# Accession Number:

## AD0628329

# Title:

## THE NULL BUNDLE OF AN EINSTEIN-RIEMANN SPACE II: NORMED NULL BASES,

# Descriptive Note:

# Corporate Author:

## RAND CORP SANTA MONICA CALIF

# Personal Author(s):

# Report Date:

## 1966-02-01

# Pagination or Media Count:

## 49.0

# Abstract:

This memorandum presents the theory of normed null bases in a hyperbolic-normal, 4-dimentional metric space E i.e., the structure and properties of all systems of four linearly independent, complex-valued, null vector fields that satisfy the given normalization conditions. It is shown that the system of all normed null bases can be obtained from a given normed null basis by the action of a group of anholonomic matrices that are generated from the group of all Jacobian matrices of the Lorentz group under all mappings of the group parameters onto functions of position in E. The anholonomic frames, anholonomic connections, and objects of anholonomy arising from normed null bases are studied and a fundamental existence theorem is obtained. This existence theorem replaces all equations involving the anholonomic quantities of an unknown normed null basis by equations involving a known normed null basis and the matrices described above. These results are applied to the systems of partial differential equations obtained in a previous report, as well as to the general Einstein equations. Explicit solutions are obtained, for purposes of illustration, when a particular form of the C-matrices is assumed. These solutions represent among other things, the gravitational fields of a rod, of two parallel rods, and of two photons, and raise some interesting questions concerning gravitational radiation. Author

# Descriptors:

# Subject Categories:

- Theoretical Mathematics
- Quantum Theory and Relativity