Arthur Komar

The coordinate group symmetries of general relativity

The symmetry group of special relativistic theories, the Poincar6 group, was imposed on physical theories to mirror the symmetries of the laws of nature under point mappings of the presumed Minkowskian space-time, thought to be the arena of physics. With the advent of the general theory of relativity the equations of the gravitational field were constructed so as to be invariant under arbitrary curvilinear point transformations of the spacetime, now taken to be a four-dimensional pseudo-Riemannian manifold. Although the dynamical laws of all general relativistic theories are taken to have this enlarged symmetry group, the geometry of any particular spacetime on which all the fields are defined no longer has this symmetry. In fact, in order to facilitate handling &the field equations of general relativity, it is often convenient to exploit the lack of symmetry of generic space-times to impose coordinate conditions upon the field variables, the metric tensor of the space-time. The coordinate transformations leading to the preferred frames of reference in which the coordinate conditions are satisfied, or which preserve those conditions, in so far as they involve specific reference to the metric of the space-time, are best understood, not so much as point mappings within a given four dimensional space-time, but rather as mappings within the function space of the field variables of the theory, guv(x~). (Greek indices are taken to range from 0 to 3, while Latin indices range from 1 to 3.) The general theory of relativity is thus seen to have a much larger natural symmetry group than was initially contemplated, namely transformations of the form 2 ~ = f~(x~, gm(x~))

SPHERICALLY SYMMETRIC GRAVITATIONAL FIELDS

It is shown that the Schwarzschild solution is the only spherically symmetric solution of the Einstein vacuum field equations, even when the differentiability of the metric is weakened to the extent of permitting solutions which are C0, piecewise C1. Petrovs purported counterexample is analyzed and shown to be essentially equivalent to Schwarzschilds example.

The Quantization Program for General Relativity

The problem of the construction of a quantum theory of gravitation is attacked by a variety of methods. The fundamental epistemological difficulties are ellucidated and certain novel qualitative features of the sought-for quantum theory are described.

Qualitative Features of Quantized Gravitation

From a re-examination of a discussion by Bohr and Einstein of a gedanken experiment designed to violate the uncertainty relations, it is suggested that a qualitatively new feature of the merging quantum theory of gravitation is that it may provide a new understanding of the ‘reduction of the wave packet’ of quantum mechanics. In brief, for finitely massive observers, half of the classical dynamical variables must be employed to specify the frame of reference, whereas only the remaining half of the dynamical variables are available for unequivocal observation. Observers whose frames of reference cannot be related by definiteC-number transformations would, in general, ‘reduce wave packets’ differently.

Generalization of Weyl's gauge group

Weyl’s gauge transformations in a general n‐dimensional Riemannian manifold are extended from the conformal group to GLn(R). The gauge‐covariant field generalizing that of the Maxwell tensor is determined. The relationship between Weyl gauging and Yang–Mills gauging is developed. It is shown that the two processes are not equivalent, but can be made compatible.

GENERAL RELATIVISTIC THIN-SANDWICH THEOREM.

The configuration variables for the gravitational field gmn are assigned arbitrarily on two infinitesimally neighboring spacelike hypersurfaces. We then investigate the extent to which a solution of the vacuum Einstein field equations can be found consistent with the given assignment. A local approach, employing Diracs Hamiltonian formalism, reveals that solutions can be found locally which are nonunique and highly unstable.

Semantic Foundation of the Quantization Program

It is a well-known result of VON Neumann [1] that the representation of the commutation relations of the basic dynamical variables of quantum theory is unique modulo a unitary transformation. This result, valid only for systems having finite numbers of degrees of freedom, seems to imply that, for such mechanical systems, the procedure of quantization is not sensitive to the particular choice of classical canonical variables which one initially employs. This implication is in fact false. The choice of canonical variables which one employs to initiate a quantization of a given classical theory is quite critical.

Generalized Weyl-type gauge geometry

Weyl‐type gauge geometry based on gauging by GL(4, R) and presented in an earlier paper is developed from an alternative point of view, which emphasizes independent geometric objects as the building blocks. This approach avoids most of the elaborate calculations of the earlier paper, and thus contributes to an intuitive understanding. The new building blocks, which are the metric and two different affine connections, uniquely determine the tensorial structures of the earlier paper, and vice versa.

The General Relativistic Quantization Program

The field equations of the classical theory of gravitation, the Einstein Theory of General Relativity, admit solutions which can be interpreted as gravitational radiation. Experiments currently in progress seem to indicate the existence of such gravitational radiation. It would be remarkable if the energy transported by the gravitational waves were not emitted and absorbed in quanta in conformity with the well known Einstein relation, E=hv. Yet, to date, we do not have a satisfactory theory for the quantized transfer of gravitational energy whose classical limit corresponds to the generally accepted Einstein Theory of Gravitation.

The quantitative epistemological content of Bohr's Correspondence Principle

The basic dynamical quantities of classical mechanics, such as position, linear momentum, angular momentum and energy, obtain their fundamental epistomological content by means of their intimate relationship to the symmetries of the space-time manifold which is the arena of physics. The program of canonical quantization can be understood as a two stage process. The first stage is Bohrs Correspondence Principle, whereby the basic dynamical quantities of the quantum theory are required to retain precisely the same relationship to the symmetries of the space-time manifold as do their classical counterparts, thereby preserving their epistemological, as well as measurement-theoretic, significance. Having so identified the basic dynamical variables, functions of these may now be used to identify the subtler symmetries of the proper canonical group. The second and determining stage of the quantization program requires the establishment of a correspondence between some of these subtler symmetries of the classical theory and related symmetries of the quantum theory, the relationship being determined by a common algebraic form for their defining functions.