Jacobi's action and the recovery of time in general relativity

Journal Article · Wed Nov 15 04:00:00 EST 1989 · Physical Review (Section) D: Particles and Fields; (USA)

DOI:https://doi.org/10.1103/PhysRevD.40.3312· OSTI ID:5147410

Brown, J D; York, Jr, J W ^[1]

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1. Institute of Field Physics, Department of Physics and Astronomy, CB No. 3255, Phillips Hall, The University of North Carolina, Chapel Hill, North Carolina 27599 (US)

We argue that the usual action principle of general relativity, applied to spacetimes with closed spatial geometries, should be regarded as analogous to Jacobi's form of the principle of stationary action, in which the energy rather than a physical time is fixed. Following the paradigm of quantization based on Jacobi's action for a nonrelativistic particle, we show that the Wheeler-DeWitt equation corresponds to a time-independent Schroedinger equation. The relationship between Jacobi's and Hamilton's action principles then allows us to derive a time-dependent Wheeler-DeWitt equation of the Schroedinger type. In this equation, the role of energy is played by the cosmological constant and that of physical time by the four-volume of spacetime.

• OSTI ID: 5147410
• Journal Information: Physical Review (Section) D: Particles and Fields; (USA), Journal Name: Physical Review (Section) D: Particles and Fields; (USA) Vol. 40:10; ISSN PRVDA; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
EQUATIONS
EQUATIONS OF MOTION
FIELD THEORIES
GENERAL RELATIVITY THEORY
HAMILTONIANS
MATHEMATICAL OPERATORS
MECHANICS
METRICS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTIZATION
QUANTUM FIELD THEORY
QUANTUM GRAVITY
QUANTUM MECHANICS
QUANTUM OPERATORS
SCHROEDINGER EQUATION
SPACE-TIME
TIME DEPENDENCE
WAVE EQUATIONS