I am trying to understand the definition of stationary axisymmetric spacetimes and I am confused!(adsbygoogle = window.adsbygoogle || []).push({});

Following Wald's definition in his book "General Relativity" a spacetime is calledstationary axisymmeticwhen it exists a timilike Killing field [itex] \xi^\alpha [/itex] and a spacelike Killing field [itex] \psi^\alpha [/itex] whose. In this case we can choose a coordinate system [itex] (x^0=t,x^1=\phi,x^2,x^3) [/itex] where the Killing fields areintegral curves are closed

[tex]\xi^\alpha= \frac{\partial}{\partial\,t}, & \psi^\alpha=\frac{\partial}{\partial\,\phi} [/tex]

Calculating now the integral curves of [itex] \psi^\alpha [/itex] we have

[tex] \frac{d\,t}{d\,\lambda}=0, \quad \frac{d\,\phi}{d\,\lambda}=1, \quad \frac{d\,x^2}{d\,\lambda}=0, \quad \frac{d\,x^3}{d\,\lambda}=0 \Rightarrow[/tex]

[tex] t(\lambda)=c_t, \quad \phi(\lambda)=\lambda+c_\phi, \quad x^2(\lambda)=c_2, \quad x^3(\lambda)=c_3 [/tex]

and here is the problem. The above curve is not closed for any interval [itex] \lambda \in (\alpha,\beta) [/itex].

The only answer I can think is that someone must make somekind of identification, say

[tex] \phi(\lambda)=\lambda+c_\phi, \quad \lambda \in [0,2\,\pi) [/tex]

[tex] \phi(\lambda+2\,\pi)=\phi(\lambda), \quad \lambda \geq 2\,\pi [/tex]

in order to produce a closed curve.

Is this the correct answer?

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# Stationary axisymmetric spacetime

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