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.
References
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.
R.K. Sachs, Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time, Proc. Roy. Soc. London A 270 (1962) 103.
G.T. Horowitz and M.J. Perry, Gravitational energy cannot become negative, Phys. Rev. Lett. 48 (1982) 371 [INSPIRE].
A. Ashtekar, Asymptotic Quantization, based on 1984 Naples Lectures, Bibliopolis, Napoli Italy (1987).
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].
C. Rovelli, Partial observables, Phys. Rev. D 65 (2002) 124013 [gr-qc/0110035] [INSPIRE].
B. Dittrich, Partial and complete observables for canonical general relativity, Class. Quant. Grav. 23 (2006) 6155 [gr-qc/0507106] [INSPIRE].
R.E. Peierls, The commutation laws of relativistic field theory, Proc. Roy. Soc. London A 214 (1952) 143.
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).
J. Lee and R.M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
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].
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].
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].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
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].
S. Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge University Press, Cambridge U.K.. (2003).
S. Carlip, Conformal field theory, (2 + 1)-dimensional gravity, and the BTZ black hole, Class. Quant. Grav. 22 (2005) R85 [gr-qc/0503022] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
G. Compère and A. Fiorucci, Asymptotically flat spacetimes with BMS3 symmetry, Class. Quant. Grav. 34 (2017) 204002 [arXiv:1705.06217] [INSPIRE].
W. Wieland, Conformal boundary conditions, loop gravity and the continuum, JHEP 10 (2018) 089 [arXiv:1804.08643] [INSPIRE].
J.C. Namburi and W. Wieland, Deformed Heisenberg charges in three-dimensional gravity, JHEP 03 (2020) 175 [arXiv:1912.09514] [INSPIRE].
W. Wieland, Twistor representation of Jackiw-Teitelboim gravity, Class. Quant. Grav. 37 (2020) 195008.
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).
E. Newman and R. Penrose, An approach to gravitational radiation by a method of spin coefficients, J. Math. Phys. 3 (1962) 566.
A. Ashtekar, New variables for classical and quantum gravity, Phys. Rev. Lett. 57 (1986) 2244 [INSPIRE].
R. Arnowitt, S. Deser and C.W. Misner, Republication of: the dynamics of general relativity, Gen. Rel. Grav. 40 (2008) 1997.
R. Arnowitt, S. Deser and C.W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. 116 (1959) 1322.
D. Harlow and J.-q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616].
A. Ashtekar, New Hamiltonian formulation of general relativity, Phys. Rev. D 36 (1987) 1587 [INSPIRE].
J.F. Barbero G., Real Ashtekar variables for Lorentzian signature space times, Phys. Rev. D 51 (1995) 5507 [gr-qc/9410014] [INSPIRE].
W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].
E. De Paoli and S. Speziale, A gauge-invariant symplectic potential for tetrad general relativity, JHEP 07 (2018) 040 [arXiv:1804.09685] [INSPIRE].
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].
F. Mercati, Shape dynamics: relativity and relationalism, Oxford University Press, Oxford, U.K. (2018).
H. Gomes and T. Koslowski, The link between general relativity and shape dynamics, Class. Quant. Grav. 29 (2012) 075009 [arXiv:1101.5974] [INSPIRE].
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].
E. Frodden, A. Ghosh and A. Perez, Quasilocal first law for black hole thermodynamics, Phys. Rev. D 87 (2013) 121503 [arXiv:1110.4055] [INSPIRE].
A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Liv. Rev. Rel. 7 (2004) [gr-qc/0407042].
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].
A. Ashtekar, C. Beetle and J. Lewandowski, Mechanics of rotating isolated horizons, Phys. Rev. D 64 (2001) 044016 [gr-qc/0103026] [INSPIRE].
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].
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].
D. Pranzetti and H. Sahlmann, Horizon entropy with loop quantum gravity methods, Phys. Lett. B 746 (2015) 209 [arXiv:1412.7435] [INSPIRE].
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].
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].
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].
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].
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].
V. Chandrasekaran and K. Prabhu, Symmetries, charges and conservation laws at causal diamonds in general relativity, JHEP 10 (2019) 229 [arXiv:1908.00017] [INSPIRE].
T. De Lorenzo and A. Perez, Light cone thermodynamics, Phys. Rev. D 97 (2018) 044052 [arXiv:1707.00479] [INSPIRE].
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.
F. Hopfmüller and L. Freidel, Gravity degrees of freedom on a null surface, Phys. Rev. D 95 (2017) 104006 [arXiv:1611.03096] [INSPIRE].
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].
J.N. Goldberg, D.C. Robinson, and C. Soteriou, Null hypersurfaces and new variables, Class. Quant. Grav. 9 (1992) 1309.
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].
A. Corichi, I. Rubalcava-GarcÃa and T. VukaÅ¡inac, Actions, topological terms and boundaries in first-order gravity: a review, Int. J. Mod. Phys. D 25 (2016) 1630011 [arXiv:1604.07764] [INSPIRE].
E. De Paoli and S. Speziale, Sachs’ free data in real connection variables, JHEP 11 (2017) 205 [arXiv:1707.00667] [INSPIRE].
E. Frodden and D. Hidalgo, Surface charges toolkit for gravity, Int. J. Mod. Phys. D 29 (2020) 2050040 [arXiv:1911.07264] [INSPIRE].
J.F. Barbero G., B. DÃaz, J. Margalef-Bentabol and E.J.S. Villaseñor, Concise symplectic formulation for tetrad gravity, Phys. Rev. D 103 (2021) 024051 [arXiv:2011.00661] [INSPIRE].
W. Wieland, Generating functional for gravitational null initial data, Class. Quant. Grav. 36 (2019) 235007 [arXiv:1905.06357] [INSPIRE].
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].
P.A. Höhn, A.R.H. Smith and M.P.E. Lock, The trinity of relational quantum dynamics, arXiv:1912.00033 [INSPIRE].
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].
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].
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.
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].
A. Fiorucci and R. Ruzziconi, Charge algebra in Al(A)dSn Spacetimes, arXiv:2011.02002 [INSPIRE].
G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
C. Troessaert, Hamiltonian surface charges using external sources, J. Math. Phys. 57 (2016) 053507 [arXiv:1509.09094] [INSPIRE].
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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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DOI: https://doi.org/10.1007/JHEP04(2021)095
Keywords
- Classical Theories of Gravity
- Models of Quantum Gravity