Curved non-relativistic spacetimes, Newtonian gravitation and massive matter
Michael Geracie, Kartik Prabhu, Matthew M. Roberts
备注：
v5: update references. matches version published in JMP v4: minor text changes. version accepted in Journal of Mathematical Physics v3: improved discussion of NR limit and added refs, v2: updated references and text. v1: 39 pages, 2 figures
There is significant recent work on coupling matter to Newton-Cartan spacetimes with the aim of investigating certain condensed matter phenomena. To this end, one needs to have a completely general spacetime consistent with local non-relativisitic symmetries which supports massive matter fields. In particular, one can not impose a priori restrictions on the geometric data if one wants to analyze matter response to a perturbed geometry. In this paper we construct such a Bargmann spacetime in complete generality without any prior restrictions on the fields specifying the geometry. The resulting spacetime structure includes the familiar Newton-Cartan structure with an additional gauge field which couples to mass. We illustrate the matter coupling with a few examples. The general spacetime we construct also includes as a special case the covariant description of Newtonian gravity, which has been thoroughly investigated in previous works. We also show how our Bargmann spacetimes arise from a suitable non-relativistic limit of Lorentzian spacetimes. In a companion paper [arXiv:1503.02680] we use this Bargmann spacetime structure to investigate the details of matter couplings, including the Noether-Ward identities, and transport phenomena and thermodynamics of non-relativistic fluids.
Einstein Equivalence Principle (EEP) requires all matter components to universally couple to gravity via a single common geometry: that of spacetime. This relates quantum theory with geometry as soon as interactions with gravity are considered. In this work, I study the geometric theory of coupling a spin-1/2 particle to gravity in a twofold expansion scheme: First with respect to the distance based on Fermi normal coordinates around a preferred worldline (e.g., that of a clock in the laboratory), second with respect to 1/c (post-Newtonian expansion). I consider the one-particle sector of a ma…
In a 4-dimensional (4D) weak field geometry governed by the linearized Einstein's equations and sourced primarily by a static, spatially localized, but otherwise arbitrary mass density $T_{00}$, it is known that the leading order tail part of the Green's functions of the minimally coupled massless scalar, Lorenz gauge photon, and de Donder gauge graviton at late times are not only time translation symmetric, but also space-translation and spherically symmetric. Only the monopole moment of the matter source is responsible for the late-time tail of the Green's functions. We provide evidence, in …
A spherically symmetric wormhole in Newtonian gravitation in curved space, enhanced with a connection between the mass density and the Ricci scalar, is presented. The wormhole, consisting of two connected asymptotically flat regions, inhabits a spherically symmetric curved space. The gravitational potential, gravitational field and the pressure that supports the fluid that permeates the Newtonian wormhole are computed. Particle dynamics and tidal effects in this geometry are studied. The possibility of having Newtonian black holes in this theory is sketched.
A second gradient generalization of Newtonian gravity is presented within the framework of gradient field theory. Weak nonlocality is introduced via first and second gradients of the gravitational field strength in the Lagrangian density. Gradient generalizations of the Poisson equation of Newtonian gravitation for the gravitational potential and of the generalized Gauss law for the gravitational field strength are presented. Such a gradient modification of Newtonian gravity provides a straightforward regularization of Newtonian gravity removing the classical Newtonian singularities. Finite gr…
Scalar-tensor (ST) gravity theories provide an appropriate theoretical framework for the variation of Newton's fundamental constant, conveyed by the dynamics of a scalar-field non-minimally coupled to the space-time geometry. The experimental scrutiny of scalar-tensor gravity theories has led to a detailed analysis of their post-newtonian features, and is encapsulated into the so-called parametrised post-newtonian formalism (PPN). Of course this approach can only be applied whenever there is a newtonian limit, and the latter is related to the GR solution that is generalized by a given ST solut…
We obtain a new form for the action of a nonrelativistic particle coupled to Newtonian gravity. The result is different from that existing in the literature which, as shown here, is riddled with problems and inconsistencies. The present derivation is based on the formalism of galilean gauge theory, introduced by us as an alternative method of analysing nonrelativistic symmetries in gravitational background.
Lorentz invariance is one of the fundamental principles of physics, and, as such, it must be experimentally tested. The purpose of this work is to obtain, within the Standard-Model Extension, the dynamics of a Lorentz-violating spinor in a uniform newtonian gravitational field. This is achieved by treating the spinor as a test particle and introducing the gravitational field through a uniformly accelerated observer. The nonrelativistic hamiltonian is obtained and some experimental consequences are discussed. One unexpected outcome of this work is that the gravitational field helps disentanglin…
The post-Newtonian approximation is a method for solving Einstein's field equations for physical systems in which motions are slow compared to the speed of light and where gravitational fields are weak. Yet it has proven to be remarkably effective in describing certain strong-field, fast-motion systems, including binary pulsars containing dense neutron stars and binary black hole systems inspiraling toward a final merger. The reasons for this effectiveness are largely unknown. When carried to high orders in the post-Newtonian sequence, predictions for the gravitational-wave signal from inspira…
We propose to slightly generalize the DeWitt-Schwinger adiabatic renormalization subtractions in curved space to include an arbitrary renormalization mass scale $μ$. The new predicted running for the gravitational couplings are fully consistent with decoupling of heavy massive fields. This is a somewhat improvement with respect to the more standard treatment of minimal (DeWitt-Schwinger) subtractions via dimensional regularization. We also show how the vacuum metamorphosis model emerges from the running couplings.
In the weak-field limit of General Relativity, gravitational waves obey linear equations and propagate at the speed of light. These properties of General Relativity are supported by the observation of ultra high energy cosmic rays as well as by LIGO's recent detection of gravitation waves. We argue that two existing relativistic generalizations of Modified Newtonian Dynamics, namely Generalized Einstein-Aether theory and BIMOND, display fatal inconsistencies with these observations.
We argue that if the Newtonian gravitational field of a body can mediate entanglement with another body, then it should also be possible for the body producing the Newtonian field to entangle directly with on-shell gravitons. Our arguments are made by revisiting a gedankenexperiment previously analyzed by Belenchia et al., which showed that a quantum superposition of a massive body requires both quantized gravitational radiation and local vacuum fluctuations of the spacetime metric in order to avoid contradictions with complementarity and causality. We provide a precise and rigorous descriptio…
Using the NRGR effective field theory formalism we calculate the remaining source multipole moments necessary to obtain the spin contributions to the gravitational wave amplitude to 2.5 Post-Newtonian (PN) order. We also reproduce the tail contribution to the waveform linear in spin at 2.5PN arising from the nonlinear interaction between the current quadrupole and the mass monopole.
A wormhole solution in Newtonian gravitation, enhanced through an equation relating the Ricci scalar to the mass density, is presented. The wormhole inhabits a spherically symmetric curved space, with one throat and two asymptotically flat regions. Particle dynamics in this geometry is studied, and the three distinct dynamical radii, namely, the geodesic, circumferential, and curvature radii, appear naturally in the study of circular motion. Generic motion is also analysed. A limiting case, although inconclusive, suggests the possibility of having a Newtonian black hole in a region of finite (…
C-theory provides a unified framework to study metric, metric-affine and more general theories of gravity. In the vacuum weak-field limit of these theories, the parameterized post-Newtonian (PPN) parameters $β$ and $γ$ can differ from their general relativistic values. However, there are several classes of models featuring long-distance modifications of gravity but nevertheless passing the Solar system tests. Here it is shown how to compute the PPN parameters in C-theories and also in nonminimally coupled curvature theories, correcting previous results in the literature for the latter.
Using effective field theory techniques we calculate the source multipole moments needed to obtain the spin contributions to the power radiated in gravitational waves from inspiralling compact binaries to third Post-Newtonian order (3PN). The multipoles depend linearly and quadratically on the spins and include both spin(1)spin(2) and spin(1)spin(1) components. The results in this paper provide the last missing ingredient required to determine the phase evolution to 3PN including all spin effects which we will report in a separate paper.
General Relativity is the modern theory of gravitation. It has replaced the newtonian theory in the description of the gravitational phenomena. In spite of the remarkable success of the General Relativity Theory, the newtonian gravitational theory is still largely employed, since General Relativity, in most of the cases, just makes very small corrections to the newtonian predictions. Moreover, the newtonian theory is much simpler, technically and conceptually, when compared to the relativistic theory. In this text, we discuss the possibility of extending the traditional newtonian theory in ord…
The Newtonian regime of a recent nonlocal extension of general relativity (GR) is investigated. Nonlocality is introduced via a scalar "constitutive" kernel in a special case of the translational gauge theory of gravitation, namely, the teleparallel equivalent of GR. In this theory, the nonlocal aspect of gravity simulates dark matter. A nonlocal and nonlinear generalization of Poisson's equation of Newtonian gravitation is presented. The implications of nonlocality for the gravitational physics in the solar system are briefly studied.
In the context of quantum field theories in curved spacetime, we compute the effective action of the transition amplitude from vacuum to vacuum in the presence of an external gravitational field. The imaginary part of resulted effective action determines the probability of vacuum decay via quantum tunneling process, giving the rate and spectrum of particle creations. We show that gravitational field polarizes vacuum and discretizes its spectrum for such a polarization gains gravitational energy. On the basis of gravitational vacuum polarization, we discuss the quantum origin of vacuum decay in…
We investigate the possibility of constructing a covariant Newtonian gravitational theory and find that the action describing a massless relativistic particle in a background Newtonian gravitodynamic field has a higher-dimensional extension with two times.
