On the internal state of the Schwarzschild black hole

2017 ◽  
Vol 26 (09) ◽  
pp. 1750088
Author(s):  
M. D. Pollock
Keyword(s):  
Black Hole ◽  
Internal State ◽  
Energy Condition ◽  
Momentum Tensor ◽  
Schwarzschild Black Hole ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Central Singularity ◽  
Symmetric Line ◽  
Tensor Formula

If the classical gravitational Lagrangian contains higher-derivative terms [Formula: see text], where [Formula: see text], then vacuum solutions of the Einstein–Hilbert theory [Formula: see text] are subject to modification at sufficiently large spacetime curvatures. Previously, we have calculated the effective energy–momentum tensor [Formula: see text] due to the quartic gravitational terms [Formula: see text] of the heterotic superstring theory in the four-dimensional background spacetime of the Schwarzschild black hole, obtaining an expression which satisfies the strong energy condition, and thereby suggests that the [Formula: see text] might not remove the central singularity. This conjecture was put forward from a different viewpoint by Horowitz and Myers, who argued that a non-singular black-hole interior resulting from the [Formula: see text] would be unstable, necessitating reappraisal of the notion of a singular interior spacetime. Here, we show that the chief features of the solution can be simulated by a Bardeen-type ansatz, assuming the spherically symmetric line element [Formula: see text], where [Formula: see text], which, when [Formula: see text], can explain heuristically why [Formula: see text] in the shell region [Formula: see text] of a macroscopic black hole for which [Formula: see text], while [Formula: see text] remains finite at [Formula: see text].

ON THE SUPERSTRING BLACK HOLE

2007 ◽  
Vol 16 (01) ◽  
pp. 123-140 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Energy Content ◽  
Momentum Tensor ◽  
Back Reaction ◽  
Effective Energy ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian

The effective Lagrangian for the heterotic superstring theory of Gross et al. contains higher-derivative gravitational terms [Formula: see text], n ≥ 2, which become important at large curvatures. This leads to a natural realization of the limiting-curvature hypothesis of Frolov et al., which was formulated to describe the interior of black holes. Assuming a purely geometrical, four-dimensional Schwarzschild black hole, for which all matter fields are zero, this interior consists of two regions: a shell of effective energy-density ρ immediately beyond the event horizon at r+ = 2M, due to the back reaction of the [Formula: see text] on the Schwarzschild metric, extending inward to a transition radius r0 ≈ M⅓, where the shell signature (- + - -) reverts to the exterior Lorentzian form (+ - - -), and an innermost core tending asymptotically to anti-de Sitter space as r → 0. The total mass-energy content of the hole M can be expressed in terms of the effective energy–momentum tensor Sij as the Nordström mass [Formula: see text], since the space–time is static and free of physical singularities. The conjecture that ρ N (r) is positive in the shell, which is necessary for the contribution to M N to be positive, is shown to be true for the term [Formula: see text], due to the unrenormalized [Formula: see text]. The corresponding "potential" energy–momentum tensor calculated in the Schwarzschild background is isotropic in the region r0 ≪ r ≪ r+, where [Formula: see text], while the dominant "kinetic" contribution is [Formula: see text], so that [Formula: see text].

ON THE SUPERSYMMETRIC DIRAC EQUATION AND THE HETEROTIC SUPERSTRING THEORY

2002 ◽  
Vol 11 (09) ◽  
pp. 1409-1418 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Dirac Equation ◽  
Momentum Tensor ◽  
Approximation Formula ◽  
Internal Space ◽  
Fermion Field ◽  
Linear Quadratic ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Tensor Formula ◽  
Weyl Equation

The supersymmetry transformations under which the four-dimensional massless Dirac equation for a two-component, spin-1/2 fermion field ψ (the Weyl equation) remains invariant were obtained by Volkov and Akulov, who used the result to construct the action S = a-1 ∫ |W| d4x in terms of the energy-momentum tensor [Formula: see text], where Wij = δij + aTij and a is a constant. Here, we show, in the approximation [Formula: see text], that the terms linear, quadratic and quartic in Tij are contained in the bosonic sector of the dimensionally reduced, heterotic superstring action, including higher-derivative gravitational terms up to order ℛ4. By comparison of coefficients, we derive the value B r ≈ 3.5 for the radius squared of the internal space in units of the Regge slope parameter α′, slightly greater than the Hagedorn radius squared [Formula: see text]. The cubic terms are also discussed.

scholarly journals A regular scale-dependent black hole solution

2018 ◽  
Vol 27 (03) ◽  
pp. 1850032 ◽  
Author(s):  
Ernesto Contreras ◽  
Ángel Rincón ◽  
Benjamin Koch ◽  
Pedro Bargueño
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Black Hole Solution ◽  
Energy Momentum Tensor ◽  
Energy Condition ◽  
Momentum Tensor ◽  
Scale Dependence ◽  
Weak Energy Condition ◽  
Effective Energy ◽  
Regular Black Hole

In this work, we present a regular black hole solution, in the context of scale-dependent General Relativity, satisfying the weak energy condition. The source of this solution is an anisotropic effective energy–momentum tensor which appears when the scale dependence of the theory is turned-on. In this sense, the solution can be considered as a semiclassical extension of the Schwarzschild one.

scholarly journals Duality and supersymmetry constraints on the weak gravity conjecture

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Gregory J. Loges ◽  
Toshifumi Noumi ◽  
Gary Shiu
Keyword(s):  
Black Hole ◽  
Scattering Amplitudes ◽  
Energy Condition ◽  
Null Energy Condition ◽  
Black Hole Thermodynamics ◽  
Simple Condition ◽  
Mass Ratio ◽  
Higher Derivative ◽  
Bps States ◽  
Weak Gravity

Abstract Positivity bounds coming from consistency of UV scattering amplitudes are not always sufficient to prove the weak gravity conjecture for theories beyond Einstein-Maxwell. Additional ingredients about the UV may be necessary to exclude those regions of parameter space which are naïvely in conflict with the predictions of the weak gravity conjecture. In this paper we explore the consequences of imposing additional symmetries inherited from the UV theory on higher-derivative operators for Einstein-Maxwell-dilaton-axion theory. Using black hole thermodynamics, for a preserved SL(2, ℝ) symmetry we find that the weak gravity conjecture then does follow from positivity bounds. For a preserved O(d, d; ℝ) symmetry we find a simple condition on the two Wilson coefficients which ensures the positivity of corrections to the charge-to-mass ratio and that follows from the null energy condition alone. We find that imposing supersymmetry on top of either of these symmetries gives corrections which vanish identically, as expected for BPS states.

scholarly journals Proposal for the proper gravitational energy–momentum tensor

2016 ◽  
Vol 31 (26) ◽  
pp. 1650151 ◽  
Author(s):  
Katsutaro Shimizu
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Momentum Conservation ◽  
Momentum Tensor ◽  
Gravitational Energy ◽  
Coordinate Transformations ◽  
Black Hole Mass ◽  
Schwarzschild Black Hole ◽  
Flrw Universe ◽  
Energy Momentum Conservation

We propose a gravitational energy–momentum (GEMT) tensor of the general relativity obtained using Noether’s theorem. It transforms as a tensor under general coordinate transformations. One of the two indices of the GEMT labels a local Lorentz frame that satisfies the energy–momentum conservation law. The energies for a gravitational wave, a Schwarzschild black hole and a Friedmann–Lemaitre–Robertson–Walker (FLRW) universe are calculated as examples. The gravitational energy of the Schwarzschild black hole exists only outside the horizon, its value being the negative of the black hole mass.

The Wheeler–Dewitt Equation for the Superstring World Sheet

1997 ◽  
Vol 06 (01) ◽  
pp. 91-105 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Black Hole ◽  
Ground State ◽  
Total Mass ◽  
Equation Of Motion ◽  
Schwarzschild Black Hole ◽  
World Sheet ◽  
Integral Expression ◽  
Higher Derivative ◽  
The World ◽  
The Universe

The Wheeler–DeWitt equation for the wave function ψ is obtained from the two-dimensional world-sheet action for the bosonic string and the superstring, including higher-derivative terms, as the Schrödinger equation i ∂ ψ/ ∂τ = V(τ)ψ. The potential is given by the rate at which the world-sheet area is swept out, V(τ) = dA(τ)/dτ, and is positive semi-definite, allowing the existence of a ground state, corresponding to the absence of the string, with a non-vanishing probability density ψ ψ*. Integration of this equation yields the solution [Formula: see text], where [Formula: see text] is the action, minus the higher-derivative terms [Formula: see text] (and terms involving ∊ab in the case of the superstring), which, however, are constrained to vanish semi-classically, being constructed from the square of the equation of motion for the bosonic coordinates XA derived from [Formula: see text] alone. This path-integral expression for ψ is consistent with the operator replacements for the canonical momenta used in its derivation, and forms the basis of the approach due to Polyakov of summing over random surfaces. Comparison is made with the Schrödinger equations derived previously from the reduced, four-dimensional effective action for the heterotic superstring, and for the Schwarzschild black hole (by Tomimatsu), where the potential is also positive semi-definite, being (twice) the total mass of the Universe and the mass of the black hole, respectively, showing the unity of the method.

scholarly journals Higher-derivative f(R,□R,T) theories of gravity

2017 ◽  
Vol 26 (03) ◽  
pp. 1750024 ◽  
Author(s):  
M. J. S. Houndjo ◽  
M. E. Rodrigues ◽  
N. S. Mazhari ◽  
D. Momeni ◽  
R. Myrzakulov
Keyword(s):  
Modified Gravity ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
De Sitter ◽  
Higher Derivative ◽  
Stress Energy ◽  
Derivative Term ◽  
Theories Of Gravity ◽  
Tensor Formula ◽  
Stress Energy Momentum Tensor

In literature, there is a model of modified gravity in which the matter Lagrangian is coupled to the geometry via trace of the stress–energy–momentum tensor [Formula: see text]. This type of modified gravity is denoted [Formula: see text] in which [Formula: see text] is Ricci scalar [Formula: see text]. We extend manifestly this model to include the higher derivative term [Formula: see text]. We derived equations of motion (EOM) for the model by starting from the basic variational principle. Later we investigate FLRW cosmology for our model. We show that de Sitter (dS) solution is unstable for a generic type of [Formula: see text] model. Furthermore we investigate an inflationary scenario based on this model. A graceful exit from inflation is guaranteed in this type of modified gravity.

scholarly journals The Schwarzschild black hole in f(R) exist superradiation phenomenon

2021 ◽  
Author(s):  
Wen-Xiang Chen
Keyword(s):  
Black Hole ◽  
Effective Potential ◽  
Momentum Tensor ◽  
Black Hole Thermodynamics ◽  
Incident Particle ◽  
Schwarzschild Black Hole ◽  
Hole Energy ◽  
Energy And Momentum ◽  
Relationship Of ◽  
The Relationship

For the relationship of the limit $y$ of the incident particle under the superradiance of the preset boundary (${\mu} = {y}{\omega}$)and the limit $y$ of the incident particle under the Hawking radiation of the preset boundary (${y}{\mu} ={\omega}$),we find the relationship between black hole thermodynamics and superradiation, and use boundary conditions to establish the relationship between y and R. We find that the black hole energy and momentum tensor is transformed into an effective potential. When the effective potential has a potential barrier, then we know that the Schwarzschild black hole in f(R) exist superradiation phenomenon.

scholarly journals HOLOGRAPHIC ENTROPY AND BRANE FRW DYNAMICS FROM AdS BLACK HOLE IN d5 HIGHER DERIVATIVE GRAVITY

2001 ◽  
Vol 16 (31) ◽  
pp. 5085-5099 ◽  
Author(s):  
SHIN'ICHI NOJIRI ◽  
SERGEI D. ODINTSOV ◽  
SACHIKO OGUSHI
Keyword(s):  
Black Hole ◽  
Gravitational Constant ◽  
Scale Parameter ◽  
Riemann Tensor ◽  
Schwarzschild Black Hole ◽  
Strongly Coupled ◽  
Verlinde Formula ◽  
Higher Derivative ◽  
Entropy Bounds ◽  
Tensor Square

Higher derivative bulk gravity (without Riemann tensor square term) admits AdS–Schwarzschild black hole as an exact solution. It is shown that induced brane geometry on such background is open, flat or closed FRW radiation dominated universe. Higher derivative terms contributions appear in the Hawking temperature, entropy and Hubble parameter via the redefinition of five-dimensional gravitational constant and AdS scale parameter. These higher derivative terms do not destroy the AdS-dual description of radiation represented by strongly-coupled CFT. The Cardy–Verlinde formula which expresses cosmological entropy as the square root from other parameters and entropies is derived in R2gravity. The corresponding cosmological entropy bounds are briefly discussed.

scholarly journals Accretion Flow onto Ellis–Bronnikov Wormhole

Universe ◽  
2021 ◽  
Vol 7 (6) ◽  
pp. 177
Author(s):  
Rosaliya M. Yusupova ◽  
Ramis Kh. Karimov ◽  
Ramil N. Izmailov ◽  
Kamal K. Nandi
Keyword(s):  
Black Hole ◽  
Scalar Field ◽  
Energy Condition ◽  
Null Energy Condition ◽  
Wick Rotation ◽  
Schwarzschild Black Hole ◽  
Stiff Matter ◽  
Barotropic Fluid ◽  
Accretion Flow ◽  
Opposite Behavior

Study of accretion onto wormholes is rather rare compared to that onto black holes. In this paper, we consider accretion flow of cosmological dark energy modeled by barotropic fluid onto the celebrated Ellis–Bronnikov wormhole (EBWH) built by Einstein minimally coupled scalar field ϕ, violating the null energy condition. The accreting fluid is assumed to be phantom, quintessence, dust and stiff matter. We begin by first pointing out a mathematical novelty showing how the EBWH can lead to the Schwarzschild black hole under a complex Wick rotation. Then, we analyze the profiles of fluid radial velocity, density and the rate of mass variation of the EBWH due to accretion and compare the profiles with those of the Schwarzschild black hole. We also analyze accretion to the massless EBWH that has zero ADM mass but has what we call nonzero Wheelerian mass (“mass without mass”), composed of the non-trivial scalar field, that shows gravitational effects. Our conclusion is that the mass of SBH due to phantom accretion decreases consistently with known results, while, in contrast, the mass of EBWH increases. Exactly an opposite behavior emerges for non-phantom accretion to these two objects. Accretion to massless EBWH (i.e., to nonzero Wheelerian mass) shares the same patterns as those of the massive EBWH; hence there is no way to distinguish massive and massless cases by means of accretion flow. The contrasting mass variations due to phantom accretion could be a reflection of the distinct topology of the central objects.

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