Abstract
If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.
| Original language | English |
|---|---|
| Pages (from-to) | 1485-1492 |
| Number of pages | 7 |
| Journal | Classical and Quantum Gravity |
| Volume | 23 |
| DOIs | |
| Publication status | Published - 2006 |
Keywords
- GENERAL-RELATIVITY
- COLLINEATIONS
Cite this
Some Remarks on Curvature Tensor Integrability in Spacetimes. / Hall, Graham Stanley.
In: Classical and Quantum Gravity, Vol. 23, 2006, p. 1485-1492.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Some Remarks on Curvature Tensor Integrability in Spacetimes
AU - Hall, Graham Stanley
PY - 2006
Y1 - 2006
N2 - If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.
AB - If a vector field is covariantly constant then the Ricci identity shows that it annihilates the curvature tensor and all its covariant derivatives. This paper attempts a 'best possible' converse to this result in the four-dimensional Lorentz case (the spacetime of general relativity theory) by supposing a vector field annihilates the curvature tensor and a certain number of its derivatives and showing how this leads to a covariantly constant vector field. A direct proof of these results, together with a brief description of how the proof can be obtained using holonomy theory, is presented. The two- and three-dimensional cases are also remarked upon.
KW - GENERAL-RELATIVITY
KW - COLLINEATIONS
U2 - 10.1088/0264-9381/23/5/003
DO - 10.1088/0264-9381/23/5/003
M3 - Article
VL - 23
SP - 1485
EP - 1492
JO - Classical and Quantum Gravity
JF - Classical and Quantum Gravity
SN - 0264-9381
ER -