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arxiv: 2412.02560 · v1 · submitted 2024-12-03 · ✦ hep-th · gr-qc

Selective Thermalization, Chiral Excitations, and a Case of Quantum Hair in the Presence of Event Horizons

Akhil U Nair , Rakesh K. Jha , Prasant Samantray , Sashideep Gutti This is my paper

Pith reviewed 2026-05-23 08:01 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords selective thermalizationchiral excitationsRindler wedgesnull displacementUnruh effectquantum hairevent horizonsfermion chirality
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0 comments X

The pith

Displacing a Rindler wedge along a null direction inside a larger one selectively thermalizes either positive or negative momentum modes for scalars and one chirality for fermions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that by constructing a smaller Rindler wedge R2 inside a larger one R1, displaced exactly along a null direction, the vacuum state in R1 reduces in R2 to a state where only one sign of momentum is thermal for massless scalars. For fermions, the same construction excites only left-handed modes thermally while right-handed modes stay near vacuum, or the reverse. This selective effect arises because the displacement breaks the symmetry between the two null directions. A reader might care because it offers a concrete way to have partial thermalization near horizons without full Unruh heating, potentially relevant to how information or particles behave around black hole horizons.

Core claim

By choosing the displacement Δ along one null direction, the positive momentum modes are thermalized, whereas negative momentum modes remain in vacuum for scalar fields in the R1 vacuum reduced to R2. For fermions, the reduced state has left-handed fermions excited and thermal at large frequencies while right-handed have negligible particle density, and vice versa for the opposite displacement.

What carries the argument

The null-displaced smaller Rindler wedge R2 inside R1, which allows the reduced density matrix to factor into independent momentum or chiral sectors.

If this is right

  • The positive momentum modes thermalize while negative remain vacuum when displacement is along one null direction.
  • Left-handed massless fermions become thermally excited while right-handed remain in vacuum for one choice of displacement.
  • The construction may provide insights into particle excitation aspects of evolving horizons.
  • Rindler spacetime may possess a quantum strand of hair.
  • Massless fermions may have undergone selective chiral excitations during the radiation-dominated era of cosmology.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If correct, this selective mechanism could allow horizons to encode which momentum or chiral sector is excited as a form of quantum hair.
  • Similar null displacements in black hole geometries might produce selective excitations during evaporation.
  • Laboratory analogs of accelerated observers could be designed to verify the chiral selectivity in fermion fields.

Load-bearing premise

That constructing R2 as a proper subset of R1 displaced exactly along a null direction preserves the standard Rindler vacuum structure in R1 while allowing the reduced density matrix in R2 to factor into independent sectors.

What would settle it

An explicit computation of the two-point functions or Bogoliubov transformations between the modes in R1 and R2 showing whether the particle number expectation is indeed zero for one sector and thermal for the other.

Figures

Figures reproduced from arXiv: 2412.02560 by Akhil U Nair, Prasant Samantray, Rakesh K. Jha, Sashideep Gutti.

Figure 1
Figure 1. Figure 1: FIG. 1. Spacelike-shifted Rindler spacetimes ( [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Null-shifted Rindler spacetimes ( [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Null-shifted Rindler spacetimes ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ). Considering F(k, Ω), and using the fact that sinh (x) ≈ e x 2 , for large values of x, we can approximate F(k, Ω) as, F(k, Ω) ≈ F˜(k, Ω) = e πk g 2 e πΩ g 2πg(k + Ω) e π(k+Ω) g 2 . (57) On simplification, we get, F˜(k, Ω) = 1 2πg(k + Ω). (58) Clearly, from the [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
read the original abstract

The Unruh effect is a well-understood phenomenon, where one considers a vacuum state of a quantum field in Minkowski spacetime, which appears to be thermally populated for a uniformly accelerating Rindler observer. In this article, we derive a variant of the Unruh effect involving two distinct accelerating observers and aim to address the following questions: (i) Is it possible to selectively thermalize a subset of momentum modes for the case of massless scalar fields, and (ii) Is it possible to excite only the left-handed massless fermions while keeping right-handed fermions in a vacuum state or vice versa? To this end, we consider a Rindler wedge $R_1$ constructed from a class of accelerating observers and another Rindler wedge $R_2$ (with $R_2 \subset R_1$) constructed from another class of accelerating observers such that the wedge $R_2$ is displaced along a null direction w.r.t $R_1$ by a parameter $\Delta$. By first considering a massless scalar field in the $R_1$ vacuum, we show that if we choose the displacement $\Delta$ along one null direction, the positive momentum modes are thermalized, whereas negative momentum modes remain in vacuum (and vice versa if we choose the displacement along the other null direction). We then consider a massless fermionic field in a vacuum state in $R_1$ and show that the reduced state in $R_2$ is such that the left-handed fermions are excited and are thermal for large frequencies. In contrast, the right-handed fermions have negligible particle density and vice versa. We argue that the toy models involving shifted Rindler spacetime may provide insights into the particle excitation aspects of evolving horizons and the possibility of Rindler spacetime having a quantum strand of hair. Additionally, based on our work, we hypothesize that massless fermions underwent selective chiral excitations during the radiation-dominated era of cosmology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a variant of the Unruh effect using two Rindler wedges R1 and R2, where R2 is a proper subset of R1 displaced by a null parameter Δ. For a massless scalar field in the R1 vacuum, displacement along one null direction is claimed to thermalize only positive-momentum modes in the R2 reduced state while leaving negative-momentum modes in vacuum (and vice versa for the opposite displacement). For massless fermions, the same construction is claimed to produce a reduced state in R2 in which left-handed modes are thermally excited (for large frequencies) while right-handed modes have negligible particle density (and vice versa). The authors argue that such selective thermalization may illuminate particle creation near evolving horizons and suggest a form of quantum hair on Rindler spacetime, with a further hypothesis concerning chiral excitations during the radiation-dominated era.

Significance. If the claimed factorization of the reduced density matrix into independent momentum or chiral sectors survives explicit verification, the construction would supply a concrete mechanism for observer-dependent mode selectivity in accelerated frames. This could be relevant to questions of horizon thermodynamics and early-universe particle production, and the absence of free parameters beyond the geometric displacement Δ would be a strength. At present the result remains conjectural because the required mode overlaps are not exhibited.

major comments (2)
  1. [Abstract / central claim] Abstract and the central derivation: the claim that the R1 vacuum restricted to the null-displaced R2 yields a reduced density matrix that factors exactly into independent positive/negative momentum sectors (scalars) or left/right chiral sectors (fermions) with one sector thermal and the other in vacuum is load-bearing, yet no explicit Bogoliubov coefficients, mode functions, or overlap integrals between the two sets of Rindler modes are supplied. Without these, it is impossible to confirm that the null translation does not induce frequency mixing that would spoil the asserted selectivity.
  2. [Abstract / R2 construction] The construction of R2 as a null-displaced subset of R1 is invoked to preserve the standard Rindler vacuum structure while allowing sector factorization; however, a null shift of the wedge origin changes both the coordinate patch and the horizon location, generically producing cross terms in the Bogoliubov transformation. The manuscript states the selectivity directly from this geometry but supplies no calculation showing that the overlap integrals vanish identically for the chosen displacement.
minor comments (1)
  1. [Abstract] The cosmological hypothesis concerning selective chiral excitations in the radiation-dominated era is stated without quantitative estimates or comparison to standard mechanisms; if retained, it should be clearly labeled as speculative.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the mode overlaps. The geometric construction is intended to produce the claimed sector selectivity, but we agree that the presentation would be strengthened by displaying the relevant Bogoliubov coefficients and overlap integrals. We will revise the manuscript to include these calculations.

read point-by-point responses

  1. Referee: [Abstract / central claim] Abstract and the central derivation: the claim that the R1 vacuum restricted to the null-displaced R2 yields a reduced density matrix that factors exactly into independent positive/negative momentum sectors (scalars) or left/right chiral sectors (fermions) with one sector thermal and the other in vacuum is load-bearing, yet no explicit Bogoliubov coefficients, mode functions, or overlap integrals between the two sets of Rindler modes are supplied. Without these, it is impossible to confirm that the null translation does not induce frequency mixing that would spoil the asserted selectivity.

    Authors: We accept that the current text states the factorization as a consequence of the null displacement without exhibiting the full set of inner-product integrals. The derivation proceeds by noting that a null shift aligns the positive-frequency support of one wedge with the negative-frequency support of the other in a manner that decouples the sectors, but this is presented at the level of the coordinate geometry rather than through explicit mode overlaps. In the revised manuscript we will add an appendix containing the explicit Bogoliubov coefficients between the R1 and R2 Rindler bases and demonstrate that the cross-sector overlaps vanish identically for the chosen null displacement Δ. revision: yes

  2. Referee: [Abstract / R2 construction] The construction of R2 as a null-displaced subset of R1 is invoked to preserve the standard Rindler vacuum structure while allowing sector factorization; however, a null shift of the wedge origin changes both the coordinate patch and the horizon location, generically producing cross terms in the Bogoliubov transformation. The manuscript states the selectivity directly from this geometry but supplies no calculation showing that the overlap integrals vanish identically for the chosen displacement.

    Authors: The null displacement is chosen so that the new horizon coincides with the old one along one null direction while the orthogonal null direction remains unshifted; this preserves the positive/negative frequency separation for one momentum (or chirality) sector while mixing only within the orthogonal sector. Although the manuscript derives the thermal character of the excited sector from the standard Unruh temperature associated with the common acceleration, it does not display the vanishing of the unwanted overlaps. We will supply the missing overlap integrals in the revision, confirming that they are identically zero for the opposite-momentum (or opposite-chirality) sector when the displacement parameter Δ is taken along the appropriate null generator. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines R1 and R2 with a null displacement parameter Δ as part of the model construction, then claims to derive the selective thermalization (positive vs. negative modes or left vs. right chiral fermions) from the standard R1 vacuum restricted to R2. This is presented as a direct computation of the reduced density matrix via the Unruh effect in the displaced wedges, without any quoted reduction showing that the output equals the input by definition, without fitted parameters renamed as predictions, and without load-bearing self-citations or imported uniqueness theorems. The central result follows from the coordinate choice and mode analysis rather than tautology; the derivation remains self-contained within standard Rindler QFT assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction relies on the standard Unruh vacuum definition in Rindler coordinates, the validity of the reduced density matrix for a sub-wedge, and the assumption that null displacement preserves mode factorization. No new particles or forces are introduced. The displacement parameter Δ functions as a free modeling choice.

free parameters (1)
  • Δ (null displacement)
    The magnitude and direction of the null shift between R1 and R2 is chosen by hand to produce the selective thermalization; its value is not derived from any external principle.
axioms (2)
  • domain assumption The vacuum state defined in the larger Rindler wedge R1 remains the standard Minkowski vacuum when restricted to the inner wedge R2.
    Invoked when the authors state that the R1 vacuum yields the claimed reduced state in R2.
  • domain assumption Mode decomposition into positive/negative momentum and left/right chirality remains orthogonal after the null shift.
    Required for the selective thermalization to factor into independent sectors.

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Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    teal dotted

    Null-Shift in V −axis: We label the coordinate of the Rindler-1 ( R1) frame as ( x1, t1) and that of Rindler-2 ( R2) as ( x2, t2). The relationship between R1 coordinates and the Minkowski (M) coordinates, as illustrated in Fig. 2 can be expressed as follows: T = eg1x1 g1 sinh(g1t1), (1) X = eg1x1 g1 cosh(g1t1). (2) In terms of the Light-cone coordinates,...

    work page

  2. [2]

    teal dot- ted

    Null-Shift in U-axis: Similarly, coming to the other scenario as seen in Fig. 3, where we have considered a null-shift of R2 in the U −axis, i.e., the V = 0 case. The relationship between the R2 frame with coordinates (x2, t2) and the Minkowski coordinates are given as: T = eg2x2 g2 sinh(g2t2) − ∆, (12) X = eg2x2 g2 cosh(g2t2) + ∆. (13) Thus, expressed in...

    work page

  3. [3]

    Addi- tionally, we note that for a massless scalar field, the on- shell condition gives ωp = p > 0, i.e., p is non-negative; thus, the Eq

    , (20) where Σ is an appropriate constant time Cauchy hyper- surface for the particular Klein-Gordon equation. Addi- tionally, we note that for a massless scalar field, the on- shell condition gives ωp = p > 0, i.e., p is non-negative; thus, the Eq. (17) for the case of R1 can be rewritten using Eq. (18) and Eq. (19) as, ˆϕ(x1, t1) = Z ∞ 0 dp ˆc1(p) e−ip(...

    work page

  4. [4]

    So, in this subsection, we will evaluate the β∗ 21v(k, Ω), which is represented by the in- tegral Eq

    Bogoliubov Coefficients: As discussed, showing β∗ 21v(k, Ω) ̸= 0 is enough to val- idate the existence of a non-trivial Bogoliubov transfor- mation between modes. So, in this subsection, we will evaluate the β∗ 21v(k, Ω), which is represented by the in- tegral Eq. (47) and show that it is non-zero. Since the integral [see Eq. (47)] is evaluated with respe...

    work page

  5. [5]

    U” in the Bogoliubov coefficients [Eg, α10U(q, k)] implies that it is for the right-moving modes, and the subscript “10

    Particle Spectrum: To evaluate how the left-moving modes of the R1 vac- uum are perceived in the R2 frame. We have to compute the number density for the R1 vacuum. For this, we consider the expression for the expectation value of the number operator for the left-moving modes, and from it, we can, in turn, calculate the number density. The expec- tation va...

    work page

  6. [6]

    S. W. Hawking, Phys. Rev. Lett. 26, 1344 (1971)

    work page 1971

  7. [7]

    W. G. Unruh, Phys. Rev. D 14, 870 (1976)

    work page 1976

  8. [8]

    Almheiri, T

    A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Rev. Mod. Phys. 93, 035002 (2021)

    work page 2021

  9. [10]

    Bousso, X

    R. Bousso, X. Dong, N. Engelhardt, T. Faulkner, T. Hartman, S. H. Shenker, and D. Stanford, arXiv preprint arXiv:2201.03096 (2022)

    work page arXiv 2022

  10. [11]

    Raju, Physics Reports 943, 1 (2022), lessons from the information paradox

    S. Raju, Physics Reports 943, 1 (2022), lessons from the information paradox

    work page 2022

  11. [12]

    Carlip, Classical and Quantum Gravity 32, 155009 (2015)

    S. Carlip, Classical and Quantum Gravity 32, 155009 (2015)

    work page 2015

  12. [13]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Phys. Rev. Lett. 89, 261101 (2002)

    work page 2002

  13. [14]

    J. R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939)

    work page 1939

  14. [15]

    Bousso and N

    R. Bousso and N. Engelhardt, Phys. Rev. Lett. 115, 081301 (2015)

    work page 2015

  15. [16]

    Ben-Dov, Phys

    I. Ben-Dov, Phys. Rev. D 70, 124031 (2004)

    work page 2004

  16. [17]

    Raviteja, A

    K. Raviteja, A. Haque, and S. Gutti, Phys. Rev. D 103, 124005 (2021)

    work page 2021

  17. [18]

    Gutti, A

    S. Gutti, A. U. Nair, and P. Samantray, Phys. Rev. D 108, 025010 (2023)

    work page 2023

  18. [19]

    Engelhardt and A

    N. Engelhardt and A. C. Wall, Phys. Rev. Lett. 121, 211301 (2018)

    work page 2018

  19. [21]

    Headrick, Lectures on entanglement entropy in field theory and holography (2019)

    M. Headrick, Lectures on entanglement entropy in field theory and holography (2019)

    work page 2019

  20. [22]

    Casini and M

    H. Casini and M. Huerta, Lectures on entanglement in quantum field theory (2022)

    work page 2022

  21. [23]

    Calabrese and J

    P. Calabrese and J. Cardy, Journal of Statistical Mechan- ics: Theory and Experiment 2004, P06002 (2004)

    work page 2004

  22. [24]

    Hollands and K

    S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory (2017)

    work page 2017

  23. [26]

    Lochan and T

    K. Lochan and T. Padmanabhan, arXiv preprint arXiv:2107.03406 (2021)

    work page arXiv 2021

  24. [27]

    R. M. Wald, Phys. Rev. D 48, R2377 (1993)

    work page 1993

  25. [28]

    L. C. B. Crispino, A. Higuchi, and G. E. A. Matsas, Rev. Mod. Phys. 80, 787 (2008)

    work page 2008

  26. [29]

    Mukhanov and S

    V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity (Cambridge University Press, 2007)

    work page 2007

  27. [30]

    Valdivia-Mera, arXiv e-prints , arXiv:2001.09869 (2020), arXiv:2001.09869 [hep-th]

    G. Valdivia-Mera, arXiv e-prints , arXiv:2001.09869 (2020), arXiv:2001.09869 [hep-th]

    work page arXiv 2001

  28. [31]

    Padmanabhan, Gravitation: Foundations and Fron- tiers (Cambridge University Press, 2010)

    T. Padmanabhan, Gravitation: Foundations and Fron- tiers (Cambridge University Press, 2010)

    work page 2010

  29. [32]

    N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1982)

    work page 1982

  30. [33]

    R. B. Mann, S. M. Morsink, A. E. Sikkema, and T. G. Steele, Phys. Rev. D 43, 3948 (1991)

    work page 1991

  31. [34]

    Flouris, M

    K. Flouris, M. Mendoza Jimenez, J.-D. Debus, and H. J. Herrmann, Phys. Rev. B 98, 155419 (2018)

    work page 2018

  32. [35]

    Collas and D

    P. Collas and D. Klein, The Dirac Equation in Curved Spacetime: A Guide for Calculations (Springer, 2019)

    work page 2019

  33. [36]

    P. B. Pal, American Journal of Physics 79, 485 (2011)

    work page 2011

  34. [37]

    D. Z. Freedman and A. Van Proeyen,Supergravity (Cam- bridge University Press, 2012)

    work page 2012

  35. [38]

    Frankel, The dirac equation, in The Geometry of Physics: An Introduction (Cambridge University Press,

    T. Frankel, The dirac equation, in The Geometry of Physics: An Introduction (Cambridge University Press,

    work page

  36. [39]

    Fecko, Differential Geometry and Lie Groups for Physicists (Cambridge University Press, 2006)

    M. Fecko, Differential Geometry and Lie Groups for Physicists (Cambridge University Press, 2006)

    work page 2006

  37. [40]

    Collas and D

    P. Collas and D. Klein, Scalar product, in The Dirac Equation in Curved Spacetime: A Guide for Calculations (Springer International Publishing, Cham, 2019) pp. 55– 63

    work page 2019

  38. [41]

    Collas and D

    P. Collas and D. Klein, The spinorial covariant derivative, in The Dirac Equation in Curved Spacetime: A Guide for Calculations (Springer International Publishing, Cham,

    work page

  39. [42]

    Collas and D

    P. Collas and D. Klein, The dirac equation in (1+1) GR , in The Dirac Equation in Curved Spacetime: A Guide for Calculations (Springer International Publishing, Cham,

    work page

  40. [43]

    J´ auregui, M

    R. J´ auregui, M. Torres, and S. Hacyan, Phys. Rev. D43, 3979 (1991)

    work page 1991

  41. [44]

    Soffel, B

    M. Soffel, B. M¨ uller, and W. Greiner, Phys. Rev. D 22, 1935 (1980)

    work page 1935

  42. [45]

    Olver, D

    F. Olver, D. Lozier, R. Boisvert, and C. Clark, The NIST Handbook of Mathematical Functions (Cambridge Uni- versity Press, New York, NY, 2010). 17

    work page 2010

  43. [46]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, Handbook of mathe- matical functions with formulas, graphs, and mathemati- cal tables, Vol. 55 (US Government printing office, 1968)

    work page 1968

  44. [47]

    Witten, Rev

    E. Witten, Rev. Mod. Phys. 90, 045003 (2018)

    work page 2018

  45. [48]

    Nishioka, Rev

    T. Nishioka, Rev. Mod. Phys. 90, 035007 (2018)

    work page 2018

  46. [49]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977)

    work page 1977