100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Gravitational Billiards, Dualities and Hidden Symmetries 47This is also the physical context in which we will consider the gravitationalbilliards in the following section. bHere we will not write out the complete Einstein equations for the metricansatz (17) (see, however, 5,9,11 ) but simply note that upon dimensionalreduction, the fields (∆, ˜B) with ∆ ≥ 0 coordinatize a homogeneous σ-model manifold SL(2, R)/SO(2). 44 The equation for ˜B reads∂ µ (t −1 ∆ 2 ∂ µ ˜B) = 0 (19)with the convention, in this subsection only, that µ, ν = 0, 1. Because intwo dimensions, every divergence-free vector field can be (locally) rewrittenas a curl, we can introduce the dual ‘Ehlers potential’ B(t, x) by means ofThe Ehlers potential obeys the equation of motiont∆ −2 ∂ µ B = ɛ µν ∂ ν ˜B (20)∂ µ (t∆ −2 ∂ µ B) = 0 (21)The combined equations of motion for ∆ and B can be compactly assembledinto the so-called Ernst equation 5∆∂ µ (t∂ µ E) = t∂ µ E∂ µ E (22)for the complex Ernst potential E := ∆ + iB. The pair (∆, B) againparametrizes a coset space SL(2, R)/SO(2), but different from the previousone.To write out the non-linear action of the two SL(2, R) symmetries, oneof which is the Ehlers symmetry, we use a notation that is already adaptedto the Kac Moody theory in the following chapters. The relation to the morefamiliar ‘physicist’s notation’ for the SL(2, R) generators is given below:e ∼ J + , f ∼ J − , h ∼ J 3 (23)In writing the variations of the fields, we will omit the infinitesimal parameterthat accompanies each transformation. The Ehlers group is generatedby 9,45 e 3 (∆) = 0 , e 3 (B) = −1h 3 (∆) = −2∆ , h 3 (B) = −2Bf 3 (∆) = 2∆B , f 3 (B) = B 2 − ∆ 2 (24)b If the Weyl coordinate ρ is taken to be spacelike, we would be dealing with a generalizationof the so-called Einstein-Rosen waves.