Pitts, J. Brian (2014) Change in Hamiltonian General Relativity from the Lack of a Timelike Killing Vector Field. [Preprint]
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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 firstclass constraints and a boundary term and thus supposedly generates gauge transformations. Attention to the gauge generator $G$ of Rosenfeld, Anderson, Bergmann, Castellani \emph{et al.}, a specially \emph{tuned sum} of firstclass constraints, facilitates seeing that a solitary firstclass constraint in fact generates not a gauge transformation, but a bad physical change in electromagnetism (changing the electric field) or General Relativity. The change spoils the Lagrangian constraints, Gauss's law or the GaussCodazzi relations describing embedding of space into spacetime, in terms of the physically relevant velocities rather than auxiliary canonical momenta. While Maudlin and Healey have defended change in GR much as G. E. Moore resisted skepticism, there remains a need to exhibit the technical flaws in the nochange argument.
Insistence on HamiltonianLagrangian 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, one recalls that there is change in vacuum GR just in case there is no timelike vector field xi^{alpha} satisfying Killing's equation pounds_{xi}g_{mu\nu}=0, because then there exists no coordinate system such that everything is independent of time. Throwing away the spatial dependence of GR for convenience, one finds explicitly that the time evolution from Hamilton's equations is real change just when there is no timelike Killing vector. The inclusion of a massive scalar field is simple. No obstruction is expected in including spatial dependence and coupling more general matter fields. Hence change is real and local even in the Hamiltonian formalism.
The considerations here resolve the EarmanMaudlin standoff over change in Hamiltonian General Relativity: 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 Lagrangianequivalent Hamiltonian analysis of change in General Relativity is compared to Belot and Earman's treatment. The more serious quantum problem of time, however, is not automatically resolved due to issues of quantum constraint imposition.
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Item Type:  Preprint  

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Keywords:  constrained Hamiltonian dynamics, General Relativity, problem of time, quantum gravity, variational principles  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Depositing User:  Dr. Dr. J. Brian Pitts  
Date Deposited:  31 May 2014 04:41  
Last Modified:  31 May 2014 04:41  
Item ID:  10699  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Date:  May 2014  
URI:  http://philsciarchive.pitt.edu/id/eprint/10699 
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Change in Hamiltonian General Relativity from the Lack of a Timelike Killing Vector Field. (deposited 14 Nov 2013 15:22)
 Change in Hamiltonian General Relativity from the Lack of a Timelike Killing Vector Field. (deposited 31 May 2014 04:41) [Currently Displayed]
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