Change in Hamiltonian general relativity from the lack of a time-like Killing vector field

Pitts, J. Brian

doi:10.1016/j.shpsb.2014.05.007

Cited by 26 publications

(39 citation statements)

References 83 publications

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“…Thus observables change by a 4-dimensional Lie derivative, not 0, under coordinate transformations, which are generated by G for solutions of Hamilton's equations. Requiring merely covariance, not invariance, under external (coordinate) transformation laws matches a conclusion drawn previously by consideration of the classical origins and meaning of the Lie derivative, especially the transport term [45]. But this bifurcation raises the question whether mixed theories such as Einstein-Maxwell receive a consistent definition of observables.…”

supporting

confidence: 68%

“…But clearly reality, observability, and gauge invariance do not require sameness at different events, even if one gives them the same coordinate value in different coordinate systems (which one can always do). The changelessness of observables has arisen as a conclusion because it has been fed in as a premise through the (weakly) 0 Poisson bracket condition in cases where G generates a Lie derivative [45]. Thus the changelessness of observables is resolved by imposing a more suitable requirement on {O, G[ξ µ ]}, namely,…”

Section: Observables and Internal Vs External Gauge Symmetries Morementioning

confidence: 99%

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What Are Observables in Hamiltonian Einstein–Maxwell Theory?

Pitts

2019

Found Phys

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Is change missing in Hamiltonian Einstein-Maxwell theory? Given the most common definition of observables (having weakly vanishing Poisson bracket with each first-class constraint), observables are constants of the motion and nonlocal. Unfortunately this definition also implies that the observables for massive electromagnetism with gauge freedom (Stueckelberg) are inequivalent to those of massive electromagnetism without gauge freedom (Proca). The alternative Pons-Salisbury-Sundermeyer definition of observables, aiming for Hamiltonian-Lagrangian equivalence, uses the gauge generator G, a tuned sum of first-class constraints, rather than each first-class constraint separately, and implies equivalent observables for equivalent massive electromagnetisms.For General Relativity, G generates 4-dimensional Lie derivatives for solutions. The Lie derivative compares different space-time points with the same coordinate value in different coordinate systems, like 1 a.m. summer time vs. 1 a.m. standard time, so a vanishing Lie derivative implies constancy rather than covariance. Requiring equivalent observables for equivalent formulations of massive gravity confirms that G must generate the 4-dimensional Lie derivative (not 0) for observables.These separate results indicate that observables are invariant under internal gauge symmetries but covariant under external gauge symmetries, but can this bifurcated definition work for mixed theories such as Einstein-Maxwell theory? Pons, Salisbury and Shepley have studied G for Einstein-Yang-Mills. For Einstein-Maxwell, both F µν and g µν are invariant under electromagnetic gauge transformations and covariant (changing by a Lie derivative) under 4dimensional coordinate transformations. Using the bifurcated definition, these ⋆ Forthcoming in Foundations of Physics

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supporting

confidence: 68%

Section: Observables and Internal Vs External Gauge Symmetries Morementioning

confidence: 99%

What Are Observables in Hamiltonian Einstein–Maxwell Theory?

Pitts

2019

Found Phys

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“…Change and local spatial variation appear for observables appear once one defines observables using G and the Lie derivative, because observables on-shell are just geometric objects (at least infinitesimally). Change is missing only when and where there is a time-like Killing vector field [75].…”

Section: Application To General Relativitymentioning

confidence: 99%

Equivalent theories redefine Hamiltonian observables to exhibit change in general relativity

Pitts

2017

Class. Quantum Grav.

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Change and local spatial variation are missing in canonical General Relativity's observables as usually defined, an aspect of the problem of time. Definitions can be tested using equivalent formulations of a theory, non-gauge and gauge, because they must have equivalent observables and everything is observable in the non-gauge formulation. Taking an observable from the non-gauge formulation and finding the equivalent in the gauge formulation, one requires that the equivalent be an observable, thus constraining definitions. For massive photons, the de Broglie-Proca non-gauge formulation observable A µ is equivalent to the Stueckelberg-Utiyama gauge formulation quantity A µ + ∂ µ φ, which must therefore be an observable. To achieve that result, observables must have 0 Poisson bracket not with each first-class constraint, but with the Rosenfeld-Anderson-Bergmann-Castellani gauge generator G, a tuned sum of first-class constraints, in accord with the Pons-Salisbury-Sundermeyer definition of observables.The definition for external gauge symmetries can be tested using massive gravity, where one can install gauge freedom by parametrization with clock fields X A . The non-gauge observable g µν has the gauge equivalent X A , µ g µν X B , ν . The Poisson bracket of X A , µ g µν X B , ν with G turns out to be not 0 but a Lie derivative. This non-zero Poisson bracket refines and systematizes Kuchař's proposal to relax the 0 Poisson bracket condition with the Hamiltonian constraint. Thus observables need covariance, not invariance, in relation to external gauge symmetries.The Lagrangian and Hamiltonian for massive gravity are those of General Relativity + Λ + 4 scalars, so the same definition of observables applies to General Relativity. Local fields such as g µν are observables. Thus observables change. Requiring equivalent observables for equivalent theories also recovers Hamiltonian-Lagrangian equivalence. Problems of Time and SpaceThere has long been a problem of missing change in observables in the constrained Hamiltonian formulation of General Relativity (GR) [1][2][3][4][5]. The typical definition is that observables have 0 Poisson bracket with all first-class constraints [6][7][8][9][10]. This problem of missing change owes much to the condition {O, H 0 } = 0. There is also a problem of space: local spatial variation is excluded by the condition for observables {O, H i } = 0, pointing to global spatial integrals instead [11].The definition of observables is not uncontested. Bergmann himself offered a variety of inequivalent definitions, essentially Hamiltonian or not, local or not [6,12,13]. Here he envisaged locality:

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“…Being gauge-invariant does not require being the same at 1 a.m. Eastern Daylight Time and 1 a.m. Eastern Standard Time an hour later. Hence covariance rather than invariance is a reasonable criterion [33].…”

Section: Internal Versus External Gauge Symmetries and Invariance Vermentioning

confidence: 99%