Static spacetimes can be defined as having no [itex]g_{tx}[/itex] component of the metric.(adsbygoogle = window.adsbygoogle || []).push({});

Alternatively we can say that they are foliated by a bunch of spacelike hypersurfaces to which the Killing vector field [itex]\frac{\partial}{\partial t}[/itex] is orthogonal.

How are these two statements consistent?

[itex]g_{tx}=0 \Rightarrow g(\frac{\partial}{\partial t}, \frac{\partial}{\partial x})=0[/itex] but I always thought this meant there was no distance between timelike and spacelike vectors rather than a statement about them being orthogonal?

Can someone please clear this up for me.

Thanks.

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# Static + Hypersurface Orthogonality

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