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What is the meaning of the change $dt^2\to-dt^2$ and $dr^2\to-dr^2$ in the Schwarzschild metric, leading to:

$$g=-c^{2}d\tau^{2}=(1-\frac{2GM}{c^{2}r})c^{2}dt^{2}-(1-\frac{2GM}{c^{2}r})^{-1}dr^{2}+r^{2}d\omega^{2}$$

Is gravity attractive or repulsive? Is there an event horizon?

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    $\begingroup$ Your metric has inconsistent signs. $\endgroup$
    – Ghoster
    Commented Jan 27 at 23:16
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    $\begingroup$ It means that your $t$ coordinate now measures distance in space while you $r$ coordinate measures time. This metric is similar to the one inside the normal black hole horizon describing a rapidly shrinking space shaped as a $3D$ hypersurface of a $4D$ hypercylinder of the spherical type (spherinder). To understand the properties of this spacetime you should solve the geodesic equations. Essentially you have flipped a black hole inside out by this transformation +1. $\endgroup$
    – safesphere
    Commented Jan 28 at 0:39

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