- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Letters

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Analytic Discrete Self-Similar Solutions of Einstein-Klein-Gordon at Large D

Christian Ecker1,*, Florian Ecker2,†, and Daniel Grumiller2,‡

  • 1Institute for Theoretical Physics, Goethe University, 60438 Frankfurt am Main, Germany
  • 2Institute for Theoretical Physics, TU Wien, Wiedner Hauptstrasse 8–10/136, A-1040 Vienna, Austria

  • *Contact author: ecker@itp.uni-frankfurt.de
  • †Contact author: fecker@hep.itp.tuwien.ac.at
  • ‡Contact author: grumil@hep.itp.tuwien.ac.at
PDF

Phys. Rev. Lett. 136, 191401 – Published 12 May, 2026

DOI: https://doi.org/10.1103/qgl5-5l3t

Abstract

Discretely self-similar solutions govern critical collapse and have been known only numerically since Choptuik’s pioneering work. Using the large-D expansion, where D is the spacetime dimension, we construct an infinite family of analytic solutions of the Einstein-massless-Klein-Gordon equations. In this limit, the field equations simplify drastically, and the solutions are encoded in a single function of time. We characterize their structure and compare them with numerical critical solutions at finite D, identifying both universal features and effects specific to large D.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (35)

  1. D. Christodoulou, Commun. Math. Phys. 105, 337 (1986).
  2. D. Christodoulou, Commun. Math. Phys. 106, 587 (1986).
  3. D. Christodoulou, Commun. Math. Phys. 109, 591 (1987).
  4. D. S. Goldwirth and T. Piran, Phys. Rev. D 36, 3575 (1987).
  5. M. W. Choptuik, Phys. Rev. Lett. 70, 9 (1993).
  6. T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett. 74, 5170 (1995).
  7. C. Gundlach, Phys. Rep. 376, 339 (2003).
  8. C. Gundlach and J. M. Martin-Garcia, Living Rev. Relativity 10, 5 (2007).
  9. C. Gundlach, Phys. Rev. Lett. 75, 3214 (1995).
  10. C. Gundlach, Phys. Rev. D 55, 695 (1997).
  11. J. M. Martin-Garcia and C. Gundlach, Phys. Rev. D 68, 024011 (2003).
  12. R. Emparan, R. Suzuki, and K. Tanabe, J. High Energy Phys. 06 (2013) 009.
  13. R. Emparan, D. Grumiller, and K. Tanabe, Phys. Rev. Lett. 110, 251102 (2013).
  14. M. Rozali and B. Way, J. High Energy Phys. 11 (2018) 106.
  15. R. Emparan and C. P. Herzog, Rev. Mod. Phys. 92, 045005 (2020).
  16. E. Sorkin and Y. Oren, Phys. Rev. D 71, 124005 (2005).
  17. J. Bland, B. Preston, M. Becker, G. Kunstatter, and V. Husain, Classical Quantum Gravity 22, 5355 (2005).
  18. R. Emparan and K. Tanabe, Phys. Rev. D 89, 064028 (2014).
  19. R. Emparan, R. Suzuki, and K. Tanabe, Phys. Rev. Lett. 115, 091102 (2015).
  20. R. Emparan, T. Shiromizu, R. Suzuki, K. Tanabe, and T. Tanaka, J. High Energy Phys. 06 (2015) 159.
  21. R. Emparan, R. Suzuki, and K. Tanabe, J. High Energy Phys. 04 (2015) 085.

  22. The idea to rescale a variable like a2−1 by a linear factor in the dimension in a large-D context is due to Rozali and Way [14]. However, they also rescaled one of the coordinates, which makes the SSH inaccessible. Therefore, we do not follow their scaling regime but establish here a different one that allows a regular SSH in the large-D expansion.

  23. The function β(τ) is equivalent to Π(τ,0) and has a physical interpretation as Lorentz factor, β=1/1−[∂ψ0(τ)/∂τ]2, where ψ0 is the LO term in the scalar field, see Eq. (S2) in Supplemental Material [24].

  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qgl5-5l3t for further details on CSS solutions, the reconstruction of the scalar field, NEC lines, extrema of the SSH function, and convexity conditions.
  25. C. R. Clark and G. L. Pimentel, J. High Energy Phys. 11 (2025) 017.
  26. C. Ecker, F. Ecker, and D. Grumiller, Phys. Rev. D 112, L021505 (2025).
  27. C. Ecker, F. Ecker, D. Grumiller, and T. Jechtl, arXiv:2602.10185.

  28. Achieving slightly lower dimensions is possible by generalizing the 2-mode ansatz (21). Allowing an arbitrary amplitude in the first term and a phase in the second term, we find that the phase needs to be close to zero and the optimal amplitude is close to 1.4. In that case, the lowest possible dimension compatible with the convexity conditions (6) turns out to be D=49.

  29. Mathematica notebook available at https://github.com/EckerChristian/CritLargeD.
  30. C. Gundlach, Adv. Theor. Math. Phys. 2, 1 (1998).
  31. M. Reiterer and E. Trubowitz, Commun. Math. Phys. 368, 143 (2019).
  32. J. B. Bland, Dimension dependence of the critical phenomena in gravitational collapse of massless scalar field, Ph.D. thesis, University of Manitoba, 2006.
  33. M. Tomašević and C.-H. Wu, arXiv:2509.03584.
  34. M. Tomašević and C.-H. Wu, arXiv:2509.03587.
  35. V. Frolov, P. Sutton, and A. Zelnikov, Phys. Rev. D 61, 024021 (2000).

Outline

Information

Phys. Rev. Lett. 136, 191401– Published 12 May, 2026
  • Received 29 January 2026
  • Accepted 22 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation