THREE-DIMENSIONAL GEOMETRY AS CARRIER OF INFORMATION ABOUT TIME

Journal Article · Fri Jun 01 04:00:00 EDT 1962 · Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D

DOI:https://doi.org/10.1103/PhysRev.126.1864· OSTI ID:4820141

Baierlein, R F; Sharp, D H; Wheeler, J A

A geometry of curved empty space which evolves in time in accordance with Einstein's field equations may be termed a geometrodynamical history.'' Such a history can be specified by giving on a 3-dimensional space-like hypersurface ( initial surface'') the geometry intrinsic to this surface and the extrinsic curvature of this surface (having to do with how the surface is imbedded, or is to be imbedded, in a yet-to-be-constructed 4-dimensional manifold). The intrinsic and extrinsic curvatures of the surface cannot be specified independently but have to satisfy the initial value equations of Foures and Lichnerowicz (analogous to div E = 0 and div B = 0 in electromagnetism). An alternative way of specifying a history is outlined in which the intrinsic geometry is given freely on each of 2 hypersurfaces, and nothing is specified as to the extrinsic curvature of either. In the special case in which the two so- specified 3-geometries are nearly alike, a procedure is outlined in order to find the following from Einstein's equations: the invariant space-time interval between an arbitrary point on one surface and a nearby point on the other surface (and thus the 4-geometry interior to the thin sandwich); the extrinsic curvature of the sandwich; hence (via the rest of Einstein's equations) the entire enveloping 4-geometry or geometrodynamical history); and the time-like separation of the original surfaces and their location in spacetime. In this sense, two 3- geometries carry latent information about time. (auth)

• Research Organization: Princeton Univ., N.J.
• NSA Number: NSA-16-024326
• OSTI ID: 4820141
• Journal Information: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D, Journal Name: Physical Review (U.S.) Superseded in part by Phys. Rev. A, Phys. Rev. B: Solid State, Phys. Rev. C, and Phys. Rev. D Vol. Vol: 126; ISSN PHRVA
• Country of Publication: Country unknown/Code not available
• Language: English

Related Subjects

DIFFERENTIAL EQUATIONS
EINSTEIN FIELD EQUATIONS
HYPERSURFACES
MATHEMATICS
PHYSICS
RELATIVITY THEORY
SURFACES