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

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 Schrödinger equation. The relationship between Jacobi's and Hamilton's action principles then allows us to derive a time-dependent Wheeler-DeWitt equation of the Schrödinger 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.