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arxiv: 2605.06813 · v1 · submitted 2026-05-07 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Towards black-hole horizons and geodesic focusing in causal sets

Astrid Eichhorn , Pedro Gamito , Nawder Stokes
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:13 UTC · model grok-4.3

classification 🌀 gr-qc
keywords causal setsblack hole horizonsnull geodesicsdiscrete spacetimeapparent horizonsquantum gravitygeodesic expansion
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0 comments X

The pith

Causal sets can identify black-hole horizons by tracking a sign change in discrete geodesic expansion using ladders and fuzzy ladders.

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

In causal set theory, spacetime is built from discrete causal relations rather than a continuous manifold, so standard definitions of black-hole horizons that rely on global causal structure or smooth geodesics must be rethought. The paper introduces local approximations: discrete timelike curves for event horizons and ladders as tracers for null geodesics to probe apparent horizons. It demonstrates that a discrete version of geodesic expansion changes sign across the horizon in a 1+1-dimensional black-hole toy model and uses fuzzy ladders to construct an explicit portion of a discrete horizon. A sympathetic reader cares because this supplies a concrete, checkable way to locate horizons inside a discrete quantum-gravity framework without presupposing a continuum limit.

Core claim

We construct a local diagnostic to approximate a global event horizon based on discrete timelike curves. We then use ladders as tracers of null geodesics and find that a discrete counterpart of the expansion changes sign across the black-hole horizon. Finally, we introduce fuzzy ladders, which enable us to track null geodesics for larger intervals of the affine parameter, and thereby construct a portion of a discrete horizon in a toy-model for a black-hole spacetime in 1+1 dimensions.

What carries the argument

Ladders and fuzzy ladders, discrete chains of points that trace null geodesics by following causal relations with controlled spacings, used to compute a discrete expansion that flips sign at the horizon.

If this is right

  • A sign change in the discrete expansion serves as a local marker for the apparent horizon without requiring knowledge of the entire spacetime.
  • Fuzzy ladders extend the range over which null geodesics can be followed, allowing construction of finite segments of discrete horizons.
  • The same ladder technique provides a discrete counterpart to geodesic focusing that can be checked directly on the causal set.
  • The method supplies a practical diagnostic that can be applied to any causal set that approximates a black-hole geometry.

Where Pith is reading between the lines

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

  • The approach could be applied to causal sets that include matter or that evolve dynamically to study horizon formation.
  • Refinements of the ladder spacing rules might reduce residual discretization effects and improve accuracy in higher-dimensional models.
  • Similar discrete tracers could be tested for other causal boundaries, such as cosmological horizons.

Load-bearing premise

The ladders and fuzzy ladders faithfully approximate continuum null geodesics and horizon properties without discretization artifacts that would invalidate the observed sign change.

What would settle it

In the 1+1D toy model, if the computed discrete expansion fails to change sign at the location where the continuum horizon is expected, or if the fuzzy-ladder construction of the horizon portion does not converge to the known continuum horizon as the causal-set density is increased.

Figures

Figures reproduced from arXiv: 2605.06813 by Astrid Eichhorn, Nawder Stokes, Pedro Gamito.

Figure 1
Figure 1. Figure 1: FIG. 1. We show a sprinkling of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. We show a sprinkling into (1+1)-dimensional Hayward spacetime. Because the sprinkling is [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Upper panel: The length of the longest timelike curves for three sprinklings into patches of [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of a trapped surface. In the left image, there is a family of observers along a spacelike [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. We compare an illustration of a ladder in which the rungs are lightlike (left side) to a ladder in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. We show ladders in a sprinkling of size [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Illustration of our discrete expansion given in Eq. [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The weighted and unweighted average of [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The dependence of [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. We show a histogram for the number of ladders of a given length (defined as the number of rungs) [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. We show two illustrations of fuzzy ladders (with [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. We show an example of a sprinkling into [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. We show a histogram for the number of fuzzy ladders, with [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Examples of fuzzy ladders approximating the horizon in sprinklings into [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Examples of the exterior analogue of our discrete expansion measurement [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Examples of the interior analogue of our discrete expansion measurement [PITH_FULL_IMAGE:figures/full_fig_p033_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The dependence of the weighted average of [PITH_FULL_IMAGE:figures/full_fig_p034_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. A study of the dependence of the discrete expansion on the choice of [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Examples of a joint configuration of two fuzzy ladders in Schwarzschild and Minkowski. Before [PITH_FULL_IMAGE:figures/full_fig_p037_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Dependence of the discrete expansion on the width of the region from which nearest-neighbors are [PITH_FULL_IMAGE:figures/full_fig_p038_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Final weighted expansion for [PITH_FULL_IMAGE:figures/full_fig_p039_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: FIG. 22. Cumulative expansion in Minkowski spacetime as a function of included runs of a squared box [PITH_FULL_IMAGE:figures/full_fig_p040_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: FIG. 23. Cumulative expansion in the interior of Schwarzschild spacetime as a function of included runs of [PITH_FULL_IMAGE:figures/full_fig_p041_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: FIG. 24. Cumulative expansion in the exterior of Schwarzschild spacetime as a function of included runs [PITH_FULL_IMAGE:figures/full_fig_p042_24.png] view at source ↗
read the original abstract

The event horizon of a black hole is arguably the most dramatic manifestation of the fact that in General Relativity, causal structure is dynamical and spacetimes can be separated into distinct regions by causal boundaries. Causal set quantum gravity is an approach to quantum gravity in which causal relations between spacetime points constitute the basic structure on which the theory is based. This raises the question how a discrete horizon can be identified in a causal set. In our paper, we first construct a local diagnostic to approximate a global concept, namely the event horizon, based on discrete timelike curves. We then turn to the concept of an apparent horizon, which is based on local properties of geodesics, rather than global properties of the entire spacetime. We undertake first steps towards detecting apparent horizons in causal sets, using so-called ladders as tracers of null geodesics. We find that a discrete counterpart of the expansion changes sign across the black-hole horizon, as it should. Finally, we introduce the notion of a fuzzy ladder, which enables us to track null geodesics for larger intervals of the affine parameter. Thereby, we construct a portion of a discrete horizon in a toy-model for a black-hole spacetime in 1+1 dimensions.

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 / 2 minor

Summary. The manuscript develops techniques to detect black-hole horizons in causal sets. It constructs a local diagnostic for event horizons using discrete timelike curves, then introduces ladders as tracers of null geodesics to define a discrete expansion for apparent horizons. In a 1+1D toy-model black-hole spacetime, the discrete expansion changes sign across the horizon as expected, and fuzzy ladders are used to construct a portion of the discrete horizon.

Significance. If the ladder-based constructions prove robust, this constitutes a meaningful advance in causal-set quantum gravity by providing explicit, local diagnostics for horizons and geodesic focusing. The constructive approach in a controlled toy model supplies a clear proof-of-principle that can be tested for convergence and extended to higher dimensions. The work directly addresses a recognized gap between causal-set structure and classical GR observables.

major comments (2)
  1. [Ladders and discrete expansion] The central claim that the discrete expansion changes sign across the horizon (abstract and the section introducing ladders) is load-bearing. The manuscript must demonstrate that this discrete expansion converges to the continuum expansion scalar in the limit of increasing sprinkling density; without such a check or analytic argument, the observed sign change could be a discretization artifact of the ladder spacing or causal-set relations.
  2. [Fuzzy ladders and horizon construction] The fuzzy-ladder construction that yields the portion of the discrete horizon (final section) relies on specific rules for allowing deviations in the causal relations. These rules must be stated precisely, and the location of the constructed horizon must be shown to be insensitive to the choice of fuzziness tolerance and sprinkling density; otherwise the construction risks being an artifact of the ad-hoc fuzzy prescription.
minor comments (2)
  1. The abstract states that a 'portion' of the discrete horizon is constructed; the main text should quantify the affine-parameter range covered relative to the full horizon in the toy model and discuss any truncation effects.
  2. Figures showing the sign change or horizon construction would benefit from multiple independent sprinklings with error bands to illustrate statistical stability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will incorporate revisions to strengthen the presentation.

read point-by-point responses

  1. Referee: The central claim that the discrete expansion changes sign across the horizon (abstract and the section introducing ladders) is load-bearing. The manuscript must demonstrate that this discrete expansion converges to the continuum expansion scalar in the limit of increasing sprinkling density; without such a check or analytic argument, the observed sign change could be a discretization artifact of the ladder spacing or causal-set relations.

    Authors: We agree that convergence to the continuum limit is essential to establish the result beyond the specific toy-model realization. The current work shows the expected sign change for a fixed sprinkling density in the 1+1D model, where direct comparison with the known continuum expansion is possible. In the revised manuscript we will add numerical checks at several higher sprinkling densities, confirming that the discrete expansion approaches the continuum scalar and that the sign change persists. We will also include a short analytic argument based on the fact that the ladder links approximate null geodesics whose expansion is recovered in the continuum limit of the causal set. revision: yes

  2. Referee: The fuzzy-ladder construction that yields the portion of the discrete horizon (final section) relies on specific rules for allowing deviations in the causal relations. These rules must be stated precisely, and the location of the constructed horizon must be shown to be insensitive to the choice of fuzziness tolerance and sprinkling density; otherwise the construction risks being an artifact of the ad-hoc fuzzy prescription.

    Authors: We acknowledge that the fuzzy-ladder rules require a more precise algorithmic statement. In the revision we will provide an explicit definition of the allowed deviations in causal relations and the precise value of the fuzziness tolerance used. We will also add numerical tests demonstrating that the location of the constructed horizon segment remains stable under moderate variations of the tolerance parameter and across different sprinkling densities, thereby showing that the construction is not an artifact of the specific choice. revision: yes

Circularity Check

0 steps flagged

No circularity: constructive definitions of discrete ladders and expansion are independent of target results

full rationale

The paper defines new discrete objects (timelike curves, ladders, fuzzy ladders) and a discrete expansion scalar directly from causal-set relations in a 1+1D toy model. The observed sign change of this expansion across the horizon and the construction of a discrete horizon portion are reported outcomes of applying these definitions, not quantities fitted to data or derived by renaming prior results. No self-citations are invoked as load-bearing uniqueness theorems, and no ansatz is smuggled in; the work remains self-contained against external benchmarks by explicit construction rather than reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The approach rests on the standard causal-set premise that discrete causal relations can approximate continuum spacetime geometry. The fuzzy ladder is a new invented construct introduced to extend geodesic tracking.

axioms (1)
  • domain assumption Causal sets provide a discrete fundamental structure whose causal relations approximate the geometry and causal structure of continuum spacetime.
    Invoked implicitly throughout the abstract as the foundation for all discrete constructions.
invented entities (1)
  • fuzzy ladder no independent evidence
    purpose: To track null geodesics over larger affine-parameter intervals in the discrete causal set by allowing limited flexibility in connections.
    New notion introduced in the paper to overcome limitations of strict ladders.

pith-pipeline@v0.9.0 · 5509 in / 1321 out tokens · 70325 ms · 2026-05-11T01:13:47.198822+00:00 · methodology

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Reference graph

Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    The continuum quantity The induced spatial distance on a constantt∗ hypersurface forr > r S in a Schwarzschild space- time is given by the spatial line element ds2 = 1− rS r −1 dr2 = 1− rS r(r∗) dr2 ∗.(15) The continuum analogue to the above discrete expansion in Eq. 14 is eΘext(t, δ) = f(t+δ)−f(t) f(t) withf(t) = Z r∗2(t) r∗1(t) dr∗ r 1− rS r(r∗) ,(16) w...

    work page

  2. [2]

    one bucket

    Discrete expansion in Minkowski spacetime We first calculatemean(E)in sprinklings into Minkowski spacetime. In the continuum, we expectE= 0, i.e., pairs of outgoing (or ingoing) null geodesics remain parallel to each other in Minkowski spacetime. In a sprinkling, our illustration in Fig. 7 already highlights that, because rung pairs do not occur equidista...

    work page

  3. [3]

    Fuzzy causal ladder

    Discrete expansion in Schwarzschild spacetime We expect mean(E)to change sign across the event horizon of a black hole. To test this hypothesis, and to discover whethermean(E)constitutes a discrete observable capable of detecting the (approximate) location of the apparent horizon, we computemean(E)separately in sprinklings into the interior and sprinkling...

    work page

  4. [4]

    for alli∈Iwithi > M, one has M−1≤ |[p i−M , pi]| ≤2M−1,

    work page

  5. [5]

    Fuzzy ladders

    for alli∈Iwithi > M, one has M−1≤ |[q i−M , qi]| ≤2M−1. “Fuzzy ladders” allow controlled violations of the rigidity conditions. Additional elements can exist between the two sides of the ladder, because the intervals betweenpi andq j withj > i+ 1 are no longer constrained. The conditions 4. and 5. permit up toMadditional elements within the causal interva...

    work page 2020

  6. [6]

    15 and only discuss the asymptotic behavior explicitly

    Exterior region of the black hole In the main text, we defined the continuum analogue of the discrete measurementEin the exterior region of the black hole by eΘext(t, δ) = f(t+δ)−f(t) f(t) withf(t) = Z r∗2(t) r∗1(t) dr∗ r 1− rS r(r∗) .(A1) Inserting the functionf, we had written eΘext(t, δ) = R r∗2+δ r∗1+δ dr∗ q 1− rS r(r∗) − R r∗2 r∗1 dr∗ q 1− rS r(r∗) R...

    work page

  7. [7]

    asymptotic silence

    Interior region of the black hole Analogously, we defined the continuum analogue of the discrete measurementEin the interior region of the black hole by eΘint(r, δ) = g(r+δ)−g(r) g(r) withg(r) = Z t2(r) t1(r) dt r rS r −1,(A10) 33 FIG. 16. Examples of the interior analogue of our discrete expansion measurementeΘint(r, δ)for various values of the parameter...

    work page

  8. [8]

    fuzzy causal ladder

    Properties of fuzzy ladders For a small subset of ladders, giving up the rigidity conditions of the original ladder definition results in problems. The first problem is that a single null geodesic may be approximated by a family of ladders, differing from one another only by a small number of points. To remove this redundancy in our results, one ladder wa...

    work page

  9. [9]

    fuzziness

    The discrete expansion with fuzzy ladders We now turn our attention towards the discrete expansion with fuzzy ladders. For fuzzy ladders, there are two competing effects on the discrete expansion. On the one hand, their greater length 37 FIG. 19. Examples of a joint configuration of two fuzzy ladders in Schwarzschild and Minkowski. Before separation, thes...

    work page

  10. [10]

    Penrose, Riv

    R. Penrose, Riv. Nuovo Cim.1, 252 (1969)

    work page 1969

  11. [11]

    Penrose (1979) pp

    R. Penrose (1979) pp. 581–638

    work page 1979

  12. [12]

    Dafermos and J

    M. Dafermos and J. Luk, Ann. Math. (2)202, 309 (2025), arXiv:1710.01722 [gr-qc]

    work page arXiv 2025

  13. [13]

    Sbierski, Invent

    J. Sbierski, Invent. Math.243, 961 (2026), arXiv:2409.18838 [gr-qc]. 43

    work page arXiv 2026

  14. [14]

    Van de Moortel, Comptes Rendus

    M. Van de Moortel, (2025), arXiv:2501.13180 [gr-qc]

    work page arXiv 2025

  15. [15]

    S. W. Hawking, Phys. Rev. D14, 2460 (1976)

    work page 1976

  16. [16]

    The Swampland: Introduction and Review

    E. Palti, Fortsch. Phys.67, 1900037 (2019), arXiv:1903.06239 [hep-th]

    work page Pith review arXiv 2019

  17. [17]

    van Beest, J

    M. van Beest, J. Calderón-Infante, D. Mirfendereski, and I. Valenzuela, Phys. Rept.989, 1 (2022), arXiv:2102.01111 [hep-th]

    work page arXiv 2022

  18. [18]

    Dou and R

    D. Dou and R. D. Sorkin, Found. Phys.33, 279 (2003), arXiv:gr-qc/0302009

    work page arXiv 2003

  19. [19]

    Barton, A

    C. Barton, A. Counsell, F. Dowker, D. S. W. Gould, I. Jubb, and G. Taylor, Phys. Rev. D100, 126008 (2019), arXiv:1909.08620 [gr-qc]

    work page arXiv 2019

  20. [20]

    A. E. Broderick, A. Loeb, and R. Narayan, Astrophys. J.701, 1357 (2009), arXiv:0903.1105 [astro- ph.HE]

    work page arXiv 2009

  21. [21]

    Is the gravitational-wave ringdown a probe of the event horizon?

    V. Cardoso, E. Franzin, and P. Pani, Phys. Rev. Lett.116, 171101 (2016), [Erratum: Phys.Rev.Lett. 117, 089902 (2016)], arXiv:1602.07309 [gr-qc]

    work page Pith review arXiv 2016

  22. [22]

    Bombelli, J

    L. Bombelli, J. Lee, D. Meyer, and R. Sorkin, Phys. Rev. Lett.59, 521 (1987)

    work page 1987

  23. [23]

    Bombelli, J

    L. Bombelli, J. Henson, and R. D. Sorkin, Mod. Phys. Lett. A24, 2579 (2009), arXiv:gr-qc/0605006

    work page arXiv 2009

  24. [24]

    Bhattacharya, A

    A. Bhattacharya, A. Mathur, and S. Surya, Gen. Rel. Grav.55, 32 (2023), arXiv:2301.06480 [gr-qc]

    work page arXiv 2023

  25. [25]

    S. W. Hawking, A. R. King, and P. J. Mccarthy, J. Math. Phys.17, 174 (1976)

    work page 1976

  26. [26]

    D. B. Malament, Journal of mathematical physics18, 1399 (1977)

    work page 1977

  27. [27]

    Dowker, Causal sets and the deep structure of spacetime, in100 Years Of Relativity: space-time structure: Einstein and beyond, edited by A

    F. Dowker, Causal sets and the deep structure of spacetime, in100 Years Of Relativity: space-time structure: Einstein and beyond, edited by A. Ashtekar (2005) pp. 445–464, arXiv:gr-qc/0508109

    work page arXiv 2005

  28. [28]

    Surya, Living Rev

    S. Surya, Living Rev. Rel.22, 5 (2019), arXiv:1903.11544 [gr-qc]

    work page arXiv 2019

  29. [29]

    Surya, Lect

    S. Surya, Lect. Notes Phys.1036, pp. (2025)

    work page 2025

  30. [30]

    Carlip, S

    P. Carlip, S. Carlip, and S. Surya, Class. Quant. Grav.40, 095004 (2023), arXiv:2209.00327 [gr-qc]

    work page arXiv 2023

  31. [31]

    Carlip, S

    P. Carlip, S. Carlip, and S. Surya, Class. Quant. Grav.41, 145005 (2024), arXiv:2311.18238 [gr-qc]

    work page arXiv 2024

  32. [32]

    Surya, Class

    S. Surya, Class. Quant. Grav.29, 132001 (2012), arXiv:1110.6244 [gr-qc]

    work page arXiv 2012

  33. [33]

    Glaser, D

    L. Glaser, D. O’Connor, and S. Surya, Class. Quant. Grav.35, 045006 (2018), arXiv:1706.06432 [gr-qc]

    work page arXiv 2018

  34. [34]

    Eichhorn, Class

    A. Eichhorn, Class. Quant. Grav.35, 044001 (2018), arXiv:1709.10419 [gr-qc]

    work page arXiv 2018

  35. [35]

    W. J. Cunningham and S. Surya, Class. Quant. Grav.37, 054002 (2020), arXiv:1908.11647 [gr-qc]

    work page arXiv 2020

  36. [36]

    Eichhorn, J

    A. Eichhorn, J. Phys. Conf. Ser.1275, 012010 (2019), arXiv:1902.00391 [gr-qc]

    work page arXiv 2019

  37. [37]

    de Boeret al., (2022), arXiv:2207.10618 [hep-th]

    J. de Boeret al., (2022), arXiv:2207.10618 [hep-th]

    work page arXiv 2022

  38. [38]

    D. P. Rideout and R. D. Sorkin, Phys. Rev. D61, 024002 (2000), arXiv:gr-qc/9904062

    work page arXiv 2000

  39. [39]

    Zalel, Covariant Growth Dynamics (2024) arXiv:2302.10582 [gr-qc]

    S. Zalel, Covariant Growth Dynamics (2024) arXiv:2302.10582 [gr-qc]

    work page arXiv 2024

  40. [40]

    Brightwell and R

    G. Brightwell and R. Gregory, Phys. Rev. Lett.66, 260 (1991)

    work page 1991

  41. [41]

    D. D. Reid, Phys. Rev. D67, 024034 (2003), arXiv:gr-qc/0207103

    work page arXiv 2003

  42. [42]

    Major, D

    S. Major, D. Rideout, and S. Surya, J. Math. Phys.48, 032501 (2007), arXiv:gr-qc/0604124

    work page arXiv 2007

  43. [43]

    R. D. Sorkin, , 26 (2007), arXiv:gr-qc/0703099

    work page arXiv 2007

  44. [44]

    Rideout and P

    D. Rideout and P. Wallden, Class. Quant. Grav.26, 155013 (2009), arXiv:0810.1768 [gr-qc]. 44

    work page arXiv 2009

  45. [45]

    Major, D

    S. Major, D. Rideout, and S. Surya, Class. Quant. Grav.26, 175008 (2009), arXiv:0902.0434 [gr-qc]

    work page arXiv 2009

  46. [46]

    D. M. T. Benincasa and F. Dowker, Phys. Rev. Lett.104, 181301 (2010), arXiv:1001.2725 [gr-qc]

    work page arXiv 2010

  47. [47]

    Aslanbeigi, M

    S. Aslanbeigi, M. Saravani, and R. D. Sorkin, JHEP06, 024, arXiv:1403.1622 [hep-th]

    work page arXiv

  48. [48]

    Eichhorn, S

    A. Eichhorn, S. Surya, and F. Versteegen, Class. Quant. Grav.36, 105005 (2019), arXiv:1809.06192 [gr-qc]

    work page arXiv 2019

  49. [49]

    G. P. de Brito, A. Eichhorn, and C. Pfeiffer, Eur. Phys. J. Plus138, 592 (2023), arXiv:2301.13525 [gr-qc]

    work page arXiv 2023

  50. [50]

    Surya, Phys

    S. Surya, Phys. Rev. D113, 024034 (2026), arXiv:2510.19403 [gr-qc]

    work page arXiv 2026

  51. [51]

    Feynman Propagator for a Free Scalar Field on a Causal Set,

    S. Johnston, Phys. Rev. Lett.103, 180401 (2009), arXiv:0909.0944 [hep-th]

    work page arXiv 2009

  52. [52]

    R. D. Sorkin, J. Phys. Conf. Ser.306, 012017 (2011), arXiv:1107.0698 [gr-qc]

    work page arXiv 2011

  53. [53]

    Albertini, F

    E. Albertini, F. Dowker, A. Nasiri, and S. Zalel, Phys. Rev. D109, 106014 (2024), arXiv:2402.08555 [hep-th]

    work page arXiv 2024

  54. [54]

    Dou, On Horizon Molecules and Entropy in Causal Sets (2024) arXiv:2307.04150 [gr-qc]

    D. Dou, On Horizon Molecules and Entropy in Causal Sets (2024) arXiv:2307.04150 [gr-qc]

    work page arXiv 2024

  55. [55]

    Y. K. Yazdi, Entanglement Entropy and Causal Set Theory (2024) arXiv:2212.13586 [hep-th]

    work page arXiv 2024

  56. [56]

    Jubb, Interacting Quantum Scalar Field Theory on a Causal Set (2024) arXiv:2306.12484 [hep-th]

    I. Jubb, Interacting Quantum Scalar Field Theory on a Causal Set (2024) arXiv:2306.12484 [hep-th]

    work page arXiv 2024

  57. [57]

    Glaser, Computer Simulations of Causal Sets (2024) arXiv:2306.09904 [gr-qc]

    L. Glaser, Computer Simulations of Causal Sets (2024) arXiv:2306.09904 [gr-qc]

    work page arXiv 2024

  58. [58]

    X., Quantum Field Theory on Causal Sets (2024) arXiv:2306.04800 [hep-th]

    N. X., Quantum Field Theory on Causal Sets (2024) arXiv:2306.04800 [hep-th]

    work page arXiv 2024

  59. [59]

    Ashmead and D

    F. Ashmead and D. D. Reid, Estimating the Manifold Dimension of Causal Sets (2024)

    work page 2024

  60. [60]

    Ahmed and H

    M. Ahmed and H. Shafi, Causal Set Cosmology (2024)

    work page 2024

  61. [61]

    Mathur, Toward the Emergence of Continuum Spacetime in Causal Set Theory (2024)

    A. Mathur, Toward the Emergence of Continuum Spacetime in Causal Set Theory (2024)

    work page 2024

  62. [62]

    He and D

    S. He and D. Rideout, Class. Quant. Grav.26, 125015 (2009), arXiv:0811.4235 [gr-qc]

    work page arXiv 2009

  63. [63]

    Homšak and S

    V. Homšak and S. Veroni, Phys. Rev. D110, 026015 (2024), arXiv:2404.11670 [gr-qc]

    work page arXiv 2024

  64. [64]

    G. J. Olmo and D. Rubiera-Garcia, (2022), arXiv:2209.05061 [gr-qc]

    work page arXiv 2022

  65. [65]

    Eichhorn and A

    A. Eichhorn and A. Held, (2022), arXiv:2212.09495 [gr-qc]

    work page arXiv 2022

  66. [66]

    Ashtekar, J

    A. Ashtekar, J. Olmedo, and P. Singh, (2023), arXiv:2301.01309 [gr-qc]

    work page arXiv 2023

  67. [67]

    Singularity-free gravitational collapse: From regular black holes to horizonless objects,

    R. Carballo-Rubio, F. Di Filippo, S. Liberati, and M. Visser, (2023), arXiv:2302.00028 [gr-qc]

    work page arXiv 2023

  68. [68]

    Towards a Non-singular Paradigm of Black Hole Physics

    R. Carballo-Rubioet al., JCAP05, 003, arXiv:2501.05505 [gr-qc]

    work page arXiv

  69. [69]

    Bueno, P

    P. Bueno, P. A. Cano, R. A. Hennigar, and Á. J. Murcia, Phys. Rev. D113, 024019 (2026), arXiv:2509.19016 [gr-qc]

    work page arXiv 2026

  70. [70]

    Eichhorn and P

    A. Eichhorn and P. G. S. Fernandes, (2025), arXiv:2508.00686 [gr-qc]

    work page arXiv 2025

  71. [71]

    Carballo-Rubio,Master field equations for spherically symmetric gravitational fields beyond general relativity,Nature Commun.17(2026), no

    R. Carballo-Rubio, Nature Commun.17, 1399 (2026), arXiv:2507.15920 [gr-qc]

    work page arXiv 2026

  72. [72]

    Borissova and R

    J. Borissova and R. Carballo-Rubio, (2026), arXiv:2602.16773 [gr-qc]

    work page arXiv 2026

  73. [73]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006), arXiv:gr-qc/0506126

    work page Pith review arXiv 2006

  74. [74]

    Geodesic incompleteness of some popular regular black holes,

    T. Zhou and L. Modesto, Phys. Rev. D107, 044016 (2023), arXiv:2208.02557 [gr-qc]

    work page arXiv 2023

  75. [75]

    Isolated and dynamical horizons and their applications

    A. Ashtekar and B. Krishnan, Living Rev. Rel.7, 10 (2004), arXiv:gr-qc/0407042

    work page Pith review arXiv 2004

  76. [76]

    Booth, Black hole boundaries, Can

    I. Booth, Can. J. Phys.83, 1073 (2005), arXiv:gr-qc/0508107

    work page arXiv 2005

  77. [77]

    Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University 45 Press, 2009)

    E. Poisson,A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics(Cambridge University 45 Press, 2009)

    work page 2009

  78. [78]

    Philpott, F

    L. Philpott, F. Dowker, and R. D. Sorkin, Phys. Rev. D79, 124047 (2009), arXiv:0810.5591 [gr-qc]

    work page arXiv 2009

  79. [79]

    Surya, N

    S. Surya, N. X, and Y. K. Yazdi, Class. Quant. Grav.38, 115001 (2021), arXiv:2008.07697 [gr-qc]

    work page arXiv 2021