Trapping Horizon Research Articles

Thermodynamics of FRW Universe: Heat engine

We assume the non-flat Friedmann-Robertson-Walker (FRW) Universe as a thermodynamical system. We assume the cosmological horizon as an inner trapping horizon which is treated as dynamical apparent horizon of FRW Universe. We write the dynamical apparent horizon radius and temperature on the apparent horizon. We assume that the fluid pressure as thermodynamical pressure of the system. Using Hayward's unified first law, Clausius relation and Friedmann equations with cosmological constant, we obtain the entropy on the apparent horizon. We assume that the cosmological constant provides the thermodynamic pressure of the system. We obtain the entropy, surface area, volume, temperature, Gibb's Helmholtz's free energies, specific heat capacity of the FRW Universe due to the thermodynamic system. We study the Joule-Thomson expansion of the FRW Universe and by evaluating the positive sign of Joule-Thomson coefficient, we determine that the FRW Universe obeys the cooling nature. We also find the inversion temperature and inversion pressure. Next we demonstrate the thermodynamical FRW Universe as heat engine. For Carnot cycle, we obtain the work done and the maximum efficiency. Also for new engine, we study the work done and its efficiency.

Towards consistent black-to-white hole bounces from matter collapse

This article presents a new model-independent constraint for bouncing black hole geometries. This constraint, applicable for a physical black hole formed by a collapsing compact object, sets a bound on the minimal allowed radius of the time-like surface of the collapsing star at the bounce. It follows from this bound that the shell always bounces in an untrapped region or precisely on a trapping horizon. This constraint is of purely kinematical origin as it descends from the continuity of the metric between the geometries describing the interior and exterior of the collapsing object so that it is completely model-independent. The second part of this work investigates the conditions under which an effective extension of the Oppenheimer-Snyder collapse can describe consistent black-to-white hole bounces. Therefore, we present a general framework in which the collapse can lead to the formation of outer and inner horizons. We show that on top of the previous kinematical constraint, an additional dynamical condition has to be satisfied in order for the model to admit well-defined black-to-white hole solutions. As expected, this second condition turns out to be model-dependent. The resulting class of models describing bouncing compact objects are characterized by three parameters for the star (mass, initial radius and density) and two quantum parameters descending from the UV-completion of the exterior and interior geometries. The solution space contains (1) bouncing stars and (2) bouncing black holes and (3) a new class of astrophysical objects which alternate between a bouncing star and a bouncing black hole. Finally, we provide an explicit construction of a black-to-white hole bounce using the techniques of spatially closed LQC and discuss the novel properties induced by the presence of the inner horizon in this new framework. This generic framework lays down the foundation for interesting phenomenological investigations concerning the astrophysical properties of these objects, as well as a novel platform for further developments.

Quantum Gravity Correction to Hawking Radiation of the 2+1-Dimensional Wormhole

We carry out the Hawking temperature of a 2+1-dimensional circularly symmetric traversable wormhole in the framework of the generalized uncertainty principle (GUP). Firstly, we introduce the modified Klein-Gordon equation of the spin-0 particle, the modified Dirac equation of the spin-1/2 particle, and the modified vector boson equation of the spin-1 particle in the wormhole background, respectively. Given these equations under the Hamilton-Jacobi approach, we analyze the GUP effect on the tunneling probability of these particles near the trapping horizon and, subsequently, on the Hawking temperature of the wormhole. Furthermore, we have found that the modified Hawking temperature of the wormhole is determined by both wormhole’s and tunneling particle’s properties and indicated that the wormhole has a positive temperature similar to that of a physical system. This case indicates that the wormhole may be supported by ordinary (nonexotic) matter. In addition, we calculate the Unruh-Verlinde temperature of the wormhole by using Kodama vectors instead of time-like Killing vectors and observe that it equals to the standard Hawking temperature of the wormhole.

Bouncing compact objects. Part I. Quantum extension of the Oppenheimer-Snyder collapse

This article proposes a generalization of the Oppenheimer-Snyder model which describes a bouncing compact object. The corrections responsible for the bounce are parameterized in a general way so as to remain agnostic about the specific mechanism of singularity resolution at play. It thus develops an effective theory based on a thin shell approach, inferring generic properties of such a UV complete gravitational collapse. The main result comes in the form of a strong constraint applicable to general UV models: if the dynamics of the collapsing star exhibits a bounce, it always occurs below, or at most at the energy threshold of horizon formation, so that only an instantaneous trapping horizon may be formed while a trapped region never forms. This conclusion relies solely on i) the assumption of continuity of the induced metric across the time-like surface of the star and ii) the assumption of a classical Schwarzschild geometry describing the (vacuum) exterior of the star. In particular, it is completely independent of the choice of corrections inside the star which leads to singularity-resolution. The present model provides thus a general framework to discuss bouncing compact objects, for which the interior geometry is modeled either by a classical or a quantum bounce. In the latter case, our no-go result regarding the formation of trapped region suggests that additional structure, such as the formation of an inner horizon, is needed to build consistent models of matter collapse describing black-to-white hole bounces. Indeed, such additional structure is needed to keep quantum gravity effects confined to the high curvature regime, in the deep interior region, providing thus a new challenge for current constructions of quantum black-to-white hole bounce models.

Some results on cosmological and astrophysical horizons and trapped surfaces

We study the evolution of horizons of black holes in the 1 + 1 + 2 covariant setting and investigate various properties intrinsic to the geometry of the foliation surfaces of these horizons. This is done by interpreting formulations of various quantities in terms of the geometric and thermodynamic quantities. We establish a causal classification for horizons in different classes of spacetimes. We have also recovered results by Ben-Dov and Senovilla which put cut-offs on the equation of state parameter , determining the spacelike, timelike and non-expanding horizons in the the Robertson–Walker class of spacetimes. We show that stability of marginally trapped surfaces (MTS) in the Robertson–Walker spacetimes is only achievable under the conditions of negative pressure, and also classify the spacelike future outer trapping horizons (SFOTH) in the Robertson–Walker spacetimes via bounds on the equation of state parameter . For the Lemaitre–Tolman–Bondi (LTB) model, it is shown that a relationship between the energy density and the electric part of the Weyl curvature, , gives the causal classification of the MTTs. It is further shown that only spacelike MTTs are foliated by stable MTS, and that this stability guarantees no shell crossing. We also provide an explicit proof of the third law of black hole thermodynamics for the LRS II class of spacetimes, and by extension, any spacetime whose outgoing and ingoing null geodesics are normal to the MTS.

Lyra black holes and Hawking radiation

Sen and Dunn constructed a field equation which is analogous to Einstein’s field equations based on the Lyra’s geometry and gave series type solution to the static vacuum field equations which correspond to black holes. In this work, the Hawking radiation near the trapping horizon of Lyra black holes is derived and the obtained results are discussed.

Trapping horizon and negative energy

Assuming spherical symmetry and the semi-classical Einstein equation, we prove that, for the observers on top of the trapping horizon, the vacuum energy-momentum tensor is always that of an ingoing negative energy flux at the speed of light with a universal energy density ℰ ≃ − 1/(2κa2), (where a is the areal radius of the trapping horizon), which is responsible for the decrease in the black hole’s mass over time. This result is independent of the composition of the collapsing matter and the details of the vacuum energy-momentum tensor. The physics behind the universality of this quantity ℰ and its surprisingly large magnitude will be discussed.

Unified first law for traversable wormholes in non-minimal coupling of curvature and matter

In this paper thermodynamics of static Morris–Thorne wormholes has been discussed in the context of f(R) gravity. The generalized surface gravity, unified first law of thermodynamics and wormhole dynamics have been studied at trapping horizons. We have investigated thermodynamics in non-minimal coupling of curvature and matter which produces very complex equations. Our results generalize the results that have already been derived in Einstein’s gravity in the absence of curvature–matter coupling.

Gravitational collapse of a massless scalar field in a periodic box

Gravitational collapse of a massless scalar field with the periodic boundary condition in a cubic box is reported. This system can be regarded as a lattice universe model. We construct the initial data for a Gaussian-like profile of the scalar field taking the integrability condition associated with the periodic boundary condition into account. For a large initial amplitude, a black hole is formed after a certain period of time. While the scalar field spreads out in the whole region for a small initial amplitude. It is shown that the expansion law in a late time approaches that of the radiation dominated universe and the matter dominated universe for the small and large initial amplitude cases, respectively. For the large initial amplitude case, the horizon is initially a past outer trapping horizon, whose area decreases with time, and after a certain period of time, it turns to a future outer trapping horizon with the increasing area.

Quasilocal horizons in inhomogeneous cosmological models

We investigate quasilocal horizons in inhomogeneous cosmological models, specifically concentrating on the notion of a trapping horizon defined by Hayward as a hypersurface foliated by marginally trapped surfaces. We calculate and analyse these quasilocally defined horizons in two dynamical spacetimes used as inhomogeneous cosmological models with perfect fluid source of non-zero pressure. In the spherically symmetric Lemaître spacetime we discover that the horizons (future and past) are both null hypersurfaces provided that the Misner–Sharp mass is constant along the horizons. Under the same assumption we come to the conclusion that the matter on the horizons is of special character—a perfect fluid with negative pressure. We also find out that they have locally the same geometry as the horizons in the Lemaître–Tolman–Bondi spacetime. We then study the Szekeres–Szafron spacetime with no symmetries, particularly its subfamily with , and we find conditions on the horizon existence in a general spacetime as well as in certain special cases.

Embedding Black Holes and Other Inhomogeneities in the Universe in Various Theories of Gravity: A Short Review

Classic black hole mechanics and thermodynamics are formulated for stationary black holes with event horizons. Alternative theories of gravity of interest for cosmology contain a built-in time-dependent cosmological “constant” and black holes are not stationary. Realistic black holes are anyway dynamical because they interact with astrophysical environments or, at a more fundamental level, because of backreaction by Hawking radiation. In these situations, the teleological concept of event horizon fails and apparent or trapping horizons are used instead. Even as toy models, black holes embedded in cosmological “backgrounds” and other inhomogeneous universes constitute an interesting class of solutions of various theories of gravity. We discuss the known phenomenology of apparent and trapping horizons in these geometries, focusing on spherically symmetric inhomogeneous universes.

Closed trapping horizons without singularity

In gravitational collapse leading to black hole formation, trapping horizons typically develop inside the contracting matter. Classically, an ingoing trapping horizon moves towards the centre where it reaches a curvature singularity, while an outgoing horizon moves towards the surface of the star where it becomes an isolated, null horizon. However, strong quantum effects at high curvature close to the centre could modify the classical picture substantially, e.g. by deflecting the ingoing horizon to larger radii, until it eventually reunites with the outgoing horizon. We here analyse some existing models of regular "black holes" of finite lifespan formed out of ingoing null shells collapsing from $\mathscr{I}^-$, after giving general conditions for the existence of (singularity-free) closed trapping horizons. We study the energy-momentum tensor of such models by solving Einstein's equations in reverse and give an explicit form of the metric to model a Hawking radiation reaching $\mathscr{I}^+$. A major flaw of the models aiming at describing the formation of black holes (with a Vaidya limit on $\mathscr{I}^-$) as well as their evaporation is finally exhibited: they necessarily violate the null energy condition up to $\mathscr{I}^-$, i.e. in a non-compact region of spacetime.

Seeking observational evidence for the formation of trapping horizons in astrophysical black holes

Black holes in general relativity are characterized by their trapping horizon, a one-way membrane that can be crossed only inwards. The existence of trapping horizons in astrophysical black holes can be tested observationally using a reductio ad absurdum argument, replacing black holes by horizonless configurations with a physical surface and looking for inconsistencies with electromagnetic and gravitational wave observations. In this approach, the radius of the horizonless object is always larger than but arbitrarily close to the position where the horizon of a black hole of the same mass would be located. Upper bounds on the radius of these alternatives have been provided using electromagnetic observations (in optical/IR band) of astronomical sources at the center of galaxies, but lower bounds were lacking, leaving unconstrained huge regions of parameter space. We show here that lower bounds on the radius of horizonless objects that do not develop trapping horizons can be placed using observations of accreting systems. This result is model-independent and relies only on the local notion of causality dictated by the spacetime geometry around the horizonless object. These observational bounds reduce considerably the previously allowed parameter space, boosting the prospects of establishing the existence of trapping horizons using electromagnetic observations.

Decreasing entropy of dynamical black holes in critical gravity

Critical gravity is a quadratic curvature gravity in four dimensions which is ghost-free around the AdS background. Constructing a Vaidya-type exact solution, we show that the area of a black hole defined by a future outer trapping horizon can shrink by injecting a charged null fluid with positive energy density, so that a black hole is no more a one-way membrane even under the null energy condition. In addition, the solution shows that the Wald-Kodama dynamical entropy of a black hole is negative and can decrease. These properties expose the pathological aspects of critical gravity at the non-perturbative level.

Complete conformal classification of the Friedmann–Lemaître–Robertson–Walker solutions with a linear equation of state

We completely classify Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature and equation of state , according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow , thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for w > −1/3 and , while there is no big-bang singularity for w < −1 and . For K = 0 or −1, −1 < w < −1/3 and , there is an initial null big-bang singularity. For each spatial curvature, there is a final spacelike future big-rip singularity for w < −1 and , with null geodesics being future complete for but incomplete for w < −5/3. For w = −1/3, the expansion speed is constant. For −1 < w < −1/3 and K = 1, the universe contracts from infinity, then bounces and expands back to infinity. For K = 0, the past boundary consists of timelike infinity and a regular null hypersurface for −5/3 < w < −1, while it consists of past timelike and past null infinities for . For w < −1 and K = 1, the spacetime contracts from an initial spacelike past big-rip singularity, then bounces and blows up at a final spacelike future big-rip singularity. For w < −1 and K = −1, the past boundary consists of a regular null hypersurface. The trapping horizons are timelike, null and spacelike for , and , respectively. A negative energy density () is possible only for K = −1. In this case, for w > −1/3, the universe contracts from infinity, then bounces and expands to infinity; for −1 < w < −1/3, it starts from a big-bang singularity and contracts to a big-crunch singularity; for w < −1, it expands from a regular null hypersurface and contracts to another regular null hypersurface.

Soft hair of dynamical black hole and Hawking radiation

Soft hair of black hole has been proposed recently to play an important role in the resolution of the black hole information paradox. Recent work has emphasized that the soft modes cannot affect the black hole S-matrix due to Weinberg soft theorems. However as soft hair is generated by supertranslation of geometry which involves an angular dependent shift of time, it must have non-trivial quantum effects. We consider supertranslation of the Vaidya black hole and construct a non-spherical symmetric dynamical spacetime with soft hair. We show that this spacetime admits a trapping horizon and is a dynamical black hole. We find that Hawking radiation is emitted from the trapping horizon of the dynamical black hole. The Hawking radiation has a spectrum which depends on the soft hair of the black hole and this is consistent with the factorization property of the black hole S-matrix.

Unified first law and some general prescription: a redefinition of surface gravity

The paper contains an extensive study of the unified first law (UFL) in the Friedmann–Robertson–Walker spacetime model. By projecting the UFL along the Kodama vector the second Friedmann equation can be obtained. Also studying the UFL on the event horizon it is found that the Clausius relation cannot be obtained from the UFL by projecting it along the tangent to the event horizon as it can be for the trapping horizon. However, it is shown in the present work that Clausius relation can be obtained by projecting the UFL along the Kodama vector on the horizon and the result is found to be true for any horizon. Finally motivated by the Unruh temperature for the Rindler observer, surface gravity is redefined and a Clausius relation is obtained from the UFL by projecting it along a vector analogous to the Kodama vector.

Nonlinear dynamics in the Einstein-Gauss-Bonnet gravity

We numerically investigated how the nonlinear dynamics depends on the dimensionality and on the higher-order curvature corrections in the form of Gauss-Bonnet (GB) terms. We especially monitored the processes of appearances of a singularity (or black hole) in two models: (i) a perturbed wormhole throat in spherically symmetric space-time, and (ii) colliding scalar pulses in plane-symmetric space-time. We used a dual-null formulation for evolving the field equations, which enables us to locate the trapping horizons directly, and also enables us to follow close to the large-curvature region due to its causal integrating scheme. We observed that the fate of a perturbed wormhole is either a black hole or an expanding throat depending on the total energy of the structure, and its threshold depends on the coupling constant of the GB terms ($\alpha_{\rm GB}$). We also observed that a collision of large scalar pulses will produce a large-curvature region, of which the magnitude also depends on $\alpha_{\rm GB}$. For both models, the normal corrections ($\alpha_{\rm GB}>0$) work for avoiding the appearance of singularity, although it is inevitable. We also found that in the critical situation for forming a black hole, the existence of the trapped region in the Einstein-GB gravity does not directly indicate the formation of a black hole.

Field dynamics on the trapping horizon in Vaidya spacetime

In this article, we shed some light on the field theoretic aspect of the spherically symmetric trapping horizon in Vaidya spacetime. The effective field equations are that of a Chern-Simons theory coupled to bulk sources through two different couplings, one is purely geometric and the other is matter dependent. This is an effective generalization of the equilibrium horizon scenario where the Chern-Simons theory is coupled to the bulk geometry through a constant matter-independent coupling. Further, we note that the field equations pulled-back to a cross-section of the horizon is inadequate to manifest the nature of the horizon. Hence, contrary to the usual practice, the evolution equation needs to be considered at least while passing on to the quantum theory.

Causal nature and dynamics of trapping horizons in black hole collapse

In calculations of gravitational collapse to form black holes, trapping horizons (foliated by marginally trapped surfaces) make their first appearance either within the collapsing matter or where it joins on to a vacuum exterior. Those which then move outwards with respect to the matter have been proposed for use in defining black holes, replacing the global concept of an ‘event horizon’ which has some serious drawbacks for practical applications. We here present results from a study of the properties of both outgoing and ingoing trapping horizons, assuming strict spherical symmetry throughout. We have investigated their causal nature (i.e. whether they are spacelike, timelike or null), making contact with the Misner–Sharp–Hernandez formalism, which has often been used for numerical calculations of spherical collapse. We follow two different approaches, one using a geometrical quantity related to expansions of null geodesic congruences, and the other using the horizon velocity measured with respect to the collapsing matter. After an introduction to these concepts, we then implement them within numerical simulations of stellar collapse, revisiting pioneering calculations from the 1960s where some features of the emergence and subsequent behaviour of trapping horizons could already be seen. Our presentation here is aimed firmly at ‘real world’ applications of interest to astrophysicists and includes the effects of pressure, which may be important for the asymptotic behaviour of the ingoing horizon.