Entropic force, holography and thermodynamics for static space-times

Recently Verlinde has suggested a new approach to gravity which interprets gravitational interaction as a kind of entropic force. The new approach uses the holographic principle by stating that the information is kept on the holographic screens which coincide with equipotential surfaces. Motivated by this new interpretation of gravity (but not being limited by it) we study equipotential surfaces, the Unruh–Verlinde temperature, energy and acceleration for various static space-times: generic spherically symmetric solutions, axially symmetric black holes immersed in a magnetic field, traversable spherically symmetric wormholes of an arbitrary shape function, system of two and more extremely charged black holes in equilibrium. In particular, we have shown that the Unruh–Verlinde temperature of the holographic screen reaches absolute zero on the wormhole throat independently of the particular form of the wormhole solution.     

Modified Friedmann equations from Debye entropic gravity

A remarkable new idea on the origin of gravity was recently proposed by Verlinde who claimed that the laws of gravitation are no longer fundamental, but rather emerge naturally as an entropic force. In Verlinde derivation, the equipartition law of energy on the holographic screen plays a crucial role. However, the equipartition law of energy fails at the very low temperature. Therefore, the formalism of the entropic force should be modified while the temperature of the holographic screen is very low. Considering the Debye entropic gravity and following the strategy of Verlinde, we derive the modified Newton’s law of gravitation and the corresponding Friedmann equations which are valid in all range of temperature. In the limit of strong gravitational field, i.e. high temperature compared to Debye temperature, T » T _D , one recovers the standard Newton’s law and Friedmann equations. We also generalize our study to the entropy corrected area law and derive the dynamical cosmological equations for all range of temperature. Some limits of the obtained results are also studied.     

Temperature and Energy of 4-Dimensional Axisymmetric Black Holes from Entropic Force

We investigate the temperature and energy on holographic screens for 4-dimensional axisymmetric black holes with the entropic force idea proposed by Verlinde. According to the principle of thermal equilibrium, the location of holographic screen outside the axisymmetric black hole horizon is not a equivalent radius surface. The location of isothermal holographic screen outside the axisymmetric black hole horizon is obtained. Using the equipartition rule, we derive the correction expression of energy of isothermal holographic screen. When holographic screens are far away the black hole horizon, the entropic force of charged rotating particles can be expressed as Newton’s law of gravity. When the screen crosses the event horizon, the temperature of the screen agrees with the Hawking temperature and the entropic force gives rise to the surface gravity for both of the black holes.     

Remarks on the Equipartition Rule and Thermodynamics of Reissner-Nordstrom Black Holes

In Verlinde’s work, gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. In this paper, we investigate the thermodynamic property of Reissner-Nordstrom black holes from the equipartition rule and holographic scenario. As a result, the first law of thermodynamics of the black holes is recovered.     

Thoughts on entropic gravity in the Parikh–Wilczek tunneling model of Hawking radiation

In this letter, we use the Parikh–Wilczek tunneling model of Hawking radiation to illustrate that a reformulation of Verlinde’s entropic gravity is needed to derive Newton’s law for a temperature-varying screen, required by the conservation of energy. Furthermore, the entropy stored in the holographic screen is shown to be additive and its temperature dependence can be obtained.     

Debye Entropic Force and Modified Newtonian Dynamics

Verlinde has suggested that the gravity has an entropic origin, and agravitational system could be regarded as a thermodynamical system. It is well-known that the equipartition law of energy is invalid at very low temperature. Therefore, entropic force should be modified while the temperature of the holographic screen is very low. It is shown that the modified entropic force is proportional to the square of the acceleration, while the temperature of the holographic screen is much lower than the Debye temperature T_D. The modified entropic force returns to the Newton's law of gravitation while the temperature of the holographic screen is much higher than the Debye temperature. The modified entropic force is connected with modified Newtonian dynamics (MOND). The constant a₀ involved in MOND is linear in the Debye frequency ω_D, which can be regarded as the largest frequency of the bits in screen.We find that there do have a strong connection between MOND and cosmology in the framework of Verlinde's entropic force, if the holographic screen is takento be bound of the Universe. The Debye frequency is linear in the Hubble constant H₀.     

Modified entropic gravity revisited

Inspired by Verlinde’s idea,some modified versions of entropic gravity have been suggested.Extending them in a unified formalism,herein we derive the generalized gravitational equations accordingly.From gravitational equations,the energy-momentum conservation law and cosmological equations are investigated.The covariant conservation law of energy-momentum tensor severely constrains viable modifications of entropic gravity.A discrepancy arises when two independent methods are applied to the homogeneous isotropic universe,posing a serious challenge to modified models of entropic gravity.     

Modified entropic gravitation in superconductors

Verlinde recently developed a theoretical account of gravitation in terms of an entropic force. The central element in Verlinde’s derivation is information and its relation with entropy through the holographic principle. The application of this approach to the case of superconductors requires to take into account that information associated with superconductor’s quantum vacuum energy is not stored on Planck size surface elements, but in four volume cells with Planck-Einstein size. This has profound consequences on the type of gravitational force generated by the quantum vacuum condensate in superconductors, which is closely related with the cosmological repulsive acceleration responsible for the accelerated expansion of the Universe. Remarkably this new gravitational type force depends on the level of breaking of the weak equivalence principle for cooper pairs in a given superconducting material, which was previously derived by the author starting from similar principles. It is also shown that this new gravitational force can be interpreted as a surface force. The experimental detection of this new repulsive gravitational-type force appears to be challenging.     

A Note on Friedmann Equation of FRW Universe in DeformedHo\v{r}ava-Lifshitz Gravity from Entropic Force

With entropic interpretation of gravity proposed by Verlinde, we obtain the Friedmann equation of the Friedmann-Robertson-Walker universe for the deformed Ho\v{r}ava-Lifshitz gravity. It is shown that, when the parameter of Ho\v{r}ava-Lifshitz gravityω→∝, the modified Friedmann equation will go back to the one in Einstein gravity. This results may imply that the entropic interpretation of gravity is effective for the deformed Ho\v{r}ava-Lifshitz gravity.     

Holographic considerations on a Machian Universe

MOND theory explains the rotation curves of the galaxies. Verlinde’s ideas establish an entropic origin for gravitational forces and Tsallis principle generalizes the theory of Boltzmann–Gibbs. In this work we have promoted a connection between these recent approaches, that at first sight seemed to have few or no points in common, using the Mach’s principle as the background. In this way we have used Tsallis formalism to calculate the main parameters of the Machian Universe including the Hubble parameter and the age of the Universe. After that, we have also obtained a new value for the Tsallis parameter via Mach’s principle. Using Verlinde’s entropic gravity we have obtained new forms for MOND’s well established ingredients. Finally, based on the relations between particles and bits obtained here, we have discussed the idea of bits entanglement in the holographic screen.     

Cosmological Consequences of the Holographic Scenario

In this paper we discuss some consequences of Verlinde’s holographic gravity model. Among other things, it yields the observed acceleration of the universe and the inflationary period at early universe obviating the Dark energy. From the Verlinde’s theory of gravity the first phenomenological Modified Newtonian dynamics obviating the Dark matter can be deduced. Moreover through the connection with the Modification of inertia resulting from a Hubble-scale Casimir effect (MiHsC) of McCulloch the model gives a promising possible explanation to the Pioneer anomaly, the flyby anomalies, the Tajmar effect and the minimum mass observed in the disc galaxies.     

Planck-scale corrections to Friedmann equation

Recently, Verlinde proposed that gravity is an emergent phenomenon which originates from an entropic force. In this work, we extend Verlinde’s proposal to accommodate generalized uncertainty principles (GUP), which are suggested by some approaches to quantum gravity such as string theory, black hole physics and doubly special relativity (DSR). Using Verlinde’s proposal and two known models of GUPs, we obtain modifications to Newton’s law of gravitation as well as the Friedmann equation. Our modification to the Friedmann equation includes higher powers of the Hubble parameter which is used to obtain a corresponding Raychaudhuri equation. Solving this equation, we obtain a leading Planck-scale correction to Friedmann-Robertson-Walker (FRW) solutions for the p = ωp equation of state.     

The dark-baryonic matter mass relation for observational verification in Verlinde’s emergent gravity

Recently, a new interesting idea of origin of gravity has been developed by Verlinde. In this scheme of emergent gravity, where horizon entropy, microscopic de Sitter states and relevant contribution to gravity are involved, an entropy displacement resulting from matter behaves as a memory effect and can be exhibited at sub-Hubble scales, namely, the entropy displacement and its “elastic” response would lead to emergent gravity, which gives rise to an extra gravitational force. Then galactic dark matter effects may origin from such extra emergent gravity. We discuss some concepts in Verlinde’s theory of emergent gravity and point out some possible problems or issues, e.g., the gravitational potential caused by Verlinde’s emergent apparent dark matter may no longer be continuous in spatial distribution at ordinary matter boundary (such as a massive sphere surface). In order to avoid the unnatural discontinuity of the extra emergent gravity of Verlinde’s apparent dark matter, we suggest a modified dark-baryonic mass relation (a formula relating Verlinde’s apparent dark matter mass to ordinary baryonic matter mass) within this framework of emergent gravity. The modified mass relation is consistent with Verlinde’s result at relatively small scales (e.g., \(R<3h_{70}^{-1}\) Mpc). However, it seems that, compared with Verlinde’s relation, at large scales (e.g., gravitating systems with \(R>3h_{70}^{-1}\) Mpc), the modified dark-baryonic mass relation presented here might be in better agreement with the experimental curves of weak lensing analysis in the recent work of Brouwer et al. Galactic rotation curves are compared between Verlinde’s emergent gravity and McGaugh’s recent model of MOND (Modified Newtonian Dynamics established based on recent galaxy observations). It can be found that Verlinde rotational curves deviate far from those of McGaugh MOND model when the MOND effect (or emergent dark matter) dominates. Some applications of the modified dark-baryonic mass relation inspired by Verlinde’s emergent gravity will be addressed for galactic and solar scales. Potential possibilities to test this dark-baryonic mass relation as well as apparent dark matter effects, e.g., planetary perihelion precession at Solar System scale, will be considered. This may enable to place some constraints on the magnitudes of the MOND characteristic acceleration at the small solar scale.     

Entropy bounds and Cardy–Verlinde formula in Yang–Mills theory

Using gauge formulation of gravity the three-dimensional SU(2) YM theory equations of motion are presented in equivalent form as FRW cosmological equations. With the radiation, the particular (periodic, big bang – big crunch) three-dimensional universe is constructed. Cosmological entropy bounds (so-called Cardy–Verlinde formula) have the standard form in such universe. Mapping such universe back to YM formulation we got the thermal solution of YM theory. The corresponding holographic entropy bounds (Cardy–Verlinde formula) in YM theory are constructed. This indicates to universal character of holographic relations.     

Noncommutative geometry inspired entropic inflation

Recently Verlinde proposed that gravity can be described as an emergent phenomena arising from changes in the information associated with the positions of material bodies. By using noncommutative geometry as a way to describe the microscopic microstructure of quantum spacetime, we derive modified Friedmann equation in this setup and study the entropic force modifications to the inflationary dynamics of early universe.     

Minimal Length, Maximal Momentum and the Entropic Force Law

Different candidates of quantum gravity proposal such as string theory, noncommutative geometry, loop quantum gravity and doubly special relativity, all predict the existence of a minimum observable length and/or a maximal momentum which modify the standard Heisenberg uncertainty principle. In this paper, we study the effects of minimal length and maximal momentum on the entropic force law formulated recently by E. Verlinde.     

Unruh–Verlinde Temperature and Energy of Charged Black Hole via Entropic Force

Verlinde’s recent work, which showed that gravity may be explained as entropic force caused by the changes of information associated with the positions of material bodies, is extended to study the Unruh–Verlinde temperature and energy of a static spherically symmetric charged black hole. The results indicate that the Unruh–Verlinde temperature is equal to the Hawking temperature at the outer and inner horizons. The energy is dependent on the radius of the screen, which is also a consequence of the Gauss’ laws of gravity and electrostatics.     

Generalized uncertainty principle and black hole entropy

Recently, there has been much attention devoted to resolving the quantum corrections to the Bekenstein–Hawking black hole entropy. In particular, many researchers have expressed a vested interest in the coefficient of the logarithmic term of the black hole entropy correction term. In this Letter, we calculate the correction value of the black hole entropy by utilizing the generalized uncertainty principle and obtain the correction terms of entropy, temperature and energy caused by the generalized uncertainty principle. We calculate Cardy–Verlinde formula after considering the correction. In our calculation, we only think that the Bekenstein–Hawking area theorem is still valid after considering the generalized uncertainty principle and do not introduce any assumption. In the whole process, the physics idea is clear and calculation is simple. It offers a new way for studying the corrections caused by the generalized uncertainty principle to the black hole thermodynamic quantity of the complicated spacetime.