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Lanczos spintensor for the Gödel metric. (English) Zbl 0933.83027

From the text: A Lanczos potential in Gödel geometry is obtained.
C. Lanczos [Rev. Mod. Phys. 34, 379-389 (1962; Zbl 0106.41701)] showed the existence of a potential tensor \(K_{abc}\) with the symmetries \[ K_{ijr}= -K_{jir}, \qquad K_{ijr}+ K_{jri}+ K_{rij}= 0, \] which generates the Weyl tensor via the expression \[ \begin{aligned} C_{abcd}= &K_{abc;d}- K_{abd;c}+ K_{cda;b}- K_{cdb;a}+ \tfrac 12 [g_{ad} (K_{bc}+ K_{cb})- g_{ac}(K_{bd}+ K_{db})\\ &+ g_{bc} (K_{ad}+ K_{da})- g_{bd} (K_{ac}+ K_{ca})]+ \tfrac 23 (g_{ac} g_{bd}- g_{ad} g-{bc}) K^{pq}_{p;q}, \end{aligned} \] where \(K_{ij}= K_i{}^p{}_{j;p}- K_i{}^p{}_{p;j}\).
The Gödel spacetime is \[ ds^2= -(dx^1)^2- 2\exp(x^4) dx^1 dx^2- \tfrac 12 \exp(2x^4) (dx^2)^2+ (dx^3)^2+ (dx^4)^2 \] and this metric accepts a Lanczos potential via \[ K_{ijr}= A_{ri;j}- A_{rj;i}, \qquad A_{ab}= A_{ba}, \] with \(A_{ij}= -\frac 19 R_{ij}\).

MSC:

83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory

Citations:

Zbl 0106.41701
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References:

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