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 -