General reference frames and their associated space manifolds

Abstract

We propose a formal definition of a general reference frame in a general spacetime, as an equivalence class of charts. This formal definition corresponds with the notion of a reference frame as being a (fictitious) deformable body, but we assume, moreover, that the time coordinate is fixed. This is necessary for quantum mechanics, because the Hamiltonian operator depends on the choice of the time coordinate. Our definition allows us to associate rigorously with each reference frame F, a unique "space" (a three-dimensional differentiable manifold), which is the set of the world lines bound to F. This also is very useful for quantum mechanics. We briefly discuss the application of these concepts to Gödel's universe.

14 pages in standard 12pt format. v2: Discussion Section 4 reinforced, now includes an application to Gödel's universe.

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Cited by in corpus (9)

• Equivalent forms of Dirac equations in curved spacetimes and generalized de Broglie relations
• A non-uniqueness problem of the Dirac theory in a curved spacetime
• A solution of the non-uniqueness problem of the Dirac Hamiltonian and energy operators
• A simpler solution of the non-uniqueness problem of the covariant Dirac theory
• Four-vector vs. four-scalar representation of the Dirac wave function
• Some remarks on quantum mechanics in a curved spacetime, especially for a Dirac particle
• Should there be a spin-rotation coupling for a Dirac particle?
• On the Hamiltonian and energy operators in a curved spacetime, especially for a Dirac particle
• Defining the space in a general spacetime