Komar Mass Research Articles

The electromagnetic energy with the truncation method for a five-dimensional rotating black hole in a uniform magnetic field

We consider a charged five-dimensional Myers–Perry black hole in a uniform magnetic (test) field. Using the Komar mass formula, we calculate the total energy of the electromagnetic field within the truncation three-sphere for a five-dimensional rotating black hole with two equal-rotation parameters and two equal-magnetic field strengths. We show that the total electromagnetic energy takes the minimum value when the five-dimensional rotating black hole acquires a non-zero net electric charge Q.

Energy Definition and Dark Energy: A Thermodynamic Analysis

Accepting the Komar mass definition of a source with energy-momentum tensor Tμν and using the thermodynamic pressure definition, we find a relaxed energy-momentum conservation law. Thereinafter, we study some cosmological consequences of the obtained energy-momentum conservation law. It has been found out that the dark sectors of cosmos are unifiable into one cosmic fluid in our setup. While this cosmic fluid impels the universe to enter an accelerated expansion phase, it may even show a baryonic behavior by itself during the cosmos evolution. Indeed, in this manner, while Tμν behaves baryonically, a part of it, namely, Tμν(e) which is satisfying the ordinary energy-momentum conservation law, is responsible for the current accelerated expansion.

Topological sources of soliton mass and supersymmetry breaking

We derive the Smarr formulae for two five-dimensional solutions of supergravity, which are asymptotically ; in particular, one has a magnetic ‘bolt’ in its center, and one is a two-center solution. We show for both spacetimes that supersymmetry—and so the BPS-bound—is broken by the holonomy and how each topological feature of a space-like hypersurface enters Smarr’s mass formula, with emphasis on the ones that give rise to the stated violation of the BPS-bound. In this light, we question if any violating extra-mass term in a spacetime with such asymptotics is only evident in the ADM mass while the Komar mass per se ‘tries’ to preserve BPS. Finally, we derive the cohomological fluxes for each situation and examine in a more general fashion how the breaking of supersymmetry—and so the BPS-bound violation—is associated with their topologies. In the second (and more complicated) scenario, we especially focus on the compact cycle linking the centers, and the contribution of non-vanishing bulk terms in the mass formula to the breaking of supersymmetry.

Energy in higher-dimensional spacetimes

We derive expressions for the total Hamiltonian energy of gravitating systems in higher dimensional theories in terms of the Riemann tensor, allowing a cosmological constant $\Lambda \in \mathbb{R}$. Our analysis covers asymptotically anti-de Sitter spacetimes, asymptotically flat spacetimes, as well as Kaluza-Klein asymptotically flat spacetimes. We show that the Komar mass equals the ADM mass in asymptotically flat space-times in all dimensions, generalising the four-dimensional result of Beig, and that this is not true anymore with Kaluza-Klein asymptotics. We show that the Hamiltonian mass does not necessarily coincide with the ADM mass in Kaluza-Klein asymptotically flat space-times, and that the Witten positivity argument provides a lower bound for the Hamiltonian mass, and not for the ADM mass, in terms of the electric charge. We illustrate our results on the Rasheed Kaluza-Klein vacuum metrics, which we study in some detail, pointing out restrictions that arise from the requirement of regularity, seemingly unnoticed so far in the literature.

A note on the relations between thermodynamics, energy definitions and Friedmann equations

We investigate the relation between the Friedmann and thermodynamic pressure equations, through solving the Friedmann and thermodynamic pressure equations simultaneously. Our investigation shows that a perfect fluid, as a suitable solution for the Friedmann equations leading to the standard modeling of the universe expansion history, cannot simultaneously satisfy the thermodynamic pressure equation and those of Friedmann. Moreover, we consider various energy definitions, such as the Komar mass, and solve the Friedmann and thermodynamic pressure equations simultaneously to get some models for dark energy fluids. The cosmological consequences of obtained solutions are also addressed. Our results indicate that some of obtained solutions may unify the dominated fluid in both the primary inflationary and current accelerating eras into one model. In addition, by taking into account a cosmic fluid of a known equation of state (EoS), and combining it with the Friedmann and thermodynamic pressure equations, we obtain the corresponding energy of these cosmic fluids and face their limitations. Finally, we point out the cosmological features of this cosmic fluid and also study its observational constraints.

A Modified Thermodynamics Method to Generate Exact Solutions of Einstein Equations**Supported by the National Natural Science Foundation of China under Grant Nos. 11273009 and 11303006

We modify the method to generate the exact solutions of the Einstein equations basing on the laws of thermodynamics. Firstly, the Komar mass is used to take the place of the Misner-Sharp energy, which is used in the original methods, and then several exact solutions of Einstein equations are obtained, including the black hole solution which is surrounded by quintessence. Moreover, the geometry surface gravity defined by Komar mass is also constructed. Secondly, we use both the Komar mass and the ADM mass to modify such method, and the similar results are obtained. Moreover, with some generalize addition to the definition of the ADM mass, our method can be generalized to global monopole spacetime.

Metric of two balancing Kerr particles in physical parametrization

The present paper aims at elaborating a completely physical representation for the general 4-parameter family of the extended double-Kerr spacetimes describing two spinning sources in gravitational equilibrium. This involved problem is solved in a concise analytical form by using the individual Komar masses and angular momenta as arbitrary parameters, and the simplest equatorially symmetric specialization of the general expressions obtained by us yields the physical representation for the well-known Dietz-Hoenselaers superextreme case of two balancing identical Kerr constituents. The existence of the physically meaningful "black hole-superextreme object" equilibrium configurations permitted by the general solution may be considered as a clear indication that the spin-spin repulsion force might actually be by far stronger than expected earlier, when only the balance between two superextreme Kerr sources was thought possible. We also present the explicit analytical formulas relating the equilibrium states in the double-Kerr and double-Reissner-Nordstr\"om configurations.

From the Komar Mass and Entropic Force Scenarios to the Einstein Field Equations on the Ads Brane

By bearing the Komar's definition for the mass, together with the entropic origin of gravity in mind, we find the Einstein field equations in $(n+1$)-dimensional spacetime. Then, by reflecting the ($4+1$)-dimensional Einstein equations on the ($3+1$)-hypersurface, we get the Einstein equations onto the $3$-brane. The corresponding energy conditions are also addressed. Since the higher dimensional considerations modify the Einstein field equations in the ($3+1$)-dimensions and thus the energy-momentum tensor, we get a relation for the Komar mass on the brane. In addition, the strongness of this relation compared with existing definition for the Komar mass on the brane is addressed.

Symmetry inheritance of scalar fields

Matter fields do not necessarily have to share the symmetries with the spacetime they live in. When this happens, we speak of the symmetry inheritance of fields. In this paper we classify the obstructions of symmetry inheritance by the scalar fields, both real and complex, and look more closely at the special cases of stationary and axially symmetric spacetimes. Since the symmetry noninheritance is present in the scalar fields of boson stars and may enable the existence of the black hole scalar hair, our results narrow the possible classes of such solutions. Finally, we define and analyse the symmetry noninheritance contributions to the Komar mass and angular momentum of the black hole scalar hair.

Stress-energy tensor of the quantized massive fields in Schwarzschild-Tangherlini spacetimes: The backreaction

We construct and study the approximate stress-energy tensor of the quantized massive scalar field in higher dimensional Schwarzschild-Tangherlini spacetimes. The stress-energy tensor is calculated within the framework of the Schwinger-DeWitt approach. It is shown that in $N$-dimensional spacetime the main approximation can be obtained from the effective action constructed form the coincidence limit of the Hadamard-DeWitt coefficient $a_{k},$ where $k-1$ is the integer part of $N/2$. The back reaction of the quantized field upon the black hole spacetime is analyzed and the quantum-corrected Komar mass and the Hawking temperature is calculated. It is shown that for the minimal and conformal coupling the increase of the Komar mass of the quantum corrected black hole leads to the decrease of its Hawking temperature. This is not generally true for more exotic values of the coupling parameter. The general formula describing the vacuum polarization, $ \langle \phi^{2} \rangle,$ is constructed and briefly examined.

Generalized black diholes

A 5-parametric exact solution, describing a binary system composed of identical counter-rotating black holes endowed with opposite electromagnetic charges, is constructed. The addition of the angular momentum parameter to the static Emparan-Teo dihole model introduces magnetic charges into this two-body system. The solution can be considered as an extended model for describing generalized black diholes as dyons. We derive the explicit functional form of the horizon half-length parameter $\sigma$ as a function of the Komar parameters: Komar mass $M$, electric/magnetic charge $Q_{E }/ Q_{B}$, angular momentum $J$, and a coordinate distance $R$, where the parameters $(M, J, Q_E, Q_B, R)$ characterize the upper constituent of the system, while $(M, -J, -Q_E, -Q_B, R)$ are associated with the lower one. The addition of magnetic charges enhances the standard Smarr mass formula in order to take into account their contribution to the mass. The solution contains, as particular cases, two solutions already discussed in the literature.

Opposite charged two-body system of identical counterrotating black holes

A 4-parametric exact solution describing a two-body system of identical Kerr-Newman counter-rotating black holes endowed with opposite electric/magnetic charges is presented. The axis conditions are solved in order to really describe two black holes separated by a massless strut. Moreover, the explicit form of the horizon half length parameter sigma in terms of physical Komar parameters, i.e., the Komar mass M, electric charge QE, angular momentum J, and a coordinate distance R is derived. Additionally, magnetic charges QB arise from the rotation of electrically charged black holes. As a consequence, in order to account for the contribution to the mass of the magnetic charge, the usual Smarr mass formula should be generalized, as it is proposed by A. Tomimatsu, Prog. Theor. Phys. 72, 73 (1984).

Exact solution for a binary system of unequal counter-rotating black holes

A complete solution describing a binary system constituted by two unequal counter-rotating black holes with a massless strut in between is presented. It is expressed in terms of four arbitrary parameters: the half length of the two rods representing the black hole horizons σ1 and σ2, the total mass M and the relative distance R between the centers of the horizons. The explicit parametrization of this solution in terms of physical parameters, i.e., the Komar masses M1 and M2, the Komar angular momenta J1 and J2 (having J1 and J2 opposite signs) and the coordinate distance R, led us to a four-parameter subclass in which the five physical parameters satisfy a simple algebraic relation, which generalizes the two statements made by Bonnor, in order to remove the additional contributions from the massless spinning rods outside the black holes. Moreover, the interaction force turns out to be of the same form as in the double-Schwarzschild static case.

Schwarzschild black hole levitating in the hyperextreme Kerr field

The equilibrium configurations between a Schwarzschild black hole and a hyperextreme Kerr object are shown to be described by a three-parameter subfamily of the extended double-Kerr solution. For this subfamily, its Ernst potential and corresponding metric functions, we provide a physical representation which employs as arbitrary parameters the individual Komar masses and relative coordinate distance between the sources. The calculation of horizon's local angular velocity induced in the Schwarzschild black hole by the Kerr constituent yields a simple expression inversely proportional to the square of the distance parameter.

Two-body equilibrium configurations involving one extreme black hole: the electrovacuum case

The present paper fills in the final gap in the search and description of different equilibrium states in the two-body systems consisting of one extreme and one non-extreme components. In the ‘extreme-non-extreme’ case of charged spinning masses we obtain, by making use of an appropriate exact solution of the Einstein-Maxwell equations and solving numerically the corresponding balance equations, the first examples of the ‘extreme-hyperextreme’ equilibrium configurations characterized by positive Komar masses of both Kerr–Newman constituents. Furthermore, we demonstrate that equilibrium in the ‘extreme-subextreme’ stationary electrovac systems is also possible, but all particular equilibrium configurations of this type found by us involve negative mass of one of the constituents. In the electrostatic case which admits a purely analytic treatment we give a rigorous proof of the non-existence of the ‘extreme-non-extreme’ equilibrium configurations in the framework of the double-Reissner–Nordstrom solution. At the same time, the electrostatic equilibrium between an extreme and a subextreme black holes can be achieved in the uniform external field, provided the two constituents form a specific dihole with zero net charge.

Spontaneously induced general relativity: Holographic interior for Reissner-Nordstrom exterior

If general relativity is spontaneously induced, that is if the reciprocal Newton constant serves as a VEV, the electrically charged black hole limit is governed by a Davidson-Gurwich phase transition which occurs precisely at the would have been outer horizon. The transition profile which connects the exterior Reissner-Nordstrom solution with the novel interior is analytically derived. The inner core is characterized by a vanishing spatial volume and constant surface gravity, and in some respects, resembles a maximally stretched horizon. The Komar mass residing inside any concentric interior sphere is proportional to the surface area of that sphere, and consequently, is non-negative definite and furthermore non-singular at the origin. The Kruskal structure is recovered, admitting the exact Hawking imaginary time periodicity, but unconventionally, with the conic defect defused at the origin. The corresponding holographic entropy packing locally saturates the 't Hooft-Susskind-Bousso holographic bound, thus making the core Nature's ultimate information storage.

First law of thermodynamics on holographic screens in entropic force frame

Imposing a mathematical definition of holographic screen, in the spirit of Verlindeʼs entropic force proposal (E.P. Verlinde, arXiv:1001.0785), we give the differential and integral form of the first law of thermodynamics on the holographic screen enclosing a spherical symmetric black hole. It is consistent with equipartition principle and the form of Komar mass. There are also other version of first law, which are equivalent up to a Legendre transformation. The holographic screen thermodynamics is defined in a quasi-local form, which is the main difference to black hole thermodynamics. Thus, the physical interpretation of holographic screen thermodynamics might be different from black hole thermodynamics. We argue that the entropy of the holographic screen determines its area, i.e. S=A4. And the metric can be expressed by thermodynamics variables, which is an illustration of how the space is foliated by the thermodynamical potentials.

Holographic Entropy Packing inside a Black Hole

If general relativity is spontaneously induced, the black hole limit is governed by a phase transition which occurs precisely at the would-have-been horizon. The exterior Schwarzschild solution then connects with a novel core of vanishing spatial volume. The Kruskal structure, admitting the exact Hawking imaginary time periodicity, is recovered, with the conic defect defused at the origin, rather than at the horizon. The entropy stored inside any interior sphere is universal, equal to a quarter of its surface area, thus locally saturating the 't Hooft-Susskind holographic bound. The associated Komar mass and material energy functions are nonsingular.

Thermodynamics of black holes from equipartition of energy and holography

A gravitational potential in the relativistic case is introduced as an alternative to Wald's potential used by Verlinde, which reproduces the familiar entropy/area relation S=A/4 (in the natural units) when Verlinde's idea is applied to the black hole case. Upon using the equipartition rule, the correct form of the Komar mass (energy) can also be obtained, which leads to the Einstein equations. It is explicitly shown that our entropy formula agrees with Verlinde's entropy variation formula in spherical cases. The stationary space-times, especially the Kerr-Newman black hole, are then discussed, where it is shown that the equipartition rule involves the reduced mass, instead of the ADM mass, on the horizon of the black hole.

Relativistic Euler’s three-body problem, optical geometry, and the golden ratio

A Weyl solution describing two Schwarzschild black holes is considered. We focus on the ${\mathbb{Z}}_{2}$ invariant solution, with Arnowitt-Deser-Misner mass ${M}_{\mathrm{ADM}}=2{M}_{\mathrm{K}}$, where ${M}_{\mathrm{K}}$ is the Komar mass of each black hole. For this solution the set of fixed points of the discrete symmetry is a totally geodesic submanifold. The existence and radii of circular photon orbits in this submanifold are studied, as functions of the distance $2L$ between the two black holes. For $L\ensuremath{\rightarrow}0$ there are two such orbits, corresponding to $r=3{M}_{\mathrm{ADM}}$ and $r=2{M}_{\mathrm{ADM}}$ in Schwarzschild coordinates. As the distance increases, it is shown that the two photon orbits approach one another and merge when ${M}_{\mathrm{K}}=\ensuremath{\varphi}L$, where $\ensuremath{\varphi}$ is the golden ratio. Beyond this distance there exist no circular photon orbits. The two null orbits delimit a forbidden band for timelike circular orbits, which is interpreted in terms of optical geometry. For large $L$, timelike circular orbits are allowed everywhere, as in the analogous Newtonian problem. The analysis is generalized by considering a ${\mathbb{Z}}_{2}$ invariant Weyl solution with an array of $N$ black holes and also by charging the black holes, which connects the Weyl solution to a Majumdar-Papapetrou spacetime.