Royal Society Publishing

The Theory of Gravitation in Hamiltonian Form

P. A. M. Dirac

Abstract

The author's generalized procedure for putting a theory into Hamiltonian form is applied to Einstein's theory of gravitation. It is shown that one can make a change in the action density, not affecting the equations of motion, which causes four of the ten degrees of freedom associated with the ten g$_{\mu \nu}$ to drop out of the Hamiltonian formalism. This simplification can be achieved only at the expense of abandoning four-dimensional symmetry. In the weak field approximation one can make a Fourier resolution of the field quantities, and one then gets a clean separation of those degrees of freedom whose variables depend on the system of co-ordinates from those whose variables do not. There are four of the former and two of the latter for each Fourier component. The two latter correspond to gravitational waves with two independent states of polarization. One of the four former is responsible for the Newtonian attraction between masses and also gives a negative gravitational self-energy for each mass.

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