arXiv subjects
Publications and source records attributed to Jarmo Makela.
It was recently pointed out by Barrau, Martineau and Renevey that the mass of the Schwarzschild black hole immersed in a heat bath tends to infity in a finite time. We show that this singularity problem in the black hole mass may be avoided, if the stretched horizon of the black hole is constructed out of a fixed, finite number of discrete constituents, and an appropriate counting is applied to the energy eigenstates of the hole.
We consider a microscopic model of a stretched horizon of the Reissner-Nordström black hole. In our model the stretched horizon consists of discrete constituents. Using our model we obtain an explicit, analytic expression for the partition function of the hole. Our partition function implies, among other things, the Hawking effect, and provides it with a microscopic explanation as a phase transition taking place at the stretched horizon. The partition function also implies the Bekenstein-Hawking entropy law. The model and its consequences are similar to those obtained previously for the Schwarzschild black hole.
We consider a microscopic model of spacetime, where spacetime is assumed to be a specific graph with Planck size quantum black holes on its vertices. As a thermodynamical system under consideration we take a certain uniformly accelerating, spacelike two-surface of spacetime which we call, for the sake of brevity and simplicity, as {\it acceleration surface}. Using our model we manage to obtain an explicit and surprisingly simple expression for the partition function of an acceleration surface. Our partition function implies, among other things, the Unruh and the Hawking effects. It turns out that the Unruh and the Hawking effects are consequences of a specific phase transition, which takes place in spacetime, when the temperature of spacetime equals, from the point of view of an observer at rest with respect to an acceleration surface, to the Unruh temperature measured by that observer. When constructing the partition function of an acceleration surface we are forced to introduce a quantity which plays the role of thermal energy of the surface. An interpretation of that quantity as energy in a normal manner yields Einstein's field equation with a vanishing cosmological constant for general matter fields.
We perform an analysis where Einstein's field equation is derived by means of very simple thermodynamical arguments. Our derivation is based on a consideration of the properties of a very small, spacelike two-plane in a uniformly accelerating motion.
We consider a possibility to construct a quantum-mechanical model of spacetime, where Planck size quantum black holes act as the fundamental constituents of space and time. Spacetime is assumed to be a graph, where black holes lie on the vertices. Our model implies that area has a discrete spectrum with equal spacing. At macroscopic length scales our model reproduces Einstein's field equation with a vanishing cosmological constant as a sort of thermodynamical equation of state of spacetime and matter fields. In the low temperature limit, where most black holes are assumed to be in the ground state, our model implies the Unruh and the Hawking effects, whereas in the high temperature limit we find, among other things, that black hole entropy depends logarithmically on the event horizon area, instead of being proportional to the area.
We consider an explicit example of a process, where the entropy carried by radiation through an accelerating two-plane is proportional to the decrease in the area of that two-plane even when the two-plane is not a part of any horizon of spacetime. Our results seem to support the view that entropy proportional to area is possessed not only by horizons but by all spacelike two-surfaces of spacetime.
We consider a spacetime foam model of the Schwarzschild horizon, where the horizon consists of Planck size black holes. According to our model the entropy of the Schwarzschild black hole is proportional to the area of its event horizon. It is possible to express geometrical arguments to the effect that the constant of proportionality is, in natural units, equal to one quarter.
We suggest that all horizons of spacetime, no matter whether they are black hole, Rindler or de Sitter horizons, have certain microscopic properties in common. We propose that these propertues may be used as the starting points, or postulates, of a microscopic theory of gravity.
We describe a general method of obtaining the constraints between area variables in one approach to area Regge calculus, and illustrate it with a simple example. The simplicial complex is the simplest tessellation of the 4-sphere. The number of independent constraints on the variations of the triangle areas is shown to equal the difference between the numbers of triangles and edges, and a general method of choosing independent constraints is described. The constraints chosen by using our method are shown to imply the Regge equations of motion in our example.
We consider the possibility to use the areas of two-simplexes, instead of lengths of edges, as the dynamical variables of Regge calculus. We show that if the action of Regge calculus is varied with respect to the areas of two-simplexes, and appropriate constraints are imposed between the variations, the Einstein-Regge equations are recovered.
We consider a Hamiltonian quantum theory of spherically symmetric, asymptotically flat electrovacuum spacetimes. The physical phase space of such spacetimes is spanned by the mass and the charge parameters $M$ and $Q$ of the Reissner-Nordström black hole, together with the corresponding canonical momenta. In this four-dimensional phase space, we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Reissner-Nordström black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator, and an eigenvalue equation for the ADM mass of the hole, from the point of view of a distant observer at rest, is obtained. Our eigenvalue equation implies that the ADM mass and the electric charge spectra of the hole are discrete, and the mass spectrum is bounded below. Moreover, the spectrum of the quantity $M^2-Q^2$ is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of the quantity $\sqrt{M^2-Q^2}$ are of the form $\sqrt{2n}$, where $n$ is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.
We consider a Hamiltonian theory of spherically symmetric vacuum Einstein gravity under Kruskal-like boundary conditions in variables associated with the Einstein-Rosen wormhole throat. The configuration variable in the reduced classical theory is the radius of the throat, in a foliation that is frozen at the left hand side infinity but asymptotically Minkowski at the right hand side infinity, and such that the proper time at the throat agrees with the right hand side Minkowski time. The classical Hamiltonian is numerically equal to the Schwarzschild mass. Within a class of Hamiltonian quantizations, we show that the spectrum of the Hamiltonian operator is discrete and bounded below, and can be made positive definite. The large eigenvalues behave asymptotically as~$\sqrt{2k}$, where $k$ is an integer. The resulting area spectrum agrees with that proposed by Bekenstein and others. Analogous results hold in the presence of a negative cosmological constant and electric charge. The classical input that led to the quantum results is discussed.
We formulate a qualitative argument, based on Heisenberg's uncertainty principle, to support the claim that when the effects of matter fields are assumed to overshadow the effects of quantum mechanics of spacetime, the discrete spectrum of black hole radiation, as such as predicted by Bekenstein's proposal for a discrete black hole area spectrum, reduces to Hawking's black-body spectrum.
We describe the gravitational degrees of freedom of the Schwarzschild black hole by one free variable. We introduce an equation which we suggest to be the Schroedinger equation of the Schwarzschild black hole corresponding to this model. We solve the Schroedinger equation explicitly and obtain the mass spectrum of the black hole as such as it can be observed by an observer very far away and at rest relative to the black hole. Our equation implies that there is no singularity inside the Schwarzschild black hole, and that the black hole has a certain ground state in which its mass is non-zero.