TY - JOUR
T1 - Gravitational dynamics—a novel shift in the hamiltonian paradigm
AU - Ashtekar, Abhay
AU - Varadarajan, Madhavan
N1 - Funding Information:
The United States NSF grants PHY-1505411 and PHYS-1806356.
Funding Information:
Acknowledgments: This work was supported by the NSF grants PHY-1505411 and PHY-1806356 and the Eberly Chair funds of Penn State. M.V. would like to thank Fernando Barbero and Eduardo Villaseñor for discussions in the initial stage of this work. A.A. is grateful to Alok Laddha for a number of discussions, both on LQG and the double copy program, and for his comments on the manuscript.
Publisher Copyright:
© 2021 by the authors. Licensee MDPI, Basel, Switzerland.
PY - 2021/1
Y1 - 2021/1
N2 - It is well known that Einstein’s equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that time evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in spatial directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomor-phism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the ‘integrable’ (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the ‘double copy’ ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.
AB - It is well known that Einstein’s equations assume a simple polynomial form in the Hamiltonian framework based on a Yang-Mills phase space. We re-examine the gravitational dynamics in this framework and show that time evolution of the gravitational field can be re-expressed as (a gauge covariant generalization of) the Lie derivative along a novel shift vector field in spatial directions. Thus, the canonical transformation generated by the Hamiltonian constraint acquires a geometrical interpretation on the Yang-Mills phase space, similar to that generated by the diffeomor-phism constraint. In classical general relativity this geometrical interpretation significantly simplifies calculations and also illuminates the relation between dynamics in the ‘integrable’ (anti)self-dual sector and in the full theory. For quantum gravity, it provides a point of departure to complete the Dirac quantization program for general relativity in a more satisfactory fashion. This gauge theory perspective may also be helpful in extending the ‘double copy’ ideas relating the Einstein and Yang-Mills dynamics to a non-perturbative regime. Finally, the notion of generalized, gauge covariant Lie derivative may also be of interest to the mathematical physics community as it hints at some potentially rich structures that have not been explored.
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U2 - 10.3390/universe7010013
DO - 10.3390/universe7010013
M3 - Article
AN - SCOPUS:85102067006
VL - 7
JO - Universe
JF - Universe
SN - 2218-1997
IS - 1
M1 - 13
ER -