So last week, I presented a talk on the Cauchy Problem in General relativity, at the science student research showcase run by my university. Today I wanted to do a brief overview of the problem and the developments made in it since Einstein published his theory of General Relativity.

The Einstein equations are given, in tensor notation by

(1)   \begin{equation*} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi T_{\mu\nu} \end{equation*}

where

  • R_{\mu\nu} is the Ricci curvature tensor of the metric g
  • R is the Ricci Scalar defined as R:=R_{\mu\nu}g^{\mu\nu}
  • T_{\mu\nu} is the energy momentum tensor of matter

This is a set of 10 partial differential equations which can be solved for the metric, which tells you how your spacetime is curved. The idea is that we want to be able to solve this set of partial differential equations, and find what our metric is, given a some initial data. We will limit ourselves to the vacuum Einstein equations, where we set T_{\mu\nu}=0 and 1 reduces to

(2)   \begin{equation*} R_{\mu\nu}=0. \end{equation*}

Now, the Cauchy problem in general relativity is that given some initial data set, can we find a stable solution to 2. Now what do I mean by an initial data set?

  • A metric, \bar{g}
  • The induced second fundamental form of g, K
  • Both defined on a 3-manifold (hypersurface), \Sigma

The initial values that we require to solve for a metric will be some

Furthermore, these initial conditions must satisfy the vacuum  constraint equations

    \[\bar{R}+|{K}^2|+(tr(K))^2=0\]

and

    \[\nabla^jK_{ij}-\nabla_{i}tr(K)=0.\]

These constraint equations ensure that \bar{g} and K form the induced metric and second fundamental form of g, and come from the Gauss and Codazzi equations.

So what are the milestones reached in addressing this problem of solving these equations?

  • 1920, de Donder discovered the harmonic Gauge
  • 1935, Schauder, Existence and Uniqueness of the Cauchy Problem of the Harmonic Gauge
  • Stellmacher built on this to prove uniqueness of the Cauchy Problem
  • 1958, Choquet-Bruhat, Local Existence
  • 1969, Choquet-Bruhat and Geroch, Global Existence

Now let’s look at each of these developments in detail.

 

The Harmonic Gauge

 

The Harmonic gauge or a harmonic co-ordinate system is one in which each of the co-ordinate functions must satisfy the wave equation,

(3)   \begin{equation*} \Box x^{\alpha}=0, \end{equation*}

where this is the box operator. In this harmonic co-ordinate system, the Einstein Vacuum equations reduce to

(4)   \begin{equation*} \Box_{g}g_{\mu\nu}=N_{\mu\nu}(g,\partial g) , \end{equation*}

where \Box_g=g^{\mu\nu}\partial^2_{\mu\nu}. This is a system of quasilinear wave equations, which is an easier problem to look at, then the partial differential equations present in the Einstein equations.
In fact, the existence and uniqueness of solutions for the Cauchy problem of this equation was proved by Schauder in 1935. Then, only using this, Stellmacher proved uniqueness for the Einstein Vacuum Equations. But to prove existence, it is not enough to have that solutions exist to the reduced Einstein equations, and this leads us to the work of Choquet-Bruhat in the 1950’s and the theorems that she proved.

Local Existence

 

Now we ask, when do solutions of the reduced Einstein equations give solutions of the Einstein equations?
The answer is when the solution also satisfies the harmonic gauge condition.
However, Choquet-Bruhat found that the harmonic gauge is automatically satisfied given that it is satisfied on \Sigma and that the initial conditions satisfy the constraint equations. This means choosing a local co-ordinate system that satisfies the harmonic gauge will also give us solutions to the Einstein Equations.
Now this was just a local existence result, in general though we would like to be able to talk about a maximal solution, that is true over the largest possible domain.

Global Existence

 

And She did that along with Geroch in 1969, and she proved that for some initial data set there exists a unique solution, that is maximal, in the sense that we mentioned before, i.e. any globally hyperbolic development of the same initial data can be extended to this maximal one. Now globally hyperbolic is a geometric quality that is required to ensure that our solutions are domain independent and so is then necessary in order to prove uniqueness of solutions. In terms of geometry, globally hyperbolic means that any point in spacetime admits a Cauchy surface.

Now this maximal solution is called the Maximal Cauchy development of the data set (\Sigma, \bar{g}, K), and is a central object in the study of general relativity. It allows us to formulate the fundamental problem of solving these Einstein equations as one of dynamics. Recall from Newtonian physics, the question is that given an initial position and velocity, what is the future? Now we ask, given the initial data set, what is the future? Where the future is manifested as this maximal Cauchy development.

The PDF Slides of my talk can be downloaded form here

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