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General Relativistic Lagrangian Continuum Theories Part I: Reduced Variational Principles and Junction Conditions for Hydrodynamics and Elasticity

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Abstract

We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of the Hamilton principle for the relativistic particle, we deduce two classes of reduced variational principles that are associated to either spacetime covariance, which is an axiom of the continuum theory, or material covariance, which is related to particular properties of the system such as isotropy. The covariance hypotheses and the Lagrangian reduction process are efficiently formulated by making explicit the dependence of the theory on given material and spacetime tensor fields that are transported by the world-tube of the continuum via the push-forward and pull-back operations. It is shown that the variational formulation, when augmented with the Gibbons–Hawking–York (GHY) boundary terms, also yields the Israel–Darmois junction conditions between the solution at the interior of the relativistic continua and the solution describing the gravity field produced outside from it. The expression of the first variation of the GHY term with respect to the hypersurface involves some extensions of previous results that we also derive in the paper. We consider in detail the application of the variational framework to relativistic fluids and relativistic elasticity. For the latter case, our setting also allows to clarify the relation between formulations of relativistic elasticity based on the relativistic right Cauchy-Green tensor or on the relativistic Cauchy deformation tensor. The setting developed here will be further exploited for modeling purpose in subsequent parts of the paper.

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Notes

  1. Note the different notation used for the Riemannian metric g on the ambient space \( \mathcal {S} \) in Sect. 2 and the Lorentzian metric \(\textsf{g}\) on spacetime \( \mathcal {M} \) used here.

  2. Explicitly, for \( \alpha ^1,..., \alpha ^r \in T^*_X \mathcal {D} \) and \( u _1,...,u_s \in T_X \mathcal {D} \), we have

    $$\begin{aligned} \Gamma (X)( \alpha ^1,..., \alpha ^r, u_1,...,u_s)= \gamma ( \Phi (X))\left( (T^*_X \Phi )^{-1} ( \alpha ^1 ),... (T^*_X \Phi )^{-1} ( \alpha ^r ), T_X \Phi (u_1),..., T_X \Phi (u_s)\right) \end{aligned}$$

    with \(T_X \Phi :T_X \mathcal {D} \rightarrow T_{ \Phi (X)} \mathcal {M} \) and \(T^*_X \Phi : T_{ \Phi (X)}^* \mathcal {M} \rightarrow T_X ^*\mathcal {D}\) isomorphisms and \(T_X \mathcal {M} =T_{ \Phi (X)}[ \Phi ( \mathcal {D} )]\) for all \(X \in \mathcal {D} \).

  3. No confusion should arise with the same symbol \( \delta \) also used for variations.

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  1. Division of Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    François Gay-Balmaz

  2. CNRS and LMD, Ecole Normale Supérieure, Paris, France

    François Gay-Balmaz

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Gay-Balmaz, F. General Relativistic Lagrangian Continuum Theories Part I: Reduced Variational Principles and Junction Conditions for Hydrodynamics and Elasticity. J Nonlinear Sci 34, 46 (2024). https://doi.org/10.1007/s00332-024-10019-5

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