REVIEW
Accelerated Detectors and Temperature in (Anti) de Sitter Spaces
General Relativity and Quantum Cosmology · Submitted 6 Jun 1997 · 5 references
Pith's verdict: Short, honest letter: the AdS Unruh temperature is a real extension and the case-A derivation holds; the stress-test's trajectory tension dissolves once you notice constant-ρ observers have imaginary a₅.
T0 · AI review
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Referee report
The paper
The paper claims
A detector's temperature in de Sitter or anti-de Sitter space depends only on its acceleration in the flat five-dimensional space in which those curved spaces are embedded.
The reading
What carries the argument
The load-bearing device is the embedding-space rewriting: both de Sitter and anti-de Sitter are represented as hyperplanes $η_{AB}z^Az^B = ∓R^2$ in a flat five-dimensional space with $R^2 = 3/|Λ|$, so that a curved-space trajectory becomes an ordinary trajectory in flat space. A detector held at constant $z^4 = Z$ traces a hyperbola $(z^1)^2 - (z^0)^2 = ±(R^2 - Z^2)$, the same shape as a uniformly accelerated detector in Minkowski space, and the flat-space Rindler result can be applied directly with the five-dimensional acceleration $a_5 = (Λ/3 + a^2)^{1/2}$ in place of the ordinary one. The second device is the Wightman function of a massless scalar field in anti-de Sitter, computed for three boundary conditions (transparent, and two reflective choices) inherited from the conformal map to the static Einstein universe, which converts the question into whether the detector's response integral produces a Planck factor.
Extended reading notes
Core claim
The central claim is that for a detector with constant four-acceleration $a$ in a space with cosmological constant $Λ$, the measured temperature is $2πT = (Λ/3 + a^2)^{1/2}$, and that this quantity is exactly the detector's acceleration $a_5$ in the flat five-dimensional embedding space of the curved geometry. In de Sitter, where the sign is positive, this recovers the known result of Narnhofer, Peter and Thirring; in anti-de Sitter, where $Λ$ is negative, the same expression is claimed to hold and to be well defined down to the critical value $a_5 = 0$. Below that critical acceleration the temperature would be imaginary, but the authors point out that the corresponding orbits are spacelike, so the imaginary case is never physically realized. The claim is then checked by an Unruh-type calculation: for the hyperbolic embedding trajectories the Wightman function reduces to a function of proper time of the form $X = a_5^2R^2[1-��cosh a_5(s-s') - iε]^{-1}$, and the resulting transition rate has the standard Planck denominator $[exp(2πΔE/a_5) - 1]^{-1}$. The authors also show that detectors sitting at constant spatial coordinate $ρ$ in anti-de Sitter, whose embedding orbits are circles rather than hyperbolas, get a vanishing transition rate, i.e. zero temperature, which they read as confirmation that the hyperbolic five-dimensional motion is what carries the thermal effect.
Load-bearing premise
For the two reflective boundary conditions in anti-de Sitter, the paper assumes without a quantitative bound that the extra oscillatory terms in the transition rate are always small next to the Planck term, which is what lets it claim a thermal spectrum for all three field definitions rather than only the transparent one.
Editorial extensions
If this is right
- A detector's thermal response in a curved background is fixed by a single embedding-space acceleration, so the Unruh and de Sitter temperature formulas become two cases of one statement rather than separate results.
- In anti-de Sitter there is a minimum four-acceleration below which no temperature is defined, and the reason is kinematic rather than dynamical: the required orbit is spacelike.
- Detectors at constant spatial coordinate in anti-de Sitter, whose embedding orbits close into circles, see no thermal bath at all, which separates thermal from merely accelerated motion.
- Because the leading Planck factor is the same for all three boundary conditions considered, the notion of temperature in anti-de Sitter is claimed not to depend on how the ambiguity in defining the field is resolved.
- In the flat-space limit $R → ∞$ the extra terms in the reflective boundary-condition rates vanish, so the construction reduces to the standard Minkowski Unruh result.
Reading between the lines
- If the temperature really depends only on the five-dimensional acceleration, then any two quite different motions with the same $a_5$ should be thermodynamically indistinguishable to the detector, which is a sharper statement than the paper needs and could be tested against other trajectories in these backgrounds.
- The vanishing result for constant-$ρ$ detectors suggests a clean operational criterion: a trajectory is thermal precisely when its embedding orbit is hyperbolic, which would let one classify detector motions in other curved spaces without computing Wightman functions at all.
- The anti-de Sitter critical acceleration $a_5 = 0$ may be connected to the known tension between anti-de Sitter's lack of global hyperbolicity and any global thermal notion, so a reader might expect the boundary-condition dependence to reappear at next order rather than disappear.
- The claim that the extra non-Planckian terms are subdominant is stated rather than bounded, so a natural extension is to map out, as a function of detector energy splitting and $a_5R$, the region where those terms would be comparable to the Planck term.
Original abstract
We show, in complete accord with the usual Rindler picture, that detectors with constant acceleration $a$ in de Sitter (dS) and Anti de Sitter (AdS) spaces with cosmological constants $\Lambda$ measure temperatures $2\pi T=(\Lambda/3+a^{2})^{1/2}\equiv a_{5}$, the detector "5-acceleration" in the embedding flat 5-space. For dS, this recovers a known result; in AdS, where $\Lambda$ is negative, the temperature is well defined down to the critical value $a_{5}=0$, again in accord with the underlying kinematics. The existence of a thermal spectrum is also demonstrated for a variety of candidate wave functions in AdS backgrounds.
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
We show, in complete accord with the usual Rindler picture, that detectors with constant acceleration $a$ in de Sitter (dS) and Anti de Sitter (AdS) spaces with cosmological constants $\Lambda$ measure temperatures $2\pi T=(\Lambda/3+a^{2})^{1/2}\equiv a_{5}$, the detector "5-acceleration" in the embedding flat 5-space. For dS, this recovers a known result; in AdS, where $\Lambda$ is negative, the temperature is well defined down to the critical value $a_{5}=0$, again in accord with the underlying kinematics. The existence of a thermal spectrum is also demonstrated for a variety of candidate wave functions in AdS backgrounds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We show, in complete accord with the usual Rindler picture, that detectors with constant acceleration $a$ in de Sitter (dS) and Anti de Sitter (AdS) spaces with cosmological constants $\Lambda$ measure temperatures $2\pi T=(\Lambda/3+a^{2})^{1/2}\equiv a_{5}$, the detector "5-acceleration" in the embedding flat 5-space. | result 0.98 | 0.91 | 0.87 | asserted 1.00 |
| s1 | For dS, this recovers a known result; in AdS, where $\Lambda$ is negative, the temperature is well defined down to the critical value $a_{5}=0$, again in accord with the underlying kinematics. | result 0.88 | 0.72 | 0.14 | asserted 1.00 |
| s2 | The existence of a thermal spectrum is also demonstrated for a variety of candidate wave functions in AdS backgrounds. | result 0.85 | 0.90 | 0.07 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v1
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- relations to other papers
- P abstract
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Figures
Figures reproduced from arXiv:gr-qc/9706018 by the authors.
References
How later papers used it
50 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it.
Compares with it 1 of 4 papers with this label
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Connection with GEMS: For (A)dS, our result is exact and reproduces the the famous result by Deser and Levin [ref] for (anti) de-Sitter spacetimes considered as embedded sub-manifolds of a five dimensional flat spacetime (the so called GEMS approach).
Effect of tidal curvature on dynamics of accelerated probes (2021) · Implications and Discussion · 0.92
How the literature uses it
44 arXiv papers cite this in 70 sentences
Papers citing it · 51 sentences without a confident label
Contests it 1 paper
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We will consider three boundary conditions on the timelike boundary of AdS, enumerated according to the value $ε$, as in [ref] : $ε=-1$ is Dirichlet, $ε=1$ is Neumann, and $ε=0$ is “transparent” (i.e. considering AdS embedded within an Einstein static universe [ref] ); we note that in [ref] the latter two boundary conditions were erroneously transposed.
Unruh-DeWitt detectors and entanglement: the anti-de Sitter space · Global AdS modes · contest 0.79
Uses its method or tool 4 papers
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Performing the global embedding of the SAdS spacetime into the higher dimensional Minkowski spacetime, a free-fall observer on the SAdS black hole could be identified with the accelerated observer in the higher dimensional Rindler spacetime [ref] , so that the Rindler observer could find the Unruh temperature as [ref] [equation] where $a$ is the proper acceleration of the observer in the higher dimensional Minkowski spacetime.
Proper temperature of Schwarzschild AdS black hole revisited · Proper temperature from the GEMS · 0.97
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It will be shown that this embedding is enough to derive the Hawking result using the Deser-Levin approach [ref] , instead of the full embedding of the curved space-time.
A New Global Embedding Approach to Study Hawking and Unruh Effects · Introduction · 0.90
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It will be shown that this embedding is enough to derive the Hawking result using the Deser-Levin approach [ref] , instead of the full embedding of the curved space-time.
Quantum Tunneling in Black Holes · Discussions · 0.84
Compares with it 4 papers
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([ref]) and ([ref]) with the effective temperature for an accelerating observer in $dS_2$ and $AdS_2$, which is given by [ref] [equation] where $R$ denotes the scalar curvature in $dS_2$ and $AdS_2$.
One-Loop Effective Action and Schwinger Effect in (Anti-) de Sitter Space · Effective Temperature for Schwinger Effect in $dS_2$ and $AdS_2$ · 0.99
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In Sec. [ref], the proper temperature will be obtained in the two-dimensional SAdS black hole by using the GEMS [ref] in the self-contained manner in comparison with our result.
Proper temperature of Schwarzschild AdS black hole revisited · Introduction · 0.97
-
Connection with GEMS: For (A)dS, our result is exact and reproduces the the famous result by Deser and Levin [ref] for (anti) de-Sitter spacetimes considered as embedded sub-manifolds of a five dimensional flat spacetime (the so called GEMS approach).
Effect of tidal curvature on dynamics of accelerated probes · Implications and Discussion · 0.92
Cites it as background 8 papers
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A unified description of them was first put forwarded by Deser and Levin [ref] which was a sequel to an earlier attempt [ref] .
A New Global Embedding Approach to Study Hawking and Unruh Effects · Introduction · 0.90
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A selection of original papers are [ref] and a review with further references can be found in [ref] .
How often does the Unruh-DeWitt detector click beyond four dimensions? · Introduction · 0.86
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Early studies that considered the Unruh effect in the context of de Sitter spacetime and assigned thermal properties to the de Sitter geometry are [ref] .
Particle detectors in superposition in de Sitter spacetime · Introduction · 0.97
Discussion
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