A Modified Variational Principle in Relativistic Hydrodynamics. II. Variations of the vector field and the projection tensor in the general case and under definite assumptions

TL;DR

This paper advances a projection variational approach in relativistic hydrodynamics by deriving a general relation for the variation of vector field length, incorporating the influence of the reference system within a non-commuting variation framework.

Contribution

It introduces a more general variational relation for vector fields in relativistic hydrodynamics, accounting for the influence of the reference system and non-commuting variations.

Findings

Derived a relation linking vector length variation to projected Christoffel connection.

Generalized known affine differential geometry formulae within the relativistic framework.

Incorporated the influence of the reference system on vector field length change.

Abstract

The purpose of the paper is to develop further a projection variational approach in relativistic hydrodynamics. The approach, previously proposed in [gr-qc/9908032], is based on the variation of the vector field and the projection tensor (instead of the given metric tensor) and their first partial derivatives. The previously proved property of non-commutativity of the variation and the partial derivative in respect to the projection tensor has been used to find all the variations. Subsequently, motivated by some analogy with the well-known (3+1) ADM projection formalism, an assumption has been made about a zero-covariant derivative of the projection tensor in respect to the projection connection. The combination of the equations for the variations of the projective tensor with covariant and contravariant indices has lead to the derivation of an important and concisely written relation: the derivative of the vector field length is equal to the ''twice'' projected along the vector field initial Christoffell connection. The result is of interest due to the following reasons: 1. It is a more general one and contains in itself a well-known formulae in affine differential geometry for the so called equiaffine connections (admitting covariantly conserved tensor fields), for which the trace of the connection is equal to the gradient of the logarithm of the vector field length. 2. The additional term is the projected (with the projection tensor) initially given connection and accounts for the influence of the reference system on the change of the vector field's length, measured in this system. 3. The formulae has been obtained within the proposed formalism of non-commuting variation and partial derivative.