Finding and using exact solutions of the Einstein equations

The evolution of the methods used to find solutions of Einstein's field equations during the last 100 years is described. Early papers used assumptions on the coordinate forms of the metrics. Since the 1950s more invariant methods have been deployed in most new papers. The uses to which the solutions found have been put are discussed, and it is shown that they have played an important role in the development of many aspects, both mathematical and physical, of general relativity.Comment: 15 pages, LaTeX2e, aipproc.cls, invited lecture to appear in the Proceedings of ERE05 (the Spanish Relativity Meeting), Oviedo, September 2005, to be published by the American Institute of Physics. v2: Remarks on black hole entropy corrected. Other minor change

Hamiltonian thermodynamics of a Lovelock black hole

We consider the Hamiltonian dynamics and thermodynamics of spherically symmetric spacetimes within a one-parameter family of five-dimensional Lovelock theories. We adopt boundary conditions that make every classical solution part of a black hole exterior, with the spacelike hypersurfaces extending from the horizon bifurcation three-sphere to a timelike boundary with fixed intrinsic metric. The constraints are simplified by a Kucha\v{r}-type canonical transformation, and the theory is reduced to its true dynamical degrees of freedom. After quantization, the trace of the analytically continued Lorentzian time evolution operator is interpreted as the partition function of a thermodynamical canonical ensemble. Whenever the partition function is dominated by a Euclidean black hole solution, the entropy is given by the Lovelock analogue of the Bekenstein-Hawking entropy; in particular, in the low temperature limit the system exhibits a dominant classical solution that has no counterpart in Einstein's theory. The asymptotically flat space limit of the partition function does not exist. The results indicate qualitative robustness of the thermodynamics of five-dimensional Einstein theory upon the addition of a nontrivial Lovelock term.Comment: 22 pages, REVTeX v3.

A perturbative perspective on self-supporting wormholes

We describe a class of wormholes that generically become traversable after incorporating gravitational back-reaction from linear quantum fields satisfying appropriate (periodic or anti-periodic) boundary conditions around a non-contractible cycle, but with natural boundary conditions at infinity (i.e., without additional boundary interactions). The class includes both asymptotically flat and asymptotically AdS examples. Simple asymptotically AdS$_3$ or asymptotically AdS$_3 \times S^1$ examples with a single periodic scalar field are then studied in detail. When the examples admit a smooth extremal limit, our perturbative analysis indicates the back-reacted wormhole remains traversable at later and later times as this limit is approached. This suggests that a fully non-perturbative treatment would find a self-supporting eternal traversable wormhole. While the general case remains to be analyzed in detail, the likely relation of the above effect to other known instabilities of extreme black holes may make the construction of eternal traversable wormholes more straightforward than previously expected.Comment: Minor corrections (including fixing a factor of 2 in several formulas/plots

Causal particle detectors and topology

We investigate particle detector responses in some topologically non-trivial spacetimes. We extend a recently proposed regularization of the massless scalar field Wightman function in 4-dimensional Minkowski space to arbitrary dimension, to the massive scalar field, to quotients of Minkowski space under discrete isometry groups and to the massless Dirac field. We investigate in detail the transition rate of inertial and uniformly accelerated detectors on the quotient spaces under groups generated by $(t,x,y,z)\mapsto(t,x,y,z+2a)$, $(t,x,y,z)\mapsto(t,-x,y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z)$, $(t,x,y,z)\mapsto(t,-x,-y,z+a)$ and some higher dimensional generalizations. For motions in at constant $y$ and $z$ on the latter three spaces the response is time dependent. We also discuss the response of static detectors on the RP^3 geon and inertial detectors on RP^3 de Sitter space via their associated global embedding Minkowski spaces (GEMS). The response on RP^3 de Sitter space, found both directly and in its GEMS, provides support for the validity of applying the GEMS procedure to detector responses and to quotient spaces such as RP^3 de Sitter space and the RP^3 geon where the embedding spaces are Minkowski spaces with suitable identifications.Comment: 47 pages, 9 figure

The Veblen functions for computability theorists

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is epsilon_X", and (2) "If X is a well-ordering, then so is phi(alpha,X)", where alpha is a fixed computable ordinal and phi the two-placed Veblen function. For the former statement, we show that omega iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ACA_0^+ over RCA_0. To prove the latter statement we need to use omega^alpha iterations of the Turing jump, and we show that the statement is equivalent to Pi^0_{omega^alpha}-CA_0. Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if X is a well-ordering, then so is phi(X,0)" is equivalent to ATR_0 over RCA_0.Comment: 26 pages, 3 figures, to appear in Journal of Symbolic Logi