How does matter couple to space-time so that space-time becomes curved?

Asked by: Dennis J. Plews

Answer

According to Riemann's theory of curved manifolds, the geometry of space-time is completely described by the metric tensor g_(ab), which you can think of as a 4x4 symmetric matrix, so it has 10 algebraically independent components at each event (point). Furthermore, the curvature of space-time at each event is completely described by a multilinear operator (a generalization of a linear operator) called the Riemann curvature tensor, which has 20 algebraically independent components at each event. The components of the Riemann tensor identically satisfy a differential equation (the Bianchi identity), which is why the metric tensor (ten algebraically independent components at each event) can and does completely determine the Riemann curvature tensor (20 algebraically independent components at each event).

The Riemann tensor R_(abcd) can be decomposed into two pieces, the Ricci tensor R_(ab) and the Weyl tensor C_(abcd), in a manner analogous to decomposing a matrix into trace and tracefree parts. If the Riemann tensor vanishes on a neighborhood of space-time, this neighborhood is locally isometric to ('same distance relations as') Minkowski space-time; it is 'locally flat'. Otherwise, if the Weyl tensor vanishes on a neighborhood of space-time, the neighborhood is locally conformally equivalent to ('same angular relations as') Minkowski space-time. Thus, the Riemann, Ricci, and Weyl tensors all have geometric meaning independent of any physical interpretation.

Physics enters via the stress-energy tensor T_(ab), which you can think of as a 4x4 symmetric matrix (so it has 10 algebraically independent components at each event); this tensor completely describes the amount of (non-gravitational) mass-energy at each event, and also any momentum (mass-energy flow) and stresses (such as the pressures in a fluid). Einstein's field equation states that

R_(ab) = 8 pi { T_(ab) - 1/2 g_(ab) T }

where T is the 'trace' of the stress-energy tensor. Thus, the Ricci curvature is directly coupled to the immediate presence of matter at a given event. If there is no mass-energy at a given event, the Ricci tensor vanishes. If it were not for the Weyl tensor, this would mean that matter here could not have a gravitational influence on distant matter separated by a void (a vacuum free of mass-energy). Thus, the Weyl tensor represents that part of space-time curvature which can propagate across and curve up a void (vacuum region of space-time).

Interestingly enough, two physically important solutions of Einstein's field equation represent two extremes of curvature. The Kerr vacuum solution, which models space-time outside a rotating body such as a star, has zero Ricci curvature but nonzero Weyl curvature at each event. The Friedmann dust solution, which models the universe on a very large scale, has zero Weyl curvature but nonzero Ricci curvature at each event.

The Weyl tensor turns out to be analogous in many ways to the electromagnetic field tensor, which you can think of as an antisymmetric four by four matrix (6 algebraically independent components at each event). With respect to the world line of a given observer, the electromagnetic field tensor decomposes into two vectors, the electric and magnetic field vectors (3 components each). Similarly, the Weyl tensor (10 algebraically independent components) decomposes into two tensors which you can think of as two 3x3 symmetric traceless matrices (5 algebraically independent components each). This permits one to rewrite Einstein's field equation as a set of equations formally resembling Maxwell's field equations, but much more complicated-- in particular, the gravitational field equations are nonlinear.

Another important way in which the gravitational field is analogous to the electromagnetic field is that, just as small disturbances in the electromagnetic field at a given even propagate outward as an electromagnetic wave, so too small disturbances in the the gravitational field at a given event propagate outward as a gravitational wave. Such waves have never been directly detected, although astronomers have strong indirect evidence that they do exist and carry energy just as general relativity predicts. In the next few years, new and fantastically sensitive detectors are expected to directly confirm the existence of gravitational waves.
Answered by: Chris Hillman, Ph.D., Mathematics, University of Washington