pith. sign in

arxiv: 2505.16480 · v2 · pith:XFW3V66Anew · submitted 2025-05-22 · 🌀 gr-qc · astro-ph.GA· hep-th· math-ph· math.MP

Gravitational collapse of matter fields in de Sitter spacetimes

Akriti Garg , Ayan Chatterjee This is my paper

Pith reviewed 2026-05-22 13:51 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.GAhep-thmath-phmath.MP
keywords gravitational collapsede Sitter spacetimetrapped surfacesblack hole horizonscosmological horizonsmarginally trapped surfacesquasilocal formalismspherically symmetric collapse
0
0 comments X

The pith

In de Sitter spacetimes, black hole and cosmological horizons develop as the time evolution of marginally trapped surfaces during gravitational collapse of matter fields.

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

This paper studies the spherically symmetric collapse of matter such as dust, perfect fluids, and viscous fluids inside de Sitter spacetime. Using a combination of exact solutions and numerical evolution under varied initial velocity and density profiles, the work follows the appearance and motion of spherical marginally trapped surfaces. The central result is that the quasilocal trapped-surface description supplies a natural way to follow how both black-hole and cosmological horizons form and change. A reader would care because the same local surfaces account for horizon behavior across many different matter models in an expanding background.

Core claim

The quasilocal formalism of trapped surfaces provides an ideal framework to study the evolution of horizons. More precisely, black hole and cosmological horizons may be viewed as the time development of marginally trapped surfaces. This holds for a wide class of energy-momentum tensors including dust, perfect fluids with equations of state, fluids with radial and tangential pressure, and fluids with bulk and shear viscosity, under different initial conditions on velocity and density profiles.

What carries the argument

Spherical marginally trapped surfaces whose time development corresponds to the formation and growth of black-hole and cosmological horizons.

If this is right

  • Horizon formation can be followed uniformly for dust, perfect fluids, and viscous fluids using the same trapped-surface evolution.
  • Both black-hole and cosmological horizons arise from the same class of marginally trapped surfaces under the examined initial data.
  • Analytical and numerical methods together suffice to track the surfaces throughout the collapse.
  • The quasilocal description applies across the listed matter models without requiring global event-horizon definitions.

Where Pith is reading between the lines

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

  • The same trapped-surface tracking might be examined in numerical relativity codes for cosmological collapse scenarios.
  • The framework could be tested by relaxing spherical symmetry or by including additional matter couplings not covered in the present cases.
  • It suggests that local trapped-surface conditions, rather than global causal boundaries, govern horizon dynamics in de Sitter-like expansion.

Load-bearing premise

The chosen initial conditions on velocity and density profiles, together with the broad range of energy-momentum tensors considered, capture the generic behavior of collapse and horizon formation in de Sitter spacetime.

What would settle it

A concrete counter-example in which a black-hole or cosmological horizon appears during collapse but does not arise as the time development of a marginally trapped surface would refute the claim.

Figures

Figures reproduced from arXiv: 2505.16480 by Akriti Garg, Ayan Chatterjee.

Figure 1
Figure 1. Figure 1: The figure (a) is the Penrose diagram describing the formation of black hole due to [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The figure corresponds to the initial density distribution referred in eq. (37). The [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The graphs show the values of C corresponding to the BH and the CH for the density distribution in eqn. (37). The negative values of C implies, from (19) and (15) that the MTTs are both timelike and hence, are unstable. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The figure in (a) shows the density profile of the matter density given in (38). The [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The graphs show the nature of MTTs for the example in (38). Note that the MTT [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The figure in (a) shows the density profile of the matter density given in (39). [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The nature of MTTs are plotted here. The black hole MTT is spacelike while that [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The figure in (a) shows the density profile of the matter density given in (39). [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The MTT corresponding to the black hole is spacelike while that of the cosmological [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The plots corresponding to eqn. (41) is given here (a). Since a black hole already [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The graph shows the formation of black hole and cosmological horizons along [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The graph shows the formation of black hole and cosmological horizons along [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The graph shows the formation of black hole and cosmological horizons along [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The graph shows the formation of black hole and cosmological horizons along [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The graph shows the formation of black hole and cosmological horizons along [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
read the original abstract

In this paper, we discuss the spherically symmetric gravitational collapse of matter fields in the de Sitter universe. The energy-momentum tensor of the matter field is assumed to admit a wide variety including dust, perfect fluids with equations of state, fluids with tangential and radial pressure, and with bulk and shear viscosity. Under different initial conditions imposed on the velocity and the density profiles, and by combining the results from exact analytical methods with those obtained from numerical techniques, we track the formation and evolution of spherical marginally trapped spheres as the matter suffers continual gravitational collapse. We show that the quasilocal formalism of trapped surfaces provides an ideal framework to study the evolution of horizons. More precisely, black hole and cosmological horizons may be viewed as the time development of marginally trapped surfaces.

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 studies spherically symmetric gravitational collapse of matter fields (dust, perfect fluids with various equations of state, fluids with radial/tangential pressure, and viscous fluids) in de Sitter spacetime. Under a range of initial velocity and density profiles, analytic and numerical methods are combined to follow the formation and evolution of spherical marginally trapped surfaces. The central claim is that the quasilocal trapped-surface formalism supplies an ideal framework for horizon dynamics, with black-hole and cosmological horizons appearing as the time development of marginally trapped surfaces.

Significance. If the results are robust, the work would supply a concrete quasilocal description linking marginally trapped surfaces to both black-hole and cosmological horizons in an expanding de Sitter background. This could clarify the dynamical relationship between trapped surfaces and apparent horizons for a family of matter models. The combination of exact solutions and numerical evolution is a positive feature, but the significance is limited by the absence of demonstrated insensitivity to the specific initial data and viscosity parameters chosen.

major comments (2)
  1. [Abstract / initial-conditions paragraph] Abstract and the section discussing initial conditions: the assertion that the quasilocal formalism provides an 'ideal framework' for the generic evolution of horizons is supported only by results for a finite collection of velocity/density profiles and the enumerated energy-momentum tensors (dust, perfect fluids, viscous fluids). No explicit test is reported showing that the observed MTS-to-horizon correspondence persists under changes in the initial profiles or in the viscosity coefficients; without such a check the stronger claim of generality remains conditional on the modeling choices.
  2. [Numerical methods section] The manuscript states that analytic and numerical techniques are combined to track trapped surfaces, yet the provided description contains no explicit convergence tests, error estimates, or resolution studies for the numerical component. Because the central claim concerns the time development of marginally trapped surfaces, the absence of these diagnostics leaves the reliability of the numerical horizon-tracking results unquantified.
minor comments (2)
  1. [Section 2] Notation for the energy-momentum tensor components and the definition of the marginally trapped surface condition should be stated explicitly at the first appearance rather than assumed from standard references.
  2. [Figures] Figure captions for the evolution plots should include the specific initial velocity and density parameters used in each run to allow direct comparison with the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract / initial-conditions paragraph] Abstract and the section discussing initial conditions: the assertion that the quasilocal formalism provides an 'ideal framework' for the generic evolution of horizons is supported only by results for a finite collection of velocity/density profiles and the enumerated energy-momentum tensors (dust, perfect fluids, viscous fluids). No explicit test is reported showing that the observed MTS-to-horizon correspondence persists under changes in the initial profiles or in the viscosity coefficients; without such a check the stronger claim of generality remains conditional on the modeling choices.

    Authors: We acknowledge that the study examines a representative but finite set of matter models (dust, perfect fluids with various equations of state, fluids with radial/tangential pressure, and viscous fluids) and a selection of initial velocity and density profiles. The consistent formation and evolution of marginally trapped surfaces into black-hole and cosmological horizons across these cases supports the utility of the quasilocal approach in de Sitter spacetime. However, we did not perform systematic sensitivity tests to arbitrary variations in viscosity coefficients or additional profile families. In the revised manuscript we will adjust the abstract and the initial-conditions discussion to state that the results hold for the considered class of models and initial data, thereby qualifying the claim of generality without overstating the scope of the present work. revision: partial

  2. Referee: [Numerical methods section] The manuscript states that analytic and numerical techniques are combined to track trapped surfaces, yet the provided description contains no explicit convergence tests, error estimates, or resolution studies for the numerical component. Because the central claim concerns the time development of marginally trapped surfaces, the absence of these diagnostics leaves the reliability of the numerical horizon-tracking results unquantified.

    Authors: We agree that explicit numerical validation is necessary to quantify the reliability of the horizon-tracking results. Although the original manuscript combines analytic and numerical methods, it does not report convergence tests, grid resolutions, or error estimates. In the revised version we will add a subsection describing the numerical scheme, the resolutions employed, results of convergence studies under successive grid refinements, and estimates of numerical errors in the locations and time evolution of the marginally trapped surfaces. revision: yes

Circularity Check

0 steps flagged

No circularity: direct tracking of marginally trapped surfaces via analytics and numerics

full rationale

The derivation proceeds by imposing standard GR definitions of marginally trapped surfaces on spherically symmetric metrics in de Sitter, then solving the Einstein equations analytically for selected matter models (dust, perfect fluids, viscous fluids) and numerically for chosen velocity/density profiles. The observation that black-hole and cosmological horizons arise as the time development of these surfaces is extracted from the resulting trajectories of the apparent horizons; it is not presupposed by redefining the surfaces or by fitting parameters whose output is then relabeled a prediction. No load-bearing self-citations, uniqueness theorems imported from prior work by the same authors, or ansatzes smuggled via citation appear in the abstract or the described methodology. The representativeness of the chosen initial data is a question of scope, not a circular reduction of the central claim to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of general relativity plus specific modeling choices for matter and symmetry; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Spherically symmetric spacetime and matter distribution
    Invoked throughout the abstract when discussing collapse and trapped surfaces.
  • domain assumption Energy-momentum tensor admits dust, perfect fluids, tangential/radial pressure, and viscous terms
    Stated explicitly as the class of matter fields considered.

pith-pipeline@v0.9.0 · 5665 in / 1347 out tokens · 29293 ms · 2026-05-22T13:51:21.334003+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

99 extracted references · 99 canonical work pages · 1 internal anchor

  1. [1]

    The Large Scale Structure of Spacetime

    S.W. Hawking and G.F.R. Ellis, “The Large Scale Structure of Spacetime”, Cambridge University Press, Cambridge 1975

    work page 1975

  2. [2]

    Wald, General Relativity, University of Chicago Press, 1984

    Robert M. Wald, General Relativity, University of Chicago Press, 1984

    work page 1984

  3. [3]

    Landau and E.M

    L.D. Landau and E.M. Lifshitz, The Classical Theory of Fields, Pergamon Press, 1975

    work page 1975

  4. [4]

    Joshi, Gravitational Collapse and Spacetime Singularities, Cambridge University Press, Cambridge, 2007

    P.S. Joshi, Gravitational Collapse and Spacetime Singularities, Cambridge University Press, Cambridge, 2007

    work page 2007

  5. [5]

    A. M. Abrahams and C. R. Evans, Phys. Rev. D49, 3998-4003 (1994) doi:10.1103/PhysRevD.49.3998

    work page doi:10.1103/physrevd.49.3998 1994

  6. [6]

    Rostworowski, [arXiv:2503.17549 [gr-qc]]

    A. Rostworowski, [arXiv:2503.17549 [gr-qc]]

    work page arXiv

  7. [7]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett.14, 57 (1965)

    work page 1965

  8. [8]

    Penrose, Riv

    R. Penrose, Riv. Nuovo Cim.1, 252 (1969) [Gen. Rel. Grav.34, 1141 (2002)]

    work page 1969

  9. [9]

    Singularities and time asymmetry,

    R. Penrose, “Singularities and time asymmetry,” S. W. Hawking and E. Israel, “General Relativity: An Einstein Centenary Survey, Pt.2,” Univ. Pr., 2010

    work page 2010

  10. [10]

    Penrose, J

    R. Penrose, J. Astrophys. Astron.20, 233-248 (1999)

    work page 1999

  11. [11]

    Landsman, Found

    K. Landsman, Found. Phys.51, no.2, 42 (2021)

    work page 2021

  12. [12]

    Landsman, Gen

    K. Landsman, Gen. Rel. Grav.54, no.10, 115 (2022)

    work page 2022

  13. [13]

    P. S. Joshi and D. Malafarina, Int. J. Mod. Phys. D20, 2641-2729 (2011)

    work page 2011

  14. [14]

    Datt, Zs

    S. Datt, Zs. f. Phys. 108, 314 (1938)

    work page 1938

  15. [15]

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

    work page 1939

  16. [16]

    Lemaitre, Gen

    G. Lemaitre, Gen. Rel. Grav.29, 641 (1997)

    work page 1997

  17. [17]

    R. C. Tolman, Proc. Nat. Acad. Sci.20, 169 (1934) [Gen. Rel. Grav.29, 935 (1997)]

    work page 1934

  18. [18]

    Bondi, Mon

    H. Bondi, Mon. Not. Roy. Astron. Soc.107, 410 (1947)

    work page 1947

  19. [19]

    C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964)

    work page 1964

  20. [20]

    W. C. Hernandez and C. W. Misner, Astrophys. J.143, 452 (1966). 28

    work page 1966

  21. [21]

    Exact solutions of Einstein’s field equations,

    H. Stephani, D. Kramer, M. A. H. MacCallum, C. Hoenselaers and E. Herlt, “Exact solutions of Einstein’s field equations,” Cambridge Univ. Press, 2003

    work page 2003

  22. [22]

    G. C. Omer, Proc. Nat. Acad. Sci. USA53, 1 (1965)

    work page 1965

  23. [23]

    Krasinski, Cambridge Univ

    A. Krasinski, Cambridge Univ. Press, 2011, ISBN 978-0-511-88754-3

    work page 2011

  24. [24]

    An introduction to general relativity and cosmology,

    J. Plebanski and A. Krasinski, “An introduction to general relativity and cosmology,” Cambridge Univ. Press, (2006)

    work page 2006

  25. [25]

    Lake, Phys

    K. Lake, Phys. Rev. D62, 027301 (2000)

    work page 2000

  26. [26]

    S. S. Deshingkar, A. Chamorro, S. Jhingan and P. S. Joshi, Phys. Rev. D63, 124005 (2001)

    work page 2001

  27. [27]

    Markovic and S

    D. Markovic and S. Shapiro, Phys. Rev. D61, 084029 (2000)

    work page 2000

  28. [28]

    Gravitational dust collapse with cosmological constant

    M. Cissoko, J. C. Fabris, J. Gariel, G. Le Denmat and N. O. Santos, [arXiv:gr-qc/9809057 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  29. [29]

    Debnath, S

    U. Debnath, S. Nath and S. Chakraborty, Mon. Not. Roy. Astron. Soc.369, 1961-1964 (2006)

    work page 1961

  30. [30]

    D. M. Eardley and L. Smarr, Phys. Rev. D19, 2239 (1979)

    work page 1979

  31. [31]

    Christodoulou, Commun

    D. Christodoulou, Commun. Math. Phys.,93, 171 (1984)

    work page 1984

  32. [32]

    Papapetrou, ‘Formation of a singularity and causality’

    A. Papapetrou, ‘Formation of a singularity and causality’. In A Random Walk in Relativ- ity and Cosmology, eds N. Dadhich, J. Krishna Rao, J. V. Narlikar, C. V. Vishveshwara. New Delhi: Wiley Eastern (1985)

    work page 1985

  33. [33]

    R. P. A. C. Newman, Class. Quant. Grav.3, 527 (1986)

    work page 1986

  34. [34]

    Ori and T

    A. Ori and T. Piran, Phys. Rev. Lett.59, 2137 (1987)

    work page 1987

  35. [35]

    R. P. A. C. Newman and P. S. Joshi, Ann. Phys.182, 112 (1988)

    work page 1988

  36. [36]

    I. H. Dwivedi and P. S. Joshi, Class. Quant. Grav.6, 1599 (1989)

    work page 1989

  37. [37]

    Ori and T

    A. Ori and T. Piran, Phys. Rev. D42, 1068 (1990)

    work page 1990

  38. [38]

    Lake and T

    K. Lake and T. Zannias, Phys. Rev. D43(1991) no.6, 1798

    work page 1991

  39. [39]

    Lake, Phys

    K. Lake, Phys. Rev. D43, no. 4, 1416 (1991)

    work page 1991

  40. [40]

    P. S. Joshi and I. H. Dwivedi, Phys. Rev. D47, 5357 (1993)

    work page 1993

  41. [41]

    Christodoulou, Ann

    D. Christodoulou, Ann. Maths.,140, 607 (1994)

    work page 1994

  42. [42]

    I. H. Dwivedi and P. S. Joshi, Commun. Math. Phys.166, 117 (1994)

    work page 1994

  43. [43]

    T. P. Singh and P. S. Joshi, Class. Quant. Grav.13, 559 (1996)

    work page 1996

  44. [44]

    Jhingan, P

    S. Jhingan, P. S. Joshi and T. P. Singh, Class. Quant. Grav.13, 3057 (1996)

    work page 1996

  45. [45]

    Harada, H

    T. Harada, H. Iguchi and K. i. Nakao, Phys. Rev. D58, 041502 (1998)

    work page 1998

  46. [46]

    Harada, Phys

    T. Harada, Phys. Rev. D58, 104015 (1998). 29

    work page 1998

  47. [47]

    Maeda, T

    K. Maeda, T. Torii and M. Narita, Phys. Rev. Lett.81, 5270 (1998)

    work page 1998

  48. [48]

    P. S. Joshi and I. H. Dwivedi, Class. Quant. Grav.16, 41 (1999)

    work page 1999

  49. [49]

    Maeda, Phys

    H. Maeda, Phys. Rev. D73, 104004 (2006)

    work page 2006

  50. [50]

    Goswami and P

    R. Goswami and P. S. Joshi, Phys. Rev. D76, 084026 (2007)

    work page 2007

  51. [51]

    Hassannejad, A

    R. Hassannejad, A. Sadeghi and F. Shojai, Class. Quant. Grav.40, no.7, 075002 (2023)

    work page 2023

  52. [52]

    Shojai, A

    F. Shojai, A. Sadeghi and R. Hassannejad, Class. Quant. Grav.39, no.8, 085003 (2022)

    work page 2022

  53. [53]

    Kottler, Ann

    F. Kottler, Ann. Physik,56, 410 (1918)

    work page 1918

  54. [54]

    Nariai, reprinted, with historical comments in Gen

    H. Nariai, reprinted, with historical comments in Gen. Rel. Grav.31, 951 (1999)

    work page 1999

  55. [55]

    Dreyer, B

    O. Dreyer, B. Krishnan, D. Shoemaker and E. Schnetter, Phys. Rev. D67, 024018 (2003)

    work page 2003

  56. [56]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Living Rev. Rel.7, 10 (2004)

    work page 2004

  57. [57]

    Booth, Can

    I. Booth, Can. J. Phys.83, 1073 (2005)

    work page 2005

  58. [58]

    Schnetter, B

    E. Schnetter, B. Krishnan and F. Beyer, Phys. Rev. D74, 024028 (2006)

    work page 2006

  59. [59]

    Booth, L

    I. Booth, L. Brits, J. A. Gonzalez and C. Van Den Broeck, Class. Quant. Grav.23, 413 (2006)

    work page 2006

  60. [60]

    Booth and S

    I. Booth and S. Fairhurst, Phys. Rev. D75, 084019 (2007)

    work page 2007

  61. [61]

    S. L. Shapiro and S. A. Teukolsky, New York, USA: Wiley (1983)

    work page 1983

  62. [62]

    Numerical Relativity: Solving Einstein’s Equations on the Computer,

    T. W. Baumgarte and S. L. Shapiro, “Numerical Relativity: Solving Einstein’s Equations on the Computer,” Cambridge University Press (2010)

    work page 2010

  63. [63]

    S. A. Hayward, Phys. Rev. D49, 6467 (1994)

    work page 1994

  64. [64]

    Chatterjee, B

    A. Chatterjee, B. Chatterjee and A. Ghosh, Phys. Rev. D87, no. 8, 084051 (2013)

    work page 2013

  65. [65]

    Ashtekar, C

    A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski and J. Wis- niewski, Phys. Rev. Lett.85, 3564 (2000)

    work page 2000

  66. [66]

    Ashtekar, S

    A. Ashtekar, S. Fairhurst and B. Krishnan, Phys. Rev. D62, 104025 (2000)

    work page 2000

  67. [67]

    Ashtekar and B

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

    work page 2002

  68. [68]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Phys. Rev. D68, 104030 (2003)

    work page 2003

  69. [69]

    Ashtekar, J

    A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett.80, 904 (1998)

    work page 1998

  70. [70]

    Chatterjee and A

    A. Chatterjee and A. Ghosh, Phys. Rev. D91, 064054 (2015)

    work page 2015

  71. [71]

    Chatterjee and A

    A. Chatterjee and A. Ghosh, Phys. Rev. D92, no. 4, 044003 (2015)

    work page 2015

  72. [72]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, [arXiv:2502.11825 [gr-qc]]

    work page arXiv

  73. [73]

    Ashtekar and G

    A. Ashtekar and G. J. Galloway, Adv. Theor. Math. Phys.9, no. 1, 1 (2005)

    work page 2005

  74. [74]

    Andersson, M

    L. Andersson, M. Mars and W. Simon, Phys. Rev. Lett.95, 111102 (2005). 30

    work page 2005

  75. [75]

    Andersson, M

    L. Andersson, M. Mars and W. Simon, Adv. Theor. Math. Phys.12, no. 4, 853 (2008)

    work page 2008

  76. [76]

    Booth and J

    I. Booth and J. Martin, Phys. Rev. D82, 124046 (2010)

    work page 2010

  77. [77]

    Bengtsson and J

    I. Bengtsson and J. M. M. Senovilla, Phys. Rev. D79, 024027 (2009)

    work page 2009

  78. [78]

    Bengtsson and J

    I. Bengtsson and J. M. M. Senovilla, Phys. Rev. D83, 044012 (2011)

    work page 2011

  79. [79]

    Bengtsson, E

    I. Bengtsson, E. Jakobsson and J. M. M. Senovilla, Phys. Rev. D88, 064012 (2013)

    work page 2013

  80. [80]

    Booth and D

    I. Booth and D. W. Tian, Class. Quant. Grav.30, 145008 (2013)

    work page 2013

Showing first 80 references.