1+3 -Newton-Cartan system and Newton-Cartan cosmology

Abstract : We perform a covariant 1+3 split of the Newton-Cartan equations. The resulting 3-dimensional system of equations, called the 1+3-Newton-Cartan equations, is structurally equivalent to the 1+3-Einstein equations. In particular it features the momentum constraint, and a choice of adapted coordinates corresponds to a choice of shift vector. We show that these equations reduce to the classical Newton equations without the need for special Galilean coordinates. The solutions to the 1+3-Newton-Cartan equations are equivalent to the solutions of the classical Newton equations if space is assumed to be compact or if fall-off conditions at infinity are assumed. We then show that space expansion arises as a fundamental field in Newton-Cartan theory, and not by construction as in the classical formulation of Newtonian cosmology. We recover the Buchert-Ehlers theorem for the general expansion law in Newtonian cosmology.
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PhysRevD.103.064064.pdf
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Quentin Vigneron. 1+3 -Newton-Cartan system and Newton-Cartan cosmology. Physical Review D, American Physical Society, 2021, 103 (6), pp.064064. ⟨10.1103/PhysRevD.103.064064⟩. ⟨hal-03115845⟩

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