Change in Hamiltonian General Relativity from the Lack of a Time-like Killing Vector Field

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. Attention to the gauge generator G of Rosenfeld, Anderson, Bergmann, Castellani et al., a specially tuned sum of first-class constraints, facilitates seeing that a solitary first-class constraint in fact generates not a gauge transformation, but a bad physical change in electromagnetism (changing E) or GR. The change spoils the Lagrangian constraints in terms of the physically relevant velocities rather than auxiliary canonical momenta. While Maudlin has defended change in GR much as G. E. Moore resisted skepticism, there remains a need to exhibit the technical flaws in the argument. Insistence on Hamiltonian-Lagrangian equivalence, a theme emphasized by Mukunda, Castellani, Sugano, Pons, Salisbury, Shepley and Sundermeyer among others, holds the key. Taking objective change to be ineliminable time dependence, there is change in vacuum GR just in case there is no time-like vector field satisfying Killing's equation. Throwing away the spatial dependence of GR for convenience, one finds that the time evolution from Hamilton's equations is real change just when there is no time-like Killing vector. Hence change is real and local even in the Hamiltonian formalism. The considerations here resolve the Earman-Maudlin standoff: the Hamiltonian formalism is helpful, and, suitably reformed, it does not have absurd consequences for change. Hence the classical problem of time is resolved, apart from the issue of observables, for which the solution is outlined. The quantum problem of time, however, is not automatically resolved due to issues of quantum constraint imposition.