The idea of Verlinde that gravity is an entropic force caused by information changes associated with the positions of material bodies, is used in the present work for the FRW model of the Universe. Using modified Hawking temperature, the Friedmann equations are derived on any horizon. For the validity of the first law of thermodynamics (i.e., Clausius relation) it is found that there is modification of Bekenstein entropy on the horizon. However, using equipartition law of energy, Bekenstein entropy is recovered.
In this paper we study Hawking-like radiation in a FRW-Universe using semi-classical tunnelling approach and the Hamilton-Jacobi method. Radial null geodesics are used to picture the process as a "tunnelling" of particles from behind the apparent horizon and Hawking -like temperature has been calculated. Quantum corrections have been evaluated in the Hamilton-Jacobi Method by solving Klein-Gordan wave equation and the temperature agrees at the semiclassical level. Also it is found that Hamilton-Jacobi formalism does not depend on the choice of the coordinate system. Finally, leading order corr…
In the paper [arXiv:0809.1554], the scalar particles' Hawking radiation from the apparent horizon of Friedmann-Robertson-Walker(FRW) universe was investigated by using the tunneling formalism. They obtained the Hawking temperature associated with the apparent horizon, which was extensively applied in investigating the relationship between the first law of thermodynamics and Friedmann equations. In this paper, we calculate Fermions' Hawking radiation from the apparent horizon of FRW universe via tunneling formalism. Applying WKB approximation to the general covariant Dirac equation in FRW space…
In an accelerated expanding universe, one can expect the existence of an event horizon. It may be interesting to study the thermodynamics of the Friedmann-Robertson-Walker (FRW) universe at the event horizon. Considering the usual Hawking temperature, the first law of thermodynamics does not hold on the event horizon. To satisfy the first law of thermodynamics, it is necessary to redefine Hawking temperature. In this paper, using the redefinition of Hawking temperature and applying the first law of thermodynamics on the event horizon, the Friedmann equations are obtained in f(R) gravity from t…
In this paper, we investigate the thermodynamics especially the Hawking-Page-like phase transition of the McVittie space-time. We formulate the first law of thermodynamics for the McVittie black hole, and find that the work density $W$ of the perfect fluid plays the role of the thermodynamic pressure, i.e. $P$:=$-W$. We also construct the thermodynamic equation of state for the McVittie black hole. Most importantly, by analysing the Gibbs free energy, we find that the Hawking-Page-like phase transition from FRW spacetime to McVittie black hole is possible in the case $P>0$.
It is well known that there is a Hawking radiation from the cosmological horizon of the de-sitter spacetime, and the de-sitter spacetime can be a special case of a FRW universe. Therefore, there may be a corresponding Hawking radiation in a FRW universe. Indeed, there have been several clues showing that there is a Hawking radiation from the apparent horizon of a FRW universe. In our paper, however, we find that the Hawking radiation may come from the cosmological horizon. Moreover, we also find that the Hawking radiation from the apparent horizon of a FRW universe in some previous works can b…
In this article we investigate the Hawking radiation of vector particles from the apparent horizon of a Friedmann-Robertson-Walker (FRW) universe in the framework of quantum tunneling method. Furthermore we use Proca equation, a relativistic wave equation for a massive/massless spin-1 particle (massless $γ$ photons, weak massive $W^{\pm}$ and $Z^{0}$ bosons, strong massless gluons and $ρ$ and $ω$ mesons) together with a Painlevé spacetime metric for the FRW universe. We solve the Proca equation via Hamilton-Jacobi (HJ) equation and the WKB approximation method. We recover the same result for t…
Hawking radiation is an important quantum phenomenon of black hole, which is closely related to the existence of event horizon of black hole. The cosmological event horizon of de Sitter space is also of the Hawking radiation with thermal spectrum. By use of the tunneling approach, we show that there is indeed a Hawking radiation with temperature, $T=1/2π\tilde r_A$, for locally defined apparent horizon of a Friedmann-Robertson-Walker universe with any spatial curvature, where $\tilde r_A$ is the apparent horizon radius. Thus we fill in the gap existing in the literature investigating the relat…
We consider the claim of Hawking and Page that the canonical measure applied to a FRW universe with a massive scalar field can solve the flatness problem regardless of inflation. By considering a general potential we are able to understand how the ambiguity for $Ω$ found by Page in the $R^2$ theory is present when the potential is bounded from above. We suggests reasons why such potentials are realistic. The ambiguity for the possibility of inflation present in the Gibbons- Hawking- Stewart measure can be resolved by an input from Quantum cosmology. We contrast the resulting measure obtained i…
We consider a freely falling holographic screen for the Schwarzschild and Reissner-Nordström black holes and evaluate the entropic force à la Verlinde. When the screen crosses the event horizon, the temperature of the screen agrees to the Hawking temperature and the entropic force gives rise to the surface gravity for both of the black holes.
I present a new derivation of Hawking radiation by applying the Crooks fluctuation theorem to the laws of black hole thermodynamics. Then, by analogy with the quantum fluctuation theorem, this allows one to identify microstates contributing to the black hole entropy. These microstates have an evolution operator on the horizon between initial and final states that is related to the entropic gravity proposal. Their full calculation perhaps requires a deeper understanding of nonperturbative Hawking radiation.
The Friedman-Robertson-Walker (FRW) space-time exhibits particle creation similar to Hawking radiation of a black hole. In this essay we show that this FRW Hawking radiation leads to an effective negative pressure fluid which can drive an inflationary period of exponential expansion in the early Universe. Since the Hawking temperature of the FRW space-time decreases as the Universe expands this mechanism naturally turns off and the inflationary stage transitions to a power law expansion associated with an ordinary radiation dominated Universe.
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 the Newton's law for a temperature-varying screen, demanded 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.
We study Hawking-like radiation in a Friedmann-Robertson-Walker (FRW) universe using the quasi-classical WKB/tunneling method which pictures this process as a "tunneling" of particles from behind the apparent horizon. The correct temperature of the Hawking-like radiation from the FRW spacetime is obtained using a canonical invariant tunneling amplitude. In contrast to the usual quantum mechanical WKB/tunneling problem where the tunneling amplitude has only a spatial contribution, we find that the tunneling amplitude for FRW spacetime (i.e. the imaginary part of the action) has both spatial and…
We revisit the derivation of the Hawking-Moss transition rate. Using the static coordinates, we show that the Euclidean action is entirely determined by the contribution of the entropy of de Sitter space which is proportional to the surface area of the horizon. This holographic feature is common to any static spacetime with a horizon on which the shift vector vanishes.
Assuming that time exists, a new, effective formulation of gravity is introduced, which lies in between the Wheeler-DeWitt approach and ordinary QFT. Remarkably, the Penrose-Hawking singularity of usual Friedman-Robertson-Walker cosmologies is naturally avoided there. The theory is made explicit via specific examples, and compared with loop quantum cosmology. It is argued that it is the regularization of the classical Hamiltonian performed in this last theory what avoid the singularity, rather than quantum effects as in our case.
In this article we have used the recently introduced redefined Hawking temperature on the event horizon and investigated whether the generalised second law of thermodynamics (GSLT) and thermodynamic equilibrium holds for both the event and the apparent horizons. Here we have considered FRW universe and examined the GSLT and thermodynamic equilibrium with three examples. Finally, we have concluded that from the thermodynamic viewpoint, the universe bounded by the event horizon is more realistic than that by the apparent horizon at least for some examples.
In this paper we revisit the topic of Hawking radiation as tunneling. We show that the imaginary part of the action of the tunneling particle should be reconstructed in a covariant way, as a line integral along the classical forbidden trajectory of tunneling particles. As the quantum tunneling phenomenon, the probability of tunneling is related to the imaginary part of the action for the classical forbidden trajectory. We do the calculations for massless and massive particles, in Schwarzschild coordinate and Painlevé coordinate. The construction of particle action is invariant under coordinate…
We consider dynamical black hole formation from a collapsing fluid described by a symmetric and flat FRW metric. Using the Hamilton-Jacobi method the local Hawking temperature for the formed trapping/apparent horizon is calculated. The local Hawking temperature depends on the tunneling path, which we take to be along a null direction $(Δs=0)$. We find that the local Hawking temperature depends directly on the equation of state of the collapsing fluid. We argue that Hawking radiation by quantum tunnelling from future inner and future outer trapping horizons is possible. However, only radiation …
We present a phenomenological model for baryogenesis based on particle creation in the Friedman-Robertson-Walker (FRW) spacetime. This study is a continuation of our proposal that Hawking-like radiation in FRW space-time explains several physical aspects of the early Universe including inflation. In this model we study a coupling between the FRW space-time, in the form of the derivative of the Ricci scalar, and the $B-L$ current, $J^μ _{B-L}$, which leads to a different chemical potential between baryons and anti-baryons resulting in an excess of baryons over anti-baryons with the right order …
We investigate the validity of the generalized second law of gravitational thermodynamics in a non-flat FRW universe containing the interacting new agegraphic dark energy with cold dark matter. The boundary of the universe is assumed to be enclosed by the dynamical apparent horizon. We show that for this model, the equation of state parameter can cross the phantom divide. We also present that for the selected model under thermal equilibrium with the Hawking radiation, the generalized second law is always satisfied throughout the history of the universe. Whereas, the evolution of the entropy of…
In this study, we investigate the effects of the one and two-parameters deformed systems on the Friedmann equations of the Friedmann-Robertson-Walker (FRW) universe in the context of the entropic gravity approach. We give simplified forms for the deformed Unruh temperature and Einstein field equations for three different deformed systems. Based on these compact equations, we derive the Friedmann equations with the effective gravitational and cosmological constant.
To explore the mechanism for the entropic force proposal in Entropic Gravity, we propose a specific thermodynamic process for states thermalized in local Hawking Temperature. We find when Casini's version of the Bekenstein bound is saturated, the thermodynamic force derived in the entanglement first law matches the local inertial force for the Schwarzschild solution, except for a negligible statistics-dependent factor. We argue the gravity viewed by static observers may have observable effects emerged from quantum statistics. The successful detailed calculation in this simple model inspires an…
Heisenberg's uncertainty principle is a fundamental element in quantum mechanics. It sets a bound on our ability to predict the measurement outcomes of two incompatible observables simultaneously. In quantum information theory, the uncertainty principle can be expressed using entropic measures. The entropic uncertainty relation can be improved by considering an additional particle as a memory particle. The presence of quantum correlation between the memory particle and the measured particle reduces the uncertainty. In a curved space-time, the presence of the Hawking radiation can reduce quantu…
Recently, a Hamilton-Jacobi method beyond semiclassical approximation in black hole physics was developed by \emph{Banerjee} and \emph{Majhi}\cite{beyond0}. In this paper, we generalize their analysis of black holes to the case of Friedmann-Robertson-Walker (FRW) universe. It is shown that all the higher order quantum corrections in the single particle action are proportional to the usual semiclassical contribution. The corrections to the Hawking-like temperature and entropy of apparent horizon for FRW universe are also obtained. In the corrected entropy, the area law involves logarithmic area…
In this paper, we propose a model in which an additional pressure due to the effects of the entropic force is added to the ideal fluid. Furthermore, we obtain the dynamic equation in the FRW universe which contains the quantum gravitational effects based on the description of entropic force and emergence of space. Our model can well explain the age of the universe and the effect of the current accelerating expansion. We give the relation between the luminosity distance and the redshift factor, and compare this relation with that of lambda cold dark matter model($ΛCDM$ model).
The Friedmann equation in the Friedmann-Robertson-Walker(FRW) universe with any spatial curvature is derived from the first law of thermodynamics on the event horizon. The key idea is to redefine a Hawking temperature on the event horizon. Furthermore, we obtain the evolution equations of the universe including the quantum correction and explore the evolution of the universe in the $f(R)$ gravity. In addition, we also investigate the generalized second law of thermodynamics in Einstein gravity and the $f(R)$ gravity. This perspective also implies that the first law of thermodynamics on the eve…
In the Big Rip cosmological scenario, a FRW universe containing dark energy with $w < -1$ in its equation of state (phantom energy) expands in such way that, in a finite time span, the scale factor diverges to infinity and the size of the cosmic horizon goes to zero. Here we revisit this scenario in light of the fact that Hawking radiation is expected to be generated at the apparent horizon of a FRW universe, and show that the energy density and temperature of that radiation both diverge at the Big Rip. We then use this fact to propose a new variant of Penrose's conformal cyclic cosmology mode…
We investigate the temperature and energy on holographic screens for 4-dimensional black holes with the entropic force idea proposed by Verlinde. We find that the "Unruh-Verlinde temperature" is equal to the Hawking temperature on the horizon and can be considered as a generalized Hawking temperature on the holographic screen outside the horizons. The energy on the holographic screen is not the black hole mass $M$ but the reduced mass $M_0$, which is related to the black hole parameters. With the replacement of the black hole mass $M$ by the reduced mass $M_0$, the entropic force can be writte…
We argue certain nonviolent local quantum field theory (LQFT) modification considered at the global horizon ($r=2M$) of a static spherically-symmetric black hole can lead to adiabatic leakage of quantum information in the form of Hawking particles. The source of the modification is (i) smooth at $r=2M$ and (ii) rapidly vanishing at $r\gg2M$. Furthermore, we restore the unitary evolution by introducing extra quanta which departs slightly from the generic Hawking emission without changing the experience of an infalling observer (no drama). Also, we suggest that a possible interpretation of the B…
We use the ideas of entropic gravity to derive the FRW cosmological model and show that for late time evolution we have an effective cosmological constant. By using the first law of thermodynamics and the modified entropy area relationship derived from the supersymmetric Wheeler-DeWitt equation of the Schwarzschild black hole, we obtain modifications to the Friedmann equations that in the late time regime gives an effective positive cosmological constant. Therefore, this simple model can account for the dark energy component of the universe by providing an entropic origin to the cosmological c…
Newtonian gravitation with some slight modifications, along with some highly simplified ideas from quantum field theory allow us to reproduce, at least at the level of back-of-the-envelope calculations, many results of black hole physics. We consider particle production by a black hole, the Newtonian equivalent of the Hawking temperature, and the Bekenstein entropy. Also, we are able to deduce Newtonian field equations from entropy. We finally study higher-order Newtonian theories under the same assumptions used for ordinary Newtonian theory. In a companion article we will look at entropic for…
We investigate the validity of generalized second law of thermodynamics of a physical system comprising of newly proposed dark energy model called Ricci Gauss-Bonnet and cold dark matter enveloped by apparent horizon and event horizon in flat Friedmann-Robertson-Walker (FRW) universe. For this purpose, Bekenstein entropy, Renyi, logarithmic and power law entropic corrections are used. It is found that this law exhibits the validity on both apparent and event horizons except for the case of logarithmic entropic correction at apparent horizon. Also, we check the thermodynamical equilibrium condi…
Does a dynamical black hole embedded in a cosmological FRW background emit Hawking radiation where a globally defined event horizon does not exist? What are the differences to the Schwarzschild black hole? What about the first law of black hole mechanics? We face these questions using the LTB cosmological black hole model recently published. Using the Hamilton-Jacobi and radial null geodesic-methods suitable for dynamical cases, we show that it is the apparent horizon which contributes to the Hawking radiation and not the event horizon. The Hawking temperature is calculated using the two diffe…
We analyze the thermodynamical consistency of entropic-force cosmological models. Our analysis is based on a generalized entropy scaling with an arbitrary power of the Hubble radius. The Bekenstein-Hawking entropy, proportional to the area, and the nonadditive $S_{δ=3/2}$-entropy, proportional to the volume, are particular cases. One of the points to be solved by entropic-force cosmology for being taken as a serious alternative to mainstream cosmology is to provide a physical principle that points out what entropy and temperature have to be used. We determine the temperature of the universe ho…
We discuss the evolution of radiation and Bekenstein-Hawking entropies in expanding isotropic universes. We establish a general relation which shows why it is inevitable that there is currently a huge difference in the numerical values of these two entropies. Some anthropic constraints on their values are given and other aspects of the cosmological 'entropy gap' problem are discussed. The coincidence of the classical and quantum entropies for black holes with Hawking lifetime equal to the age of the universe, and hence of radius equal to the proton size, is shown to be identical to the conditi…
The paper deals with universal thermodynamics for FRW model of the universe bounded by apparent (or event) horizon. Assuming Hawking temperature on the horizon, the unified first law is examined on the horizon for different gravity theories. The results show that equilibrium configuration is preserved with a modification to Bekenstein entropy on the horizon.
We discuss the emergence of relativistic effects in the Machian universe with a global preferred frame and use thermodynamic considerations to clarify the origin of gravity as an entropic force and the origin of dark energy/cosmic acceleration as related to the Hawking-Unruh temperature at the universe's horizon.
In the previous analyses in the literature about the generalized second law (GSL) in an accelerated expanding universe the usual area relation for the entropy, i.e. $S={A\over 4G}$, was used for the cosmological horizon entropy. But this entropy relation may be modified due to thermal and quantum fluctuations or corrections motivated by loop quantum gravity giving rise to S={A\over 4}+πα\ln({A\over 4})+γ, where $α$ and $γ$ are some constants whose the values are still in debate in the literature. Our aim is to study the constraints that GSL puts on these parameters. Besides, we investigate the…
We discuss consequences of an entropic view of gravity, which differs in details from the original proposals of Jacobson and Verlinde. We assume the entropy is localized in the degrees of freedom of the systems interacting gravitationally. We then find there is a density-dependent minimum length for Newtonian gravity. This makes renormalizability issues irrelevant. It also follows that at nanometer scales Newton's universal law fails, black holes with mass $M_{BH} \lesssim 10^{-13} M_{\odot}$ do not exist and all Hawking radiation is at unobservably low temperature.
A toy model of Einstein gravity with a Gauss-Bonnet classically "entropic" term mimicking a quantum correction is considered. The static black hole solution due to Tomozawa is found and generalized with the inclusion of non trivial horizon topology, and its entropy evaluated deriving the first law by equations of motion. As a result the Bekenstein-Hawking area law turns to be corrected by a logarithmic area term. A Misner-Sharp expression for the mass of black hole is found. Within a Friedmann-Lemaître-Robertson-Walker (FLRW) cosmological setting, the model is used in order to derive modified …
Modak and Singleton [Phys. Rev. D 86, 123515 (2012)] have presented Hawking-like radiation for cosmological inflation which has a natural "turn on" and a natural "turn off" mechanism. This Hawking-like radiation results in an effective negative pressure "fluid" which leads to a rapid period of expansion in the very early Universe. We discuss that the turn on mechanism can not happen for FRW model in early universe because its horizon is apparent horizon not event horizon. Hence, we cannot apply geometric optic approximation which is a necessary condition for tunneling method. It was shown that…
It has been proposed that the entropy of any object must satisfy fundamental (holographic or Bekenstein) bounds set by the object's size and perhaps its energy. However, most discussions of these bounds have ignored the possibility that objects violating the putative bounds could themselves become important components of Hawking radiation. We show that this possibility cannot a priori be neglected in existing derivations of the bounds. Thus this effect could potentially invalidate these derivations; but it might also lead to observational evidence for the bounds themselves.
Universal thermodynamics for FRW model of the Universe bounded by apparent/event horizon has been considered for massive gravity theory. Assuming Hawking temperature and using the unified first law of thermodynamics on the horizon, modified entropy on the horizon has been determined. For simple perfect fluid with constant equation of state, generalized second law of thermodynamics and thermodynamical equilibrium have been examined on both the horizons.
Recently, the vacuum energy of the QCD ghost in a time-dependent background is proposed as a kind of dark energy candidate to explain the acceleration of the Universe. In this model, the energy density of the dark energy is proportional to the Hubble parameter $H$, which is the Hawking temperature on the Hubble horizon of the Friedmann-Robertson-Walker (FRW) Universe. In this paper, we generalized this model and choice the Hawking temperature on the so-called trapping horizon, which will coincides with the Hubble temperature in the context of flat FRW Universe dominated by the dark energy comp…
From the entropy argument for the dS swampland conjecture which connects the Gibbons-Hawking entropy bound with the distance conjecture, we find the entropic quasi-dS instability time given by $1/(\sqrt{ε_H}H)\log(m_{\rm Pl}/H)$ as the lifetime of quasi-dS spacetime. It depends on the slow-roll parameter, and contains the logarithmic factor $\log(m_{\rm Pl}/H)$ which can be found in the scrambling (or decoherence) time as well. Such a logarithmic factor enhances the geodesic distance of the modulus from the mere Planck scale, and also possibly relaxes the bound on $m_{\rm Pl}^2 \nabla^2 V/V$.
A study of Universal thermodynamics is done in Lanczos-Lovelock gravity. The Universe is chosen as FRW model bounded by apparent or event horizon. The unified first law is examined, assuming extended Hawking temperature on the horizon. As a result there is a modification of Bekenstein entropy on the horizons. Further the validity of generalized second law of thermodynamics and thermodynamical equilibrium are also investigated for perfect fluid (with constant equation of state), modified chaplygin gas model and holographic dark energy model.
One proposal by Verlinde \cite{Verlinde:2010hp} is that gravity is not a fundamental, but an entropic force. In this way, Verlinde has provide us with a way to derive the Newton's law of gravitation from the Bekenstein-Hawking entropy-area formula. On the other hand, since it has been demonstrated that this formula is susceptible to quantum gravity corrections, one may hope that these corrections could be inherited by the Newton's law. In this way, the entropic interpretation of Newton's law could be a prolific way in order to get verifiable or falsifiable quantum corrections to ordinary gravi…
A study of Universal thermodynamics is done in the frame work of RSII brane model and DGP brane scenario. The Universe is chosen as FRW model bounded by apparent or event horizon. Assuming extended Hawking temperature on the horizon, the unified first law is examined for perfect fluid (with constant equation of state) and modified Chaplygin gas model. As a result there is a modification of Bekenstein entropy on the horizons. Further the validity of the generalized second law of thermodynamics and thermodynamical equilibrium are also investigated.
We investigate entropic force cosmological models with the possibility of matter creation and energy exchange between the bulk and the horizon of a homogeneous and an isotropic flat Universe. We consider three different kinds of entropy, Bekenstein's, the non-extensive Tsallis-Cirto's and the quartic entropy, plus some phenomenological functional forms for matter creation rate to model different entropic force models and put the observational constraints on them. We show that while most of them are basically indistinguishable from a standard $Λ$CDM scenario, the Bekenstein entropic force model…
A possible source of black hole entropy could be the entanglement of quantum fields in and out the horizon. The entanglement entropy of the ground state obeys the area law. However, a correction term proportional to a fractional power of area results when the field is in a superposition of ground and excited states. Inspired by the power-law corrections to entropy and adopting the viewpoint that gravity emerges as an entropic force, we derive modified Newton's law of gravitation as well as the corrections to Friedmann equations. In a different approach, we obtained power-law corrected Friedman…
In this paper, we study the thermodynamic quantities of Friedmann-Robertson-Walker (FRW) universe by using the tunneling formalism beyond semiclassical approximation developed by \emph{Banerjee} and \emph{Majhi}\cite{beyond0}. For this we first calculate the corrected Hawking-like temperature on apparent horizon by considering both scalar particle and fermion tunneling. With this corrected Hawking-like temperature, the explicit expressions of the corrected entropy of apparent horizon for various gravity theories including Einstein gravity, Gauss-Bonnet gravity, Lovelock gravity, $f(R)$ gravity…
The uncertainty principle restricts our ability to simultaneously predict the measurement outcomes of two incompatible observables of a quantum particle. However, this uncertainty could be reduced and quantified by a new Entropic Uncertainty Relation (EUR). By the open quantum system approach, we explore how the nature of de Sitter space affects the EUR. When the quantum memory $A$ freely falls in the de Sitter space, we demonstrate that the entropic uncertainty acquires an increase resulting from a thermal bath with the Gibbons-Hawking temperature. And for the static case, we find that the te…
Considering the quantum statistics of the degrees of freedom on the holographic screen it is shown that the ratio of the number of excited bulk degrees of freedom to the number of excited surface degrees of freedom, is given by the MOND interpolating function μ. This relationship is shown to hold also in AQUAL, and in the extension of MOND to de Sitter space. Based on the relationship between the entropy, and the number of degrees of freedom on the holographic screen, a simple expression, relating the MOND interpolating function to the ratio of the two-dimensional entropy to Bekenstein-Hawking…
We give an exact solution for the static force between two black holes at the turning points in their binary motion. The results are derived by Gibbs' principle and the Bekenstein-Hawking entropy applied to the apparent horizon surfaces in time-symmetric initial data. New power laws are derived for the entropy jump in mergers, while Newton's law is shown to derive from a new adiabatic variational principle for the Hilbert action in the presence of apparent horizon surfaces. In this approach, entropy is strictly monotonic such that gravity is attractive for all separations including mergers, an…
