Einstein equations

[Note: This is an immense topic, and we do not even begin to do it justice with this very brief selection of results. Further references or expansion of this article will, of course, be very much appreciated.]

The (vacuum) Einstein equations take the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R_{ \alpha \beta } = C R g_{ \alpha \beta }}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g} is the metric for a 3+1-dimensional manifold, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R} is the Ricci curvature tensor, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C} is an absolute constant. The Cauchy data for this problem is thus a three-dimensional Riemannian manifold together with the second fundamental form of this manifold (roughly speaking, this is like the initial position and initial velocity for the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g} ). However, these two quantities are not completely independent; they must obey certain constraint equations. These equations are now known to be well behaved for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 3/2} Max-p, Max2005 (see also earlier work in higher regularities in RenFri2000, Ren2002).

Because of the diffeomorphism invariance of the Einstein equations, these equations are not hyperbolic as stated. However, this can be remedied by choosing an appropriate choice of co-ordinate system (which is the analog of a gauge transformation in gauge theory). One popular choice is harmonic co-ordinates or wave co-ordinates, where the co-ordinate functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x^a } are assumed to obey the wave equation $\Box _{g}x^{a}=0$ with respect to the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g} . In this case the Einstein equations take a form which (in gross caricature) looks something like

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Box_g g = \Gamma (g) Q(dg, dg) + } lower order terms

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Q} is some quadratic form of the first two derivatives. In other words, it becomes a quasilinear wave equation. One would then specify initial data on the initial surface Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x = 0} ; the co-ordinate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} plays the role of time, locally at least.

• The critical regularity is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s_c = 3/2} . Thus energy is super-critical, which seems to make a large data global theory extremely difficult.
• LWP is known in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^s} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 5/2} by energy estimates (see HuKaMar1977, AnMc-p; for smooth data Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s > 4} this is in Cq1952) - given that the initial data obeys the constraint equations, of course.
  • This result can be improved to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s>2} by the recent quasilinear theory (see in particular KlRo-p3, KlRo-p4, KlRo-p5).
  • This result has now been improved further to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s=2} (KlRo-p6, KlRo-p7, KlRo-p8).
  • For smooth data, one has a (possibly geodesically incomplete) maximal Cauchy development (CqGc1969).
• GWP for small smooth asymptotically flat data was shown in CdKl1993 (see also CdKl1990). In other words, Minkowski space is stable.
  • Another proof using the double null foliation is in KlNi2003, KlNi-p
  • Another proof of this fact (using the Lorenz gauge, and assuming Schwarzschild metric outside of a compact set) is in LbRo-p (see also LbRo2003 for a treatment of the asymptotic dynamics)
  • Singularities must form if there is a trapped surface (Pn1965).
• Many special solutions (Schwarzschild space, Kerr space, etc.) The stability of these spaces is a very interesting (and difficult) question.
• The equations can simplify under additional symmetry assumptions. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U(1)} -symmetric case reduces to a system of equations which closely resembles the two-dimensional wave maps equation (with the target manifold being hyperbolic space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle H^2} ).

Open problems

• Cosmic Censorship Hypothesis

• BKL Conjecture

Further reading

• For more detail, we recommend the very nice survey on existence and global dynamics of the Einstein equations by Alan Rendall.

References

[Max-p] D. Maxwell. Rough solutions of the Einstein constraint equations. J. Reine Angew. Math., 590 (2006), 1-29. MathSciNet, arXiv.

[Max2005] D. Maxwell. Rough solutions of the Einstein constraint equations on compact manifolds. J. Hyperbolic Diff. Eq., 2 (2005), 521-546. MathSciNet, arXiv.

[RenFri2000] A. Rendall and H. Friedrich. The Cauchy problem for the Einstein equations. In B. G. Schmidt (eds.) Einstein's Field Equations and Their Physical Implications. Lecture Notes in Physics 540 Springer Berlin (2000). MathSciNet, arXiv.

[Ren2002] A. Rendall. Theorems on existence and global dynamics for the Einstein equations. Living Rev. Relativ., 8 (2005), 6. MathSciNet, arXiv.