8.7 The Vacuum Einstein Equations

In discussing the structure of space, we considered how the volumes of geodesic balls grow with radius. In discussing spacetime, we want to consider the analogous question of how the volumes of light cones [1:p1052]) grow with time. But to do this, we have to say what we mean by time, since—as we saw in the previous subsection—different foliations can lead to different identifications.

Any particular foliation—with its sequence of spacelike hypersurfaces—provides at every point a timelike vector that defines a time direction in spacetime. So if we start at any point in the causal graph, we can look at the forward light cone from this point, and follow the connections in the causal graph until we have gone a proper time t in the time direction we have defined. Then we can ask how many nodes we have reached in the causal graph.

The result will depend on the underlying rule for the system. But if in the limit it is going to correspond to flat (d + 1)-dimensional spacetime, at any spacetime position X it must grow like:

If we include the possibility of curvature, we get to first order

where R_μν is the spacetime Ricci tensor, and δt^μ δt^νR_μν is effectively its projection along the infinitesimal timelike vector δt^μ.

For any particular underlying rule, C_t(X) will take on a definite form. But in making connections with traditional continuum spacetime, we are interested in its limiting behavior.

Assume, to begin, that we have scaled t to be measured relative to the size of the whole causal graph. Then for small t we can expand C_t(X) to get the expression involving curvature above. But now imagine scaling up t. Eventually it is inevitable that the curvature term has the potential to affect the overall t dependence, and potentially change the effective exponent of t. But if the overall continuum limit is going to correspond to a (d + 1)-dimensional spacetime, this cannot happen. And what this means is that at least a suitably averaged version of the curvature term must not in fact grow [1:9.15].

The details are slightly complicated [113], but suffice it to say here that the constraint on R_μν is obtained by averaging over directions, then averaging over positions with a weighting determined by the volume element associated with the metric g_μν defined by our choice of hypersurfaces. The requirement that this average not grow when t is scaled up can then be expressed as the vanishing of the variation of , which is precisely the usual Einstein–Hilbert action—thereby leading to the conclusion that R_μν must satisfy exactly the usual vacuum Einstein equations [114][115][75][116]:

A full derivation of this is given in [113]. Causal invariance plays a crucial role, ensuring for example that timelike directions tⁱ associated with different foliations give invariant results. Much like in the derivation of continuum fluid behavior from microscopic molecular dynamics (e.g. [110]), one also needs to take a variety of fairly subtle limits, and one needs sufficient intrinsic generation of effective randomness [1:7.5] to justify the use of certain statistical averages.

But there is a fairly simple interpretation of the result above. Imagine all the geodesics that start at a particular point in the causal graph. The further we go, the more possible geodesic paths there will be in the graph. To achieve a power law corresponding to a definite dimension, the geodesics must in a sense just “stream outwards”, evenly distributed in direction.

But the Ricci tensor specifically measures the rate at which bundles of geodesics change their cross-sectional area. And as soon as this change is nonzero, it will inevitably change the local density of geodesics and eventually grow to disrupt the power law. And so the only way a fixed limiting dimension can be achieved is for the Ricci curvature to vanish, just as it does according to the vacuum Einstein equations. (Note that higher-order terms, involving for example the Weyl tensor and other components of the Riemann tensor, yield changes in the shape of bundles of geodesics, but not in their cross-sectional area, and are therefore not constrained by the requirement of fixed limiting dimension.)