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The general spherically symmetric, static solution of +"T""=0 in the exterior Schwarzschild metric is expressed in terms of two integration constants and two arbitrary functions, one of which is the trace of T",. One constant is the magnitude of T,"at infinity, and the other is determined if the physically normalized components of T", are finite on the future horizon. The trace of the stress tensor of a conformally invariant quantum field theory may be nonzero (anomalous), but must be proportional (here) to the Wey) scalar, 48M'r; we fix the coefficient for the scalar field by indirect arguments to be (2880m') '. In the twodimensional analog, the magnitude of the Hawking blackbody effect at infinity is directly proportional to the magnitude of the anomalous trace (a multiple of the curvature scalar); a knowledge of either number completely determines the stress tensor outside a body in the final state of collapse. In four dimensions, one obtains instead a relation constraining the remaining undetermined function, which we choose as T f)-T /4. This, plus additional physical and mathematical considerations, leads us to a fairly definite, physically convincing qualitative picture of (T",). Groundwork is laid for explicit calculations of (T",).

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%'e calculate the vacuum expectation value, T"", of the energy-momentum tensor of a massless scalar field in a general two-dimensional spacetime and evaluate it in a two-dimensional model of gravitational collapse. In two dimensions, quantum radiation production is incompatible with a conserved and traceless T"".We therefore resolve an ambiguity in our expression for T~", regularized by a geodesic point-separation procedure, by demanding conservation but allowing a trace. In the collapse model, the results support that picture of black-hole evaporation in which pairs of particles are created outside the horizon (and not entirely in the collapsing matter), one of which carries negative energy into the future horizon of the black hole, while the other contributes to the thermal flux at infinity.

Renormalization theory in quantum gravity, among other applications, continues to stimulate many attempts to calculate asymptotic expansions of heat kernels and other Green functions of differential operators. Computer algebra systems now make it possible to carry these calculations to high orders, where the number of terms is very large. To be understandable and usable, the result of the calculation must be put into a standard form; because of the subtleties of tensor symmetry, to specify a basis set of independent terms is a non-trivial problem. This problem can be solved by applying some representation theory of the symmetric, general linear and orthogonal groups. In this work the authors treat the case of scalars or tensors formed from the Riemann tensor (of a torsionless, metric-compatible connection) by covariant differentiation, multiplication and contraction. (The same methods may be applied readily to other tensors.) The authors have determined the number of independent homogeneous scalar monomials of each order and degree up to order twelve in derivatives of the metric, and exhibited a basis for these invariants up through order eight. For tensors of higher rank, they present bases through order six; in that case some effort is required to match the familiar classical tensor expressions (usually supporting reducible representations) against the lists of irreducible representations provided by the more abstract group theory. Finally, the analysis yields (more easily for scalars than for tensors) an understanding of linear dependences in low dimensions among otherwise distinct tensors.

The energy-momentum tensor is calculated in the two dimensional quantum theory of a massless scalar field influenced by the motion of a perfectly reflecting boundary (mirror). This simple model system evidently can provide insight into more sophisticated processes, such as particle production in cosmological models and exploding black holes. In spite of the conformally static nature of the problem, the vacuum expectation value of the tensor for an arbitrary mirror trajectory exhibits a non-vanishing radiation flux (which may be readily computed). The expectation value of the instantaneous energy flux is negative when the proper acceleration of the mirror is increasing, but the total energy radiated during a bounded mirror motion is positive. A uniformly accelerating mirror does not radiate; however, our quantization does not coincide with the treatment of that system as a ‘static universe’. The calculation of the expectation value requires a regularization procedure of covariant separation of points (in products of field operators) along time-like geodesics; more naïve methods do not yield the same answers. A striking example involving two mirrors clarifies the significance of the conformal anomaly.

In the point-splitting prescription for renormalizing the stress-energy tensor of a scalar field in curved spacetime, it is assumed that the anticommutator expectation value G(x, x') = <φ(x)φ(x') + φ(x')φ(x)> has a singularity of the Hadamard form as x-»x'. We prove here that if G(x, x') has the Hadamard singularity structure in an open neighborhood of a Cauchy surface, then it does so everywhere, i.e., Cauchy evolution preserves the Hadamard singularity structure. In particular, in a spacetime which is flat below a Cauchy surface, for the "in" vacuum state G(x, x') is of the Hadamard form everywhere, and thus the point-splitting prescription in this case has been rigorously shown to give meaningful, finite answers.