Abstract
The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.
Highlights
Current Status of Hamiltonian TheoriesA fundamental aspect of the standard formulation of General Relativity (SF-GR), i.e., the Einstein field equations [1,2], concerns its Hamiltonian representation
Background on the Cosmological Constant. It is well-known that the cosmological constant (CC) Λ was considered for some time by Einstein as meaningless despite the fact that in his original formulation of General Relativity (GR), referred to as standard formulation of GR (SF-GR) [1,2], its introduction was crucial to warrant the existence of stationary cosmological solutions to his namesake tensor field equation and corresponding set of tensor field components [3,7]
Long-term experimental evidence based on astrophysical observations of the large-scale structure of the universe [27] has shown that the CC can be given a definite value, so that its inclusion in the same equations has become nowadays a well-established part of GR theory
Summary
A fundamental aspect of the standard formulation of General Relativity (SF-GR), i.e., the Einstein field equations [1,2], concerns its Hamiltonian representation. The same CC acquires the connotation of a 4-scalar gauge function which at the classical level remains undetermined Both gauge properties indicated here, and in particular the property of the CC which motivates the present paper, are features which, as explained below, depart in several respects from previous non-manifestly covariant and non-gauge invariant variational approaches to the Einstein equations [1,2]. CDS [25], whereby the state of the same CDS, while being generally non-canonical, can be represented in terms of two suitably-coupled Hamiltonian systems The consequences of such a type of setting are serious: (1) At the classical level the correct gauge properties of SF-GR, which usually hold in classical field theory, are prohibited (see [26] and Section 2). The consequences of such a type of setting are serious: (1) At the classical level the correct gauge properties of SF-GR, which usually hold in classical field theory, are prohibited (see [26] and Section 2). (2) Standard canonical quantization methods become inapplicable. (3) Both at classical and quantum levels the so-called principle of objectivity is violated, namely, the fundamental requisite of retaining the same (tensorial) form in arbitrary coordinate systems (GR-frames) is not fulfilled any more
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