Vacuum polarization on topological black holes
Abstract: We investigate quantum effects on topological black hole space-times within the framework of quantum field theory on curved space-times. Considering a quantum scalar field, we extend a recent mode-sum regularization prescription for the computation of the renormalized vacuum polarization to asymptotically anti-de Sitter black holes with nonspherical event horizon topology. In particular, we calculate the vacuum polarization for a massless, conformally-coupled scalar field on a four-dimensional topological Schwarzschild-anti-de Sitter black hole background, comparing our results with those for a spherically-symmetric black hole.
Summary (3 min read)
- In quantum field theory on curved space-time , the renormalized stress-energy tensor (RSET) 〈T̂µν〉 is an object of central importance.
- Their method makes heavy use of WKB approximations and the RSET is given as a sum of two parts, the first of which is analytic and the second of which requires numerical computation.
- Nonetheless, the VP shares many features with the RSET, for example, if the VP diverges on a horizon, then it is likely that the RSET will also diverge there.
- The corresponding calculations on the three-dimensional, asymptotically adS BTZ black hole [41–43] are rather simpler than those for four-dimensional black holes and both the VP and RSET for a conformally-coupled scalar field can be found in closed form (both when there is a black hole event horizon and in the naked singularity case [44–49]).
- These closed-form expressions have been used to study the back-reaction of the quantum field on the space-time geometry via the semi-classical Einstein equations (1.1) [50–52].
2. Topological black holes
- The authors consider static black hole solutions of the vacuum Einstein equations with a negative cosmological constant Λ: Gµν + Λgµν = 0. (2.1).
- The metric in this case is the usual Schwarzschild anti-de Sitter spacetime.
- Here θ again takes on all positive real values.
- The authors will parameterize their black holes by (k, α,M) rather than (k, L,M).
3. Quantum scalar fields on topological black hole space-times
- The authors wish to consider the VP for a quantum Klein-Gordon field in the Hartle-Hawking state propagating on the classical topological black hole space-times discussed in the previous section.
- In the quantum theory, the field gets promoted to an operator-valued distribution, φ(x) → φ̂(x), upon which is imposed a set of canonical commutation relations (see, for example, ).
- Extracting a meaningful finite quantity from this is the essence of the regularization problem which the authors discuss in the following section.
- For the remainder of this section, the authors will focus on constructing the Green’s function in the Hartle-Hawking state |HH〉.
- An important point in the construction of the mode-sum representation of the Green’s function is that the pathological behaviour in the coincidence limit manifests as the nonconvergence of the mode-sums in (3.6).
4. Hadamard regularization
- The näıve expression for the VP (3.5) is ill-defined and requires a prescription that (i) removes the singular terms (a process called regularization) and (ii) absorbs the terms introduced in order to cure the divergences into some other parameters in the theory (a process called renormalization).
- The conceptual framework for achieving this in a curved space-time, known as the point-splitting scheme, dates back to seminal work by DeWitt and Christensen [1–3].
- A more general axiomatic variant of their original point-splitting prescription, known as Hadamard regularization [4–6], is the approach adopted here.
4.4. k = −1 regularization parameters
- Factoring out the time dependence using an appropriate addition theorem is a little more subtle than in the other two cases.
- This is unlike the other cases where the appropriate addition theorems involved convergent sums, albeit sums that are only slowly converging.
- The result is however the same as that obtained by using (4.37) with ψ = ψ′ and ignoring the fact that the sum does not converge.
5. Renormalized vacuum polarization
- To compute the renormalized VP numerically, the authors therefore need to find the mode functions pnλ(ζ), qnλ(ζ), the normalization constants Nnλ, and the regularization parameters Ψnλ(i, j|r), before combining all these quantities into the sums in (5.1).
- The authors describe their numerical methodology for each of these parts before discussing their results.
5.1. Radial functions
- Since the ODE (3.9) is singular at the horizon, the authors start their integration close to the horizon and use the Frobenius series (5.2) to give the initial values of pnλ and its first derivative.
- Unlike the asymptotically flat case, infinity is a regular singular point of the ODE (3.9).
- On pure adS space-time, either transparent or reflective boundary conditions can be imposed on a massless, conformally-coupled scalar field at time-like infinity .
- The effect on the renormalized VP of choosing alternative boundary conditions will be explored elsewhere  (see also  for the effect of boundary conditions on the renormalized VP on a BTZ black hole).
5.2. Mode sums
- Having found the radial functions pnλ and qnλ, the regularization parameters Ψnλ(i, j|r) (4.40) are readily computed in Mathematica since the authors have analytic expressions for these quantities.
- The relative error between these two mode sums is typically O(10−9) or smaller.
- Second, approximating the integrand by an interpolating function between the grid points in λ also introduces an error into the evaluation of the integral.
- Here δλ and ǫλ are constants which are estimated by fitting this power law (using the Mathematica function FindFit) to the values of the integrand computed numerically for λ ∈ (40, 50).
- Once the sum/integral over λ has been performed, the final, trivial, part of the computation of the renormalized VP is to subtract the term −f ′(r)/48π2r in (5.1).
5.3. Results for the vacuum polarization
- The authors now present their results for the renormalized VP (5.1) on a selection of topological black holes.
- In all their plots below, dots denote values computed numerically, while the curves interpolate between these values.
- Alternatively, for large α and small L, the black hole is large compared with the adS curvature length-scale.
- For all values of α, the authors find that the renormalized VP is monotonically decreasing from its value on the horizon to the asymptotic value (5.6).
- 8), both of which leave the dimensionless radial coordinate ζ unchanged.
- Conformally-coupled scalar field on an asymptotically adS topological black hole space-time, for which the event horizon is a two-surface of constant curvature (either positive, negative, or zero).the authors.
- This method extends in an elegant way to topological black holes having an event horizon with either zero or negative curvature, and enables the Hadamard parametrix to be written as a mode sum.
- This monotonically decreasing behaviour as the radial distance from the event horizon increases is the same as that observed for a massless conformallycoupled scalar field in the Hartle-Hawking state on an asymptotically flat Schwarzschild black hole .
- Either of these extensions to the work presented here would involve finding a representation of the tail part of the Hadamard parametrix, which can be ignored when calculating the renormalized VP for a massless, conformally-coupled scalar field.
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Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "Vacuum polarization on topological black holes" ?
The authors investigate quantum effects on topological black hole space-times within the framework of quantum field theory on curved space-times.
Q2. What future works have the authors mentioned in the paper "Vacuum polarization on topological black holes" ?
Here the authors have restricted their attention to a massless, conformally-coupled scalar field, and an open question remains whether these effects of the event horizon topology on the VP extend to the massive case or more general couplings to the space-time curvature. Extending this method to horizons with zero or negative curvature is likely to be rather involved, so the authors leave this for future work. Either of these extensions to the work presented here would involve finding a representation of the tail part of the Hadamard parametrix, which can be ignored when calculating the renormalized VP for a massless, conformally-coupled scalar field. The extended coordinate method can be used to find a mode-sum representation for the tail [ 64 ] when the event horizon has positive curvature.