as an accessory for the care of its claws.
The left hand side of the Einstein equations is universal within general relativity – it is always the same, no matter which type of physical situation is to be described. On the other hand the nature of the matter fields depends very much on what physical situation is to be described and what aspects of it are to be included in the description. It is necessary to make a choice of matter model. What is remarkable is that there is a large variety of choices which, in conjunction with the Einstein equations, lead to a consistent closed system of equations which bears no traces of the fact that other physical effects have been omitted. In fact there are three related choices which have to be made to set up the mathematical model in any given case. The first is the matter fields themselves – what kind of geometrical objects are they? The second is the expression which defines the energy-momentum tensor in terms of the matter fields and the metric. The third is the system of equations of motion which describe the dynamics of the matter. Note that in general the energy-momentum tensor depends explicitly on the metric. It is not possible to define an energy-momentum tensor unless the spacetime geometry is given. The same is true in the case of the equations of motion of the matter. They also contain the metric explicitly. Without the metric even the nature of the matter fields themselves can become ambiguous. Which positions should we choose for the indices of a tensor occurring in the description of the matter fields? From a physical point of view it is clear why the metric is necessary in so many ways. The mathematical model must be given a physical interpretation which involves the consideration of measurements. In the absence of a given geometry there is no way to talk about measurements.
When a matter model has been chosen the basic equations which are to be solved are the Einstein-matter equations, i.e the Einstein equations coupled to the equations of motion for the chosen type of matter. The unknowns are the metric and the basic matter fields. For any reasonable choice of matter fields the energy-momentum tensor has zero divergence as a consequence of the equations of motion. However the equations of motion in general contain more information than the divergence-free property of the energy-momentum tensor. For more discussion of these things together with examples see Chapter 3 of my book. I emphasize that solving the equations describing the physical situation within the given model means solving both the Einstein equations and the equations of motion of the matter. This is too often neglected in the literature. A particular danger occurs when the solutions under consideration are of low regularity. If the Einstein equations do not make sense pointwise then it should be checked that they hold in the sense of distributions. For solutions which lack regularity on a hypersurface this is expressed in the junction conditions and it is common in the literature to check that they hold. The equations of motion should also be satisfied in the sense of distributions and this is often ignored. When I use the phrase ‘in the sense of distributions’ here this is just a shorthand since the equations are nonlinear. The correct statement is that it is necessary to think carefully about the sense in which the equations are satisfied.
An example may help to make the importance of the issue clear. At the GR12 conference in Boulder in 1989 there was a heated discussion of the question, whether colliding plane waves can give rise to spontaneous creation of matter. (I emphasize that this discussion was in a purely classical context. Quantum theory was not being taken into account.) This kind of creation of matter sounds ridiculous from a physical point of view. Nevertheless people exhibited ‘solutions’ which showed this type of effect. Their mistake was that they had only verified those things which I said above were usual in the literature. They had not considered whether the equations of motion of matter were satisfied. If the equations of motion are ignored, it is not surprising that arbitrary things can happen.