REVIEW
Spontaneous breaking of Lorentz invariance, black holes and perpetuum mobile of the 2nd kind
High Energy Physics - Theory · Submitted 21 Mar 2006 · 29 references
Pith's verdict: A clever and significant paradox—species-dependent Hawking temperature in Lorentz-violating theories—but the written derivation of the key formula is muddled and the conclusion remains conditional on trans-Planckian behavior.
T0 · AI review
Why Pith says this
Highlight any text to comment on it.
Referee report
The paper
The paper claims
A Lorentz-violating ghost condensate makes black-hole temperature species-dependent, so a two-shell device can transfer heat from a colder shell to a hotter one and violate the second law of thermodynamics.
The reading
What carries the argument
The load-bearing object is the species-dependent effective metric $\tilde g_{\mu\nu} = g_{\mu\nu} + \varepsilon\,\partial_\mu\phi\,\partial_\nu\phi$, which lets each particle species experience a different causal geometry. For the ghost-condensate background $\phi = t$ and the black-hole metric in Lemaitre coordinates, this effective metric is exactly a Schwarzschild metric with mass $\tilde M = (1+\varepsilon)M$ after rescaling time, radius, and mass. That rescaling converts the standard Hawking temperature $(8\pi\tilde M)^{-1}$ in rescaled time into $T_\psi = (1+\varepsilon)^{-3/2} T_H = v^3 T_H$ in physical time. The same metric is then used to build the thermodynamic cycle: two shells that couple exclusively to fast and slow species exchange energy fluxes tuned to cancel, transferring heat from the colder shell to the hotter shell while the black hole mass remains constant.
Extended reading notes
Core claim
The central discovery is that in a Lorentz-violating background of the ghost-condensate type, the notions of black hole temperature and entropy are not universal. The hidden-sector field has a time-dependent vacuum expectation value $\phi = t$, breaking Lorentz invariance while preserving a time-translation symmetry. A visible-sector field $\psi$ coupled through the leading operator $(\partial_\mu\phi\,\partial^\mu\psi)^2$ propagates in the effective metric $\tilde g_{\mu\nu} = g_{\mu\nu} + \varepsilon\,\partial_\mu\phi\,\partial_\nu\phi$, so its maximal velocity is $v = (1+\varepsilon)^{-1/2}$. In the black-hole background this effective metric is again a Schwarzschild metric written in Lemaitre coordinates, but with rescaled mass $\tilde M = (1+\varepsilon)M$. Matching the standard Hawking derivation to the physical time coordinate gives $T_\psi = v^3 T_H$. With two species at different velocities, a black hole surrounded by two shells at temperatures between the two species' radiation temperatures transfers net energy from the colder to the hotter shell while keeping the black hole mass constant, so the second law fails within the effective theory.
Load-bearing premise
The argument assumes that the standard low-energy Hawking derivation can be applied to each species' effective metric, so that trans-Planckian modes and ghost-condensate caustics do not change the answer; the authors explicitly flag this as the possible loophole they call option (ii).
Editorial extensions
If this is right
- If the low-energy derivation holds, a black hole does not have a single temperature; the temperature read by a given species is $v^3$ times the usual Hawking temperature, so superluminal species see a hotter, larger horizon and subluminal species a colder, smaller one.
- The second law of thermodynamics can be violated within the effective theory by the two-shell arrangement, with the black hole serving as a passive intermediary whose mass never changes.
- The black-hole entropy-area relation cannot retain its standard form, because the horizon area itself depends on which species defines the geometry.
- A UV completion of a ghost-condensate-type theory must either restore a universal radiation temperature, introduce new long-lived low-energy degrees of freedom that store the entropy, or be radically nonlocal or unusual in its time structure.
- The argument suggests that the breakdown of black-hole thermodynamics is generic for spontaneously Lorentz-violating effective theories, not an artifact of the specific ghost-condensate action.
Reading between the lines
- One testable extension would be to repeat the derivation in the gauged ghost condensate, which resolves the caustics; if the species-dependent temperature survives, the invalid-derivation loophole is weakened.
- The infinite-tower scenario the authors sketch implies that any theory with arbitrarily fast species has no black-hole horizons at all, so the second law could be restored by direct accounting of the interior — a concrete design criterion for UV completions.
- If the violation is real in the low-energy theory, the paper's bound on transferred energy, $E < \sqrt{|\varepsilon|}/\Lambda$, gives an observable upper limit on how long a Lorentz-violating sector can masquerade as equilibrium thermodynamics.
- The same effective-metric logic should apply to other Lorentz-violating condensates, such as vector-type backgrounds, where the effective metric for each sector is determined by the condensate's alignment; checking whether $T \propto v^3$ arises there would test the genericity claim.
Original abstract
We study the effect of spontaneous breaking of Lorentz invariance on black hole thermodynamics. We consider a scenario where Lorentz symmetry breaking manifests itself by the difference of maximal velocities attainable by particles of different species in a preferred reference frame. The Lorentz breaking sector is represented by the ghost condensate. We find that the notions of black hole entropy and temperature loose their universal meaning. In particular, the standard derivation of the Hawking radiation yields that a black hole does emit thermal radiation in any given particle species, but with temperature depending on the maximal attainable velocity of this species. We demonstrate that this property implies violation of the second law of thermodynamics, and hence, allows construction of a perpetuum mobile of the 2nd kind. We discuss possible interpretation of these results.
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 study the effect of spontaneous breaking of Lorentz invariance on black hole thermodynamics. We consider a scenario where Lorentz symmetry breaking manifests itself by the difference of maximal velocities attainable by particles of different species in a preferred reference frame. The Lorentz breaking sector is represented by the ghost condensate. We find that the notions of black hole entropy and temperature loose their universal meaning. In particular, the standard derivation of the Hawking radiation yields that a black hole does emit thermal radiation in any given particle species, but with temperature depending on the maximal attainable velocity of this species. We demonstrate that this property implies violation of the second law of thermodynamics, and hence, allows construction of a perpetuum mobile of the 2nd kind. We discuss possible interpretation of these results.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the effect of spontaneous breaking of Lorentz invariance on black hole thermodynamics. | contribution 0.95 | 0.86 | 0.03 | unclear 0.65 |
| s1 | We consider a scenario where Lorentz symmetry breaking manifests itself by the difference of maximal velocities attainable by particles of different species in a preferred reference frame. | method 0.75 | 0.32 | 0.03 | asserted 0.78 |
| s2 | The Lorentz breaking sector is represented by the ghost condensate. | method 0.95 | 0.18 | 0.02 | asserted 1.00 |
| s3 | We find that the notions of black hole entropy and temperature loose their universal meaning. | result 0.96 | 0.93 | 0.13 | asserted 1.00 |
| s4 | In particular, the standard derivation of the Hawking radiation yields that a black hole does emit thermal radiation in any given particle species, but with temperature depending on the maximal attainable velocity of this species. | result 1.00 | 0.91 | 0.12 | asserted 1.00 |
| s5 | We demonstrate that this property implies violation of the second law of thermodynamics, and hence, allows construction of a perpetuum mobile of the 2nd kind. | interpretation 0.57 | 0.92 | 0.87 | asserted 1.00 |
| s6 | We discuss possible interpretation of these results. | interpretation 0.94 | 0.61 | 0.02 | unclear 0.94 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
References
How the literature uses it
73 arXiv papers cite this in 124 sentences
Papers citing it · 101 sentences without a confident label
Compares with it 5 papers
-
For the Schwarzschild metric, we will study the null geodesics finding that particles with different energies will still have the same horizon, in contrast with the LIV case [ref] .
Phenomenological consequences of a geometry in the cotangent bundle · Introduction · 0.98
-
The nonstandard photons perceive a horizon outside the standard Schwarzschild event horizon at $r=r_Schw≡2M$.[The effective background [ref] agrees with the effective metric obtained in Ref. [ref] for a minimally coupled scalar field interacting with the ghost condensate [ref] .
Lorentz violation and black-hole thermodynamics: Compton scattering process · Effective Schwarzschild metric for the photons · 0.90
-
Analogously to Lorentz-violating gravitational black holes [ref] , the effective Hawking temperature of the noncommutativity acoustic black holes now is not universal for all species of particles.
Supersonic Velocities in Noncommutative Acoustic Black Holes · Introduction · 0.85
Cites it as support 2 papers
-
An argument suggesting that it may be impossible is the break-down of the black hole thermodynamics in the ghost condensate model [ref] , and more generally, in theories with low-energy violation of Lorentz invariance [ref] .
On the Extra Mode and Inconsistency of Horava Gravity · Conclusions and discussion · 0.99
-
Small violations of Lorentz invariance also lead to problems with black hole thermodynamics: unless black holes simply do not exist in the underlying theory, such effects generically violate the generalized second law of thermodynamics, allowing perpetual motion machines [ref] .
Challenges for Emergent Gravity · Lorentz invariance · 0.99
Cites it as background 14 papers
-
In [ref] , Dubovsky and Sibiryakov discussed a relationship between theories with spontaneous violation of Lorentz invariance, and violation of the generalized second law of thermodynamics (GSL) [ref] in the presence of black holes.
Spontaneous Lorentz Violation, Negative Energy and the Second Law of Thermodynamics · Introduction · 0.92
-
In the context of the ghost condensate models [ref] this question has been raised in Refs. [ref] .
Black Hole Thermodynamics and Massive Gravity · Introduction · 0.80
-
Recently, Dubovsky and Sibiryakov [ref] investigated a specific Lorentz-violating theory, based on the ghost-condensate model of Ref. [ref] , which allows for different maximal velocities of different species of particles in a preferred reference frame.
Lorentz violation and black-hole thermodynamics · Introduction · 0.99
Discussion
Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.