# Do photons slow down as they approach the event horizon of a black hole?

So, if a black hole is so dense, that light cannot escape, what happens to the photons as the approach and go past the event horizon (or some other horizon, about which I am ignorant)? Do photons slow down at some point, or is it somehow binary, where at one moment they are moving at the speed of light and the next they aren't moving at all? Or, do they continue to move at the speed of light, just never moving beyond the black hole?

• And you measure speed according to which observer? One close by? Or a distant one? Always, ALWAYS specify the observer when asking such questions about relativity. Often, if you do specify the observer, the answer becomes obvious. See the reply by Rob Jeffries below. May 5, 2019 at 22:44

Photons travel on null geodesics through space time. In the curved space-time near a non-spinning black hole, a distant observer could infer that the speed of light (travelling on a radial path) is given by $$v_{\rm light} = c \left(1 - \frac{r_s}{r}\right),$$ where $$r_s$$ is the Schwarzschild radius.

As $$r \rightarrow r_s$$ then we see that $$v_{\rm light} \rightarrow 0$$. So yes, according to an observer far from a black hole, the speed of light approaching a black hole appears to slow down and asymptotically approaches zero at the event horizon.

However, the locally measured speed of light is always $$c$$.

• I belive a distant observer would find that the speed of light radially inwards/outwards from the black hole is exactly as you right but that the speed of light in a plane transverse to the radial direction is $v_{light}=c\sqrt{1-r_s/r}$. May 4, 2019 at 12:57
• "So yes, according to an observer far from a black hole, the speed of light approaching a black hole appears to slow down" -> so the speed of light in a vacuum in an inertial frame of reference is not a constant but depends on where the light we are measuring is located at? May 11, 2019 at 17:36
• @Pathfinder The speed of light is $c$ when measured locally in an inertial frame of reference. In GR, non-local measurements may return other values. That is why, for example, light is bent in non-uniform gravitational fields. May 11, 2019 at 18:09

The last option

do they continue to move at the speed of light, just never moving beyond the black hole?

is the most nearly correct. But it is essential to realise that in GR questions like "how fast is something moving" cannot be answered until you know where the observer is and how they are moving. There simply is no universal answer.

One way to think of it is that space-time is constantly rushing into the black hole, dragged in by its gravity.

• I like that "waterfall" model, but it applies to everything, not just black holes. Eg, spacetime is falling towards the centre of the Earth, and the ground has to accelerate us upwards at $9.81 m/s^2$ just to keep us in place. May 2, 2019 at 21:56
• @PM2Ring It would be pretty weird and awesome if some version of quantum gravity showed that spacetime emerging from a web of quantum entanglement or something actually does flow into the source of gravity. Like, there was an actual flow of whatever relationships underpin that which we call spacetime. Anyway, showerthoughts. May 5, 2019 at 22:42
• @Florin I said the other day on Physics, spacetime isn't a kind of stuff. IMHO, too often people get led astray by the rubber sheet model, or aether-like conceptions. OTOH, you're correct that quantum gravity may give us a spacetime with some kind of structure, but even if it does, trying to imagine it with our clunky macroscopic intuitions will probably be misleading. ;) May 6, 2019 at 3:14

I believe it just appears that way, the photons racing away reach a point of equilibrium where their speed getting away from the black hole matches the speed of the space following into it.

Since the light speed limit only applies to the speed which everything inside the universe can move within it. It doesn’t apply to the universe or space itself. That can move much quicker than the speed of light. At some it all flows back into the singularity faster than anything can move away from it.

You could still get away from it if you could somehow go faster than light but since you can’t you’re screwed along with the photons.