Abstract
In General Relativity in Hamiltonian form, change has seemed to be missing, defined only asymptotically, or otherwise obscured at best, because the Hamiltonian is a sum of first-class constraints and a boundary term and thus supposedly generates gauge transformations. By construing change as essential time dependence (lack of a time-like Killing vector field), one can find change locally in vacuum GR in the Hamiltonian formulation just where it should be. But what if spinors are present? This paper is motivated by the tendency in space-time philosophy tends to slight fermionic/spinorial matter, the tendency in Hamiltonian GR to misplace changes of time coordinate, and the tendency in treatments of the Einstein-Dirac equation to include a gratuitous local Lorentz gauge symmetry along with the physically significant coordinate freedom. Spatial dependence is dropped in most of the paper, both restricting the physical situation and largely fixing the spatial coordinates. In the interest of including all and only the coordinate freedom, the Einstein-Dirac equation is investigated using the Schwinger time gauge and Kibble-Deser symmetric triad condition are employed as a 3+1 version of the DeWitt-Ogievetsky-Polubarinov nonlinear group realization formalism that dispenses with a tetrad and local Lorentz gauge freedom. Change is the lack of a time-like stronger-than-Killing field for which the Lie derivative of the metric-spinor complex vanishes. An appropriate 3+1-friendly form of the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first class-constraints, is shown to change the canonical Lagrangian by a total derivative, implying the preservation of Hamilton’s equations. Given the essential presence of second-class constraints with spinors and their lack of resemblance to a gauge theory (unlike, say, massive photons), it is useful to have an explicit physically interesting example. This gauge generator implements changes of time coordinate for solutions of the equations of motion, showing that the gauge generator makes sense even with spinors.
Highlights
1.1 Hamiltonian Change Seems Missing but Lagrangian Change is NotIt has been argued that General Relativity, at least in Hamiltonian form, lacks change, has change only asymptotically and only for certain topologies, or appears to lack change with no clear answer in sight (e.g., [1,2,3,4,5,6])
6 Role of Second‐Class Constraints in Gauge Generator G?. This example might be a useful one to examine in the context of the 1990s controversy about what role second-class constraints might play in the gauge generator
We prove that the secondclass constraints do not contribute to the local-symmetry transformation law and, the transformation generator is a linear combination of only the firstclass constraints. [95]
Summary
It has been argued that General Relativity, at least in Hamiltonian form, lacks change, has change only asymptotically and only for certain topologies, or appears to lack change with no clear answer in sight (e.g., [1,2,3,4,5,6]). During the later 1950s–60s it became common in Hamiltonian GR to discard a spatio-temporal viewpoint that retained the freedom to change time coordinates [30,31,32], obscuring one of the most conceptually interesting features of GR; this bargain was viewed as an aid to quantization. This paper, aims to think foundationally about spinors, to include freedom to change time coordinates and more of a spatio-temporal view point than much work in Hamiltonian GR, and to avoid gratuitous gauge symmetry from the tetrad by making use of nonlinear group realizations, including all the physically significant gauge freedom (including changes of time coordinates) and no physically insignificant gauge freedom ( excluding/fixing every part of the tetrad that carries more information than the metric). If the time-space components of the metric (the ADM shift vector) disappear from the formalism altogether, as happens in the spatially homogeneous toy theory studied here, the distinction between the two nonlinear formalisms disappears
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