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# The Anti-Hawking effect on Hyperbolic Black Holes

Posted 26 days ago

The Anti-Hawking Effect is not manifest for a thermal state on massless hyperbolic black holes of three or four dimensions. Lissa de Souza Campos University of Pavia - Italy For criticism, suggestions or requests, please contact me via lissa.campos@alumni.usp.br Black holes radiate; Hawking radiation is the thermal radiation emitted by a black hole. One way of formalizing this---Hawking effect---is using the concept of a particle detector. A two-level system is a good, simple model of a particle detector, called an Unruh-DeWitt detector. (Imagine an electron of a Hydrogen atom that can be excited or de-excited.) If we put a theoretical particle detector interacting with a quantum state Ψ of a free scalar quantum field theory on a black hole spacetime, we can compute its probabilities of excitation and de-excitation. Probabilities, in turn, can give us thermality. If the transition probabilities of a system behave like an ideal gas in thermal equilibrium, which is Boltzmann Statistics, it yields a temperature, it can be called thermal. When Ψ is a KMS state, aka thermal state, we do obtain a Boltzmann factor for the detector. And this is the Hawking effect: an Unruh-DeWitt detector with energy gap Ω, following static trajectories on Schwarzschild spacetime and interacting with the Hartle-Hawking state, responds such that the ratio between the transition rates of de-excitation per excitation is ℱ de-exc. ℱ exc. Ω/ T H ( 1 )We say the detector thermalized at the Hawking temperature T H The anti-Hawking effect does not work against the Hawking effect; it does not opposes the Hawking effect. It is a negative differential effect, with a somewhat misleading name. It is not an effect that a detector would measure, but it occurs when we compare the response of different detectors in different positions. Usually, the closest to the black hole horizon, the higher the Hawking temperature measured, and there is a one to one correspondence between the fixed spatial coordinates and T H T H ∂ ℱ ∂ T H ( 2 )Classically, we could think of the anti-Hawking effect as: “if the temperature is higher, there is more energy around, more collisions, a particle detector should click more, the transition rate should increase.” In this sense, it is a counter-intuitive effect. However, if we are talking about quantum field theory, we can base our intuition with what happens in Minkowski spacetime. From Jorma’s and Lee’s paper, the transition rate of an Unruh-DeWitt detector following Rindler trajectories and coupled to the ground state of a massless scalar field on the four-dimensional Minkowski spacetime is given by: In[]:= TransitionMink[4,Egap_,a_]= 1 2π Egap 2πEgap a ℱ Mink 4 ℱ Mink 4 Out[]= , The proper acceleration of each Rindler trajectory a In[]:= TransitionMink[3,Egap_,a_]= 1 2 1 2πEgap a ℱ Mink 3 ℱ Mink 3 Out[]= , In this notebook, we study the transition rate of an Unruh-DeWitt detector coupled to a ground and to a thermal state of a free scalar massless conformally-coupled quantum field theory on massless hyperbolic black holes of three and four dimensions. We show that it is not manifest for the thermal states in both dimensions and that it is manifest for the ground-state only in three dimensions; summarized in one picture-matrix, we show that: ( 3 ) The results match the behavior in Minkowski. But why is it manifest at all? What does it mean? Why is it manifest in three but not four dimensions? Why is it manifest for the ground state but not for the thermal state? Does it happen in classical physics as well? Maybe in quantum mechanics? Statistics have meaning; I think there is something here to be understood. If we model the interaction between an Unruh-DeWitt detector, with an energy gap Ω, and a quantum state Ψ by a monopole-type Hamiltonian operator, if we assume the detector is following a static trajectory γ parametrized by its proper time τ, and if the interaction is "turned on" for an infinite proper time interval, then, within first-order of perturbation theory, the instantaneous transition rates of excitation (Ω>0) or de-excitation (Ω<0) are given by the Fourier transform of the two-point function's pullback along the detector's trajectory 2 Ψ Ground and thermal states for the Klein-Gordon field on a massless hyperbolic black hole with applications to the anti-Hawking effect ”. Here, we include the numerical analysis to obtain the results, and part of the computations that are nicely done in Mathematica: checking that the line-elements are solutions of Einstein’s field equations; showing that the Klein-Gordon equation reduces to a hypergeometric equation; and checking the poles of the radial Green function.
Contents ■ Chapter 0: “Definitions (initialization cells)”: contains the definitions and documentation of all the functions and symbols used in this notebook. The main function is called SetFunctions, with which one can re-set the values of the position of the horizon and the AdS radius. For more information, run all initialization cells and use Help[SetFunctions] or Options[SetFunctions]. ■ Chapter 1: “The spacetimes”: contains the line-element of massless hyperbolic black holes and, using the package EinsteinTensor ■ Chapter 2: “The radial equation”: shows that the radial equation (the equation we are left with after separation of variables on the Klein-Gordon equation) can be written as a hypergeometric equation. This chapter uses the packages: ScalarWaveEquation ODEsTransformations ■ Chapter 3: “The poles of the Green function”: illustrates the behavior of the Green function with respect to the frequency ω in the complex plane for different boundary conditions. ■ Chapter 4: “The transition rate on Massless Hyperbolic Black Holes”: this is the main chapter. It contains the numerical analysis to get the plots of the transition rate of the Unruh-DeWitt detector on massless hyperbolic black holes of three and four dimensions. Some computations take time, and for that some data were save in external .txt files.
0. Definitions (initialization cells)
1. The spacetimes Massless hyperbolic black holes are static n ( T μν G μν Λg μν ( 4 )Their line element, in Schwarzschild-like coordinates, are given by: 2 ds 2 r 2 L 2 r h 2 L 2 dt 1 2 r 2 L 2 r h 2 L 2 dr 2 r 2 dΣ n-2 φ 1 φ n-3 ( 5 )L is called the “AdS radius”, the horizon is located at r= r h r h r h Using the package EinsteinTensor In[]:= EinsteinTensor[metric_,x_]:= Block[ {Dim,Metric, PreChristoffel, Christoffel, Riemann, PreRiemann, Ricci,sigma, mu, nu, alpha, beta, gamma}, Dim = Length[x]; Metric = Simplify[Inverse[metric]]; (* Metric with upper indices *) PreChristoffel = Table[ D[metric[[gamma,alpha]],x[[beta]]] + D[metric[[beta,gamma]],x[[alpha]]] - D[metric[[alpha,beta]],x[[gamma]]], {gamma,Dim}, {alpha,Dim}, {beta,Dim} ]; (* The "lower index part" of Christoffel symbols *) PreChristoffel = Simplify[PreChristoffel]; Christoffel = (1/2) Metric . PreChristoffel; (* The full Christoffel symbols *) Christoffel = Simplify[Christoffel]; PreRiemann = Table[ D[Christoffel[[sigma,alpha,nu]],x[[mu]]] + Sum[Christoffel[[gamma,alpha,nu]] Christoffel[[sigma,gamma,mu]], {gamma,Dim} ], {sigma,Dim}, {alpha,Dim}, {mu,Dim}, {nu,Dim} ]; (* PreRiemann has to be antisymmetrized to yield Riemann tensor: *) Riemann = Table[ PreRiemann[[sigma,alpha,mu,nu]] - PreRiemann[[sigma,alpha,nu,mu]], {sigma,Dim}, {alpha,Dim}, {mu,Dim}, {nu,Dim} ]; Ricci = Table[ Sum[Riemann[[sigma,alpha,sigma,beta]], {sigma,Dim}], {alpha,Dim}, {beta,Dim} ]; CurvatureScalar = Sum[ Metric[[alpha,beta]] Ricci[[alpha,beta]], {alpha,Dim}, {beta,Dim} ]; (* Return Einstein tensor: *) Ricci - (1/2) CurvatureScalar metric ]
Three dimensional case: g=
The metric g above solves (1) if and only if Λ= -1 2 L In[]:= EinsteinEQs=EinsteinTensor[g,x]+Λg//FullSimplify Out[]= - (r-rh)(r+rh)(1+ 2 L 4 L 1+ 2 L 2 r 2 rh 2 r 1 2 L In[]:= Solve[EinsteinEQsConstantArray[0,{3,3}],Λ] Out[]= Λ- 1 2 L And its scalar curvature is: In[]:= CurvatureScalar//FullSimplify Out[]= - 6 2 L
Four dimensional case: g=
The metric g above solves (1) if and only if r h Λ=- 3 2 L In[]:= EinsteinEQs=EinsteinTensor[g,x]+Λg//FullSimplify Out[]= - (r-rh)(r+rh)(3 2 r 2 rh 2 L 2 r 4 L 2 r 2 L 2 r 2 rh 2 L 2 r 4 r 2 r 2 rh 2 r 3 2 L 2 r 2 L 2 Sinh[θ] 2 L In[]:= Solve[EinsteinEQs[[1,1]]0,Λ] Out[]= Λ - 2 L 2 r 2 rh 2 L 2 r The cosmological constant must be a constant in r, hence for g to solve (1), the horizon must be at r h In[]:= SolveD - 2 L 2 r 2 rh 2 L 2 r Out[]= {{rh-L},{rhL}} Then the cosmological constant is: In[]:= - 2 L 2 r 2 rh 2 L 2 r Out[]= - 3 2 L And its scalar curvature is: In[]:= CurvatureScalar/.rhL//FullSimplify Out[]= - 12 2 L 2. The Klein-Gordon equation On a spacetime with metric tensor g 1 |Det[g]| ∂ μ |Det[g]| μν g ∂ ν 2 m ( 6 )Let’s import the package ScalarWaveEquation f[r_]= 2 r 2 L 2 rh 2 L
Import["ScalarWaveEquation.m"] With the ansatz Ψ(t,r,θ,ϕ)= -ωt
( 7 )where Y[θ,ϕ] are the eigenfunctions of the Laplacian operator on Σ n-2 In[]:= ScalarWaveEquation["metric"g3,"ansatz" -ωt Out[]= - 2 m 2 L 2 ω 2 r 2 rh (3 2 r 2 rh ′ R ′′ R 2 L ′′ Y 2 r In[]:= ScalarWaveEquation["metric"g4,"ansatz" -ωt Out[]= - 2 m 2 L 2 ω 2 r 2 rh (4 2 r 2 rh ′ R ′′ R 2 L Coth[θ] (1,0) Y (2,0) Y 2 r Let λ be the eigenvalue of Y. Both equations above reduce to the following O.D.E, called the radial equation: radialEQ=f[r]R''[r]+ (n-2)f[r] r 2 ω f[r] λ 2 r 2 μ To manipulate the radial equation, let' s import this auxiliary package, ODEsTransformations Import["ODEsTransformations.m"] First, we show that it can be reduced to a hypergeometric equation. Then, we obtain the measure with respect to which the solutions must be square-integrable.
As a hypergeometric equation First, applying the transformation on the independent variable: r↦z:= 2 r 2 r h 2 r ( 8 )we get: In[]:= auxEQ=TransformEQradialEQ,R,r,1,R, 2 r 2 rh 2 r - 2 rh (z-1) Out[]= 1 4 2 μ -1+z 2 ω z ′ R ′′ R The radial equation is z zEQ=(1-z)z ′′ R (n-5) 2 ′ R 2 L 4 2 L 2 ω z 2 r h λ 2 r h 2 μ (1-z) ( 9 )Asymptotically, we have: In[]:= AsymptoticDSolveValue[zEQ0,R,z0] Out[]= 2 L 2rh z z 2 L 4rh 2 L 2 L 2 rh 2 μ 4 L 2 ω 4 2 rh 1+ 2 L 2rh 4 L 2 ω 4 2 rh 2 L 2 L 2rh 2rh 1 - 2 L 2rh z z 2 L 4rh 2 L 2 L 2 rh 2 μ 4 L 2 ω 4 2 rh 1- 2 L 2rh 4 L 2 ω 4 2 rh 2 L 2 L 2rh 2rh 2 In[]:= AsymptoticDSolveValue[zEQ0,R,z1] Out[]= 1 4 1-2n+ 2 n 2 L 2 μ (-1+z) 1 4 1-2n+ +2 n 2 L 2 μ - 2 L 2 L 2 rh 2 μ 4 L 2 ω 4 2 rh 1 4 2 L 2 μ 1 2 1 4 1-2n+ 2 n 2 L 2 μ 1 4 1-2n+ 2 n 2 L 2 μ 1 4 1-2n+ 2 n 2 L 2 μ 1 1 4 1-2n+ 2 n 2 L 2 μ (-1+z) 1 4 1-2n+ +2 n 2 L 2 μ - 2 L 2 L 2 rh 2 μ 4 L 2 ω 4 2 rh 1 4 2 L 2 μ 1 2 1 4 1-2n+ 2 n 2 L 2 μ 1 4 1-2n+ 2 n 2 L 2 μ 1 4 1-2n+ 2 n 2 L 2 μ 2 Defining the auxiliary parameters: In[]:= α[ω_,rh_]= 2 L 2rh 1 4 1-2n+ ;2 n 2 L 2 μ And considering the ansatz: R[r(z)]= α z β (1-z) ( 10 )auxEQ=TransformEQ[zEQ,R,z, α[ω,rh] z β[n,μ] (1-z) we obtain that (9) solves the radial equation if and only if h[z] solve the hypergeometric equation z(1-z)h''[z]+(c-(a+b+1)z)h'[z]-abh[z]0 ( 11 )with coefficients given by: In[]:= Clear[a,b,c]HyperEQ=z(1-z)h''[z]+(c-(a+b+1)z)h'[z]-abh[z];test=coefshEQ[[1]]-z(1-z)//FullSimplify;test0 Out[]= True From the first derivative term is easy to extract c In[]:= c=Coefficient[coefshEQ[[2]],z,0]//FullSimplify Out[]= 1+ 2 L rh Now we must solve a system to determine a and b: In[]:= solsab=Solve[{a+b+1-Coefficient[coefshEQ[[2]],z],a*b-coefshEQ[[3]]},{a,b}]//FullSimplify;a=solsab[[1,1,2]]b=solsab[[1,2,2]] Out[]= rh 2 (-3+n) 4 2 L 2 rh 2 (-1+n) 2 L 2 μ 2 L 4rh Out[]= rh 2 (-3+n) 4 2 L 2 rh 2 (-1+n) 2 L 2 μ 2 L 4rh With the auxiliary definitions: ν= 2 (-1+n) 2 L 2 μ 2 1 4 2 (-3+n) 4 2 L 2 rh we write the parameters of the Hypergeometric equation as: a= 1 2 |