We construct 2d Jackiw-Teitelboim (JT) gravity in the framework of symmetric teleparallel gravity based on a non-metricity tensor. In symmetric teleparallel gravity, we often use the scalar quantity <italic>Q</italic> composed of the bilinear terms of the non-metricity tensor but <italic>Q</italic> is not a unique scalar quantity from the viewpoint of covariance. However, other scalar quantities composed of the non-metricity tensor generate ghosts in general. In two dimensions, because a propagating gravitational mode does not exist, if we consider general scalar quantities, there might appear any propagating modes although these modes are often ghosts. We consider a general combination of the terms given by the bi-linear of the non-metricity tensors. After finding the condition that the conformal form of the metric is compatible with the coincident gauge, where connections vanish, we determine the combination so that a solution with constant curvature exists. The obtained model corresponds to 2d JT gravity in the framework of the symmetric teleparallel gravity. The conformal mode propagates as a scalar in this theory and we also consider the condition that the scalar mode is not a ghost.

Japan

Shin'ichi Nojiri

Spain

S.D. Odintsov

Non-metricity approach to Jackiw-Teitelboim gravity

We show analytically that there exist compact stellar objects akin to neutron stars whose radius is smaller than the Schwarzschild radius defined by Arnowitt-Deser-Misner (ADM) mass. The radius of the compact object is defined by the radius where the energy density and the pressure of ordinary matter vanish, while clouds of scalar(s) can extend beyond this radius – a situation that is often encountered in modified gravity theories, like <math><mi>F</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity and the scalar–Einstein–Gauss-Bonnet gravity. The clouds of scalar mode(s) give additional contributions to the ADM mass and as a result, the corresponding Schwarzschild radius given by the ADM mass can be larger than that of the compact object.

Sergei D. Odintsov

Germany

Poland

Armen Sedrakian

Extremely small stars in scalar-tensor gravity: When stellar radius is less than Schwarzschild one

We investigate gravitational waves in the <math><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math> gravity, i.e., a geometric theory of gravity described by a non-metric compatible connection, free from torsion and curvature, known as symmetric-teleparallel gravity. We show that <math><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math> gravity exhibits only two massless and tensor modes. Their polarizations are transverse with helicity equal to two, exactly reproducing the plus and cross tensor modes typical of General Relativity. In order to analyze these gravitational waves, we first obtain the deviation equation of two trajectories followed by nearby freely falling point-like particles and we find it to coincide with the geodesic deviation of General Relativity. This is because the energy-momentum tensor of matter and field equations are Levi-Civita covariantly conserved and, therefore, free structure-less particles follow, also in <math><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math> gravity, the General Relativity geodesics. Equivalently, it is possible to show that the curves are solutions of a force equation, where an extra force term of geometric origin, due to non-metricity, modifies the autoparallel curves with respect to the non-metric connection. In summary, gravitational waves produced in non-metricity-based <math><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo></math> gravity behave as those in torsion-based <math><mi>f</mi><mo>(</mo><mi>T</mi><mo>)</mo></math> gravity and it is not possible to distinguish them from those of General Relativity only by wave polarization measurements. This shows that the situation is different with respect to the curvature-based <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity where an additional scalar mode is always present for <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>≠</mo><mi>R</mi></math>.

Italy

Salvatore Capozziello

Maurizio Capriolo

Gravitational waves in ( ) Non-metric Gravity via Geodesic Deviation

The propagation speed of the gravitational wave in scalar–Einstein–Gauss-Bonnet (sEGB) gravity is generally different from that of light. Using differential equation conditions for the speed of gravitational waves to coincide with the light speed in the expanding universe, we constructed a general class of sEGB gravities where this condition is satisfied and realistic inflation occurs. It is demonstrated that the condition that the speed of gravitational wave coincides with that of the light in the Friedmann-Lemaître-Robertson-Walker (FRLW) universe is always different from the condition for gravitational wave speed in the sEGB black hole background. Moreover, it is shown that when gravitational wave speed in sEGB black hole is equal to the speed of light the black hole spacetime geometry is changing too so that formally there is no solution for such sEGB black hole. This may indicate that sEGB black holes hardly can be considered as realistic black holes unless some reasonable scenario to make gravitational wave speed to be equal to that of light is proposed, at least asymptotically.

Propagation speed of gravitational wave in scalar–Einstein–Gauss-Bonnet gravity

Generalized entropy, that has been recently proposed, puts all the known and apparently different entropies like The Tsallis, the Rényi, the Barrow, the Kaniadakis, the Sharma-Mittal and the loop quantum gravity entropy within a single umbrella. However, the microscopic origin of such generalized entropy as well as its relation to thermodynamic system(s) is not clear. In the present work, we will provide a microscopic thermodynamic explanation of generalized entropy(ies) from canonical and grand-canonical ensembles. It turns out that in both the canonical and grand-canonical descriptions, the generalized entropies can be interpreted as the statistical ensemble average of a series of microscopic quantity(ies) given by various powers of <math><msup><mrow><mo>(</mo><mo>−</mo><mi>k</mi><mi>ln</mi><mo>⁡</mo><mi>ρ</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup></math> (with n being a positive integer and ρ symbolizes the phase space density of the respective ensemble), along with a term representing the fluctuation of Hamiltonian and number of particles of the system under consideration (in case of canonical ensemble, the fluctuation on the particle number vanishes).

India

Tanmoy Paul

Microscopic interpretation of generalized entropy

Few parameters dependent generalised entropy includes Tsallis entropy, Rényi entropy, Sharma-Mittal entropy, Barrow entropy, Kaniadakis entropy, etc as particular representatives. Its relation to physical systems is not always clear. In this paper, we propose the microscopic thermodynamic description for an arbitrary generalised entropy in terms of the particle system. It is shown that the change in the volume of the phase space of the particle system in the micro-canonical description or the difference in the integration measure in the phase space in the canonical description may lead to generalised entropy. Our consideration may help us understand the structure of quantum gravity.

Micro-canonical and canonical description for generalised entropy

In the formalism of generalized holographic dark energy, the infrared cut-off <math><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub></math> is generalized to the form, <math><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>IR</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¨</mo></mrow></mover></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><msub><mrow><mover><mrow><mi>L</mi></mrow><mrow><mo>¨</mo></mrow></mover></mrow><mrow><mi>f</mi></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>a</mi><mo>,</mo><mi>H</mi><mo>,</mo><mover><mrow><mi>H</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>,</mo><mover><mrow><mi>H</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>,</mo><mo>⋯</mo><mo>)</mo></mrow></math>, where <math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math> and <math><msub><mrow><mi>L</mi></mrow><mrow><mi>f</mi></mrow></msub></math> are the particle horizon and the future horizon, respectively (moreover, a is the scale factor and H is the Hubble parameter of the universe). Based on such formalism, we establish a holographic realization of constant roll inflation during the early universe, where the corresponding cut-off depends on the Hubble parameter and its derivatives (up to the second order). The viability of this holographic constant roll inflation with respect to the Planck data in turn puts a certain bound on the infrared cut-off at the time of horizon crossing. Such holographic correspondence of constant roll inflation is extended to the scenario where the infrared cut-off is corrected by the ultraviolet one, which may originate due to quantum effects. Besides the mere inflation, we further propose the holographic realization of an <math><mi>u</mi><mi>n</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>d</mi></math> cosmic scenario from constant roll inflation (at the early time) to the dark energy era (at the late time) with an intermediate radiation dominated era followed by a Kamionkowski like reheating stage. In such a unified holographic scenario, the inflationary quantities (like the scalar spectral index and the tensor-to-scalar ratio) and the dark energy quantities (like the dark energy EoS parameter and the present Hubble rate) prove to be simultaneously compatible with observable constraints for suitable ranges of the infrared cut-off and the other model parameters. Moreover the curvature perturbations at super-Hubble scale prove to be a constant (with time) during the entire cosmic era, which in turn ensures the stability of the model under consideration.

Holographic realization of constant roll inflation and dark energy: An unified scenario

In the realm of the Bekenstein-Hawking entropy, the thermodynamics of apparent horizon bridges with the usual FLRW (Friedmann-Lemaître-Robertson-Walker) equation only for a special case where the matter field is given by a perfect fluid having equation of state (EoS) parameter =−1, i.e. <math><mi>p</mi><mo>=</mo><mo>−</mo><mi>ρ</mi></math> with ρ and p representing the energy density and the pressure of the fluid, respectively. To include the case <math><mi>p</mi><mo>≠</mo><mo>−</mo><mi>ρ</mi></math>, we consider the modification of the Bekenstein-Hawking entropy in the present work. In particular, we develop an entropy function that leads to the usual FLRW equations, for a <math><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>l</mi></math> EoS of the matter fluid given by <math><mi>p</mi><mo>=</mo><mi>w</mi><mi>ρ</mi></math>, directly from the thermodynamics of the apparent horizon. The newly developed entropy acquires a correction over the Bekenstein-Hawking entropy and differs from the known entropies like the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis, and Loop Quantum Gravity entropies proposed so far. Based on this finding, we examine how the Friedmann equations of the apparent horizon cosmology are accordingly modified if one starts with a general entropy depending on the Bekenstein-Hawking entropy. This results in some interesting cosmological consequences during the early and late stages of the universe.

Modified cosmology from the thermodynamics of apparent horizon

We show that the mimetic theory with the constraint <math><msup><mrow><mi>g</mi></mrow><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><msub><mrow><mo>∂</mo></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ϕ</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>σ</mi></mrow></msub><mi>ϕ</mi><mo>=</mo><mn>1</mn></math> cannot realize the black hole geometry with the horizon(s). To overcome such issue, we may change the mimetic constraint a little bit by <math><mi>ω</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo><msup><mrow><mi>g</mi></mrow><mrow><mi>ρ</mi><mi>σ</mi></mrow></msup><msub><mrow><mo>∂</mo></mrow><mrow><mi>ρ</mi></mrow></msub><mi>ϕ</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>σ</mi></mrow></msub><mi>ϕ</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, where <math><mi>ω</mi><mo>(</mo><mi>ϕ</mi><mo>)</mo></math> is a function of the scalar field ϕ. As an example, we consider <math><mo>(</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>)</mo></math>-dimensional mimetic gravity with the mimetic potential and construct black hole (BH) solutions by using this modified constraint. We study three different classes: In the first class, we assume the Lagrange multiplier and mimetic potential are vanishing and obtain a BH solution that fully matches the BH of GR despite the non-triviality of the mimetic field which ensures the study presented in Nashed et al. (2019) [42]. In the second class, we obtain a BH having constant mimetic potential and a non-trivial form of the Lagrange multiplier. In the third class, we obtain a new BH solution with non-vanishing values of the mimetic field, the Lagrange multiplier, and the mimetic potential. In any case, the solutions correspond to the space-time with only one horizon but we show that the formalism for the constraint works.

Egypt

G.G.L. Nashed

Consistency between black hole and mimetic gravity – Case of (2 + 1)-dimensional gravity

We propose a new four-parameter entropy function that generalizes the Tsallis, Rényi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable limits of the parameters. Consequently, we address the early and late universe cosmology corresponding to the proposed four-parameter entropy function. As a result, it turns out that the entropic cosmology from the generalized entropy function can unify the early inflation to the late dark energy era of the universe. In such a unified scenario, we find that – (1) the inflation era is described by a quasi de-Sitter evolution of the Hubble parameter, which has an exit at around 58 e-folding number, (2) the inflationary observable quantities like the spectral index for primordial scalar perturbation and the tensor-to-scalar ratio are simultaneously compatible with the recent Planck data, and (3) regarding the late time cosmology, the dark energy EoS parameter is found to be consistent with the Planck result for the same values of the entropy parameters that lead to the viable inflation during the early universe. Furthermore, we show that the entropic cosmology from the proposed entropy function is equivalent to holographic cosmology, where the respective holographic cut-offs are determined in terms of either particle horizon and its derivative or future horizon and its derivative.