October 09, 2013

Real Change Happens in Hamiltonian General Relativity; Just Ask the Lagrangian (about Time-like Killing Vectors, First-Class Constraints and Observables)

37 minutes

J. Brian Pitts (Cambridge) gives a talk at the MCMP workshop "Quantum Gravity in Perspective" (31 May-1 June, 2013) titled "Real Change Happens in Hamiltonian General Relativity; Just Ask the Lagrangian (about Time-like Killing Vectors, First-Class Constraints and Observables)". Abstract: In Hamiltonian GR, change has seemed absent. Attention to the gauge generator G facilitates a neglected calculation: a first-class constraint generates a bad physical change in electromagnetism and GR, spoiling the constraints, Gauss's law or the momentum and Hamiltonian constraints in the (physically relevant) velocities. Only as a team G do first-calss constraints generate a gauge transformation. To find change, insist on Hamiltonian-Lagrangian equivalence. Change is ineliminable time dependence; in vaccum GR it is the absence of a time-like Killing vector field. Neglecting spatial dependence, invariantly something depends on time via Hamilton's equations iff there is no time-like Killing vector. According to Bergmann, reality is not confined to observables, defined as both gauge invariant (hence real) and economical (Cauchy data on space). Thus change can exist outside observables. Bergmanns lemma that observables have vanishing Poisson brackets for gauge transformations was imported by analogy to electromagnetism, neglecting the external vs. internal distinction and Hamiltonaian-Lagrangian equivalence. The resulting implausible Killing-type condition lacks the local examples required by Bergmann. Taking observables to be geometric objects (tensors, etc.) as usual in the 4-dimensional Lagrangian formalism makes the Poisson bracket of G with an observable the Lie derivative of a geometric object (on-shell): covariance, not invariance.

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