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Gielen, Steffen; Wise, Derek K.

Lifting general relativity to observer space. (English) Zbl 1285.83005

J. Math. Phys. 54, No. 5, 052501, 29 p. (2013).
Summary: The “observer space” of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein’s equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.{
©2013 American Institute of Physics}

Cited in 4 Documents

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
53Z05 Applications of differential geometry to physics
81P15 Quantum measurement theory, state operations, state preparations
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism

Keywords:

general relativity; observer space; future-timelike unit tangent vectors; Cartan geometry; Einstein’s equations
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References:

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