The Cauchy Problem on a Characteristic Cone for the Einstein Equations in Arbitrary Dimensions

Abstract

We derive explicit formulae for a set of constraints for the Einstein equations on a null hypersurface, in arbitrary dimensions. We solve these constraints and show that they provide necessary and sufficient conditions so that a spacetime solution of the Cauchy problem on a characteristic cone for the hyperbolic system of the reduced Einstein equations in wave-map gauge also satisfies the full Einstein equations. We prove a geometric uniqueness theorem for this Cauchy problem in the vacuum case.

Conformally Covariant Systems of Wave Equations and their Equivalence to Einstein’s Field Equations

“…That is the reason why we cannot immediately deduce H σ = 0 as in [5] supposing that this is initially the case. Given two covariant derivative operators ∇ and∇ (associated to the metrics g andĝ, respectively), there exists a tensor field C σ µν = C σ νµ , which depends on g, ∂g,ĝ and ∂ĝ, such that…”

“…1 ≡ r > 0 parameterizes the null rays emanating from i − , and x A , A = 2, 3, are local coordinates on the level sets {r = const, u = 0} ∼ = S 2 (note that these coordinates are singular at the tip, see [5] for more details). First we shall sketch how the constraint equations are obtained in a generalized wave-map gauge with arbitrary gauge functions.…”

“…It should be emphasized that, on a light-cone, the initial data for the ODEs cannot be specified freely but follow from regularity conditions at the vertex. For sufficiently regular gauges the behaviour of the relevant fields near the vertex is computed in [5]. When stating this behaviour below we shall always tacitly assume that the gauge is sufficiently regular.…”

“…In the following we shall frequently make use of the formulae (A.8)-(A.25) in [5] for the Christoffel symbols computed in adapted null coordinates on a cone.…”

“…(note that H 0 = 0 implies κ = Γ 1 11 [5]). Differentiating (4.10) and inserting the result into (4.9) we obtain an equation for τ ,…”