I write for the space-time manifold

7.1 |

where I take as the basis in my four-dimensional manifold one vector orthogonal to the initial hypersurface, so that

7.2 |

7.3 |

Then the differential relations for the field can be written in the form

7.4 |

7.5 |

where
is the covariant derivative, and
is what Lichnerowicz

7.6 |

The coefficients are the coefficients of the second fundamental quadratic form of the hypersurface .

We are dealing with the purely gravitational case. To solve the differential equations I choose on the hypersurface three particular vectors, the eigenvector of the tensor , that is to say we will have

7.7 |

, , are unknowns. The metric of the hypersurface is

7.8 |

If I set
, *et cycl.*, and if I use Lichnerowicz’s

7.9 |

then I obtain the following system of equations:

7.10 |

7.11 |

where
is the Beltrami operator

7.12 |

“It is possible to solve this system of equations which are linear with respect to the highest derivatives. For instance, I can solve them by giving the values of the unknown
in my three-dimensional space
on a two-dimensional variety
, and I can solve then the Cauchy

WHEELER

MME. FOURES,

DE WITT asked if the Cauchy

MISNER ^{1} On the surface
you put

7.13 |

and being distances measured from two points in space. Then you assign initial conditions for the eigenvalues . You want to specify in such a way that the velocities of the two particles are initially like that:

Then you try to use the same method which was used for the general proofs to find particular solutions not just on
but through all space. These can be interpreted as two particles which are non-singular, or they can be thought of as the kind of
type singularity of which one ordinarily thinks in gravitational theory. These partial differential equations, although very difficult, can then in principle be put on a computer.”

DE WITT pointed out some difficulties encountered in high speed computational techniques. “Singularities are of course difficult to handle. Secondly, any non-linear hydrodynamic

BERGMANN

ANDERSON

Discussion of these points, which are closely related with the problem of quantization, was postponed until later.

WHEELER

“I would like to raise in this connection the question of the use of the variational principle itself. The variational principle in general relativity

7.14 |

How do we tell what to fix at the endpoints of the time interval? We do this by performing an integration by parts:

7.15 |

From this we draw two separate conclusions:

**1**.

**2** at the two endpoints of the motion, i.e., we must specify the coordinate
at the times 1 and 2. This is a time-symmetric specification. One is so accustomed to this pattern of thinking that when general relativity presents us with a different pattern we are not prepared for that. Therefore, let me introduce a problem which does have the pattern characteristic of general relativity, and let us see how we would face up to it.
Consider this:

7.16 |

Traditionally one abhors these second derivatives in the problem, so one quickly transforms it into the previous problem. I propose to deal rather directly with the new problem like this:

7.17 |

The conclusions would now be:

**1**.

**2**,
i.e., we are to specify the value of
at the two endpoints of the motion.

“In other words, we are led here to a different kind of specification of the problem. I merely want to point out that the variational problem itself has the property if we don’t monkey with it, to tell us what it is that we should deal with.”

“In the case of electromagnetism we can also let the variational principle tell us what the quantity is that we are concerned with. Note the equation:^{2}

7.18 |

The volume integral gives us the usual field equations. The boundary term tells us that we should specify
at the limits. If you specify
itself, rather than its transverse part, you would be taking too seriously the demand that the boundary term must vanish. Clearly you don’t have to be that hard on
. Analogous considerations can and have been applied to the general relativity by Misner

MISNER

7.19 |

From this one gets the surface term of form:

The suggestion is that if you look at the surface term carefully you will find what the correct variables to use are, and what you can specify on the initial and final surface.”

MISNER *problem of the initial values*, to a smaller computer first, before you start on what Lichnerowicz *evolutionary problem*. The small computer would prepare the initial conditions for the big one. Then the theory, while not guaranteeing solutions for the whole future, says that it will be some finite time before anything blows up.”

LICHNEROWICZ