Pitts, J. Brian (2013) 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 et al., a specially _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. But the resemblance between the gauge generator G and the Hamiltonian H leaves still unclear where objective change is in GR.
Insistence on HamiltonianLagrangian equivalence, a theme emphasized by 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^a satisfying Killing's equation L_xi g_mn=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 and observables. Hence the classical problem of time is resolved. 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:  14 Nov 2013 15:22  
Last Modified:  31 May 2014 04:41  
Item ID:  10094  
Subjects:  Specific Sciences > Physics > Cosmology Specific Sciences > Physics > Relativity Theory Specific Sciences > Physics > Symmetries/Invariances 

Date:  October 2013  
URI:  https://philsciarchive.pitt.edu/id/eprint/10094 
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 Change in Hamiltonian General Relativity from the Lack of a Timelike Killing Vector Field. (deposited 14 Nov 2013 15:22) [Currently Displayed]
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