“Suppose we consider the following set of equations:

6.1 |

6.2 |

6.3 |

6.4 |

where and

We study these equations in order of difficulty. Eq.(6.1) has as its most general solution:

6.5 |

where has vanishing divergence and also vanishing curl:

6.6 |

but

6.7 |

This is only possible if “wormholes”

6.8 |

for every Killing vector (generating a group of motions) satisfying

6.9 |

Whether this has any significant applications I do not know. About Eq. (6.4) I can say nothing.

“l conjecture that the homogeneous equation could be solved by the use of potentials, provided we use a rather broad idea of a potential. By this I mean a solution which somehow involves the use of an arbitrary function. To illustrate the idea I give a particular class of solutions to the equation , namely:

6.10 |

where is an arbitrary function of the scalar curvature on the surface, and is the derivative of . is the covariant derivative.

“Besides the problems of topology in the small associated with “flares” and wormholes, or with the symmetric Einstein-Rosen

“Another point is this: Once you take the course indicated here and decide to make a unified field theory

Fig. 6.1

From the work of ^{1} one can show that there are four-dimensional manifolds which have these two surfaces as boundaries. The question whether a continuous metric of Lorentz signature can be put on such manifolds can, I believe, be answered. In such a metric can you find solutions to any hyperbolic equations, or better, Maxwell’s equations? Anyone trying to study this would be much aided by certain qualitative classifications of metrics.

MISNER ^{2} and others: “Instead of the string of components of the Ricci tensor, one looks at all the invariants which can be constructed from these. One possible use of these might be as variables in discussing the quantization of general relativity, since quantization seems to handle invariant quantities better than tensor components. Also it would be interesting to see *Gedankenexperiment*

BERGMANN

MME. FOURES