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

'We are all in the gutter, but some of us are looking at the stars.'