Non-Relativistic Gravity and its Coupling to Matter

Abstract

We study the non-relativistic expansion of general relativity coupled to matter. This is done by expanding the metric and matter fields analytically in powers of $1/c^2$ where $c$ is the speed of light. In order to perform this expansion it is shown to be very convenient to rewrite general relativity in terms of a timelike vielbein and a spatial metric. This expansion can be performed covariantly and off shell. We study the expansion of the Einstein-Hilbert action up to next-to-next-to-leading order. We couple this to different forms of matter: point particles, perfect fluids, scalar fields (including an off-shell derivation of the Schrödinger-Newton equation) and electrodynamics (both its electric and magnetic limits). We find that the role of matter is crucial in order to understand the properties of the Newton-Cartan geometry that emerges from the expansion of the metric. It turns out to be the matter that decides what type of clock form is allowed, i.e. whether we have absolute time or a global foliation of constant time hypersurfaces. We end by studying a variety of solutions of non-relativistic gravity coupled to perfect fluids. This includes the Schwarzschild geometry, the Tolman-Oppenheimer-Volkoff solution for a fluid star, the FLRW cosmological solutions and anti-de Sitter spacetimes.

Figures (3)

• Figure 1: Structure of the equations of motion in the $1/c^2$ expansion, of which many will enter in the Lagrangian at subleading orders. Because of the way the EH Lagrangian is expanded and the property \ref{['eq:NNLO_Nvar_NLO_var']} there will only be two new EOMs at each order to solve, the remaining ones being recursively equal to those of the previous order. Notice that when we impose TTNC off shell, all the outermost equations are zero since the LO EOMs are $\propto \tau\wedge \mathrm{d}\tau$ as explained in Section \ref{['sec:Lagrangian_NNLO_1c2_expansion']}.
• Figure 2: Structure of the currents in the $1/c^2$ expansion of the matter Lagrangian similar to Figure \ref{['fig:structure_EOMs_expansion']}, but with $\tau\wedge\mathrm{d}\tau=0$ imposed off shell. For this to be consistent, the leading order currents and those related to them by variational calculus identities must be zero.
• Figure 3: Structure of the wi (): At each order there are LO WIs generated by the LO vector field $\xi^\mu$ through the Lie derivative $\mathcal{L}_\xi$. The subleading vector field $\zeta^\mu$ generates a WI through $\mathcal{L}_\zeta$, which is equivalent to the LO WI at LO. Similarly $\mathcal{L}_\zeta$ at NNLO generates a WI which is equivalent to the LO WI at NLO. This works similarly for subsubleading vector field $\Xi_{(4)}^\mu$ and is systematically extended to higher orders in the expansion. Hence, when working at a particular order, energy--momentum conservation of the previous orders in the expansion is always included at that given order.