Consistency of the Cauchy initial value problem in a nonsymmetric theory of gravitation

Abstract

The consistency of the Cauchy initial value problem in a theory of gravity based on a nonsymmetric metric is investigated. It is demonstrated that the dynamical equations preserve the constraint equations as the evolution of the initial data occurs. The consistency is shown rigorously using the full field equations rather than the series expansion used previously.

Dirac constraint analysis of a linearised theory of gravitation

“…Note that this analysis is consistent with a previous discussion of the Cauchy initial value problem in NGT (Moffat 1980, McDow andMoffat 1982), in which, with the gauge choice WO = Wt,, = 0 , a complete set of initial data, to nth order, was shown to be g:ij), g{$, g[$ and Win'. When this set of data is supplemented by the constraint g / z $ i =0, which is preserved in time, we see that there are precisely ten pieces of independent initial data.…”

“…The Lagrangian in equation ( 1) is known to describe a single dynamical degree of freedom, as can be seen either from a helicity analysis (Ogievetskii and Polubarinov 1967) of the Euler-Lagrange equations or from the application (Kaul 1978) of Dirac's canonical constraint formalism (Dirac 1964). Antisymmetric tensor fields also appear in the Lagrangian of Moffat's nonsymmetric theory of gravitation (NGT) (Moffat 1979(Moffat , 1982:…”

“…The Lagrangian of the full nonlinear non-symmetric theory of gravitation is (Moffat 1979(Moffat , 1982 (4) with…”

Massive NGT and spherically symmetric systems

“…This analysis correctly represents the asymptotic behavior of the fields (W, h), and is equivalent to equation ( 18) of [25], where the higher order pole resides in the projection operator: P (1 + ). One also sees the true propagating nature of W , and this is borne out by the analysis in [26,27] where there are five degrees of freedom evolving from each Cauchy surface, the extra two of which are associated with the field W . That a Lagrange multiplier is propagating merely signifies that it is a determined multiplier, with its evolution derived from the field equations [15] and not freely fixable as was done in [28,29] and in the next to last section of [24] where ad hoc constraints were imposed on the linearized theory in order to obtain the dynamics of a Kalb-Ramond theory.…”

“…The field equations will be written without the source terms for simplicity although it is straightforward to include them and relate the constants of integration to properties of the source. First reviewing how the symmetric (in this case identically GR) perturbations become static, it is simplest to begin with the field equation: 1 R (01) = 0, which implies: (27) immediately showing that h 11 must be static. By considering 1 R 22 = 0, it is determined to be:…”