Homogeneous plane waves

Abstract

Motivated by the search for potentially exactly solvable time-dependent string backgrounds, we determine all homogeneous plane wave (HPW) metrics in any dimension and find one family of HPWs with geodesically complete metrics and another with metrics containing null singularities. The former generalises both the Cahen-Wallach (constant A ij ) metrics to time-dependent HPWs, A ij (x + ), and the Ozsvath-Schücking anti-Mach metric to arbitrary dimensions. The latter is a generalisation of the known homogeneous metrics with A ij ∼ 1/(x + ) 2 to a more complicated time-dependence. We display these metrics in various coordinate systems, show how to embed them into string theory, and determine the isometry algebra of a general HPW and the associated conserved charges. We review the Lewis-Riesenfeld theory of invariants of time-dependent harmonic oscillators and show how it can be deduced from the geometry of plane waves. We advocate the use of the invariant associated with the extra (timelike) isometry of HPWs for lightcone quantisation, and illustrate the procedure in some examples.

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

“…A small refinement of this argument also shows that this homogeneous plane wave must be smooth (corresponding to one of the two families of homogeneous plane waves found in [10]). For definiteness we phrase the argument in the context of eleven-dimensional supergravity, but it is clearly more general than that.…”

“…In particular, it is not obvious at this point that this is really a homogeneous plane wave. To exhibit this, we now show that we can put the above metric into the general form of a smooth homogeneous plane wave in stationary coordinates, namely [10] …”

“…In particular, this one-parameter family of homogeneous plane waves is precisely of the anti-Mach kind [14] discussed in [10,12] in the sense that in stationary coordinates one of the frequencies is zero, reflecting an additional commuting isometry. In summary: the Penrose limit of a Gödel metric is an anti-Mach homogeneous plane wave.…”

“…they are homogeneous [10]. In [10] it was shown that there are two families of homogeneous plane waves, smooth homogeneous plane waves which generalise the symmetric (Cahen-Wallach, constant A ij ) plane waves but are generically time-dependent, A ij = A ij (u), and singular homogeneous plane waves which generalise the metrics with A ij ∼ u −2 . In Brinkmann coordinates the extra Killing vector K has the form…”

“…For example, while any plane wave solution of supergravity preserves half of the supersymmetries [5], these time-independent plane waves have been shown to realise various exotic > 1/2 fractions of unbroken supersymmetries [6]- [9]. In another development [10] attention was drawn to homogeneous plane waves. These generalise the time-independent (and symmetric homogeneous) plane waves to time-dependent plane waves in a way which does not destroy the homogeneity of the metric.…”