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From: Stephen Wolfram, A New Kind of Science
Notes for Chapter 9: Fundamental Physics
Section: The Phenomenon of Gravity
Page 1053

Pure gravity. In the absence of matter, the Einstein equations always admit ordinary flat Minkowski space as a solution. But they also admit other solutions that in effect represent configurations of pure gravitational field. And in fact the 4D vacuum Einstein equations are already a sophisticated set of nonlinear partial differential equations that can support all sorts of complex behavior. Several tens of families of solutions to the equations have been found - some with obvious physical interpretations, others without.

Already in 1916 Karl Schwarzschild gave the solution for a spherically symmetric gravitational field. He imagined that this field itself existed in a vacuum - but that it was produced by a mass such as a star at its center. In its original form the metric becomes singular at radius 2 G m/c^2 (or 3 m km with m in solar masses). At first it was assumed that this would always be inside a star, where the vacuum Einstein equations would not apply. But in the 1930s it was suggested that stars could collapse to concentrate their mass in a smaller radius. The singularity was then interpreted as an event horizon that separates the interior of a black hole from the ordinary space around it. In 1960 it was realized, however, that appropriate coordinates allowed smooth continuation across the event horizon - and that the only genuine singularity was infinite curvature at a single point at the center. Some× it was said that this must reflect the presence of a point mass, but soon it was typically just said to be a point at which the Einstein equations - for whatever reason - do not apply. Different choices of coordinates led to different apparent locations and forms of the singularity, and by the late 1970s the most common representation was just a smooth manifold with a topology reflecting the removal of a point - and without any specific reference to the presence of matter.

Appealing to ideas of Ernst Mach from the late 1800s it has often been assumed that to get curvature in space always eventually requires the presence of matter. But in fact even the vacuum Einstein equations for complete universes (with no points left out) have solutions that show curvature. If one assumes that space is both homogeneous and isotropic then it turns out that only ordinary flat Minkowski space is allowed. (When matter or a cosmological term is present one gets different solutions - that always expand or contract, and are much studied in cosmology.) If anisotropy is present, however, then there can be all sorts of solutions - classified for example as having different Bianchi symmetry types. And a variety of inhomogeneous solutions with no singularities are also known - an example being the 1962 Ozsváth-Schücking rotating vacuum. But in all cases the structure is too simple to capture much that seems relevant for our present universe.

One form of solution to the vacuum Einstein equations is a gravitational wave consisting of a small perturbation propagating through flat space. No solutions have yet been found that represent complete universes containing emitters and absorbers of such waves (or even for example just two massive bodies). But it is known that combinations of gravitational waves can be set up that will for example evolve to generate singularities. And I suspect that nonlinear interactions between such waves will also inevitably lead to the analog of turbulence for pure gravity. (Numerical simulations often show all sorts of complex behavior - but in the past this has normally been attributed just to the approximations used. Note that for example Bianchi type IX solutions for a complete universe show sensitive dependence on initial conditions - and no doubt this can also happen with nonlinear gravitational waves.)

As mentioned on page 1032, Albert Einstein considered the possibility that particles of matter might somehow just be localized structures in gravitational and electromagnetic fields. And in the mid-1950s John Wheeler studied explicit simple examples of such so-called geons. But in all cases they were found to be unstable - decaying into ordinary gravitational waves. The idea of having purely gravitational localized structures has also occasionally been considered - but so far no stable field configuration has been found. (And no purely repetitive solutions can exist.)

The equivalence principle (see page 1052) might suggest that anything with mass - or energy - should affect the curvature of space in the same way. But in the Einstein equations the energy-momentum tensor is not supposed to include contributions from the gravitational field. (There are alternative and seemingly inelegant theories of gravity that work differently - and notably do not yield black holes. The setup is also somewhat different in recent versions of string theory.) The very definition of energy for the gravitational field is not particularly straightforward in general relativity. But perhaps a definition could be found that would allow localized structures in the gravitational field to make effective contributions to the energy-momentum tensor that would mimic those from explicit particles of matter. Nevertheless, there are quite a few phenomena associated with particles that seem difficult to reproduce with pure gravity - at least say without extra dimensions. One example is parity violation; another is the presence of long-range forces other than gravity.

Stephen Wolfram, A New Kind of Science (Wolfram Media, 2002), page 1053.
© 2002, Stephen Wolfram, LLC