Abstract
In recent previous work, the authors proposed a new approach extending the framework of statistical mechanics to reparametrization-invariant systems with no additional gauges. In this paper, the approach is generalized to systems defined by more than one Hamiltonian constraint (multi-fingered time). We show how well-known features as the Ehrenfest–Tolman effect and the Jüttner distribution for the relativistic gas can be consistently recovered from a covariant approach in the multi-fingered framework. Eventually, the crucial role played by the interaction in the definition of a global notion of equilibrium is discussed.
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Notes
If \(C^b\) is not the only conserved quantity for \(\mathcal {S}^b\), the statistical state should include additional “\(\delta\)-functions”.
The split \(C= C^a+C^b=0\) determines a foliation of the presymplectic surface
$$\begin{aligned} \varSigma = \bigsqcup _{I^a+I^b=0} \varSigma ^a_{I^a} \times \varSigma ^b_{I^b}, \end{aligned}$$where \(X^\alpha = \bigsqcup _{I^\alpha } \varSigma ^\alpha _{I^\alpha }\) and \(I^\alpha\) is the value of \(C^\alpha\) on the leave. Each \(\left( \varSigma ^\alpha _{I^\alpha }, \omega _{X^\alpha }|_{\varSigma ^\alpha _{I^\alpha }} \right)\) is the presymplectic space that would describe the subsystem \(\mathcal {S}^\alpha\) if it was isolated. The existence of the split implies the existence of the conserved quantity \(I=I^b=-I^a\), defined up to clock reparametrization \(I \rightarrow f(I)\). See cf. [8] for full details.
This fact was actually glimpsed by Einstein [12], even before General Relativity was completed!
So that the local gravitational field is approximately uniform.
V can be thought as an effective potential coming from a (Lorentz invariant) interaction with an (relativistic) ambient field.
In the absence of an external potential, the total 4-momentum is conserved, insuring Lorentz covariance of the statistical state.
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Acknowledgements
The authors are grateful to Carlo Rovelli for careful reading of the draft and useful comments, and to the Quantum Gravity Group at the Centre de Physique Théorique de Luminy, where this project was started. G.C. is grateful to Daniele Oriti and Hermann Nicolai for the support at the Max Planck Institute for Gravitational Physics in Potsdam, where the work was partially completed.
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Chirco, G., Josset, T. Statistical Mechanics of Covariant Systems with Multi-fingered Time. Found Phys 51, 3 (2021). https://doi.org/10.1007/s10701-021-00406-3
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DOI: https://doi.org/10.1007/s10701-021-00406-3