Weak-field Limit Of General Relativity Research Articles

Momentum-space gravity from the quantum geometry and entropy of Bloch electrons

Quantum geometry is a key quantity that distinguishes electrons in a crystal from those in the vacuum. Its study continues to provide insights into quantum materials, uncovering new design principles for their discovery. However, unlike the Berry curvature, an intuitive understanding of the quantum metric is lacking. Here, we show that the quantum metric of Bloch electrons leads to a momentum-space gravity. In particular, by extending the semiclassical formulation of electron dynamics to second order, we find that the resulting velocity is modified by a geodesic term and becomes the momentum-space dual of the Lorentz force in curved space. We calculate this geodesic response for magic-angle twisted bilayer graphene and show that moir\'e systems with flat bands are ideal candidates to observe this effect. Extending this analogy with gravity further, we find that the momentum-space dual of the Einstein field equations remains sourceless for pure states, while for mixed states it acquires a source term that depends on the von Neumann entropy for small entropies. We compare this stress-energy equation with the weak-field limit of general relativity, and we conclude that the von Neumann entropy is the momentum-space dual of the gravitational potential. Consequently, the momentum-space geodesic equation for mixed states is modified by a term resembling an entropic force. Our results highlight connections between quantum geometry, momentum-space gravity, and quantum information, prompting further exploration of this dual gravity in quantum materials.

Gravitomagnetic vorticity generation in black hole accretion discs: a potential spatial constraint on plasma flow stability

ABSTRACT We calculate the vorticity generation rate in the accretion disc near a slowly rotating black hole in the low-velocity, weak-field limit of general relativity. Specifically, we find that the frame-dragging effect due to the black hole’s rotation – manifested through the gravitomagnetic field – can generate vorticity in a moving plasma in the accretion disc. The mechanism remains operational as long as the accretion disc has non-negligible vertical height and is independent of the exact thermodynamical profile of the disc. The enstrophy density generation rate, as a measure of turbulence and dissipation, is presented, which indicates that the frame-dragging effect can disrupt the stability of the disc away from the z = 0 plane.

Gravitational effects in macroscopic quantum systems: a first-principles analysis

We analyze the weak-field limit of general relativity with matter and its possible quantisations. This analysis aims toward a predictive quantum theory to provide a first-principles description of gravitational effects in macroscopic quantum systems. This includes recently proposed experiments on the generation of (Newtonian) gravitational forces from quantum distributions of matter, and phenomena like gravity-induced entanglement, gravitational cat states, gravity-induced Rabi oscillations, and quantum causal orderings of events. Our main results include: (i) the demonstration that these phenomena do not involve true gravitational degrees of freedom. (ii) We show that, unlike full general relativity, weak gravity with matter is a parameterised field theory, i.e., a theory obtained by promoting spacetime coordinates to ‘dynamical’ variables. (iii) Quantisation via gauge-fixing leads to an effective field theory that account for some phenomena, but at the price of gauge dependence that manifests more strongly on spacetime observables. This ambiguity is a manifestation of the problem of time that persists even in weak gravity. (iv) A consistent quantisation of parameterised field theories is essential for a predictive and spacetime covariant theory of weak gravity that describes gravitational effects in macroscopic quantum systems. We also discuss the implication of our results to gravitational decoherence theories, the notion of locality in gravity vis-a-vis quantum information theory, and the intriguing possibility that proposed solutions to the problem of time can be tested in weak-gravity quantum experiments.

Domain walls and other defects in Eddington-inspired Born-Infeld gravity

We investigate domain wall and other defect solutions in the weak-field limit of Eddington-inspired Born-Infeld gravity as a function of $\kappa$, the only additional parameter of the theory with respect to General Relativity. We determine, both analytically and numerically, the internal structure of domain walls, quantifying its dependency on $\kappa$ as well as the impact of such dependency on the value of the tension measured by an outside observer. We find that the pressure in the direction perpendicular to the domain wall can be, in contrast to the weak-field limit of General Relativity, significantly greater or smaller than zero, depending, respectively, on whether $\kappa$ is positive or negative. We further show that the generalized von Laue condition, which states that the average value of the perpendicular pressure is approximately equal to zero in the weak-field limit of General Relativity, does not generally hold in EiBI gravity not only for domain walls, but also in the case cosmic strings and spherically symmetric particles. We argue that a violation of the generalized von Laue condition should in general be expected in any theory of gravity whenever geometry plays a significant role in determining the defect structure.

Probing the post-Minkowskian approximation using recursive addition of self-interactions

We address the problem of deriving the post-Minkowskian approximation, widely used in current gravitational wave literature by investigating a possible deduction out of the recursive N\"other coupling approach, from the Pauli-Fierz spin-2 theory in flat spacetime. We find that this approach yields the post-Minkowskian approximation correctly to the first three orders, without invoking any weak-field limit of general relativity. This connection thus establishes that the post-Minkowskian approximation has a connotation independent of a weak-field expansion of general relativity, which is the manner usually presented in the literature. As a consequence, a link manifests between the recursive N\"other coupling approach to deriving general relativity from a linear spin-2 theory in flat spacetime and theoretical analyses of recent detection of gravitational wave events.

A relativistic interpretation of bias in Newtonian simulations

Observables of cosmic structures are usually not the underlying matter field but biased tracers of matter, such as galaxies or halos. We show how the bias found in Newtonian N-body simulations can be interpreted in terms of the weak-field limit of General Relativity (GR) . For this we employ standard Newtonian simulations of cold dark matter and incorporate GR/radiation via a weak-field dictionary that we have recently developed. We find that even when a simple local biasing scheme is employed in the Newtonian simulation, the relativistic bias becomes inherently scale-dependent due to the presence of radiation and GR corrections. This scale-dependence could be in principle observed on large scales in upcoming surveys. As a working example, we apply our methodology to Newtonian simulations for the spherical collapse and recover permille-level agreement between the approaches for extracting the relativistic bias on all considered scales.

General relativistic weak-field limit and Newtonian N-body simulations

We show how standard Newtonian N-body simulations can be interpreted in terms of the weak-field limit of general relativity by employing the recently developed Newtonian motion gauge. Our framework allows the inclusion of radiation perturbations and the non-linear evolution of matter. We show how to construct the weak-field metric by combining Newtonian simulations with results from Einstein-Boltzmann codes. We discuss observational effects on weak lensing and ray tracing, identifying important relativistic corrections.

The (weak) gravitational field of a Dirac monopole

We establish the gravitational detectability of a Dirac monopole using a weak-field limit of general relativity, which can be developed from the Newtonian gravitational potential by including energy as a source. The resulting potential matches (by construction) the weak-field limit of two different solutions to Einstein’s equations of general relativity: one associated with the magnetically monopolar spray of field lines emerging from the half-infinite solenoid that makes up the Dirac monopole, the other associated with the field-energetic source of the solenoid itself (the Dirac string). The string’s gravitational effect dominates, and we suggest that the primary strong-field contribution of the Dirac configuration is that of a half-infinite line of energy, whose GR solution is known.

Constraining Relativistic Generalizations of Modified Newtonian Dynamics with Gravitational Waves.

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 ultrahigh-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, the generalized Einstein-aether theory and bimetric modified Newtonian dynamics, display fatal inconsistencies with these observations.

Probing theories of gravity with phase space-inferred potentials of galaxy clusters

Modified theories of gravity provide us with a unique opportunity to generate innovative tests of gravity. In Chameleon f(R) gravity, the gravitational potential differs from the weak-field limit of general relativity (GR) in a mass dependent way. We develop a probe of gravity which compares high mass clusters, where Chameleon effects are weak, to low mass clusters, where the effects can be strong. We utilize the escape velocity edges in the radius/velocity phase space to infer the gravitational potential profiles on scales of 0.3-1 virial radii. We show that the escape edges of low mass clusters are enhanced compared to GR, where the magnitude of the difference depends on the background field value |fR0|. We validate our probe using N-body simulations and simulated light cone galaxy data. For a DESI (Dark Energy Spectroscopic Instrument) Bright Galaxy Sample, including observational systematics, projection effects, and cosmic variance, our test can differentiate between GR and Chameleon f(R) gravity models, |fR0| = 4e-6 (2e-6) at > 5{\sigma} (> 2{\sigma}), more than an order of magnitude better than current cluster-scale constraints.

Post-Newtonian cosmological modelling

We develop a new approach to building cosmological models, in which small pieces of perturbed Minkowski space are joined together at reflection-symmetric boundaries in order to form a global, dynamical space-time. Each piece of this patchwork universe is described using post-Newtonian gravitational physics, with the large-scale expansion of the Universe being an emergent phenomenon. This approach to cosmology does not require any assumptions about nonlocal averaging processes. Our framework clarifies the relation between the weak-field limit of general relativity, and the cosmological solutions that result from solving Einstein's equations with a set of symmetry assumptions. It also allows the effects of structure formation on the large-scale expansion of the Universe to be investigated without averaging anything. As an explicit example, we use this formalism to investigate the cosmological behavior of a large number of regularly arranged pointlike masses. In this case we find that the large-scale expansion is well modelled by a Friedmann-like equation that contains terms that take the form of dust, radiation, and spatial curvature. The radiation term, while small compared to the dust term, is purely a result of the nonlinearity of Einstein's equations.

Recovering MOND from extended metric theories of gravity

We show that the Modified Newtonian Dynamics (MOND) regime can be fully recovered as the weak-field limit of a particular theory of gravity formulated in the metric approach. This is possible when Milgrom's acceleration constant is taken as a fundamental quantity which couples to the theory in a very consistent manner. As a consequence, the scale invariance of the gravitational interaction is naturally broken. In this sense, Newtonian gravity is the weak-field limit of general relativity and MOND is the weak-field limit of that particular extended theory of gravity. We also prove that a Noether's symmetry approach to the problem yields a conserved quantity coherent with this relativistic MONDian extension.

The mass-zero spin-two field and gravitational theory

It is shown that the conventional theory of the mass-zero spin-two field with sources introduces extraneous non-spin-two field components in source regions and fails to be covariant under the full or restricted conformal group. A modified theory is given, expressed in terms of the physical components of the mass-zero spin-two field rather than in terms of « potentials », which has no extraneous components inside or outside sources and which is covariant under the full conformai group. For a proper choice of source term, this modified theory has the correct Newtonian limit and automatically implies that a symmetric second-rank source tensor has zero divergence. The development given suggests that the weak-field limit of general relativity is not a satisfactory spin-two theory in the interior of sources, but it is shown that possibly a generally covariant form of the spin-two theory derived here can be constructed to agree with general relativity in all currently accessible experimental situations. The completion of the theory of the interacting mass-zero spin-two field and a construction of its generally covariant form will require the identification of a suitable potential for the field, a problem which has not yet been solved.