scholarly journals Axisymmetric black holes allowing for separation of variables in the Klein-Gordon and Hamilton-Jacobi equations

2018 ◽  
Vol 97 (8) ◽  
Author(s):  
R. A. Konoplya ◽  
Z. Stuchlík ◽  
A. Zhidenko
Keyword(s):  
Black Holes ◽  
Separation Of Variables ◽  
Klein Gordon ◽  
Jacobi Equations

scholarly journals Universe as Klein–Gordon eigenstates

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Marco Matone
Keyword(s):  
Linear Form ◽  
Schwarzian Derivative ◽  
Eigenvalue Problems ◽  
Measurement Problem ◽  
Linear Equations ◽  
Flat Space ◽  
Curvature Term ◽  
Klein Gordon ◽  
Linear Differential ◽  
Jacobi Equations

AbstractWe formulate Friedmann’s equations as second-order linear differential equations. This is done using techniques related to the Schwarzian derivative that selects the$$\beta $$ β -times $$t_\beta :=\int ^t a^{-2\beta }$$ t β : = ∫ t a - 2 β , where a is the scale factor. In particular, it turns out that Friedmann’s equations are equivalent to the eigenvalue problems $$\begin{aligned} O_{1/2} \Psi =\frac{\Lambda }{12}\Psi , \quad O_1 a =-\frac{\Lambda }{3} a , \end{aligned}$$ O 1 / 2 Ψ = Λ 12 Ψ , O 1 a = - Λ 3 a , which is suggestive of a measurement problem. $$O_{\beta }(\rho ,p)$$ O β ( ρ , p ) are space-independent Klein–Gordon operators, depending only on energy density and pressure, and related to the Klein–Gordon Hamilton–Jacobi equations. The $$O_\beta $$ O β ’s are also independent of the spatial curvature, labeled by k, and absorbed in $$\begin{aligned} \Psi =\sqrt{a} e^{\frac{i}{2}\sqrt{k}\eta } . \end{aligned}$$ Ψ = a e i 2 k η . The above pair of equations is the unique possible linear form of Friedmann’s equations unless $$k=0$$ k = 0 , in which case there are infinitely many pairs of linear equations. Such a uniqueness just selects the conformal time $$\eta \equiv t_{1/2}$$ η ≡ t 1 / 2 among the $$t_\beta $$ t β ’s, which is the key to absorb the curvature term. An immediate consequence of the linear form is that it reveals a new symmetry of Friedmann’s equations in flat space.

scholarly journals Instability of charged massive scalar fields in bound states around Kerr–Sen black holes

2015 ◽  
Vol 24 (14) ◽  
pp. 1550102 ◽  
Author(s):  
Haryanto M. Siahaan
Keyword(s):  
Black Holes ◽  
Bound States ◽  
Gordon Equation ◽  
Scalar Fields ◽  
Bound State ◽  
Imaginary Component ◽  
Scalar Perturbation ◽  
Massive Scalar ◽  
Massive Scalar Field ◽  
Klein Gordon

In this paper, we show the instability of a charged massive scalar field in bound states around Kerr–Sen black holes. By matching the near and far region solutions of the radial part in the corresponding Klein–Gordon equation, one can show that the frequency of bound state scalar fields contains an imaginary component which gives rise to an amplification factor for the fields. Hence, the unstable modes for a charged and massive scalar perturbation in Kerr–Sen background can be shown.

scholarly journals Hawking radiation from cubic and quartic black holes via tunneling of GUP corrected scalar and fermion particles

2019 ◽  
Vol 34 (09) ◽  
pp. 1950057 ◽  
Author(s):  
Wajiha Javed ◽  
Rimsha Babar ◽  
Ali Övgün
Keyword(s):  
Black Holes ◽  
Hawking Radiation ◽  
Generalized Uncertainty Principle ◽  
Hawking Temperature ◽  
Jacobi Method ◽  
Dirac Equations ◽  
Tensor Theory ◽  
Klein Gordon ◽  
The Given ◽  
Do So

We analyze the effect of the generalized uncertainty principle (GUP) on the Hawking radiation from the hairy black hole in U(1) gauge-invariant scalar–vector–tensor theory by utilizing the semiclassical Hamilton–Jacobi method. To do so, we evaluate the tunneling probabilities and Hawking temperature for scalar and fermion particles for the given spacetime of the black holes with cubic and quartic interactions. For this purpose, we utilize the modified Klein–Gordon equation for the Boson particles and then Dirac equations for the fermion particles, respectively. Next, we examine that the Hawking temperature of the black holes do not depend on the properties of tunneling particles. Moreover, we present the corrected Hawking temperature of scalar and fermion particles which look similar in both interactions, but there are different mass and momentum relationships for scalar and fermion particles in cubic and quartic interactions.

scholarly journals Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 386 ◽  
Author(s):  
Andrei D. Polyanin
Keyword(s):  
Exact Solutions ◽  
Direct Method ◽  
Separation Of Variables ◽  
Surface Condition ◽  
Reaction Diffusion ◽  
Variable Coefficients ◽  
Nonlinear Pdes ◽  
Boundary Layer Equations ◽  
Compatibility Analysis ◽  
Klein Gordon

The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

scholarly journals Minimal Length Effects on Tunnelling from Spherically Symmetric Black Holes

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Benrong Mu ◽  
Peng Wang ◽  
Haitang Yang
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Angular Momentum ◽  
Minimal Length ◽  
Jacobi Method ◽  
Spherically Symmetric ◽  
Infinite Time ◽  
Quantum Tunnelling ◽  
Zero Mass ◽  
Jacobi Equations

We investigate effects of the minimal length on quantum tunnelling from spherically symmetric black holes using the Hamilton-Jacobi method incorporating the minimal length. We first derive the deformed Hamilton-Jacobi equations for scalars and fermions, both of which have the same expressions. The minimal length correction to the Hawking temperature is found to depend on the black hole’s mass and the mass and angular momentum of emitted particles. Finally, we calculate a Schwarzschild black hole's luminosity and find the black hole evaporates to zero mass in infinite time.

scholarly journals Functional separable solutions of two classes of nonlinear mathematical physics equations

2019 ◽  
Vol 486 (3) ◽  
pp. 287-291
Author(s):  
A. D. Polyanin ◽  
A. I. Zhurov
Keyword(s):  
Separation Of Variables ◽  
Functional Differential Equations ◽  
Traveling Wave Solutions ◽  
Reaction Diffusion ◽  
Mathematical Physics ◽  
Variable Coefficients ◽  
Reaction Diffusion Equations ◽  
Equations Of Mathematical Physics ◽  
Klein Gordon ◽  
Functional Differential

The study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions; the specific expressions of these functions are determined in the subsequent analysis of the arising functional differential equations. The effectiveness of the method is illustrated by examples of nonlinear reaction-diffusion equations and Klein-Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained.

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