Fermi Coordinates and Penrose Limits

We propose a formulation of the Penrose plane wave limit in terms of null Fermi coordinates. This provides a physically intuitive (Fermi coordinates are direct measures of geodesic distance in space-time) and manifestly covariant description of the expansion around the plane wave metric in terms of components of the curvature tensor of the original metric, and generalises the covariant description of the lowest order Penrose limit metric itself, obtained in hep-th/0312029. We describe in some detail the construction of null Fermi coordinates and the corresponding expansion of the metric, and then study various aspects of the higher order corrections to the Penrose limit. In particular, we observe that in general the first-order corrected metric is such that it admits a light-cone gauge description in string theory. We also establish a formal analogue of the Weyl tensor peeling theorem for the Penrose limit expansion in any dimension, and we give a simple derivation of the leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5.Comment: 25 page

The Refractive Index of Curved Spacetime II: QED, Penrose Limits and Black Holes

This work considers the way that quantum loop effects modify the propagation of light in curved space. The calculation of the refractive index for scalar QED is reviewed and then extended for the first time to QED with spinor particles in the loop. It is shown how, in both cases, the low frequency phase velocity can be greater than c, as found originally by Drummond and Hathrell, but causality is respected in the sense that retarded Green functions vanish outside the lightcone. A "phenomenology" of the refractive index is then presented for black holes, FRW universes and gravitational waves. In some cases, some of the polarization states propagate with a refractive index having a negative imaginary part indicating a potential breakdown of the optical theorem in curved space and possible instabilities.Comment: 62 pages, 14 figures, some signs corrected in formulae and graph

Penrose Limits and Spacetime Singularities

We give a covariant characterisation of the Penrose plane wave limit: the plane wave profile matrix $A(u)$ is the restriction of the null geodesic deviation matrix (curvature tensor) of the original spacetime metric to the null geodesic, evaluated in a comoving frame. We also consider the Penrose limits of spacetime singularities and show that for a large class of black hole, cosmological and null singularities (of Szekeres-Iyer ``power-law type''), including those of the FRW and Schwarzschild metrics, the result is a singular homogeneous plane wave with profile $A(u)\sim u^{-2}$, the scale invariance of the latter reflecting the power-law behaviour of the singularities.Comment: 9 pages, LaTeX2e; v2: additional references and cosmetic correction

BF Theories and Group-Level Duality

It is known that the partition function and correlators of the two-dimensional topological field theory $G_K(N)/ G_K(N)$ on the Riemann surface $\Sigma_{g,s}$ is given by Verlinde numbers, dim($V_{g,s,K}$) and that the large $K$ limit of dim($V_{g,s,K}$) gives Vol(${\cal M}_s$), the volume of the moduli space of flat connections of gauge group $G(N)$ on $\Sigma_{g,s}$, up to a power of $K$. Given this relationship, we complete the computation of Vol(${\cal M}_s$) using only algebraic results from conformal field theory. The group-level duality of $G(N)_K$ is used to show that if $G(N)$ is a classical group, then $\displaystyle \lim_{N\rightarrow \infty} G_K(N) / G_K(N)$ is a BF theory with gauge group $G(K)$. Therefore this limit computes Vol(${\cal M}^\prime_s$), the volume of the moduli space of flat connections of gauge group $G(K)$

Goedel, Penrose, anti-Mach: extra supersymmetries of time-dependent plane waves

We prove that M-theory plane waves with extra supersymmetries are necessarily homogeneous (but possibly time-dependent), and we show by explicit construction that such time-dependent plane waves can admit extra supersymmetries. To that end we study the Penrose limits of Goedel-like metrics, show that the Penrose limit of the M-theory Goedel metric (with 20 supercharges) is generically a time-dependent homogeneous plane wave of the anti-Mach type, and display the four extra Killings spinors in that case. We conclude with some general remarks on the Killing spinor equations for homogeneous plane waves.Comment: 20 pages, LaTeX2

PP-waves on Superbrane Backgrounds

In this paper we discuss a method of generating supersymmetric solutions of the Einstein equations. The method involves the embedding of one supersymmetric spacetime into another. We present two examples with constituent spacetimes which support "charges", one of which was known previously and the other of which is new. Both examples have PP-waves as one of the embedding constituents.Comment: 6 pages no figure

Penrose Limits of the Baryonic D5-brane

The Penrose limits of a D5-brane wrapped on the sphere of AdS_5 x S^5 and connected to the boundary by M fundamental strings, which is dual to the baryon vertex of the N=4 SU(M) super Yang-Mills theory, are investigated. It is shown that, for null geodesics that lead to the maximally supersymmetric Hpp-wave background, the resulting D5-brane is a 1/2-supersymmetric null brane. For an appropriate choice of radial geodesic, however, the limiting configuration is 1/4-supersymmetric and closely related to the Penrose limit of a flat space BIon.Comment: LaTeX, 1+18 pages, 1 figure; v2: obvious misquotation of the number of preserved supersymmetries correcte