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I've been trying to express the Komar mass formula in component notation for a general static metric.

I'm finding that the expression for

[tex]

\nabla_c \xi_d

[/tex]

is reasonably simple, where [itex]\xi^{\mu}[/itex] is a timelike Killing vector, but the formula calls for

[tex]

\epsilon_{abcd} \nabla^c \xi^d

[/tex]

and this is very messy.

(We have to integrate the above two-form over some surface to get the mass and multiply by an appropriate constant).

Is it kosher to re-write the formula for the Komar mass as

[tex]

-\frac{1}{8 \pi} \int_S \epsilon^{abcd} \nabla_c \xi_d

[/tex]

and to do so, would I be expressing the surface to be integrated by one-forms rather than vectors?

I'm finding that the expression for

[tex]

\nabla_c \xi_d

[/tex]

is reasonably simple, where [itex]\xi^{\mu}[/itex] is a timelike Killing vector, but the formula calls for

[tex]

\epsilon_{abcd} \nabla^c \xi^d

[/tex]

and this is very messy.

(We have to integrate the above two-form over some surface to get the mass and multiply by an appropriate constant).

Is it kosher to re-write the formula for the Komar mass as

[tex]

-\frac{1}{8 \pi} \int_S \epsilon^{abcd} \nabla_c \xi_d

[/tex]

and to do so, would I be expressing the surface to be integrated by one-forms rather than vectors?

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