- #1

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## Main Question or Discussion Point

Lets consider some kind of metrics:

\begin{equation}

ds^2 = dt^2 - \frac{dr^2}{1-\frac{2M}{r}} - r^2(d\theta^2 + \sin^2\theta d\phi^2).

\end{equation}

here ##r = l/2\pi## is the radial coordinte like in Schwarzschild metrics.

As far as I know, this metrics is describe the whormhole.

Lets put ##\theta=\pi/2##.

First, I can find the velocities:

\begin{equation}

\frac{dr}{ds} = \sqrt{1- \frac{2M}{r}} \left[ (E^2 - 1) - \frac{L^2}{r^2}\right].

\end{equation}

whrere the conserved quantities are ##E = \frac{dt}{ds} = \frac{1}{\sqrt{1-v^2/c^2}} = \gamma## and ##L = r^2\frac{d\phi}{ds} ##.

Also, the velocity of any particle measured by stationary observer ##v = \frac{proper\, distance = \frac{dr}{\sqrt{1-\frac{2M}{r}}}}{proper\, time = dt}##is constant:

\begin{equation}

\frac{dl}{dt} = \sqrt{\frac{E^2 - 1}{E^2}} = v.

\end{equation}

But, If I have not read a literature, how do I know is it a wormhole metric? What properties should lead me to the conclusion that it's a wormhole?

\begin{equation}

ds^2 = dt^2 - \frac{dr^2}{1-\frac{2M}{r}} - r^2(d\theta^2 + \sin^2\theta d\phi^2).

\end{equation}

here ##r = l/2\pi## is the radial coordinte like in Schwarzschild metrics.

As far as I know, this metrics is describe the whormhole.

Lets put ##\theta=\pi/2##.

First, I can find the velocities:

\begin{equation}

\frac{dr}{ds} = \sqrt{1- \frac{2M}{r}} \left[ (E^2 - 1) - \frac{L^2}{r^2}\right].

\end{equation}

whrere the conserved quantities are ##E = \frac{dt}{ds} = \frac{1}{\sqrt{1-v^2/c^2}} = \gamma## and ##L = r^2\frac{d\phi}{ds} ##.

Also, the velocity of any particle measured by stationary observer ##v = \frac{proper\, distance = \frac{dr}{\sqrt{1-\frac{2M}{r}}}}{proper\, time = dt}##is constant:

\begin{equation}

\frac{dl}{dt} = \sqrt{\frac{E^2 - 1}{E^2}} = v.

\end{equation}

But, If I have not read a literature, how do I know is it a wormhole metric? What properties should lead me to the conclusion that it's a wormhole?

Last edited: