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Null infinity as an open Hamiltonian system
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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 12 April 2021

Null infinity as an open Hamiltonian system

  • Wolfgang Wieland  ORCID: orcid.org/0000-0003-1371-34321 

Journal of High Energy Physics volume 2021, Article number: 95 (2021) Cite this article

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  • 15 Citations

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A preprint version of the article is available at arXiv.

Abstract

When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time-dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman-Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald-Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium.

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References

  1. H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity, VII. Waves from axi-symmetric isolated system, Proc. Roy. Soc. London A 269 (1962) 21.

  2. R.K. Sachs, Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London A 270 (1962) 103.

  3. G.T. Horowitz and M.J. Perry, Gravitational energy cannot become negative, Phys. Rev. Lett. 48 (1982) 371 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. A. Ashtekar, Asymptotic Quantization, based on 1984 Naples Lectures, Bibliopolis, Napoli Italy (1987).

  5. A. Ashtekar, Geometry and Physics of Null Infinity, in Surveys in differential geometry — One hundred years of general relativity, L. Bieri and S.T. Yau eds., International Press of Boston, U.S.A. (2015), arXiv:1409.1800 [INSPIRE].

  6. C. Rovelli, Partial observables, Phys. Rev. D 65 (2002) 124013 [gr-qc/0110035] [INSPIRE].

  7. B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23 (2006) 6155 [gr-qc/0507106] [INSPIRE].

  8. R.E. Peierls, The commutation laws of relativistic field theory, Proc. Roy. Soc. London A 214 (1952) 143.

    Article  MathSciNet  ADS  Google Scholar 

  9. A. Ashtekar, L. Bombelli and O. Reula, The covariant phase space of asymptotically flat gravitational fields, in Mechanics, analysis and geometry: 200 years after Lagrange, M. Francaviglia and D. Holm eds., North Holland, Amsterdam The Netherlands (1990).

  10. J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  11. V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].

  12. R.M. Wald and A. Zoupas, A general definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [INSPIRE].

  13. A.P. Balachandran, L. Chandar and A. Momen, Edge states in gravity and black hole physics, Nucl. Phys. B 461 (1996) 581 [gr-qc/9412019] [INSPIRE].

  14. A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  15. M. Bañados, T. Brotz and M.E. Ortiz, Boundary dynamics and the statistical mechanics of the (2+1)-dimensional black hole, Nucl. Phys. B 545 (1999) 340 [hep-th/9802076] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge University Press, Cambridge U.K.. (2003).

    MATH  Google Scholar 

  17. S. Carlip, Conformal field theory, (2 + 1)-dimensional gravity, and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [INSPIRE].

  18. H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. G. Compère and A. Fiorucci, Asymptotically flat spacetimes with BMS3 symmetry, Class. Quant. Grav. 34 (2017) 204002 [arXiv:1705.06217] [INSPIRE].

    Article  ADS  Google Scholar 

  20. W. Wieland, Conformal boundary conditions, loop gravity and the continuum, JHEP 10 (2018) 089 [arXiv:1804.08643] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. J.C. Namburi and W. Wieland, Deformed Heisenberg charges in three-dimensional gravity, JHEP 03 (2020) 175 [arXiv:1912.09514] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. W. Wieland, Twistor representation of Jackiw-Teitelboim gravity, Class. Quant. Grav. 37 (2020) 195008.

    Article  MathSciNet  ADS  Google Scholar 

  23. R. Penrose and W. Rindler, Spinors and space-time, two-spinor calculus and relativistic fields, volumes 1 and 2, Cambridge University Press, Cambridge U.K. (1984).

  24. E. Newman and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566.

    Article  MathSciNet  ADS  Google Scholar 

  25. A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  26. R. Arnowitt, S. Deser and C.W. Misner, Republication of: the dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997.

    Article  ADS  Google Scholar 

  27. R. Arnowitt, S. Deser and C.W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. 116 (1959) 1322.

    Article  MathSciNet  ADS  Google Scholar 

  28. D. Harlow and J.-q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616].

    Article  MathSciNet  ADS  Google Scholar 

  29. A. Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D 36 (1987) 1587 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  30. J.F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D 51 (1995) 5507 [gr-qc/9410014] [INSPIRE].

  31. W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. E. De Paoli and S. Speziale, A gauge-invariant symplectic potential for tetrad general relativity, JHEP 07 (2018) 040 [arXiv:1804.09685] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  33. J. Isenberg, The initial value problem in general relativity, in Springer handbook of spacetime, A. Ashtekar and V. Petkov eds., Springer, Germany (2013) [arXiv:1304.1960] [INSPIRE].

  34. F. Mercati, Shape dynamics: relativity and relationalism, Oxford University Press, Oxford, U.K. (2018).

    Book  Google Scholar 

  35. H. Gomes and T. Koslowski, The link between general relativity and shape dynamics, Class. Quant. Grav. 29 (2012) 075009 [arXiv:1101.5974] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].

  37. E. Frodden, A. Ghosh and A. Perez, Quasilocal first law for black hole thermodynamics, Phys. Rev. D 87 (2013) 121503 [arXiv:1110.4055] [INSPIRE].

    Article  ADS  Google Scholar 

  38. A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Liv. Rev. Rel. 7 (2004) [gr-qc/0407042].

  39. A. Ashtekar, C. Beetle and S. Fairhurst, Isolated horizons: a generalization of black hole mechanics, Class. Quant. Grav. 16 (1999) L1 [gr-qc/9812065] [INSPIRE].

  40. A. Ashtekar, C. Beetle and J. Lewandowski, Mechanics of rotating isolated horizons, Phys. Rev. D 64 (2001) 044016 [gr-qc/0103026] [INSPIRE].

  41. A. Ashtekar, J. Engle, T. Pawlowski and C. Van Den Broeck, Multipole moments of isolated horizons, Class. Quant. Grav. 21 (2004) 2549 [gr-qc/0401114] [INSPIRE].

  42. N. Bodendorfer, T. Thiemann and A. Thurn, New variables for classical and quantum gravity in all dimensions V. Isolated horizon boundary degrees of freedom, Class. Quant. Grav. 31 (2014) 055002 [arXiv:1304.2679] [INSPIRE].

  43. D. Pranzetti and H. Sahlmann, Horizon entropy with loop quantum gravity methods, Phys. Lett. B 746 (2015) 209 [arXiv:1412.7435] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. B. Dittrich, C. Goeller, E.R. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity, Class. Quant. Grav. 35 (2018) 13LT01 [arXiv:1803.02759] [INSPIRE].

  45. B. Dittrich, C. Goeller, E. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity I — Convergence of multiple approaches and examples of Ponzano-Regge statistical duals, Nucl. Phys. B 938 (2019) 807 [arXiv:1710.04202] [INSPIRE].

    Article  ADS  Google Scholar 

  46. B. Dittrich, C. Goeller, E.R. Livine and A. Riello, Quasi-local holographic dualities in non-perturbative 3d quantum gravity II — From coherent quantum boundaries to BMS3 characters, Nucl. Phys. B 938 (2019) 878 [arXiv:1710.04237] [INSPIRE].

    Article  ADS  Google Scholar 

  47. W. Wieland, Fock representation of gravitational boundary modes and the discreteness of the area spectrum, Annales Henri Poincaré 18 (2017) 3695 [arXiv:1706.00479] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  48. N. Lashkari, J. Lin, H. Ooguri, B. Stoica and M. Van Raamsdonk, Gravitational positive energy theorems from information inequalities, PTEP 2016 (2016) 12C109 [arXiv:1605.01075] [INSPIRE].

  49. V. Chandrasekaran and K. Prabhu, Symmetries, charges and conservation laws at causal diamonds in general relativity, JHEP 10 (2019) 229 [arXiv:1908.00017] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  50. T. De Lorenzo and A. Perez, Light cone thermodynamics, Phys. Rev. D 97 (2018) 044052 [arXiv:1707.00479] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  51. S. Chakraborty and T. Padmanabhan, Boundary term in the gravitational action is the heat content of the null surfaces, Phys. Rev. D 101 (2020) 064023.

    Article  MathSciNet  ADS  Google Scholar 

  52. F. Hopfmüller and L. Freidel, Gravity degrees of freedom on a null surface, Phys. Rev. D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  53. V. Chandrasekaran, E.E. Flanagan and K. Prabhu, Symmetries and charges of general relativity at null boundaries, JHEP 11 (2018) 125 [arXiv:1807.11499] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  54. J.N. Goldberg, D.C. Robinson, and C. Soteriou, Null hypersurfaces and new variables, Class. Quant. Grav. 9 (1992) 1309.

    Article  MathSciNet  ADS  Google Scholar 

  55. J.N. Goldberg and C. Soteriou, Canonical general relativity on a null surface with coordinate and gauge fixing, Class. Quant. Grav. 12 (1995) 2779 [gr-qc/9504043] [INSPIRE].

  56. A. Corichi, I. Rubalcava-García and T. Vukašinac, Actions, topological terms and boundaries in first-order gravity: a review, Int. J. Mod. Phys. D 25 (2016) 1630011 [arXiv:1604.07764] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. E. De Paoli and S. Speziale, Sachs’ free data in real connection variables, JHEP 11 (2017) 205 [arXiv:1707.00667] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  58. E. Frodden and D. Hidalgo, Surface charges toolkit for gravity, Int. J. Mod. Phys. D 29 (2020) 2050040 [arXiv:1911.07264] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. J.F. Barbero G., B. Díaz, J. Margalef-Bentabol and E.J.S. Villaseñor, Concise symplectic formulation for tetrad gravity, Phys. Rev. D 103 (2021) 024051 [arXiv:2011.00661] [INSPIRE].

  60. W. Wieland, Generating functional for gravitational null initial data, Class. Quant. Grav. 36 (2019) 235007 [arXiv:1905.06357] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  61. A. Vanrietvelde, P.A. Hoehn, F. Giacomini and E. Castro-Ruiz, A change of perspective: switching quantum reference frames via a perspective-neutral framework, Quantum 4 (2020) 225 [arXiv:1809.00556] [INSPIRE].

    Article  Google Scholar 

  62. P.A. Höhn, A.R.H. Smith and M.P.E. Lock, The trinity of relational quantum dynamics, arXiv:1912.00033 [INSPIRE].

  63. E. Castro-Ruiz, F. Giacomini, A. Belenchia and v. Brukner, Quantum clocks and the temporal localisability of events in the presence of gravitating quantum systems, Nature Commun. 11 (2020) 2672 [arXiv:1908.10165] [INSPIRE].

  64. F. Giacomini, E. Castro-Ruiz and v. Brukner, Relativistic quantum reference frames: the operational meaning of spin, Phys. Rev. Lett. 123 (2019) 090404 [arXiv:1811.08228] [INSPIRE].

  65. F. Giacomini, E. Castro-Ruiz and Č. Brukner, Quantum mechanics and the covariance of physical laws in quantum reference frames, Nature Commun. 10 (2019) 494.

    Article  ADS  Google Scholar 

  66. M. Krumm, P.A. Hoehn and M.P. Mueller, Quantum reference frame transformations as symmetries and the paradox of the third particle, arXiv:2011.01951 [INSPIRE].

  67. A. Fiorucci and R. Ruzziconi, Charge algebra in Al(A)dSn Spacetimes, arXiv:2011.02002 [INSPIRE].

  68. G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  69. C. Troessaert, Hamiltonian surface charges using external sources, J. Math. Phys. 57 (2016) 053507 [arXiv:1509.09094] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090, Vienna, Austria

    Wolfgang Wieland

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  1. Wolfgang Wieland
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Correspondence to Wolfgang Wieland.

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Wieland, W. Null infinity as an open Hamiltonian system. J. High Energ. Phys. 2021, 95 (2021). https://doi.org/10.1007/JHEP04(2021)095

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  • Received: 05 January 2021

  • Accepted: 01 March 2021

  • Published: 12 April 2021

  • DOI: https://doi.org/10.1007/JHEP04(2021)095

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Keywords

  • Classical Theories of Gravity
  • Models of Quantum Gravity
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