We revisit the decoupling limits that lead to matrix theories on D-branes. We highlight the BPS nature of these limits, in which the target space geometry becomes nonLorentzian and wrapped D-branes experience instantaneous gravitational forces. Applied to curved D-brane geometries, we show that a single BPS decoupling limit induces the bulk near-horizon limit leading to AdS/CFT. By consecutively applying two such limits, we systematically generate further examples of holography, including novel versions with non-Lorentzian bulk geometry. Uplifted to M-theory, we are led to a unified framework where each BPS decoupling limit corresponds to a Discrete Light Cone Quantisation (DLCQ). We conjecture that a DLCQ(n)/DLCQ(m) correspondence, with m > n, captures the notion of holography in string theory. In particular, AdS(5)/CFT4 can be viewed as an example of DLCQ(0)/DLCQ(1) , with the extra DLCQ on the field theory side corresponding to the near-horizon limit in the bulk geometry. We further show that undoing these BPS decoupling limits can be viewed as deformations of matrix theories. We explain how these deformations are related to the TT<overline> deformation in two dimensions. In the context of holography, this allows us to view the ten-dimensional near-horizon brane geometry as an intrinsic deformation of the flat non-Lorentzian geometry that arises asymptotically. In field theoretic terms, these generalisations lead to TT<overline>-like flow equations for the Dp-brane DBI action.

The interplay between quantum theory and general relativity remains one of the main challenges of modern physics. A renewed interest in the low -energy limit is driven by the prospect of new experiments that could probe this interface. Here we develop a covariant framework for expressing post -Newtonian corrections to Schr & ouml;dinger's equation on arbitrary gravitational backgrounds based on a 1/c2 expansion of Lorentzian geometry, where c is the speed of light. Our framework provides a generic coupling prescription of quantum systems to gravity that is valid in the intermediate regime between Newtonian gravity and General Relativity, and that retains the focus on geometry. At each order in 1/c2 this produces a nonrelativistic geometry to which quantum systems at that order couple. By considering the gauge symmetries of both the nonrelativistic geometries and the 1/c2 expansion of the complex Klein-Gordon field, we devise a prescription that allows us to derive the Schr & ouml;dinger equation and its post -Newtonian corrections on a gravitational background order -by -order in 1/c2. We also demonstrate that these results can be obtained from a 1/c2 expansion of the complex Klein-Gordon Lagrangian. We illustrate our methods by performing the 1/c2 expansion of the Kerr metric up to O(c-2), which leads to a special case of the Hartle-Thorne metric. The associated Schr & ouml;dinger equation captures novel and potentially measurable effects.

We obtain a Palatini-type formulation for the Galilei and Carroll expansions of general relativity, where the connection is promoted to a variable. Known versions of these large and small speed of light expansions are derived from the Einstein-Hilbert action and involve dynamical Newton-Cartan or Carroll geometry, along with additional gauge fields at subleading orders. The corresponding Palatini actions that we obtain in this paper are derived from an appropriate expansion of the Einstein-Palatini action, and the connection variable reduces to the Galilei- or Carroll-adapted connection on shell. In particular, we present the Palatini form for the next-to-leading-order Galilean action and recover the known equations of motion. We also compute the leading-order Palatini-type action for the Carrollian case and show that, while it depends on the connection variable, it reduces on shell to the known action of electric Carroll gravity, which only depends on extrinsic curvature.

We study various aspects of the Carroll limit in which the speed of light is sent to zero. A large part of this paper is devoted to the quantization of Carroll field theories. We show that these exhibit infinite degeneracies in the spectrum and may suffer from non-normalizable ground states. As a consequence, partition functions of Carroll systems are ill-defined and do not lead to sensible thermodynamics. These seemingly pathological properties might actually be a virtue in the context of flat space holography. Better defined is the Carroll regime, in which we consider the leading order term in an expansion around vanishing speed of light without taking the strict Carroll limit. Such an expansion may lead to sensible notions of Carroll thermodynamics. An interesting example is a gas of massless particles with an imaginary chemical potential conjugate to the momentum. In the Carroll regime we show that the partition function of such a gas leads to an equation of state with w = −1. As a separate story, we study aspects of Carroll gravity and couplings to Carrollian energy-momentum tensors. We discuss many examples of solutions to Carroll gravity, including wormholes, Maxwell fields, solutions with a cosmological constant, and discuss the structure of geodesics in a Carroll geometry. The coupling of matter to Carroll gravity also allows us to derive energy-momentum tensors for hypothetical Carroll fluids from expanding relativistic fluids as well as directly from hydrostatic partition functions.

It is well known that one can take an infinite speed of light limit that gives rise to non-relativistic strings with a relativistic worldsheet sigma model but with a non-relativistic target space geometry. In this work we systematically explore two further limits in which the worldsheet becomes non-Lorentzian. The first gives rise to a Galilean string with a Galilean structure on the worldsheet, extending previous work on Spin Matrix-related string theory limits. The second is a completely novel limit leading to a worldsheet theory with a Carrollian structure. We find the Nambu-Goto and Polyakov formulations of both limits and explore gauge fixing choices. Furthermore, we study in detail the case of the Galilean string for a class of target space geometries that are related to Spin Matrix target space geometries, for which the Nambu-Goto action (in static gauge) is quadratic in the fields.

This study reviews the history of Newton-Cartan (NC) gravity with an emphasis on recent developments, including the covariant, off-shell large speed of light expansion of general relativity. Depending on the matter content, this expansion leads to either NC geometry with absolute time or NC geometry with non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity (NRG) includes a strong field regime and goes beyond Newtonian gravity. We start by reviewing early developments in NC geometry, including the covariant description of Newtonian gravity, mainly through the works of Trautman, Dautcourt, Kunzle, and Ehlers. We then turn to more modern developments, such as the gauging of the Bargmann algebra and describe why the latter cannot be used to find an off-shell covariant description of Newtonian gravity. We review recent work on the 1/c expansion of general relativity and show that this leads to an alternative "type II" notion of NC geometry. Finally, we discuss matter couplings, solutions, and odd powers in 1/c and conclude with a brief summary of related topics.

We study the small speed of light expansion of general relativity, utilizing the modern perspective on non-Lorentzian geometry. This is an expansion around the ultra-local Car-roll limit, in which light cones close up. To this end, we first rewrite the Einstein???Hilbert action in pre-ultra-local variables, which is closely related to the 3+1 decomposition of general relativity. At leading order in the expansion, these pre-ultra-local variables yield Carroll geometry and the resulting action describes the electric Carroll limit of general relativity. We also obtain the next-to-leading order action in terms of Carroll geometry and next-to-leading order geometric fields. The leading order theory yields constraint and evolution equations, and we can solve the evolution analytically. We furthermore construct a Carroll version of Bowen???York initial data, which has associated conserved boundary linear and angular momentum charges. The notion of mass is not present at leading order and only enters at next-to-leading order. This is illustrated by considering a particular truncation of the next-to-leading order action, corresponding to the magnetic Carroll limit, where we find a solution that describes the Carroll limit of a Schwarzschild black hole. Finally, we comment on how a cosmological constant can be incorporated in our analysis.

Carroll symmetry arises from Poincare symmetry upon taking the limit of vanishing speed of light. We determine the constraints on the energy-momentum tensor implied by Carroll symmetry and show that for energy-momentum tensors of perfect fluid form, these imply an equation of state epsilon + P = 0 for energy density plus pressure. Therefore Carroll symmetry might be relevant for dark energy and inflation. In the Carroll limit, the Hubble radius goes to zero and outside it recessional velocities are naturally large compared to the speed of light. The de Sitter group of isometries, after the limit, becomes the conformal group in Euclidean flat space. We also study the Carroll limit of chaotic inflation, and show that the scalar field is naturally driven to have an equation of state with w = - 1. Finally we show that the freeze-out of scalar perturbations in the two point function at horizon crossing is a consequence of Carroll symmetry. To make the paper self-contained, we include a brief pedagogical review of Carroll symmetry, Carroll particles and Carroll field theories that contains some new material as well. In particular we show, using an expansion around speed of light going to zero, that for scalar and Maxwell type theories one can take two different Carroll limits at the level of the action. In the Maxwell case these correspond to the electric and magnetic limit. For point particles we show that there are two types of Carroll particles: those that cannot move in space and particles that cannot stand still.

We present a systematic technique to expand the Einstein-Hilbert Lagrangian in inverse powers of the speed of light squared. The corresponding result for the non-relativistic gravity Lagrangian is given up to next-to-next-to-leading order. The techniques are universal and can be used to expand any Lagrangian theory whose fields are a function of a given parameter.

We present a systematic technique to expand the Einstein-Hilbert Lagrangian in inverse powers of the speed of light squared. The corresponding result for the non-relativistic gravity Lagrangian is given up to next-to-next-to-leading order. The techniques are universal and can be used to expand any Lagrangian theory whose fields are a function of a given parameter.