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Title: Black hole and de Sitter microstructures from a semiclassical perspective

Abstract

We describe two different, but equivalent semiclassical views of black hole physics in which the equivalence principle and unitarity are both accommodated. In one, unitarity is built-in, while the black hole interior emerges only effectively as a collective phenomenon involving horizon (and possibly other) degrees of freedom. In the other, more widely studied approach, the existence of the interior is manifest, while the unitarity of the underlying dynamics can be captured only indirectly by incorporating certain nonperturbative effects of gravity. These two pictures correspond to a distant description and the description based on entanglement islands/replica wormholes, respectively. We also present a holographic description of de Sitter spacetime based on the former approach, in which the holographic theory is located on the stretched horizon of a static patch. We argue that the existence of these two approaches is rooted in the two formulations of quantum mechanics: the canonical and path integral formalisms.

Authors:
ORCiD logo; ORCiD logo; ORCiD logo
Publication Date:
Research Org.:
Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP); Ministry of Education, Culture, Sports, Science and Technology (MEXT)
OSTI Identifier:
1914338
Alternate Identifier(s):
OSTI ID: 2242447
Grant/Contract Number:  
SC0019380; AC02-05CH11231; JP20H05850; JP20H05860
Resource Type:
Published Article
Journal Name:
Physical Review D
Additional Journal Information:
Journal Name: Physical Review D Journal Volume: 107 Journal Issue: 2; Journal ID: ISSN 2470-0010
Publisher:
American Physical Society
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS; Quantum aspects of black holes; Quantum gravity; Semiclassical physics; Gravitation, Cosmology & Astrophysics; Particles & Fields

Citation Formats

Murdia, Chitraang, Nomura, Yasunori, and Ritchie, Kyle. Black hole and de Sitter microstructures from a semiclassical perspective. United States: N. p., 2023. Web. doi:10.1103/PhysRevD.107.026016.
Murdia, Chitraang, Nomura, Yasunori, & Ritchie, Kyle. Black hole and de Sitter microstructures from a semiclassical perspective. United States. https://doi.org/10.1103/PhysRevD.107.026016
Murdia, Chitraang, Nomura, Yasunori, and Ritchie, Kyle. Fri . "Black hole and de Sitter microstructures from a semiclassical perspective". United States. https://doi.org/10.1103/PhysRevD.107.026016.
@article{osti_1914338,
title = {Black hole and de Sitter microstructures from a semiclassical perspective},
author = {Murdia, Chitraang and Nomura, Yasunori and Ritchie, Kyle},
abstractNote = {We describe two different, but equivalent semiclassical views of black hole physics in which the equivalence principle and unitarity are both accommodated. In one, unitarity is built-in, while the black hole interior emerges only effectively as a collective phenomenon involving horizon (and possibly other) degrees of freedom. In the other, more widely studied approach, the existence of the interior is manifest, while the unitarity of the underlying dynamics can be captured only indirectly by incorporating certain nonperturbative effects of gravity. These two pictures correspond to a distant description and the description based on entanglement islands/replica wormholes, respectively. We also present a holographic description of de Sitter spacetime based on the former approach, in which the holographic theory is located on the stretched horizon of a static patch. We argue that the existence of these two approaches is rooted in the two formulations of quantum mechanics: the canonical and path integral formalisms.},
doi = {10.1103/PhysRevD.107.026016},
journal = {Physical Review D},
number = 2,
volume = 107,
place = {United States},
year = {Fri Jan 20 00:00:00 EST 2023},
month = {Fri Jan 20 00:00:00 EST 2023}
}

Works referenced in this record:

Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime
journal, January 2015


Tensile Strength and the Mining of Black Holes
journal, November 2013


Stability of gravity with a cosmological constant
journal, February 1982


Holographic representation of local bulk operators
journal, September 2006


Violations of the Equivalence Principle by a Nonlocally Reconstructed Vacuum at the Black Hole Horizon
journal, January 2014


Black holes as mirrors: quantum information in random subsystems
journal, September 2007


Conserved charges in general relativity
journal, April 2021

  • Aoki, Sinya; Onogi, Tetsuya; Yokoyama, Shuichi
  • International Journal of Modern Physics A, Vol. 36, Issue 10
  • DOI: 10.1142/S0217751X21500986

Computational complexity and black hole horizons: Computational complexity and black hole horizons
journal, January 2016


An alternative to the Teukolsky equation
journal, October 2021


Multiboundary wormholes and holographic entanglement
journal, September 2014


Eternal black holes in anti-de Sitter
journal, April 2003


Local subsystems in gauge theory and gravity
journal, September 2016

  • Donnelly, William; Freidel, Laurent
  • Journal of High Energy Physics, Vol. 2016, Issue 9
  • DOI: 10.1007/JHEP09(2016)102

On BMS invariance of gravitational scattering
journal, July 2014


Volume inside old black holes
journal, November 2016


Hologravity
journal, June 2005


Symmetries and strings in field theory and gravity
journal, April 2011


Replica wormholes and the black hole interior
journal, March 2022

  • Penington, Geoff; Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2022, Issue 3
  • DOI: 10.1007/JHEP03(2022)205

Entropy of a vacuum: What does the covariant entropy count?
journal, November 2014


Holographic entanglement entropy conjecture for general spacetimes
journal, October 2016


Black holes and the butterfly effect
journal, March 2014

  • Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2014, Issue 3
  • DOI: 10.1007/JHEP03(2014)067

Extremal surface barriers
journal, March 2014


4D scattering amplitudes and asymptotic symmetries from 2D CFT
journal, January 2017

  • Cheung, Clifford; de la Fuente, Anton; Sundrum, Raman
  • Journal of High Energy Physics, Vol. 2017, Issue 1
  • DOI: 10.1007/JHEP01(2017)112

An infalling observer in AdS/CFT
journal, October 2013

  • Papadodimas, Kyriakos; Raju, Suvrat
  • Journal of High Energy Physics, Vol. 2013, Issue 10
  • DOI: 10.1007/JHEP10(2013)212

The large $N$ limit of superconformal field theories and supergravity
journal, January 1998


Remarks on the necessity and implications of state-dependence in the black hole interior
journal, April 2016


Bulk locality and quantum error correction in AdS/CFT
journal, April 2015

  • Almheiri, Ahmed; Dong, Xi; Harlow, Daniel
  • Journal of High Energy Physics, Vol. 2015, Issue 4
  • DOI: 10.1007/JHEP04(2015)163

Non-local entanglement and fast scrambling in de-Sitter holography
journal, March 2021


de Sitter Space as a Resonance
journal, March 2017


Maximin surfaces, and the strong subadditivity of the covariant holographic entanglement entropy
journal, November 2014


Free energy from replica wormholes
journal, February 2021


A covariant holographic entanglement entropy proposal
journal, July 2007

  • Hubeny, Veronika E.; Rangamani, Mukund; Takayanagi, Tadashi
  • Journal of High Energy Physics, Vol. 2007, Issue 07
  • DOI: 10.1088/1126-6708/2007/07/062

Holevo information and ensemble theory of gravity
journal, February 2022

  • Qi, Xiao-Liang; Shangnan, Zhou; Yang, Zhenbin
  • Journal of High Energy Physics, Vol. 2022, Issue 2
  • DOI: 10.1007/JHEP02(2022)056

Reconstruction of Bulk Operators within the Entanglement Wedge in Gauge-Gravity Duality
journal, July 2016


Spacetime and universal soft modes: Black holes and beyond
journal, March 2020


Acceleration radiation and the generalized second law of thermodynamics
journal, February 1982


Shocks and information exchange in de Sitter space
journal, October 2021

  • Aalsma, L.; Cole, A.; Morvan, E.
  • Journal of High Energy Physics, Vol. 2021, Issue 10
  • DOI: 10.1007/JHEP10(2021)104

Relativeness in quantum gravity: limitations and frame dependence of semiclassical descriptions
journal, April 2015

  • Nomura, Yasunori; Sanches, Fabio; Weinberg, Sean J.
  • Journal of High Energy Physics, Vol. 2015, Issue 4
  • DOI: 10.1007/JHEP04(2015)158

Relative entropy equals bulk relative entropy
journal, June 2016

  • Jafferis, Daniel L.; Lewkowycz, Aitor; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2016, Issue 6
  • DOI: 10.1007/JHEP06(2016)004

Is Black-Hole Evaporation Predictable?
journal, February 1980


Disturbing Implications of a Cosmological Constant
journal, October 2002


Entropy Evaporated by a Black Hole
journal, December 1982


The gravity dual of a density matrix
journal, July 2012


Decomposition of entanglement entropy in lattice gauge theory
journal, April 2012


A remark on the AdS/CFT correspondence and the renormalization group flow
journal, December 1998


Local Operators in the Eternal Black Hole
journal, November 2015


Gravity and the crossed product
journal, October 2022


The trouble with de Sitter space
journal, July 2003


Spacetime from unentanglement
journal, May 2018


Interior of a unitarily evaporating black hole
journal, July 2020


Entanglement and the foundations of statistical mechanics
journal, October 2006

  • Popescu, Sandu; Short, Anthony J.; Winter, Andreas
  • Nature Physics, Vol. 2, Issue 11
  • DOI: 10.1038/nphys444

Sufficiency of Channels over von Neumann Algebras
journal, January 1988


The world as a hologram
journal, November 1995

  • Susskind, Leonard
  • Journal of Mathematical Physics, Vol. 36, Issue 11
  • DOI: 10.1063/1.531249

Information paradox and its resolution in de Sitter holography
journal, June 2021


Path-integral derivation of black-hole radiance
journal, April 1976


Bulk and transhorizon measurements in AdS/CFT
journal, October 2012

  • Heemskerk, Idse; Marolf, Donald; Polchinski, Joseph
  • Journal of High Energy Physics, Vol. 2012, Issue 10
  • DOI: 10.1007/JHEP10(2012)165

Generalized second law for cosmology
journal, January 2016


Holography in general space-times
journal, June 1999


Black holes: complementarity or firewalls?
journal, February 2013

  • Almheiri, Ahmed; Marolf, Donald; Polchinski, Joseph
  • Journal of High Energy Physics, Vol. 2013, Issue 2
  • DOI: 10.1007/JHEP02(2013)062

Entanglement entropy in gauge theories and the holographic principle for electric strings
journal, December 2008


Cool horizons for entangled black holes
journal, August 2013


A holographic reduction of Minkowski space–time
journal, August 2003


Black hole interior in unitary gauge construction
journal, March 2021


Nonexistence of Baryon Number for Static Black Holes
journal, March 1972


Black Holes and Entropy
journal, April 1973


The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral
journal, January 2021

  • Maxfield, Henry; Turiaci, Gustavo J.
  • Journal of High Energy Physics, Vol. 2021, Issue 1
  • DOI: 10.1007/JHEP01(2021)118

Action integrals and partition functions in quantum gravity
journal, May 1977


The central dogma and cosmological horizons
journal, January 2022


Thermo-field dynamics of black holes
journal, May 1976


Particle creation by black holes
journal, June 1976

  • Hawking, S. W.
  • Communications in Mathematical Physics, Vol. 46, Issue 2
  • DOI: 10.1007/BF01608497

Overcounting of interior excitations: a resolution to the bags of gold paradox in AdS
journal, February 2021


How big is a black hole?
journal, March 2015


The dS/dS Correspondence
conference, January 2004

  • Alishahiha, Mohsen
  • THE NEW COSMOLOGY: Conference on Strings and Cosmology; The Mitchell Symposium on Observational Cosmology, AIP Conference Proceedings
  • DOI: 10.1063/1.1848341

Seeing behind black hole horizons in SYK
journal, June 2022


Black holes and large N species solution to the hierarchy problem
journal, April 2010


Causality & holographic entanglement entropy
journal, December 2014

  • Headrick, Matthew; Hubeny, Veronika E.; Lawrence, Albion
  • Journal of High Energy Physics, Vol. 2014, Issue 12
  • DOI: 10.1007/JHEP12(2014)162

Global symmetry, Euclidean gravity, and the black hole information problem
journal, April 2021

  • Harlow, Daniel; Shaghoulian, Edgar
  • Journal of High Energy Physics, Vol. 2021, Issue 4
  • DOI: 10.1007/JHEP04(2021)175

The ghost in the radiation: robust encodings of the black hole interior
journal, June 2020

  • Kim, Isaac; Tang, Eugene; Preskill, John
  • Journal of High Energy Physics, Vol. 2020, Issue 6
  • DOI: 10.1007/JHEP06(2020)031

Constraints on Symmetries from Holography
journal, May 2019


No Page curves for the de Sitter horizon
journal, March 2022

  • Kames-King, Joshua; Verheijden, Evita M. H.; Verlinde, Erik P.
  • Journal of High Energy Physics, Vol. 2022, Issue 3
  • DOI: 10.1007/JHEP03(2022)040

Generalized gravitational entropy
journal, August 2013

  • Lewkowycz, Aitor; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2013, Issue 8
  • DOI: 10.1007/JHEP08(2013)090

On the quantum structure of a black hole
journal, January 1985


Notes on black-hole evaporation
journal, August 1976


Flat space amplitudes and conformal symmetry of the celestial sphere
journal, September 2017


Black hole entanglement and quantum error correction
journal, October 2013

  • Verlinde, Erik; Verlinde, Herman
  • Journal of High Energy Physics, Vol. 2013, Issue 10
  • DOI: 10.1007/JHEP10(2013)107

Remarks on entanglement entropy for gauge fields
journal, April 2014


Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information
journal, August 2020


Gravity factorized
journal, September 2022

  • Blommaert, Andreas; Iliesiu, Luca V.; Kruthoff, Jorrit
  • Journal of High Energy Physics, Vol. 2022, Issue 9
  • DOI: 10.1007/JHEP09(2022)080

Comment on "Entropy Evaporated by a Black Hole"
journal, March 1983


Information in black hole radiation
journal, December 1993


The four laws of black hole mechanics
journal, June 1973

  • Bardeen, J. M.; Carter, B.; Hawking, S. W.
  • Communications in Mathematical Physics, Vol. 31, Issue 2
  • DOI: 10.1007/BF01645742

Entanglement wedge reconstruction and the information paradox
journal, September 2020


Chaos and complementarity in de Sitter space
journal, May 2020


Black Hole Interior in Quantum Gravity
journal, May 2015


Negative energy, superluminosity, and holography
journal, September 1999


Homology conditions for RT surfaces in double holography
journal, March 2022


Cosmological event horizons, thermodynamics, and particle creation
journal, May 1977


The volume of the black hole interior at late times
journal, July 2022

  • Iliesiu, Luca V.; Mezei, Márk; Sárosi, Gábor
  • Journal of High Energy Physics, Vol. 2022, Issue 7
  • DOI: 10.1007/JHEP07(2022)073

Black holes, information, and Hilbert space for quantum gravity
journal, April 2013


On the entanglement entropy for gauge theories
journal, September 2015

  • Ghosh, Sudip; Soni, Ronak M.; Trivedi, Sandip P.
  • Journal of High Energy Physics, Vol. 2015, Issue 9
  • DOI: 10.1007/JHEP09(2015)069

Spacetime and the Holographic Renormalization Group
journal, November 1999


Eigenstate Thermalization and Disorder Averaging in Gravity
journal, July 2020


Coarse-graining holographic states: A semiclassical flow in general spacetimes
journal, October 2020


Average entropy of a subsystem
journal, August 1993


Breakdown of predictability in gravitational collapse
journal, November 1976


Information transfer with a gravitating bath
journal, January 2021


Holographic duality from random tensor networks
journal, November 2016

  • Hayden, Patrick; Nezami, Sepehr; Qi, Xiao-Liang
  • Journal of High Energy Physics, Vol. 2016, Issue 11
  • DOI: 10.1007/JHEP11(2016)009

Canonical Typicality
journal, February 2006


Fast scramblers
journal, October 2008


Sufficient subalgebras and the relative entropy of states of a von Neumann algebra
journal, March 1986

  • Petz, D�nes
  • Communications in Mathematical Physics, Vol. 105, Issue 1
  • DOI: 10.1007/BF01212345

The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
journal, December 2019

  • Almheiri, Ahmed; Engelhardt, Netta; Marolf, Donald
  • Journal of High Energy Physics, Vol. 2019, Issue 12
  • DOI: 10.1007/JHEP12(2019)063

Generalized second law of thermodynamics in black-hole physics
journal, June 1974


On the holographic renormalization group
journal, August 2000


Gravity/ensemble duality
journal, September 2020


Beyond toy models: distilling tensor networks in full AdS/CFT
journal, November 2019

  • Bao, Ning; Penington, Geoffrey; Sorce, Jonathan
  • Journal of High Energy Physics, Vol. 2019, Issue 11
  • DOI: 10.1007/JHEP11(2019)069

de Sitter vacua in string theory
journal, August 2003


A precision test of averaging in AdS/CFT
journal, November 2022


Entropy, extremality, euclidean variations, and the equations of motion
journal, January 2018


The dS/CFT correspondence
journal, October 2001


Hawking radiation, the Stefan–Boltzmann law, and unitarization
journal, March 2016


Cosmological Breaking of Supersymmetry?
journal, February 2001


The Page curve of Hawking radiation from semiclassical geometry
journal, March 2020

  • Almheiri, Ahmed; Mahajan, Raghu; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2020, Issue 3
  • DOI: 10.1007/JHEP03(2020)149

A note on (no) firewalls: the entropy argument
journal, July 2013


From the black hole conundrum to the structure of quantum gravity
journal, January 2021


Replica wormholes and the entropy of Hawking radiation
journal, May 2020

  • Almheiri, Ahmed; Hartman, Thomas; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2020, Issue 5
  • DOI: 10.1007/JHEP05(2020)013

Stability of a Schwarzschild Singularity
journal, November 1957


A bound on chaos
journal, August 2016

  • Maldacena, Juan; Shenker, Stephen H.; Stanford, Douglas
  • Journal of High Energy Physics, Vol. 2016, Issue 8
  • DOI: 10.1007/JHEP08(2016)106

Random statistics of OPE coefficients and Euclidean wormholes
journal, July 2021


Origin of the particles in black-hole evaporation
journal, January 1977


Particle emission rates from a black hole: Massless particles from an uncharged, nonrotating hole
journal, January 1976


The stretched horizon and black hole complementarity
journal, October 1993


Entanglement Wedge Reconstruction via Universal Recovery Channels
journal, July 2019


The black hole interpretation of string theory
journal, April 1990


Surface/state correspondence as a generalized holography
journal, July 2015

  • Miyaji, Masamichi; Takayanagi, Tadashi
  • Progress of Theoretical and Experimental Physics, Vol. 2015, Issue 7
  • DOI: 10.1093/ptep/ptv089

M theory as a matrix model: A conjecture
journal, April 1997


Ensemble from coarse graining: Reconstructing the interior of an evaporating black hole
journal, October 2020


The Holographic Nature of Null Infinity
journal, January 2021


State-dependent bulk-boundary maps and black hole complementarity
journal, April 2014


The entanglement wedge of unknown couplings
journal, August 2022


Deriving covariant holographic entanglement
journal, November 2016

  • Dong, Xi; Lewkowycz, Aitor; Rangamani, Mukund
  • Journal of High Energy Physics, Vol. 2016, Issue 11
  • DOI: 10.1007/JHEP11(2016)028

Precursors, black holes, and a locality bound
journal, December 2001


Gravitational stability and renormalization-group flow
journal, November 1999


Quantum focusing conjecture
journal, March 2016


Entanglement Entropies of Equilibrated Pure States in Quantum Many-Body Systems and Gravity
journal, March 2021


Holographic Derivation of Entanglement Entropy from the anti–de Sitter Space/Conformal Field Theory Correspondence
journal, May 2006


A covariant entropy conjecture
journal, July 1999


The holographic principle
journal, August 2002


Holographic Complexity Equals Bulk Action?
journal, May 2016


Local bulk operators in AdS/CFT correspondence: A boundary view of horizons and locality
journal, April 2006


Gravitational effects on and of vacuum decay
journal, June 1980


Learning the Alpha-bits of black holes
journal, December 2019

  • Hayden, Patrick; Penington, Geoffrey
  • Journal of High Energy Physics, Vol. 2019, Issue 12
  • DOI: 10.1007/JHEP12(2019)007

Inside the hologram: reconstructing the bulk observer’s experience
journal, March 2022

  • Jafferis, Daniel Louis; Lamprou, Lampros
  • Journal of High Energy Physics, Vol. 2022, Issue 3
  • DOI: 10.1007/JHEP03(2022)084

The information paradox: a pedagogical introduction
journal, October 2009


The statistical mechanics of near-BPS black holes
journal, December 2021

  • Heydeman, Matthew; Iliesiu, Luca V.; Turiaci, Gustavo J.
  • Journal of Physics A: Mathematical and Theoretical, Vol. 55, Issue 1
  • DOI: 10.1088/1751-8121/ac3be9

Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence
journal, June 2015

  • Pastawski, Fernando; Yoshida, Beni; Harlow, Daniel
  • Journal of High Energy Physics, Vol. 2015, Issue 6
  • DOI: 10.1007/JHEP06(2015)149

Geometric holography, the renormalization group and the c-theorem
journal, February 1999


Gravity edges modes and Hayward term
journal, February 2020

  • Takayanagi, Tadashi; Tamaoka, Kotaro
  • Journal of High Energy Physics, Vol. 2020, Issue 2
  • DOI: 10.1007/JHEP02(2020)167

Entanglement entropy and quantum field theory
journal, June 2004


Reanalyzing an evaporating black hole
journal, April 2019


On geometric entropy
journal, July 1994


Particle creation by black holes
journal, August 1975

  • Hawking, S. W.
  • Communications In Mathematical Physics, Vol. 43, Issue 3
  • DOI: 10.1007/BF02345020

Black holes, entropies, and semiclassical spacetime in quantum gravity
journal, October 2014

  • Nomura, Yasunori; Weinberg, Sean J.
  • Journal of High Energy Physics, Vol. 2014, Issue 10
  • DOI: 10.1007/JHEP10(2014)185

Toward a holographic theory for general spacetimes
journal, April 2017


Quantum corrections to holographic entanglement entropy
journal, November 2013

  • Faulkner, Thomas; Lewkowycz, Aitor; Maldacena, Juan
  • Journal of High Energy Physics, Vol. 2013, Issue 11
  • DOI: 10.1007/JHEP11(2013)074

Quantum information measures for restricted sets of observables
journal, August 2018


Entanglement in De Sitter space
journal, August 2022

  • Shaghoulian, Edgar; Susskind, Leonard
  • Journal of High Energy Physics, Vol. 2022, Issue 8
  • DOI: 10.1007/JHEP08(2022)198

Linearity of holographic entanglement entropy
journal, February 2017

  • Almheiri, Ahmed; Dong, Xi; Swingle, Brian
  • Journal of High Energy Physics, Vol. 2017, Issue 2
  • DOI: 10.1007/JHEP02(2017)074

Pulling the boundary into the bulk
journal, July 2018


A world without pythons would be so simple
journal, November 2021

  • Engelhardt, Netta; Penington, Geoff; Shahbazi-Moghaddam, Arvin
  • Classical and Quantum Gravity, Vol. 38, Issue 23
  • DOI: 10.1088/1361-6382/ac2de5

The Python’s Lunch: geometric obstructions to decoding Hawking radiation
journal, August 2020

  • Brown, Adam R.; Gharibyan, Hrant; Penington, Geoff
  • Journal of High Energy Physics, Vol. 2020, Issue 8
  • DOI: 10.1007/JHEP08(2020)121

Entanglement renormalization and holography
journal, September 2012


Quantum Field Theory in De Sitter Space: Renormalization by Point-Splitting
journal, March 1978

  • Bunch, T. S.; Davies, P. C. W.
  • Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 360, Issue 1700
  • DOI: 10.1098/rspa.1978.0060