The Reduction O(3,1)![[superset or implies]](http://scitation.aip.org/stockgif3/sup.gif)
O(2,1)O(1,1)
We set up a representation of the principal series of the group SL(2, C) in terms of the sequence of noncompact subgroups SL(2,C)SU(1,1)O(1,1). The basis functions are just the cross-basis matrix elem...
O(2,1)O(1,1)We set up a representation of the principal series of the group SL(2, C) in terms of the sequence of noncompact subgroups SL(2,C)SU(1,1)O(1,1). The basis functions are just the cross-basis matrix elem...
Diffraction Radiation by a Line Charge Moving past a Comb: A Model of Radiation Losses in an Electron Ring Accelerator
A calculation is given of the radiated-energy loss from a charged rod which moves at constant speed past an infinite set of parallel semi-infinite conducting plates of infinitesimal thickness, with th...
A calculation is given of the radiated-energy loss from a charged rod which moves at constant speed past an infinite set of parallel semi-infinite conducting plates of infinitesimal thickness, with th...
The Einstein Tensor and Its Generalizations
J. Math. Phys. 12, 498 (1971); doi:10.1063/1.1665613
Issue Date: March 1971
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The Einstein tensor Gij is symmetric, divergence free, and a concomitant of the metric tensor gab together with its first two derivatives. In this paper all tensors of valency two with these properties are displayed explicitly. The number of independent tensors of this type depends crucially on the dimension of the space, and, in the four dimensional case, the only tensors with these properties are the metric and the Einstein tensors.
©1971 The American Institute of Physics
| History: | Received 27 August 1970 |
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http://link.aip.org/link/?JMAPAQ/12/498/1 |
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REFERENCES (17)
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- Unless otherwise specified, Latin indices run from 1 to n.
- A comma denotes partial differentiation.
- The summation convention is used throughout. The vertical bar denotes covariant differentiation.
- E. Cartan, J. Math. Pure Appl. 1, 141 (1922).
- H. Weyl, Space-Time-Matter (Dover, New York, 1922), 4th ed., pp. 315ff;
- H. Vermeil, Nachr. Ges. Wiss. Göttingen, 334 (1917).
- If Xi is any contravariant vector field, then we define the Riemann curvature tensor R
, the Ricci tensor Rhj, the curvature scalar R, and the Einstein tensor Gij by
and
, respectively.
- D. Lovelock,
Aequationes Math. 4, 127 (1970) . - Reference 7, Theorem 4.
- Reference 7, Theorem 3.
- Reference 7, Corollary 1.
- For various applications of (2.6), see D. Lovelock,
Atti Accad. Nazi. Lincei 42, 187 (1967) ; -
Proc. Cambridge Phil. Soc. 68, 345 (1970) . - g = det gij. Without loss of generality, we may assume g>0.
- Reference 7, Theorem 5.
- This scalar has arisen elsewhere in an entirely different context. H. Rund, “Curvature Invariants Associated with Sets of n Fundamental Forms of Hypersurfaces of n-Dimensional Riemannian Manifolds” [to appear in Tensor (1971)].
- D. Lovelock, “Intrinsic Expressions for Curvatures of Even Order of Hypersurfaces in a Euclidean Space” [to appear in Tensor (1971)].
- H. Rund,
Abhandl. Math. Sem. Univ. Hamburg 29, 243 (1966) . - The relationship of (3.6) to Lagrangians which satisfy the Euler-Lagrange equations identically has been investigated by R. Pavelle (private communication).
- D. Lovelock,
Arch. Rati. Mech. Anal. 33, 54 (1969) .
