Effective Field Equations Research Articles

Observable strong field effects of extra spacetime dimension in the braneworld black hole

Inspired by the string theory, the braneworld picture introduces extra dimensions beyond the four that may have observable non-trivial effects in short distance (strong field) gravity experiments. A case in point is the Randall–Sundrum braneworld picture that projects the 5d bulk Weyl tensor onto the 3d brane providing a stress tensor in the effective Einstein field equations on the brane. Dadhich, Maartens, Papadopoulos and Rezania (DMPR) derived an exact braneworld black hole solution of the brane vacuum field equations. The solution formally resembles that of Reissner–Nordström but is physically different from it since the ”tidal charge” Υ in the solution is not the electric charge but an imprint from the fifth dimension allowing both signs in the power law modification ±Υ2r2 to the Schwarzschild metric Υ=0. The corresponding black holes are designated as DMPR±. We study here the effect of Υ on strong field lensing observables and compare in the eikonal limit the ring down quasinormal mode (QNM) frequencies of DMPR− with those of DMPR＋ , the two variants of tidal charge modified Schwarzschild black hole (Υ=0). It turns out that the tidal charge can significantly modify the Schwarzschild lensing observables and QNM frequencies. In particular, we find that the Pretorius–Khurana critical exponent γ of circular null orbits in the DMPR− black hole has a lower value than that for the Schwarzschild black hole, which indicates a stronger Lyapunov instability suggesting that the accretion disks of DMPR− black holes would appear brighter. The case of the SgrA* black hole is considered for a possible constraint on Υ from the EHT observation of its shadow size.

Alternate computation of gravitational effects from a single loop of inflationary scalars

We present a new computation of the renormalized graviton self-energy induced by a loop of massless, minimally coupled scalars on de Sitter background. Our result takes account of the need to include a finite renormalization of the cosmological constant, which was not included in the first analysis. We also avoid preconceptions concerning structure functions and instead express the result as a linear combination of 21 tensor differential operators. By using our result to quantum-correct the linearized effective field equation we derive logarithmic corrections to both the electric components of the Weyl tensor for gravitational radiation and to the two potentials which quantify the gravitational response to a static point mass.

Gauge independent logarithms from inflationary gravitons

Dependence on the graviton gauge enters the conventional effective field equations because they fail to account for quantum gravitational correlations with the source which excites the effective field and with the observer who measures it. Including these correlations has been shown to eliminate gauge dependence in flat space background. We generalize the technique to de Sitter background for the case of the 1-loop graviton corrections to the exchange potential of a massless, minimally coupled scalar.

The need to renormalize the cosmological constant

We consider the massless, minimally coupled scalar on de Sitter background. Although the 1-loop divergences of the graviton 1PI 2-point function are canceled by the usual Weyl (C2) and Eddington (R2) counterterms, there is still a finite, nonzero contribution to the graviton 1-point function. Unless this is canceled by a finite renormalization of the cosmological constant, the 1PI 2-point function will not be conserved, nor will the parameter “H” correspond to the actual Hubble constant. We argue that a similar finite renormalization of the cosmological constant is necessary in pure gravity, and that this must be done when solving the effective field equations for 1-loop corrections to the graviton wave function and to the force of gravity.

Anisotropic ultra-compact object in Serrano–Liska gravity model

We implement a recent model proposed by Serrano–Liska (SL) (Alonso-Serrano and Liška, JHEP, 12:196, 2020) to study the ultra-compact star properties. The matter in the interior star is modeled by the quark model, deducted from QCD theory equipped with anisotropic pressure. Anisotropy is used to increase the compactness of the stars. We intend to see the signature of the “quantum gravity” effect through the SL model in ultra-compact stars. The SL model was motivated by quantum correction appearing in the black hole by adding a logarithmic term in gravity entropy used to derive the effective Einstein field equation. We expect this correction term in the SL model also affects ultra-compact star properties. The SL model with coupling constant {tilde{c}} in logarithmic term equipped with spherically symmetric metric yields correction terms mathscr {O}({tilde{c}}) that can be expressed by a function varXi (r). The varXi (r) function vanishes at the star’s exterior. We found that the mass-radius relation prediction by the SL model with anisotropic matter deviates from the one predicted by the standard Tolman–Oppenheimer–Volkoff (TOV) equation for {tilde{c}}ge 10^7 m^2. We also have a sufficiently deep enough effective potential to produce a quasi-normal mode. We obtain the echo frequency of 15.2 kHz using maximum anisotropic pressure contribution and {tilde{c}}= 10^7 m^2. Because the corresponding effective potential is almost indistinguishable from that of GR, this echo frequency value can be indistinguishable to one of GR, but not comparable to the result from GW170817 data analysis, i.e., 72 Hz. To circumvent this problem, we can decrease the value of echo frequency by increasing the magnitude of {tilde{c}} to orders of magnitude than {tilde{c}}= 10^7 m^2. On the other hand, too large a strength from the logarithmic correction term is not physically favored because we learn from the black hole case that the logarithmic term is expected to be smaller than that of the Bekenstein term. Therefore, more precise gravitation echo measurements are crucial to understand this issue.

Unfinished business in a nonlinear sigma model on de Sitter background

Nonlinear sigma models on de Sitter background possess the same kind of derivative interactions as gravity, and show the same sorts of large spacetime logarithms in correlation functions and solutions to the effective field equations. It was recently demonstrated that these logarithms can be resummed by combining a variant of Starobinsky’s stochastic formalism with a variant of the renormalization group. This work considers one of these models and completes two pieces of analysis which were left unfinished: the evolution of the background at two loop order and the one loop beta function.

Modelling the emergence of cosmic anisotropy from non-linear structures

Astronomical observations suggest that the Universe may be anisotropic on the largest scales. In order to model this situation, we develop a new approach to cosmology that allows for large-scale anisotropy to emerge from the growth of non-linear structure. This is achieved by decomposing all relevant fields with respect to a preferred space-like direction, and then averaging the resulting scalar quantities over spatial domains. Our approach allows us to derive a set of large-scale effective field equations that govern the dynamics of any emergent large-scale anisotropy, and which (up to back-reaction terms) take the form of the field equations of the locally rotationally symmetric Bianchi cosmologies. We apply our approach to the dust-filled Farnsworth solutions, which are an interesting set of exact cosmological models that allow for both anisotropic expansion and large-scale bulk flow.

On the possibility of a novel (A)dS/CFT relationship emerging in Asymptotic Safety

Quantum Einstein Gravity (QEG), nonperturbatively renormalized by means of a certain asymptotically safe renormalization group (RG) trajectory, is explored by solving its scale dependent effective field equations and embedding the family of emerging 4-dimensional spacetimes into a single 5-dimensional manifold, which thus encodes the complete information about all scales. By construction the latter manifold is furnished with a natural foliation. Heuristically, its leaves are interpreted as physical spacetime observed on different scales of the experimental resolution. Generalizing earlier work on the embedding of d-dimensional Euclidean QEG spacetimes in (d + 1)-dimensional flat or Ricci flat manifolds, we admit Lorentzian signature in this paper and we consider embeddings in arbitrary (d + 1)-dimensional Einstein spaces. Special attention is paid to the sector of maximally symmetric metrics, and the fundamental definition of QEG in d = 4 that employs the cross-over trajectory connecting the non-Gaussian to the Gaussian RG fixed point. Concerning the embedding of the resulting family of 4D de Sitter solutions with a running Hubble parameter, we find that there are only two possible 5D spacetimes, namely the anti-de Sitter manifold AdS5 and the de Sitter manifold dS5. To arrive at this result essential use is made of the monotone scale dependence of the running cosmological constant featured by the gravitational effective average action. We show that if the scale invariance of the QEG fixed points extends to full conformal invariance, the 5D picture of the resulting geometric and field theoretic structure displays a novel kind of “AdS/CFT correspondence”. While strongly reminiscent of the usual string theory-based AdS/CFT correspondence, also clear differences are found.

Statefinder analysis of scale-dependent cosmology

We study the statefinder parameters of a cosmological model based on scale-dependent gravity. The effective Einstein field equations come from an average effective action. From the dynamical system, we derive analytical expressions that improve the convergence of the numerical solutions. We determine the statefinder parameters for moderate redshift and compare them with well-known alternatives to ΛCDM.

Cotton gravity and 84 galaxy rotation curves

Recently, as a generalization of general relativity, a gravity theory has been proposed in which gravitational field equations are described by the Cotton tensor. That theory allows an additional contribution to the gravitational potential of a point mass that rises linearly with radius as $\Phi = -GM/r + \gamma r/2$, where $G$ is the Newton constant. The coefficients $M$ and $\gamma$ are the constants of integration and should be determined individually for each physical system. When applied to galaxies, the coefficient $\gamma$, which has the dimension of acceleration, should be determined for each galaxy. This is the same as having to determine the mass $M$ for each galaxy. If $\gamma$ is small enough, the linear potential term is negligible at short distances, but can become significant at large distances. In fact, it may contribute to the extragalactic systems. In this paper, we derive the effective field equation for Cotton gravity applicable to extragalactic systems. We then use the effective field equation to numerically compute the gravitational potential of a sample of 84 rotating galaxies. The 84 galaxies span a wide range, from stellar disk-dominated spirals to gas-dominated dwarf galaxies. We do not assume the radial density profile of the stellar disk, bulge, or gas; we use only the observed data. We find that the rotation curves of 84 galaxies can be explained by the observed distribution of baryons. This is due to the flexibility of Cotton gravity to allow the integration constant $\gamma$ for each galaxy. In the context of Cotton gravity, "dark matter" is in some sense automatically included as a curvature of spacetime. Consequently, even galaxies that have been assumed to be dominated by dark matter do not need dark matter.

Brane cosmology with variable tension

We study brane-world models and demonstrate that such models do not admit self-similar solutions through the matter collineation approach. By introducing the hypothesis of variable brane tension, λ, we outline the new effective field equation (EFE) in the most simple case (symmetric embedding) under the assumption that the fundamental constants in five dimensions (5D) are constants. In this case, we find the exact form that each physical quantity may take in order that the EFE become invariant under scale transformations. By taking into account such assumptions, we find that in four dimensions the gravitational constant κ2 ∼ λ, while the cosmological constant Λ ∼ λ2 is always decreasing. These results are quite general and valid for any homogeneous self-similar metric. Nevetheless, the study of the EFE under scale symmetries suggests that ρ ∼ λ (as a functional relationship). This allows us to get a growing κ2 but, in this case, the fundamental constants in 5D must vary as well. We outline a toy model allowing such a possibility.

Large logarithms from quantum gravitational corrections to a massless, minimally coupled scalar on de Sitter

We consider single graviton loop corrections to the effective field equation of a massless, minimally coupled scalar on de Sitter background in the simplest gauge. We find a large temporal logarithm in the approach to freeze-in at late times, but no correction to the feeze-in amplitude. We also find a large spatial logarithm (at large distances) in the scalar potential generated by a point source, which can be explained using the renormalization group with one of the higher derivative counterterms regarded as a curvature-dependent field strength renormalization. We discuss how these results set the stage for a project to purge gauge dependence by including quantum gravitational corrections to the source which disturbs the effective field and to the observer who measures it.

4D Einstein-Gauss-Bonnet gravity from non-Einsteinian phase

We set up an Einstein-Gauss-Bonnet theory in four dimensions, based on the recent formulation of pure gravity with extra dimensions of vanishing metrical length [1]. In absence of torsion, the effective field equations depend only on the four-metric, reflecting a quadratic curvature nonlinearity but no higher than second order derivatives. In contrast with recent proposals to obtain 4dEGB theories through a singular rescaling of the Gauss-Bonnet coupling, this formalism requires no (classical) regularization of divergences and is inequivalent to Horndeski gravity, while being generally covariant and independent of compactification. Notably, the vacuum field equations admit FLRW cosmologies containing nonsingular bounce and self-accelerating Universe, and spherically symmetric black holes more general than Schwarzschild. The quadratic theory of gravity emerging here is shown to be preserved against the inclusion of higher Lovelock densities in 5+2n dimensions.

Two-dimensional gravity from vanishing metrical dimensions

We obtain a dynamical formulation of two-dimensional gravity from a non-Einsteinian phase in higher dimensions $(D=3+2n)$. The formalism is associated with (at least) one extra dimension of vanishing proper length, thus being inequivalent to either a Kaluza-Klein compactification or the Mann-Ross dimensional reduction defined upon a singular limit. The emergent solutions admit any arbitrary curvature in contrast with Jackiw-Teitelboim constant curvature gravity. We present the static and homogeneous solutions as explicit examples. The effective field equations are shown to remain unaffected by the inclusion of higher Lovelock terms beyond Einstein.

Anderson localization of excitations in disordered Gross-Pitaevskii lattices

We examine the one-dimensional Gross-Pitaevskii lattice at zero temperature in the presence of uncorrelated disorder. We obtain analytical expressions for the thermodynamic properties of the ground state field and compare them with numerical simulations both in the weak and strong interaction regimes. We analyze weak excitations above the ground state and compute the localization properties of Bogoliubov-de Gennes modes. In the long-wavelength limit, these modes delocalize in accordance with the extended nature of the ground state. For strong interactions, we observe and derive a divergence of their localization length at finite energy due to an effective correlated disorder induced by the weak ground state field fluctuations. We derive effective strong interaction field equations for the excitations and generalize to higher dimensions.

AdS graviton stars and differential configurational entropy

AdS graviton stars are studied in the differential configurational entropy setup, as solutions of the effective Einstein field equations that backreact to compactification. With the critical central density of AdS graviton stars, the differential configurational entropy is derived and computed, presenting global minima for a wide range of stellar mass magnitude orders. It indicates insular domains of configurational stability for AdS graviton stars near astrophysical neutron star densities. Other relevant features are also reported.

Gauge independent quantum gravitational corrections to Maxwell\u2019s equation

We consider quantum gravitational corrections to Maxwell’s equations on flat space background. Although the vacuum polarization is highly gauge dependent, we explicitly show that this gauge dependence is canceled by contributions from the source which disturbs the effective field and the observer who measures it. Our final result is a gauge independent, real and causal effective field equation that can be used in the same way as the classical Maxwell equation.

Braneworld‐Klein‐Gordon System in the Framework of Gravitational Decoupling

AbstractWe analyze the effective field equations of the Randall‐Sundrum braneworld coupled with a Klein‐Gordon scalar field through the minimal geometric deformation decoupling method (MGD‐decoupling). We introduce two different ways to apply the MGD‐decoupling method to obtain new solutions for this enlarged system. We also compare the behavior of the new solutions with those obtained directly from the Randall‐Sumdrum braneworld without coupling to the scalar field.

Effective field equations and scale-dependent couplings in gravity

A new set of field equations for a space-time dependent Newton's constant $G(x)$ and cosmological constant $\Lambda(x)$ in the presence of matter is presented. We prove that it represents the most general mathematically consistent, physically plausible, set of evolution equations assuming at most second derivatives in the dynamical variables. In the new Einstein's equations, only $\Lambda$-kinetic terms arise, while in the modified conservation equation, derivative terms of $G$ also appear. As an application, this formalism is applied in the context of the Asymptotic Safety scenario to the early universe, assuming a perfect fluid with a radiation equation of state. Cosmological solutions are obtained for all types of spatial curvature, displaying a variety of interesting cosmic evolutions. As an indication of such behaviours, bouncing solutions, recollapsing solutions or non-singular expanding solutions with a transient acceleration era are discussed in details.

Quantum corrections to slow-roll inflation: scalar and tensor modes

Inflation is often described through the dynamics of a scalar field, slow-rolling in a suitable potential. Ultimately, this inflaton must be identified with the expectation value of a quantum field, evolving in a quantum effective potential. The shape of this potential is determined by the underlying tree-level potential, dressed by quantum corrections from the scalar field itself and the metric perturbations. Following [1], we compute the effective scalar field equations and the corrected Friedmann equations to quadratic order in both scalar field, scalar metric and tensor perturbations. We identify the quantum corrections from different sources at leading order in slow-roll, and estimate their magnitude in benchmark models of inflation. We comment on the implications of non-minimal coupling to gravity in this context.