Abstract
A Hamiltonian framework for (2+1)-dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically flat context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual field theories, the Hamiltonian is bounded from above. This is a genuinely nonperturbative result. In the presence of a spacelike Killing field, (3+1)-dimensional vacuum general relativity is equivalent to (2+1)-dimensional general relativity coupled to certain matter fields. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a spacelike symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, spacelike Killing fields. In this case, our expression is related to the "c energy" in a nonpolynomial fashion. While in the weak field limit the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time translation in the full theory, and therefore represents the physical energy in the gravitational field.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 4944-4956 |
| Number of pages | 13 |
| Journal | Physical Review D |
| Volume | 50 |
| Issue number | 8 |
| DOIs | |
| State | Published - Jan 1 1994 |
Fingerprint
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics
- Physics and Astronomy (miscellaneous)
Cite this
}
Striking property of the gravitational Hamiltonian. / Ashtekar, Abhay; Varadarajan, Madhavan.
In: Physical Review D, Vol. 50, No. 8, 01.01.1994, p. 4944-4956.Research output: Contribution to journal › Article
TY - JOUR
T1 - Striking property of the gravitational Hamiltonian
AU - Ashtekar, Abhay
AU - Varadarajan, Madhavan
PY - 1994/1/1
Y1 - 1994/1/1
N2 - A Hamiltonian framework for (2+1)-dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically flat context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual field theories, the Hamiltonian is bounded from above. This is a genuinely nonperturbative result. In the presence of a spacelike Killing field, (3+1)-dimensional vacuum general relativity is equivalent to (2+1)-dimensional general relativity coupled to certain matter fields. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a spacelike symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, spacelike Killing fields. In this case, our expression is related to the "c energy" in a nonpolynomial fashion. While in the weak field limit the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time translation in the full theory, and therefore represents the physical energy in the gravitational field.
AB - A Hamiltonian framework for (2+1)-dimensional gravity coupled with matter (satisfying positive energy conditions) is considered in the asymptotically flat context. It is shown that the total energy of the system is non-negative, vanishing if and only if space-time is (globally) Minkowskian. Furthermore, contrary to one's experience with usual field theories, the Hamiltonian is bounded from above. This is a genuinely nonperturbative result. In the presence of a spacelike Killing field, (3+1)-dimensional vacuum general relativity is equivalent to (2+1)-dimensional general relativity coupled to certain matter fields. Therefore, our expression provides, in particular, a formula for energy per-unit length (along the symmetry direction) of gravitational waves with a spacelike symmetry in 3+1 dimensions. A special case is that of cylindrical waves which have two hypersurface orthogonal, spacelike Killing fields. In this case, our expression is related to the "c energy" in a nonpolynomial fashion. While in the weak field limit the two agree, in the strong field regime they differ significantly. By construction, our expression yields the generator of the time translation in the full theory, and therefore represents the physical energy in the gravitational field.
UR - http://www.scopus.com/inward/record.url?scp=0000716603&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=0000716603&partnerID=8YFLogxK
U2 - 10.1103/PhysRevD.50.4944
DO - 10.1103/PhysRevD.50.4944
M3 - Article
AN - SCOPUS:0000716603
VL - 50
SP - 4944
EP - 4956
JO - Physical Review D
JF - Physical Review D
SN - 0556-2821
IS - 8
ER -