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A time-like window into tensionless worldsheets

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Pith's one-line read This paper argues that a closed string worldsheet embedded in the future and past Kasner (Milne) wedges of Minkowski spacetime becomes tensionless and Carrollian as it approaches its null horizons, and that this limit is exactly…

desk verdict A promising extension of the Rindler-tensionless program to Kasner worldsheets, undermined by a mode-dependent contraction parameter that fails to define a single tensionless limit. read the letter →

arxiv 2412.06387 v2 pith:7VRA6EUM submitted 2024-12-09 hep-th

Sudip Karan , Bibhas Ranjan Majhi This is my paper · ORCID
classification hep-th PACS 11.25.-w04.62.+v03.65.Ud
keywords tensionlessstringsCarrollianworldsheetsKasnerworldsheetRindlerBogoliubovtransformationstimelikeUnruheffectnullstringcomplementarityHagedorntemperature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Closed strings on a time-evolving Kasner (Milne) worldsheet are claimed to become tensionless, Carrollian strings when the worldsheet approaches its null horizons, with an infinite limit on the time-evolution parameter $a/c$. The paper constructs the quantum theory of tensile Kasner worldsheets, derives Bogoliubov transformations linking their oscillators to inertial Minkowski oscillators, and shows that the near-horizon limit of these transformations reproduces exactly the known tensionless/Carrollian map on a flat worldsheet. If this identification holds, time-like entanglement between future and past Kasner wedges is not merely analogous to, but exactly equivalent to, space-like right/left Rindler entanglement at the tensionless point. That would give a second, time-like route to the same tensionless string physics, with consequences for open-string emergence, a time-like Unruh temperature, and Hagedorn-type transitions.

What carries the argument

The load-bearing object is the set of Bogoliubov transformations connecting Kasner worldsheet oscillators ($\beta$) to inertial Minkowski oscillators ($\alpha$), with coefficients built from the Kasner parameter $a$ and mode frequency $\Omega_n = 2\pi c n / L_\varphi$. Near the null horizon, $a/c \to \infty$, the hyperbolic coefficients in eq. (3.37) expand to leading order and the map becomes the tensionless/Carrollian Bogoliubov map (2.20); the single matching parameter is $\epsilon = \pi^2 n c/(\psi a)$. The same machinery produces the evolving squeezed vacuum $|0_K(a)\rangle$, whose $a \to \infty$ limit is the tensionless vacuum $|0_c\rangle$, and the timelike Unruh temperature $T_U = a/2\pi$.

What would settle it

Compute the exact algebra of the limiting operators $\tilde{\beta}^\infty_n$ and $\beta^\infty_n$ using the full near-horizon expansion of (3.37) for two distinct modes $n \neq m$ and different regularization shifts $\psi$; if the commutator develops a nonzero cross-term or any residual dependence on $\psi$ beyond the leading-order identification, the limiting oscillators do not form the tensionless algebra (2.16) and the claimed equivalence would fail.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that Kasner worldsheets open a time-like window onto tensionless strings: as $a/c \to \infty$, the worldsheet metric degenerates, the Kasner oscillators map onto the tensionless $c$-oscillators, and the vacuum becomes the tensionless vacuum $|0_c\rangle$, a squeezed state that behaves as a Dirichlet boundary state in the inertial frame. The equivalence is established at the level of the Bogoliubov transformations (3.37), whose leading-order near-horizon expansion (4.3) is exactly the flat-worldsheet tensionless map (2.20) with $\epsilon = \pi^2 n c/(\psi a)$. The concluding claim is that time-like (F-P) entanglement in Kasner worldsheets is precisely equivalent to standard (R-L) entanglement in Rindler worldsheets, viewed in causally disconnected regions.

Load-bearing premise

The load-bearing premise is that the single parameter $\epsilon = \pi^2 n c/(\psi a)$ really is a legitimate tensionless contraction parameter for the whole Kasner worldsheet; since it depends on the oscillator mode number $n$ and on the arbitrary regularization shift $\psi$, the near-horizon limit does not define one coherent worldsheet contraction unless that dependence is absorbed.

Editorial extensions

If this is right

  • The Kasner worldsheet vacuum $|0_K(a)\rangle$ is a squeezed state over the Minkowski vacuum, and in the $a \to \infty$ limit it becomes the tensionless vacuum $|0_c\rangle$, so tensionless Kasner strings inherit a built-in open-string and boundary-state description.
  • Because the tensionless vacuum $|0_c\rangle$ satisfies Dirichlet gluing conditions and the Minkowski vacuum becomes a Neumann boundary state, closed strings in this limit exhibit null string complementarity: closed-string worldsheets look like open strings, D-instantons, or space-filling D-branes.
  • The same Bogoliubov coefficients yield the timelike Unruh temperature $T_U = a/2\pi$, so a Kasner-worldsheet observer sees a Bose-Einstein spectrum; as $T_U \to \infty$ the effective tension flows to zero, reproducing Hagedorn-like physics.
  • The two claimed routes to tensionlessness, $a \to \infty$ at fixed $c$ and $c \to 0$ at fixed $a$, are complementary Carrollian contractions of the same Kasner worldsheet, giving an $\epsilon$-parameter flow from tensile to tensionless regimes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to compute the exact commutator of the limiting operators $\tilde{\beta}^\infty_n$ and $\beta^\infty_m$ for distinct modes; if the mode dependence of $\epsilon$ survives beyond leading order, the limiting algebra may not be the single standard tensionless oscillator algebra, and the equivalence would hold only per mode rather than for the full worldsheet.
  • The same near-horizon matching should occur for worldsheets embedded in spacetimes whose near-horizon slices are Kasner-like, such as black-hole interiors or Milne-like cosmologies; if so, the paper's mechanism would be a general principle that any null horizon induces Carrollian, tensionless worldsheet physics.
  • One could define a finite-$a$ effective tension $T_{\rm eff} = T \epsilon(a)$ and compare it with the Hagedorn effective-tension formula, yielding a quantitative relation between the Kasner time-evolution parameter and the worldsheet temperature that could be checked in a worldsheet thermodynamic calculation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The manuscript constructs a time-evolving (Kasner/Milne) closed-string worldsheet by reparametrizing the Minkowski worldsheet coordinates, derives its quantum mode expansion and Bogoliubov transformations, and claims that in the near-horizon limit a/c -> infinity the Bogoliubov map truncates to the flat tensionless/Carrollian contraction (2.20), making Kasner time-like entanglement exactly equivalent to Rindler space-like entanglement. It also interprets the resulting tensionless vacuum as boundary states and connects the construction to timelike Unruh and Hagedorn physics.

Significance. The question addressed is natural and timely: whether a time-like analogue of the Rindler tensionless worldsheet exists. The paper is self-contained and develops a detailed tensile Kasner worldsheet construction, explicit squeezed-state vacua, and a clear external benchmark in the flat tensionless map (2.20). If the central identification were valid, it would establish a new connection between time-like entanglement, Carrollian worldsheets, open/closed string transitions, and Hagedorn behavior. However, the load-bearing identification of the contraction parameter is not sound, and the claimed exact equivalence is not supported by the derivation as written.

major comments (5)
  1. [Sec. 4, Eq. (4.5)] The central identification epsilon = pi^2 n c/(psi a) in Eq. (4.5) is mode-dependent and depends on the arbitrary regularization shift psi introduced in Eq. (3.14). In the flat-worldsheet contraction (2.18)-(2.20), epsilon is a single parameter defining the contraction of the worldsheet coordinate tau for all modes; it cannot depend on the oscillator index n or on a coordinate reparametrization. Matching (4.3) to (2.20) therefore requires a different contraction parameter for every n and every psi, so the near-horizon Kasner limit does not define one worldsheet contraction. The remark in Sec. 4.1 that the result holds 'up to constant factors n and psi' does not resolve the problem, because epsilon itself is defined by those factors. Consequently, the conclusion that the near-horizon Kasner worldsheet is precisely equivalent to the Rindler tensionless worldsheet is not established.
  2. [Sec. 4, Eqs. (4.1)-(4.3)] The approximation Omega_n/a ~ 2 pi c n/(a psi) in Eq. (4.1) and the subsequent expansion (4.2) require pi Omega_n/(2a) << 1. For fixed a, psi, and c, this fails for sufficiently large n, so the truncation leading to Eq. (4.3) is not uniform in the mode number. Because the mode expansion (3.33) contains all n, the limiting Bogoliubov map is not obtained for the whole worldsheet; the expansion and the limit a -> infinity do not commute with the sum over n. This reinforces the mode-dependence problem in Eq. (4.5): no single tensionless parameter governs the limit for all oscillators.
  3. [Sec. 3.2, Eqs. (3.12)-(3.16)] The regularization procedure produces a complex effective periodicity. Eq. (3.12) evaluates xi at tau = 0 with negative arguments, and Eq. (3.15) explicitly sets phi = +/- (c/a) exp(-a psi) exp(+/- i pi/2), yet no branch prescription for L_phi = (1/a) ln(1 + ell/phi) is given. Since L_phi enters the mode frequency Omega_n in Eq. (3.23), the Bogoliubov coefficients in Eq. (3.40), and finally the inferred epsilon in Eq. (4.5), all physical quantities inherit a complex and branch-dependent regulator. A real, branch-independent definition of L_eff and Omega_n is needed before the tensionless limit can be meaningfully extracted.
  4. [Sec. 3.2, Eq. (3.11) and footnote 6] The effective periodicity L_eff is derived from the closedness condition at tau = 0 (footnote 6), but L_eff is then used through Eq. (3.23) to define frequencies Omega_n for modes at all times. If the identification of the Kasner and Minkowski periodicities is valid only on a single time slice, the global mode expansion (3.20) is not justified. This is a load-bearing assumption that needs to be either proven for all tau or explicitly relaxed.
  5. [Sec. 3.4, Eqs. (3.28)-(3.36)] The normalized global Kasner modes in Eq. (3.28) are presented without derivation. The orthonormality relations (3.29) and the P-wedge inner product (3.31) are stated, but the normalization constants 1/sqrt(2 sinh(pi Omega_n/a)) are not obtained from those inner products; appendix A stops at the unnormalized combinations (A.2)-(A.6). Since these modes determine the Bogoliubov coefficients (3.37), the thermal spectrum (5.8), and the tensionless map (4.3), the normalization should be verified explicitly for the Kasner construction rather than assumed by analogy with the Rindler case.
minor comments (4)
  1. [Eq. (3.17)] The expression c^2 tau^2 - (sigma^2 + phi)^2 appears to be a typo; from the shifted inversion of (3.8) one expects c^2 tau^2 - (sigma + phi)^2.
  2. [Secs. 2.3 and 3.4] The symbol beta is used both for the numerical Bogoliubov coefficients beta_+ and beta_- in Eq. (2.20) and for the worldsheet oscillator operators beta^Lambda_n in Eq. (3.20); this notation is confusing and should be changed, for example to b^Lambda_n for the Kasner oscillators.
  3. [Sec. 4, Eq. (4.8)] The claimed flow starts at a = 0, but the Kasner transformation (2.2) is singular at a = 0; the a = 0 endpoint of the flow in Eq. (4.8) is therefore not defined and requires a separate limiting prescription.
  4. [Eq. (3.15)] The passage following Eq. (3.15) states the condition 0 < |a phi/c| < 1, but phi is complex there; the inequality should be stated in terms of the modulus or the branch should be specified before imposing it.

Circularity Check

1 steps flagged · score 6.0 of 10

The central equivalence is enforced by matching: the tensionless parameter ε is read off from the known flat map (2.20), and the paper itself concedes the identification holds only 'up to constant factors n and ψ.'

  1. fitted input called prediction [Section 4, eqs. (4.3)-(4.5); Sec. 4.1 remark]
    "Comparing (2.20) and (4.3) leads us to the identification of the tensionless state of the Kasner worldsheet, ˜cµ n = ˜β∞ µ n, c µ n = β∞ µ n, ˜cµ −n = ˜β∞ µ −n, c µ −n = β∞ µ −n, (4.4) followed by inferring: ϵ = π²nc ψa , ∀ n > 0. (4.5)"

    The paper's key claim—that the near-horizon Kasner worldsheet becomes tensionless and exactly equivalent to the flat/Rindler tensionless worldsheet—rests on identifying the derived map (4.3) with the known tensionless map (2.20). That comparison does not predict ε; it defines ε = π²nc/(aψ) by matching the two maps. Eq. (4.5) is therefore the condition imposed to make (4.3) equal (2.20), not an independent output of the derivation. Moreover, ε depends on the oscillator mode number n and on the arbitrary regularization shift ψ introduced in eq. (3.14), so the identification does not define a single worldsheet contraction of the type used in (2.18). The paper itself concedes in Sec.

full rationale

Apart from eq. (4.5), the derivation is largely self-contained: the Kasner mode expansion, the Bogoliubov map (3.37), and the near-horizon asymptotic expansion (4.1)–(4.3) are performed within the paper, with refs. [19,20] (not by these authors) serving as an independent benchmark for the Rindler/flat tensionless map. There is no load-bearing self-citation. However, the paper's headline conclusion—that the Kasner near-horizon limit is 'precisely equivalent' to the flat/Rindler tensionless worldsheet—is secured by the step that compares (4.3) to (2.20) and fixes ε accordingly. Since the inferred ε is mode-number- and ψ-dependent, the matching does not define a single worldsheet contraction, and the asserted equivalence is enforced by construction rather than independently demonstrated. The geometric degeneration of the Kasner worldsheet metric, eqs. (3.4)–(3.6), gives partial independent support for a null/Carrollian limit, so the circularity is partial rather than total. Hence score 6.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The derivation rests on the standard worldsheet CFT machinery, on the Unruh mode construction borrowed from flat-space QFT, and on an ad hoc regularization scheme involving phi and psi. The central quantitative result, the identification of the tensionless parameter, depends on the arbitrary shift psi and on the mode number n, which is a load-bearing weakness. No new physical entities are introduced.

free parameters (3)
  • phi (regularization offset) = phi = +/- i (c/a) e^{-a psi} (eq. 3.15)
    Introduced as the initial value of the inertial sigma coordinate to regularize the divergent periodicity. The expression given is complex-valued, which is inconsistent with a real coordinate shift.
  • psi (Kasner coordinate shift) = free positive parameter; appears in final epsilon = pi^2 n c / (psi a)
    A regularization parameter in the worldsheet coordinate reparametrization. The central identification of the tensionless parameter depends on psi, so the physical content is gauge-dependent.
  • ell (worldsheet periodicity) = ell (arbitrary)
    Closed string periodicity in the inertial frame; enters the frequency Omega_n. It is conventional in string theory and not fitted.
assumptions (5)
  • domain assumption The Kasner worldsheet metric is conformally flat, so the mode functions satisfy the same massless Klein-Gordon equation as the Minkowski worldsheet.
    Section 3.3, eq. (3.18). Relies on conformal invariance of the worldsheet CFT.
  • domain assumption Global Unruh modes (3.28) are orthonormal under the inner products (3.30)-(3.31).
    Section 3.4 states the normalized forms without showing the normalization calculation; orthonormality is assumed.
  • ad hoc to paper The closedness condition of the string on the Kasner worldsheet is periodicity in the time-like coordinate xi with effective length L_eff, valid only at tau = 0.
    Section 3.2, eqs. (3.10)-(3.13). The equivalence of closedness conditions is stated to be valid only at tau = 0, which is an artificial restriction.
  • domain assumption The worldsheet must 'uphold the same physical structure as the target spacetime', justifying the Kasner embedding.
    Section 3.1, paragraph after eq. (3.4): 'Arguably, the worldsheet of a string must uphold the same physical structure as the target spacetime in which it is embedded.' This is a heuristic assumption.
  • ad hoc to paper Near-horizon truncation of the Bogoliubov coefficients reproduces the tensionless map (2.20) with a mode-dependent epsilon_n.
    Section 4, eqs. (4.3)-(4.5). The map is obtained by matching coefficients; the inferred epsilon depends on n and psi.

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Pith. "Pith review of A time-like window into tensionless worldsheets." pith.science (2026). https://pith.science/paper/7VRA6EUM

@misc{pith2026241206387,
  author       = {Pith},
  title        = {Pith review of: A time-like window into tensionless worldsheets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7VRA6EUM}},
  note         = {Machine review of arXiv:2412.06387}
}
read the original abstract

Rindler worldsheets are well known to transition into a tensionless regime in the infinite acceleration limit, acquiring an emergent Carrollian structure. In this work, we uncover an entirely new manifestation of such tensionless dynamics by constructing a time-evolving Milne worldsheet within the future (expanding) and past (contracting) regions of the background target spacetime. We show that as the Milne worldsheet approaches its null horizons, it reveals a deep and previously unexplored interplay between tensionless string dynamics and an emergent Carrollian structure, governed by an ultra-high frequency limit in time evolution. Our results demonstrate that non-inertial string dynamics can emerge in two structurally independent yet complementary settings--governed either by worldsheet acceleration (Rindler) or by the frequency of its intrinsic time evolution (Milne). Remarkably, we uncover a non-trivial duality-like correspondence between Rindler and Milne worldsheets near their respective null horizons, both exhibiting identical tensionless physics despite their distinct causal and dynamical origins. This discovery positions the Milne construction as a genuine time-like counterpart to the ultra-relativistic (Carrollian) or infinite-boost limit, offering new foundational insights into the deep structure of non-inertial string theory.

Figures

Figures reproduced from arXiv: 2412.06387 by the authors.

Figure 1
Figure 1. Trajectories of the Minkowski observers in future and past Kasner wedges F (ct > |x|) and P (ct < |x|) with coordinates (ξ, η) ∈ (−∞, +∞) and bounded by the null horizons J +(ξ = −∞, x = +ct, η = +∞) and J −(ξ = −∞, x = −ct, η = −∞). The space-like regions R (x > c|t|) and L (x < c|t|) are analogous right and left Rindler wedges. Therefore, at any moment, one can interpret ac as the redefined time-evolution paramete… view at source ↗
Figure 2
Figure 2. Illustration of a closed string Kasner worldsheet deforming with increasing a as seen from the Minkowski worldsheet. In the upper half, the cylindrical snapshot at a = 0 shows the inertial state. As a increases (a2 ≫ a1) keeping c constant, the worldsheet distorts into hyperboloids with growing eccentricity, eventually transforming into a light￾cone at a → ∞. The lower half shows the τ = 0 cross-sectional snapshots … view at source ↗
Figure 3
Figure 3. Illustration of a closed string on the Minkowski worldsheet deforming as ob￾served from the Kasner worldsheet along changing constant η lines, as shown in fig. 1. The circular snapshot on the left represents the pure tensile state. As the slope of the η lines decreases or increases (η2 ≫ η1), the string distorts into an ellipse, ultimately aligning with the light-cone at η → ±∞. η = 0 to η → ±∞ along a fixed ξ hyper… view at source ↗

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Cited by 1 Pith paper

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Reference graph

Works this paper leans on

42 extracted references · 14 canonical work pages · cited by 1 Pith paper

  1. [1]

    de Boer, J

    J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Carroll stories , JHEP 09 (2023) 148 [ 2307.06827]

    arXiv 2023

  2. [2]

    Hansen, N.A

    D. Hansen, N.A. Obers, G. Oling and B.T. Søgaard, Carroll Expansion of General Relativity , SciPost Phys. 13 (2022) 055 [ 2112.12684]

    arXiv 2022

  3. [3]

    L´ evy-Leblond,Une nouvelle limite non-relativiste du groupe de Poincar´ e, Annales de l’institut Henri Poincar´ e

    J.-M. L´ evy-Leblond,Une nouvelle limite non-relativiste du groupe de Poincar´ e, Annales de l’institut Henri Poincar´ e. Section A, Physique Th´ eorique3 (1965) 1

    1965

  4. [4]

    Sen Gupta, On an analogue of the Galilei group , Nuovo Cim

    N.D. Sen Gupta, On an analogue of the Galilei group , Nuovo Cim. A 44 (1966) 512

    work page 1966

  5. [5]

    Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity , JHEP 08 (2015) 069 [1505.05011]

    J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity , JHEP 08 (2015) 069 [1505.05011]

    arXiv 2015

  6. [6]

    Bagchi, Tensionless Strings and Galilean Conformal Algebra , JHEP 05 (2013) 141 [1303.0291]

    A. Bagchi, Tensionless Strings and Galilean Conformal Algebra , JHEP 05 (2013) 141 [1303.0291]

    arXiv 2013

  7. [7]

    Bagchi, S

    A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Strings from Worldsheet Symmetries , JHEP 01 (2016) 158 [ 1507.04361]

    arXiv 2016

  8. [8]

    Bagchi, A

    A. Bagchi, A. Banerjee, S. Chakrabortty, S. Dutta and P. Parekh, A tale of three — tensionless strings and vacuum structure , JHEP 04 (2020) 061 [ 2001.00354]

    arXiv 2020

Show all 42 references
  1. [9]

    Bagchi, A

    A. Bagchi, A. Banerjee, J. Hartong, E. Have, K.S. Kolekar and M. Mandlik, Strings near black holes are Carrollian , Phys. Rev. D 110 (2024) 086009 [ 2312.14240]

    2024 arXiv

  2. [10]

    Bagchi, D

    A. Bagchi, D. Grumiller and M.M. Sheikh-Jabbari, Horizon strings as 3D black hole microstates, SciPost Phys. 15 (2023) 210 [ 2210.10794]

    2023 arXiv

  3. [11]

    Schild, Classical Null Strings , Phys

    A. Schild, Classical Null Strings , Phys. Rev. D 16 (1977) 1722

    1977

  4. [12]

    Cardona, J

    B. Cardona, J. Gomis and J.M. Pons, Dynamics of Carroll Strings , JHEP 07 (2016) 050 [1605.05483]

    2016 arXiv

  5. [13]

    Bagchi, S

    A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Superstrings: View from the Worldsheet, JHEP 10 (2016) 113 [ 1606.09628]

    2016 arXiv

  6. [14]

    Bagchi, A

    A. Bagchi, A. Banerjee, S. Chakrabortty and P. Parekh, Inhomogeneous Tensionless Superstrings, JHEP 02 (2018) 065 [ 1710.03482]. – 32 –

    2018 arXiv

  7. [15]

    Bagchi, A

    A. Bagchi, A. Banerjee and P. Parekh, Tensionless Path from Closed to Open Strings , Phys. Rev. Lett. 123 (2019) 111601 [ 1905.11732]

    2019 arXiv

  8. [16]

    Banerjee, R

    A. Banerjee, R. Chatterjee and P. Pandit, Tensionless tales of compactification, JHEP 09 (2023) 050 [ 2307.01275]

    2023 arXiv

  9. [17]

    Banerjee, R

    A. Banerjee, R. Chatterjee and P. Pandit, Tensionless strings in a Kalb-Ramond background , JHEP 06 (2024) 067 [ 2404.01385]

    2024 arXiv

  10. [18]

    Francia, J

    D. Francia, J. Mourad and A. Sagnotti, Current Exchanges and Unconstrained Higher Spins , Nucl. Phys. B 773 (2007) 203 [ hep-th/0701163]

    2007 arXiv

  11. [19]

    Bagchi, A

    A. Bagchi, A. Banerjee and S. Chakrabortty, Rindler Physics on the String Worldsheet , Phys. Rev. Lett. 126 (2021) 031601 [ 2009.01408]

    2021 arXiv

  12. [20]

    Bagchi, A

    A. Bagchi, A. Banerjee, S. Chakrabortty and R. Chatterjee, A Rindler road to Carrollian worldsheets, JHEP 04 (2022) 082 [ 2111.01172]

    2022 arXiv

  13. [21]

    Atick and E

    J.J. Atick and E. Witten, The Hagedorn Transition and the Number of Degrees of Freedom of String Theory, Nucl. Phys. B 310 (1988) 291

    1988

  14. [22]

    Pisarski and O

    R.D. Pisarski and O. Alvarez, Strings at Finite Temperature and Deconfinement , Phys. Rev. D 26 (1982) 3735

    1982

  15. [23]

    Olesen, Strings, Tachyons and Deconfinement , Phys

    P. Olesen, Strings, Tachyons and Deconfinement , Phys. Lett. B 160 (1985) 408

    1985

  16. [24]

    Bowick and S.B

    M.J. Bowick and S.B. Giddings, HIGH TEMPERATURE STRINGS , Nucl. Phys. B 325 (1989) 631

    1989

  17. [25]

    Giddings, Strings at the Hagedorn Temperature , Phys

    S.B. Giddings, Strings at the Hagedorn Temperature , Phys. Lett. B 226 (1989) 55

    1989

  18. [26]

    Socolovsky, Rindler Space and Unruh Effect , 1304.2833

    M. Socolovsky, Rindler Space and Unruh Effect , 1304.2833

    arXiv

  19. [27]

    Olson and T.C

    S.J. Olson and T.C. Ralph, Entanglement between the future and past in the quantum vacuum, Phys. Rev. Lett. 106 (2011) 110404 [ 1003.0720]

    2011 arXiv

  20. [28]

    Higuchi, S

    A. Higuchi, S. Iso, K. Ueda and K. Yamamoto, Entanglement of the Vacuum between Left, Right, Future, and Past: The Origin of Entanglement-Induced Quantum Radiation , Phys. Rev. D 96 (2017) 083531 [ 1709.05757]

    2017 arXiv

  21. [29]

    Quach, T.C

    J.Q. Quach, T.C. Ralph and W.J. Munro, Berry Phase from the Entanglement of Future and Past Light Cones: Detecting the Timelike Unruh Effect , Phys. Rev. Lett. 129 (2022) 160401 [2112.00898]

    2022 arXiv

  22. [30]

    Barman, P

    S. Barman, P. Kumawat and B.R. Majhi, Timelike virtual transition in a static atom by a static mirror in Kasner Universe and in future Kruskal-Szekeres region , 2408.12378

  23. [31]

    Birrell and P.C.W

    N.D. Birrell and P.C.W. Davies, Quantum fields in curved space , Cambridge Monographs on Mathematical Physics, Cambridge University Press (1984)

    1984

  24. [32]

    Mukhanov, Physical Foundations of Cosmology , Physical Foundations of Cosmology, Cambridge University Press (2005)

    V. Mukhanov, Physical Foundations of Cosmology , Physical Foundations of Cosmology, Cambridge University Press (2005)

    2005

  25. [33]

    Polchinski, String theory

    J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string , Cambridge Monographs on Mathematical Physics, Cambridge University Press (12, 2007), 10.1017/CBO9780511816079

    2007 doi

  26. [34]

    Isberg, U

    J. Isberg, U. Lindstrom, B. Sundborg and G. Theodoridis, Classical and quantized tensionless strings , Nucl. Phys. B 411 (1994) 122 [ hep-th/9307108]. – 33 –

    1994 arXiv

  27. [35]

    Duval, G.W

    C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups , J. Phys. A 47 (2014) 335204 [ 1403.4213]

    2014 arXiv

  28. [36]

    Duval, G.W

    C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time , Class. Quant. Grav. 31 (2014) 085016 [1402.0657]

    2014 arXiv

  29. [37]

    de Boer, J

    J. de Boer, J. Hartong, N.A. Obers, W. Sybesma and S. Vandoren, Carroll Symmetry, Dark Energy and Inflation , Front. in Phys. 10 (2022) 810405 [ 2110.02319]

    2022 arXiv

  30. [38]

    Unruh, Notes on black hole evaporation , Phys

    W.G. Unruh, Notes on black hole evaporation , Phys. Rev. D 14 (1976) 870

    1976

  31. [39]

    Smaldone, Canonical transformations in Quantum Field Theory , Ph.D

    L. Smaldone, Canonical transformations in Quantum Field Theory , Ph.D. thesis, Naples U., 2018

    2018

  32. [40]

    Katagiri, Unlocking Novel Quantum States: Virasoro-Bogoliubov Transformations in Two Modes, 2312.07247

    S. Katagiri, Unlocking Novel Quantum States: Virasoro-Bogoliubov Transformations in Two Modes, 2312.07247

    arXiv

  33. [41]

    Das, Finite Temperature Field Theory, World Scientific, New York (1997)

    A.K. Das, Finite Temperature Field Theory, World Scientific, New York (1997)

    1997

  34. [42]

    Blumenhagen and E

    R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: with applications to String theory , vol. 779 (2009), 10.1007/978-3-642-00450-6. – 34 –

    2009 doi

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