5 On the Integration of the Einstein Equations

This is a survey of the main results on the integration of the Einstein equations and the problems which remain open.

5.1 The Space-Time Manifold

In any relativistic theory of the gravitational field, the main element is a differentiable manifold of four dimensions, the space-time . To be precise: I assume that this differentiable structure is “ , piecewise .” This means that in the intersection of the domains of two admissible systems of local coordinates the local coordinates of a point for one system are functions of class , i.e., possess continuous derivatives up to second order, with non-vanishing Jacobian, of the coordinates of the point in the other system. Third and fourth derivatives also exist, but are only piecewise continuous.

On we have a hyperbolic normal Riemannian metric

which is everywhere “ , piecewise .” This metric is said to be regular on . Note that the manifolds which admit hyperbolic metrics are precisely those on which there exist vector fields without zeros. Such a manifold admits global systems of time-like curves, but generally does not admit global systems of space-like hypersurfaces.

We suppose that no further specification of the differentiable structure of the manifold or of the metric has any physical meaning.

For the study of global problems assumptions belonging to the differential geometry in the large are often needed. Frequently is identified with a topological product , being the real numbers, where straight lines are time-like curves and thus trajectories of a unitary vector field . The space sections defined by are then generally either closed manifolds or complete Riemannian manifolds for the metric defined on them by

moreover, in this case admits a Minkowskian asymptotic behavior at infinity.

The problem whether these assumptions are necessary or sufficient is open.

5.2 The Cauchy Problem for the Gravitational Field

In general relativity the metric satisfies the Einstein equations:

where the energy-momentum tensor is piecewise continuous. It defines the sources of the field. In the regions where we have the “exterior case.”

Let be a local hypersurface of which is not tangent to the cones . If defines locally, we have . The Einstein equations can then be written as¹

where and are known functions on if we know the values of the potentials and their first derivatives (“Cauchy data”).

Eq.(5.5) links the Cauchy data to the sources. Eq.(5.4) gives the values on of the six second derivatives which I have called the significant derivatives. The four derivatives remain unknown and may be discontinuous at . But, according to our differentiable structure, these discontinuities have no physical or geometrical meaning. We grasp here the connection between the covariance of the formalism and the assumptions on the structure.

From Eq.(5.4) one sees that the significant derivatives for can have discontinuities only if is tangent to the fundamental cones, or if the energy-momentum tensor is discontinuous at . Gravitational waves are defined as the hypersurfaces , tangent to the fundamental cone, i.e., such that

(WHEELER pointed out, as a matter of terminology that what is called “gravitational wave” by Lichnerowicz is a purely geometrical concept, like the eikonal in optics, and not the same as the physical “gravitational waves.” The question whether gravitational radiation does or does not exist was aired thoroughly in later sessions.) Gravitational rays are defined as the characteristics of , or as the singular geodesics of the metric.

In the hydrodynamic interior case the exceptional hypersurfaces are (1) the gravitational waves, (2) the manifolds generated by streamlines, and (3) the hydrodynamic waves.

5.3 Local Solutions for the Exterior Case

In the exterior case, , the system of Einstein equations has the involution property: If a metric satisfies Eq.(5.5) and, on only, the equations , then it satisfies these equations also outside of . This is a trivial consequence of the conservation identities. Thus, for a space-like hypersurface the Cauchy problem leads to two different problems:

1The search for Cauchy data satisfying the system on . This is the problem of the initial values.

2The evolutionary problem of integrating Eq.(5.4) subject to these Cauchy data.

Assuming only differentiability, Mme. Fourès has recently solved this second problem locally by considering a difficult system of partial differential equations. She uses isothermal coordinates (for which ) and thus proves the local existence and physical uniqueness theorem for the solution of the Cauchy problem of the Einstein equations. The characteristic conoid generated by the singular geodesics which issue from a point plays the essential part, and it is clear that the values of the solution in depend only on the values of the Cauchy data in the part of interior to the characteristic conoid. We obtain thus all the results of a classical wave propagation theory.

5.4 The Problem of the Initial Values

This, most interesting, problem is simple if is a minimal hypersurface. If

we take

and we obtain for the exterior case

where is the operator of covariant differentiation, and the scalar curvature of . If we set

where we assume the metric to be known on , then it is possible to introduce the functions

For these functions a very simple system of equations holds:

and the elliptic equation:

The Dirichlet problem for Eq.(5.5) admits solutions if the domain of is sufficiently small or if and if is sufficiently small. This study can be extended to some hydrodynamic interior cases, and it is thus possible to construct examples of Cauchy data corresponding to the motion of bodies. For the two-body problem, Newton’s law is obtained approximately.

Recently Mme. Fourès has extended my method to the general case.

It would be important to obtain new global theorems for this problem, currently perhaps the most important problem of the theory.

5.5 The Cauchy Problem for the Asymmetric Theory

The Cauchy problem of the asymmetric theory was studied in collaboration with Mme. Maurer.

On the differentiable manifold we have:

1An asymmetric tensor field of class “ , piecewise ,” with a non-vanishing determinant , the associated quadratic form defined by being hyperbolic normal.

2An affine connection of class “ , piecewise ;” is the torsion vector of the connection.

If we substitute into this connection the connection without torsion vector, which admits the same parallelism, the field equations deduced from the classical variational principle for the first connection give two partial systems. According to Mme. Tonnelat and Hlavatý, the first system gives the connection from and the first derivatives of the tensor. The field is now defined by which satisfy the equations

where is the Ricci tensor of . In addition we have a normalization condition for :

The Cauchy problem can now be investigated. The system obtained still possesses the involution property. The field waves are hypersurfaces tangent to one of the following two cones:

• (a): If we have the cone

where is dual to .

• (b): If we have the cone

where is dual to .

If the skewsymmetric part of is small compared to the symmetric part, then contains . itself has no wave properties in the unified field theory.

The evolutionary problem of the asymmetric theory remains unsolved.

5.6 Global Solutions and Universes

I now return to general relativity. The main question of the theory is the following: When is a gravitation problem effectively solved?

A model of the universe - or shortly, “a universe” - is a with a regular metric satisfying the Einstein equations and certain asymptotic conditions. When is discontinuous through a hypersurface we assume that is always “ , piecewise ” in the neighborhood of . The joining of the different interior material fields with the same field causes the interdependence of the motions, and the classical equations of motion are due to the continuity through of the four quantities

In this view the main problem is to construct and study universes. This being a hyperbolic non-linear global problem, it is very difficult to do this. Clearly global solutions of the problem of the initial values would be very helpful here.

Another approach might be the study of some elementary global solutions of the Einstein equations. It appears that such solutions are connected with the solutions with singularities introduced by various authors.

5.7 Global Problems

This view of the universe leads us to the following question: Is it possible to introduce in a universe new energy distributions, the interior fields of which are consistent with the exterior field of the universe? Furthermore, if a universe model is defined by a purely exterior field, regular everywhere, is the universe empty and thus locally flat?

The regularity problem is the study of the exterior fields regular everywhere under some general topological or geometrical assumptions. Good results are known only in the stationary case. We assume , and we use the explicit assumptions made at the beginning of this survey. Then we have the following results:

An exterior stationary field which is regular everywhere is locally flat

(a) if is compact, or

(b) if is complete and has a Minkowskian asymptotic behavior.

In a stationary universe the exterior field extended through continuity of the second derivative into the interior of the bodies is singular in the interior.

A stationary universe which admits a domain surrounding infinity and which in this domain has a Minkowskian asymptotic behavior and for which the streamlines are time lines, is static. There exist space sections orthogonal to the time lines. This is of interest in the Schwarzschild theory.

Much less is known in the non-stationary case, which is of course more interesting. Concerning the regularity problem, some counterexamples have been constructed, especially by Racine, Taub and Bonner. But for these examples either the existence of the solutions is certain only in a finite interval of time, or the constructed solutions do not behave Minkowskian asymptotically. The general regularity problem is still an open one.

However, the results on the problem of the initial values give new theorems under the following assumptions:

It is possible to diagonalize the matrix which defines the second fundamental quadratic form of and there exist foliations of by the corresponding system of curves, that is to say by curvature lines.

has a flat asymptotic behavior.

The tensor has a summable square on .

If these assumptions are satisfied on and also on the sections corresponding to arbitrarily small , the universe is static and thus locally flat.

***

MISNER first stressed the physical ideas in which he and Wheeler were interested and which had led them to inquire into the details of the Einstein-Maxwell equations, particularly whether or not they are singular.

“This comes about because, according to work of Rainich,² the Einstein-Maxwell equations can be interpreted as an already unified field theory. Explicit mention of the electromagnetic field is not necessary even while working with a system equivalent in all cases, except the null electromagnetic field, to the Einstein-Maxwell set of equations. One aim of unified field theory has always been the notion that fields are more fundamental than particles, and that it should be possible to construct all particles from the purely geometrical concept of the field. To allow singularities in the fields to represent the particles would be a delusion, since then the stress tensor is not merely the electromagnetic one but includes mass terms, even though idealized to delta functions. Since we wish to be careful about singularities the results of Mme. Fourès and Lichnerowicz³ have been important to us. They assure us that if we specify certain initial conditions we shall have non-singular solutions at least for a short time. We are interested in finding solutions to these initial value equations and seeing what they lead to, whether they indicate any possibility of constructing particles from these electromagnetic and gravitational fields. Notice that even though one speaks of the electromagnetic field here, one does not really have to, since one can use the ideas of Rainich to give a complete description of the electromagnetic field in terms of metric quantities alone. The influence of the electromagnetic stress energy tensor upon the gravitational field is sufficiently specific so that from the curvature quantities one can work backwards to the electromagnetic field and have a purely metric description of both electromagnetism and gravitation.

“We have studied a certain class of solutions to the problem of initial values. These solutions are global but by no means the most general. They serve as examples. These solutions are always continuous and instantaneously static. We find that there exist exact solutions of the Einstein-Maxwell equations for which the fields on a certain surface are given by:

where satisfy

This metric together with the electric field gives a solution of the initial value equations. In the neighborhood of the surface we have a solution of the time-dependent equation but do not know what it is. Perhaps one should use a high speed computing machine. The regularity conditions of Lichnerowicz, that the three-dimensional manifold should be either compact or complete but asymptotically flat, lead to fields which are completely free of singularities. The functions and can only have the form:

where

These functions appear to contain singularities, but they really do not. This brings us into the study of the topology of this situation. The point at which the metric seems to become singular can be thrown away. This is consistent with the requirement of completeness, but would not be consistent if the point which is thrown away were one at which the metric did not become singular. A picture can be drawn to indicate what the space so obtained looks like. Near the point the space, which is flat further away, curves and flares out and goes over into another roughly flat portion like this:

We have here as many regions where the space is asymptotically flat as we have particles. The next problem is to put these portions together. One should then try to construct the initial values which would give rise to the wormhole picture of Wheeler (see Session 1). All these things are possible; they just have to be worked out.

“One additional advantage of the solutions exhibited here over introducing point singularities into the theory is that by using Lichnerowicz’s regularity conditions you eliminate automatically all negative masses, mass dipoles and higher multipoles. It also, unfortunately, implies that for any particle or cluster of particles described by these deformities the total charge and mass have a ratio , while for an electron in these units, where , . The reason for this too large mass, for a given electron charge, is that you don’t have enough of a cut-off for the electromagnetic mass. Perhaps one might hope that quantum mechanics might help here. In any event, we have here strong indications that the Einstein-Maxwell theory is a good model for a unified field theory. It can tell us what a unified field theory may have to say about physics.”

In the discussion, PIRANI asked for the definition of “completeness.” MISNER said that a manifold is complete if all geodesics can be continued to infinite length.

BERGMANN compared the present topology with the symmetric one of Einstein and Rosen.⁴

DE WITT noted the similarity between Misner’s model and the Schwarzschild solution for a charged mass particle. MISNER agreed and said that he had been led to his solutions by the observation that the space part of the Schwarzschild metric can be written in the form

MlSNER then added some remarks on global solutions.

Footnotes