REVIEW
Hawking Radiation from Charged Black Holes via Gauge and Gravitational Anomalies
High Energy Physics - Theory · Submitted 15 Feb 2006 · 18 references
Pith's verdict: A clean, correct-looking extension of the anomaly derivation to charged black holes; the only real soft spot is the horizon boundary condition, which the authors themselves flag.
T0 · AI review
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Referee report
The paper
The paper claims
Requiring that gauge invariance and general covariance survive at the horizon of a charged black hole forces the outgoing flux of charge and energy in each partial wave to equal that of a one-dimensional blackbody at the Hawking temperature
The reading
What carries the argument
The engine is the distinction between consistent and covariant anomalies in two dimensions. The consistent anomaly satisfies the Wess-Zumino condition but its current transforms non-covariantly; the Bardeen-Zumino shift converts it into a covariant current whose anomaly coefficient is twice as large. Writing the current and the energy-momentum tensor as an exterior piece plus a near-horizon piece joined by a step function, the variation of the effective action produces a delta function at the junction plus a total derivative. Cancelling the delta function relates the integration constants of the two regions, and cancelling the total derivative is attributed to the quantum effect of the ingoing modes that were dropped. A Ward identity that keeps the dilaton coupling and the gauge-anomaly term supplies the classical piece, and the gravitational anomaly contributes a purely timelike term $N^{rt} = (f'^2 + f f'')/192pi$ whose value at the horizon is $kappa^2/24pi$.
Extended reading notes
Core claim
Reducing each partial wave of a charged field in a Reissner-Nordström background to a massless $(1+1)$-dimensional theory near the horizon, the authors solve the anomalous conservation laws for the current and for the energy-momentum tensor in the region where only outgoing modes propagate. Matching the exterior region to the near-horizon region across a thin shell, and imposing that the covariant current and covariant energy-momentum tensor vanish at the horizon, fixes the charge flux to $c_o = e^2Q/(2pi r_+)$ and the energy flux to $a_o = e^2Q^2/(4pi r_+^2) + pi/(12beta^2)$. These coincide with the fluxes obtained by integrating a Fermi-Dirac distribution with chemical potential $c = eQ/r_+$ at inverse temperature $beta$, which is the statement that the flux which cancels the anomaly is the Hawking flux.
Load-bearing premise
The numerical coefficients come from requiring the covariant current and the covariant energy-momentum tensor to vanish at the horizon, and the authors state in their Discussion that they do not fully understand why the covariant rather than the consistent forms are the correct boundary condition.
Editorial extensions
If this is right
- The derivation fixes only the flux at the horizon; the spectrum seen at infinity follows by propagating that flux through the effective potential of the curved exterior, giving a greybody at the Hawking temperature.
- For charged fermions the same argument predicts a nonzero charge flux even in the extremal limit $M = Q$, because the low-frequency occupation numbers approach one rather than vanishing.
- For bosons the negative-frequency distributions correspond to superradiance rather than a plain thermal flux, so the clean, unambiguous statement of the result is the fermionic one.
- Because only horizon data enter, the result is independent of the detailed dynamics outside the horizon; the $nu = r$ component of the Ward identity, by contrast, would require microscopic input and is left aside.
Reading between the lines
- The same bookkeeping plausibly extends to rotating black holes, where the relevant near-horizon symmetry is a mixed gauge-gravitational one for an axial current; the appearance of the standard $pi/(12beta^2)$ coefficient suggests the derivation is really a statement about the central charge of the near-horizon modes rather than about anything inside the horizon.
- If the covariant-vanishing boundary condition is the right one, it amounts to saying that a freely falling observer at the horizon measures no local charge or energy density, which is close in content to the statement that the Unruh vacuum is regular there.
- The equality can be read backwards as a constraint on any candidate microscopic description of the near-horizon regime: a model predicting a different horizon flux would violate anomaly matching and would have to explain where the would-be anomaly goes.
- The authors compare only against a free-field blackbody; whether the matching survives interactions that couple the partial waves away from the horizon is not tested here and would be the natural next check.
Original abstract
Extending gr-qc/0502074, we show that in order to avoid a breakdown of general covariance and gauge invariance at the quantum level the total flux of charge and energy in each outgoing partial wave of a charged quantum field in a Reissner-Nordstrom black hole background must be equal to that of a (1+1) dimensional blackbody at the Hawking temperature with the appropriate chemical potential.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
Extending gr-qc/0502074, we show that in order to avoid a breakdown of general covariance and gauge invariance at the quantum level the total flux of charge and energy in each outgoing partial wave of a charged quantum field in a Reissner-Nordstrom black hole background must be equal to that of a (1+1) dimensional blackbody at the Hawking temperature with the appropriate chemical potential.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Extending gr-qc/0502074, we show that in order to avoid a breakdown of general covariance and gauge invariance at the quantum level the total flux of charge and energy in each outgoing partial wave of a charged quantum field in a Reissner-Nordstrom black hole background must be equal to that of a (1+1) dimensional blackbody at the Hawking temperature with the appropriate chemical potential. | result 0.79 | 0.90 | 0.91 | asserted 1.00 |
What was examined, and what was not
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References
How later papers used it
50 later papers cite it
2 later papers by other authors are listed below: 2 used its method or tool.
Uses its method or tool 2 of 12 papers with this label
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Considered the problem of Hawking radiation following the approach based on the anomaly cancellation proposed in this paper.
Hawking Radiation from Non-Extremal D1-D5 Black Hole via Anomalies (2007) · 0.98 · paraphrased: the licence does not allow quoting
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Introduced a new current as this paper did, satisfying the conservation law with covariant anomaly.
Higher Derivative Correction to the Hawking Flux via Trace Anomaly (2008) · 0.93 · paraphrased: the licence does not allow quoting
How the literature uses it
47 arXiv papers cite this in 135 sentences
Papers citing it · 86 sentences without a confident label
Builds on it 3 papers
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In this section, we give a direct generalization of the anomaly method [ref] to spin 3 and 4 fluxes.
Hawking Radiation via Higher-spin Gauge Anomalies · Anomaly method for Hawking radiation · 0.95
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Its extension to charged and rotating black holes has been done in [ref] , where the original idea by Robinson and Wilczek has been elaborated.
Anomaly Analysis of Hawking Radiation from Acoustic Black Hole · Introduction · 0.81
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By applying these anomaly equations to black holes, it gives a generalization of the anomaly method [ref] (see also appendix of [ref] and [ref] ).
Higher-spin Gauge and Trace Anomalies in Two-dimensional Backgrounds · Introduction · 0.92
Uses its method or tool 11 papers
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Having the two-dimensional effective field theory near the horizon ([ref]) and the two-dimensional background ([ref]), we move on to the problem of Hawking radiation following the approach based on the anomaly cancellation proposed in [ref] .
Anomaly Analysis of Hawking Radiation from Acoustic Black Hole · Introduction · 0.93
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Having the two-dimensional effective field theory near the horizon ([ref]), we consider the problem of Hawking radiation following the approach based on the anomaly cancellation proposed in [ref] .
Hawking Radiation from Non-Extremal D1-D5 Black Hole via Anomalies · Anomalies and Hawking fluxes · 0.98
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In this section, having the two-dimensional effective theory near the horizon ([ref]) and the background ([ref]), we are going to consider the problem of Hawking radiation following the approach based on the anomaly cancellation proposed in [ref] .
Anomaly and Hawking radiation from regular black holes · Anomalies and Hawking fluxes · 0.98
Compares with it 6 papers
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This result is the same with just the Schwarzschild [ref] .
Hawking flux of 4D Schwarzschild black hole with supertransition correction to second-order · Hawking flux of the energy-momentum tensors · 0.99
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Hence $t(u)$ is determined as [equation] Similarly $t̃(v)$ is determined by requiring that there is no ingoing flux from infinity as [equation] Hence the $rt$-component of the energy-momentum tensor $T^r_t$ is obtained as [equation] The asymptotic flux of the energy is given by the asymptotic value of $T^r_t$ at $r→∞$, [equation] The flux reproduces the correct flux of Hawking radiation from the charged black hole [ref] .
Fluxes of Higher-spin Currents and Hawking Radiations from Charged Black Holes · Solving conservation equations · 0.84
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In [ref] the authors obtain the Hawking fluxes using the covariant methods of [ref] and the results agree with the ones obtained here in section [ref] following the original method of [ref] .
Anomalies and Hawking fluxes from the black holes of topologically massive gravity · Conclusion-Discussion · 1.00
Cites it as background 12 papers
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Recently, a new method for calculating Hawking radiation from the horizon of black holes has been developed by Robinson and Wilczek [ref] and by Iso, Umetsu, and Wilczek [ref] .
Hawking radiation via anomaly cancelation for the black holes of five-dimensional minimal gauged supergravity · Introduction · 0.81
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Till now various approaches (e.g. path integral of gravitational action [ref] , tunneling formalism [ref] , gravitational anomaly approach [ref] , extremization of action functional for extremal black holes [ref] , Noether conserved charge [ref] , etc.) have been put forward to get better understanding of the Hawking radiation phenomenon as well as thermal behaviour of black holes.
Thermality of horizon through near horizon instability: a path integral approach · Introduction and basic formalism · 0.98
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An interesting approach in connecting Hawking radiation from black holes to quantum anomalies was proposed and developed in the papers [ref] .
Chiral anomalies in black hole spacetimes · Introduction · 0.90
Discussion
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