Abstract
We present the state-of-the-art concerning the relativistic constraints, which describe the geometry of hypersurfaces in a spacetime subject to the Einstein field equations. We review a variety of solvability results, the construction of several classes of solutions of special relevance and place results in the broader context of mathematical general relativity. Apart from providing an overview of the subject, this paper includes a selection of open questions, as well as a few complements to some significant contributions in the literature.
Highlights
Let ðL1þn; cÞ be a Lorentzian manifold of dimension n þ 1 ! 2 and signature ðÀ þ|fflfflfflÁ{ÁzÁfflfflþffl}Þ n slots solving the Einstein field equationsRicc À 1 Rcc þ Kc 1⁄4 jT: 2 ð1:1ÞHere, we let Ricc denote the Ricci curvature of c, Rc its scalar curvature and we employ the letter j for a positive constant, whose value depends on the specific conventions one adopts
When n 1⁄4 3 and adopting ‘nongeometrised units’ one would find j 1⁄4 ð8pG=c2Þ as could be checked by studying the so-called Newtonian limit of (1.1), see e.g., Misner et al (1973) or Wald (1984). As it is rather customary in the literature, T stands for the stress-energy tensor of the sources while K 2 R stands for a cosmological constant
As encoded by a notion of distance between points on M, is determined by the so-called first fundamental form g
Summary
Let ðL1þn; cÞ be a Lorentzian manifold of dimension n þ 1 ! 2 and signature ðÀ þ|fflfflfflÁ{ÁzÁfflfflþffl}Þ n slots solving the Einstein field equations. The extrinsic geometry of M, so they way M is bent inside L, is described by the so-called second fundamental form k It is a well-known fact that the triple (M, g, k) solves a system of equations that takes the form. If we let V be a timelike unit normal vector field to M we have set l 1⁄4 TðV; VÞ and J 1⁄4 TðV; ÁÞ, where it is understood that the second slot is allocated for tangent vectors to M These equations are obtained by combining (1.1) with the Gauss and Codazzi equations for the submanifold M (see e.g., O’Neill 1983 or Petersen 2006) as we will anyway review below in Sect. Before presenting a more detailed outline of the contents of this survey, let us digress on the way these equations are derived, on some heuristics behind them, and introduce a list of basic questions that may guide the reader through our discussion
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