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edwiddy
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Hi, I'm a physics undergrad working through Carroll at the moment. In the section on the Kerr black hole, he states that [itex]K= \partial_t[/itex] is a Killing vector because the coefficients of the metric are independent of [itex]t[/itex]. He then states in eq. 6.83 that [itex]K^\mu [/itex] is normalized by:
[tex]K^\mu K_\mu = - \frac{1}{\rho^2} (\Delta - a^2 \sin^2{\theta})[/tex]
where [itex] \Delta = r^2 - 2GMr + a^2[/itex] and [itex] \rho^2 = r^2 + a^2 \cos^2{\theta} [/itex].
Now I can't seem to for the life of me duplicate this from the metric. We take [itex]K^\mu = (\partial_t)^\mu = \delta ^\mu_t[/itex] right? Then:
[tex]K^\mu K_\mu = g^{\mu\nu}K_\nu K_\mu[/tex]
which is only non zero for [itex]\mu=\nu=t[/itex]...but that doesn't match up. The crossterms in the metric need to come into play, but it seems that if anyone of the indices is [itex]\phi[/itex] then it goes to zero...
Thanks.
[tex]K^\mu K_\mu = - \frac{1}{\rho^2} (\Delta - a^2 \sin^2{\theta})[/tex]
where [itex] \Delta = r^2 - 2GMr + a^2[/itex] and [itex] \rho^2 = r^2 + a^2 \cos^2{\theta} [/itex].
Now I can't seem to for the life of me duplicate this from the metric. We take [itex]K^\mu = (\partial_t)^\mu = \delta ^\mu_t[/itex] right? Then:
[tex]K^\mu K_\mu = g^{\mu\nu}K_\nu K_\mu[/tex]
which is only non zero for [itex]\mu=\nu=t[/itex]...but that doesn't match up. The crossterms in the metric need to come into play, but it seems that if anyone of the indices is [itex]\phi[/itex] then it goes to zero...
Thanks.
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