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When we set the interval(ds) for a given coordinate(i.e. Schwarzschild coordinates) to 0 to calculate for the velocity of light in a gravitational field, we can arrive at an equation such as the one John Rennie finds.

The final result is:

$$ \frac{\mathrm{d}r}{\mathrm{d}t} = v_c = \left(1-\frac{r_s}{r}\right) $$

(c=1)

But when I compare this with that of the Gullstrand-Painleve metric, we find there are two roots:

$$ \frac{\mathrm{d}r}{\mathrm{d}t_r} = v_c = ± 1 - \sqrt{\frac{r_s}{r}} $$

The plus and minus 1 is dependent on the direction that the light is traveling.

If we suppose that the Schwarzschild metric should have one more solution/root, I get the following graph: enter image description here

Since 2 is the event horizon, we can see light going in either direction slows down to a complete stop at the horizon.

While if I graph the Gullstrand-Painleve solution, I arrive at the following: enter image description here

We can see that the direction light is traveling affects the velocity seen by the coordinate, which has been described as the river model. If the photon is traveling away from the singularity, it will sit at the event horizon until the singularity gains more mass. But if it is moving the opposite direction, it will "flow with the river" and continue at 2c through the event horizon.

So my question summed up is are these in agreement? Or is my second root for the Schwarzschild solution incorrect?

Having a tough time getting my head around what seems to be a discrepancy between the two. My main point of interest is right before the event horizon where we don't have maximal time dilation, yet.

Thank you for your help.

user2299067
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1 Answers1

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Yes, your two results are in agreement.

It seems unintuitive that in one case the ingoing and outgoing velocities have the same magnitude while in the other case they don't, but this is simply because the time coordinate is different in the two cases.

It's important to realise that the Gullstrand-Painlevé velocity $dr/dt_r$ is not a velocity that any observer could measure, because the $r$ and $t_r$ coordinates are taken from the coordinates of two different observers. It is a purely abstract velocity and we have to be very careful about attaching a physical significance to it. All observers making a local measurement of the speed of light, i.e. a measurement of the speed of light at their location, will find the velocities are always $\pm c$. The difference in the ingoing and outgoing velocity is purely an artefact of the coordinates.

This is invariably glossed over in the proofs that light cannot escape from an event horizon. We tend to say oh look, $dr/dt_r=0$ so the light can't escape and leave it at that. I have to confess I did this as well in my treatment of the problem.

John Rennie
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