Tunneling of massive particles from noncommutative inspired Schwarzschild black hole

We apply the generalization of the Parikh–Wilczek method to the tunneling of massive particles from noncommutative inspired Schwarzschild black holes. By deriving the equation of radial motion of the tunneling particle directly, we calculate the emission rate which is shown to be dependent on the noncommutative parameter besides the energy and mass of the tunneling particle. After equating the emission rate to the Boltzmann factor, we obtain the modified Hawking temperature which relates to the noncommutativity and recovers the standard Hawking temperature in the commutative limit. We also discuss the entropy of the noncommutative inspired Schwarzschild black hole and its difference after and before a massive particle’s emission.     

Schwarzschild black hole in noncommutative spaces

We study the effects of noncommutative spaces on the horizon, the area spectrum and Hawking temperature of a Schwarzschild black hole. The results show deviations from the usual horizon, area spectrum and the Hawking temperature. The deviations depend on the parameter of space/space noncommutativity.     

Isotropic coordinates for Schwarzschild black hole radiation

The isotropic coordinate system of Schwarzschild spacetime has several attractive properties similar with the Painlevé–Gullstrand coordinates. The purpose for us to choose the isotropic coordinates is to resolve the ambiguities of the tunneling picture in Hawking radiation. Based on energy conservation, we investigate Hawking radiation as massless particles tunneling across the event horizon of the Schwarzschild black hole in the isotropic coordinates. Because the amplitude for a black hole to emit particles is related to the amplitude for it to absorb, we must take into account the contribution of ingoing solution to the action, ImS=ImS_out−ImS_in. It will be shown that the imaginary part of action for ingoing particles is zero (ImS_in=0) in the Painlevé–Gullstrand coordinates, so the equation ImS=ImS_out−ImS_in is valid in both the isotropic coordinates and the Painlevé–Gullstrand coordinates.     

Falling into a Schwarzschild black hole

Consider a radially freely falling observer who plunges into a Schwarzschild black hole. In contrast to a static observer, he will have a different view of the black hole and of the outer sky. Furthermore, the relationship between the proper time of the falling observer and the proper time of a distant static observer differs from the relationship between the proper times of two static observers or two freely falling observers.     

量子史瓦茨黑洞和暗物质

引入Sommerfeld作用量量子化条件来处理Schwarzschild黑洞的量子化问题. 发现此类量子化黑洞存在一个质量为m_G=123m_p的基态，处于基态的量子Schwarzschild黑洞不再存在Hawking蒸发和任何其他辐射，可名之曰暗星. 它的存在不仅可以解决信息丢失的疑难，而且极可能是构成暗物质的主要候选者. 关键词： 量子史瓦茨黑洞 暗物质     

Gravitational energy of a magnetized Schwarzschild black hole: a teleparallel approach

We investigate the distribution of gravitational energy in the spacetime of a Schwarzschild black hole immersed in a cosmic magnetic field. This is done in the context of the teleparallel equivalent of general relativity, which is an alternative geometrical formulation of general relativity, where gravity is described by a spacetime endowed with torsion rather than curvature, whose fundamental field variables are tetrad fields. We calculate the energy enclosed by a two-surface of constant radius—in particular, the energy enclosed by the event horizon of the black hole. In this case we find that the magnetic field has the effect of increasing the gravitational energy as compared to the vacuum Schwarzschild case. We also compute the energy (i) in the weak magnetic field limit, (ii) in the limit of vanishing magnetic field, and (iii) in the absence of the black hole. In all cases our results are consistent with what should be expected on physical grounds.     

Phase transition of quantum-corrected Schwarzschild black hole

We study the thermodynamic phase transition of a quantum-corrected Schwarzschild black hole. The modified metric affects the critical temperature which is slightly less than the conventional one. The space without black holes is not the hot flat space but the hot curved space due to vacuum fluctuations so that there appears a type of Gross–Perry–Yaffe phase transition even for the very small size of black hole, which is impossible for the thermodynamics of the conventional Schwarzschild black hole. We discuss physical consequences of the new phase transition in this framework.     

Un-graviton corrections to the Schwarzschild black hole

We introduce an effective action smoothly extending the standard Einstein–Hilbert action to include un-gravity effects. The improved field equations are solved for the un-graviton corrected Schwarzschild geometry reproducing the Mureika result. This is an important test to confirm the original “guess” of the form of the un-Schwarzschild metric. Instead of working in the weak field approximation and “dressing” the Newtonian potential with un-gravitons, we solve the “effective Einstein equations” including all order un-gravity effects. An unexpected “bonus” of accounting un-gravity effects is the fractalisation   of the event horizon. In the un-gravity dominated regime the event horizon thermodynamically behaves as fractal surface of dimensionality twice the scale dimension dU$d_U$.     

Statistical entropy of a Schwarzschild black hole

A model describing the internal microstates of particles is used to calculate the statistical entropy of a Schwarzschild black hole. The state of the system is described by a nonextensive entropy function which is superadditive and so fails to be concave. A strict maximum of the entropy does not exist; nonetheless, the entropy increases on merging two such systems.     

Schwarzschild black hole with global monopole charge

We derive the metric for a Schwarzschild black hole with global monopole charge by relaxing asymptotic flatness of the Schwarzschild field. We then study the effect of global monopole charge on particle orbits and the Hawking radiation. It turns out that existence, boundedness and stability of circular orbits scale up by (1−8πη ²)⁻¹, and the perihelion shift and the light bending by (1−8πη ²)^−3/2, while the Hawking temperature scales down by (1−8πη ²)² the Schwarzschild values. Hereη is the global charge.     

非对易时空下Gibbons-Maeda dilaton黑洞和Garfinkle-Horowitz-Strominger dilaton黑洞的热力学性质

以Gibbons-Maeda dilaton黑洞和Garfinkle-Horowitz-Strominger dilaton黑洞为例，研究空间的非对易性对黑洞热力学性质的影响.通过对比对易时空中Gibbons-Maeda dilaton黑洞和非对易时空中Garfinkle-Horowitz-Strominger dilaton黑洞的温度，得出如下结论：从对黑洞热力学性质产生影响这一角度来说，时空的非对易性和黑洞的荷(电荷或磁荷)有相似的作用.     

Gravitating discs around a Schwarzschild black hole II

A sequence of exact spacetimes is obtained describing the fields of a Schwarzschild black hole surrounded by stable static axisymmetric thin discs having their inner rim at the least possible radius. In the previous paper we only required stability with respect to perturbations in the disc plane, while it turns out that for discs with relative mass >0.23 the perturbations in perpendicular direction are more dangerous. The discs of the resulting sequence have their inner rims just on, or very close to, circular geodesics marginally stable with respect to either of the perturbations. Redshift from static and Keplerian observers in the disc is computed. The inverted first Morgan-Morgan counter-rotating disc, used in superpositions, has a number of satisfactory physical properties, but it has turned out to have a curvature singularity at the inner rim. However, this is only a consequence of a too steep radial start of density, not present in (inverted) “higher” Morgan-Morgan solutions. Dedicated to Professor Jiří Bičák on the occasion of his 60th birthday.     

Thermodynamical fluctuations of a massive Schwarzschild black hole

Starting from the Einstein-Boltzmann formula we determine classically the second moments for the fluctuations of a massive Schwarzschild black hole enclosed with thermal radiation in a rigid box.     

Schwarzschild generalized black hole horizon and the embedding space

The European Physical Journal C - By performing a Taylor expansion along the extra dimension of a metric describing a black hole on a brane, we explore the influence of the embedding space on the...     

Conformal scalar propagation inside the Schwarzschild black hole

The analytic expression obtained in the preceding project for the massless conformal scalar propagator in the Hartle–Hawking vacuum state for small values of the Schwarzschild radial coordinate above r = 2M is analytically extended into the interior of the Schwarzschild black hole. The result of the analytical extension coincides with the exact propagator for a small range of values of the Schwarzschild radial coordinate below r = 2M and is an analytic expression which manifestly features its dependence on the background space–time geometry. This feature as well as the absence of any assumptions and prerequisites in the derivation render this Hartle–Hawking scalar propagator in the interior of the Schwarzschild black-hole geometry distinct from previous results. The two propagators obtained in the interior and in the exterior region of the Schwarzschild black hole are matched across the event horizon. The result of that match is a massless conformal scalar propagator in the Hartle–Hawking vacuum state which is shown to describe particle production by the Schwarzschild black hole.

“The future is not what it used to be!” From Alan Parker’s film “Angel Heart”     

Schwarzschild black hole as moving puncture in isotropic coordinates

The success of the moving puncture method for the numerical simulation of black hole systems can be partially explained by the properties of stationary solutions of the 1 + log coordinate condition. We compute stationary 1 + log slices of the Schwarzschild spacetime in isotropic coordinates in order to investigate the coordinate singularity that the numerical methods have to handle at the puncture. We present an alternative integration method to obtain isotropic coordinates that simplifies numerical integration and that gives direct access to a local expansion in the isotropic radius near the puncture. Numerical results have shown that certain quantities are well approximated by a function linear in the isotropic radius near the puncture, while here we show that in some cases the isotropic radius appears with an exponent that is close to but unequal to one. This paper is dedicated to the memory of Jürgen Ehlers. I have known JE for a number of years, in particular during his time as founding director of the Albert Einstein Institute in Potsdam. JE was the mentor of my habilitation thesis in 1996, and I am deeply thankful for many insightful discussions. JE combined great breadth and physical intuition with sharp analytical thought. His example inspired me to look beyond the numerical methods and results of numerical relativity to the analytic foundations. For example, while at the AEI, S. Brandt and I introduced “puncture initial data” for the numerical construction of general multiple black hole spacetimes [3]. While the puncture construction starts with an analytic trick of the sort that numerical relativists may devise, it is fair to say that the keen interest in analytical relativity created by JE at the AEI induced us to push our analysis one step further. As a result [3] connects to [26] for an existence and uniqueness proof for such black hole initial data, using weighted Sobolev spaces (see also [4–6]). The present work and its predecessors [9–12] represent an example where numerical experiments led to the discovery of an analytic solution for the 1 + log gauge for the Schwarzschild solution, and the present result, although modest, is of the type which I believe JE would have appreciated.