Why doesn't the vertical light beam get out of a black hole?

[PeterDonis]

I beg to differ about the claim that it's impossible to have a static object.

I didn't say it was impossible to have a static object. I said that spacetime inside a black hole's horizon is not static. That is a straightforward conclusion from the metric of said spacetime, and is not in any way controversial. (In fact, the black hole interior is not even stationary, which is a stronger condition than not being static. See below.)

please note that "the curvature of spacetime" relates to the tidal force

The curvature of spacetime is tidal gravity (not "force"--tidal gravity is not a force). They are the same thing, physically.

and the second derivative of potential

Only in a stationary region of spacetime, where a "potential" can be defined.

the force of gravity relates to the first derivative of potential

Gravity is not a force in GR, and the "acceleration due to gravity"--which is better expressed as the proper acceleration required to remain at the same spatial location in a stationary spacetime--is what relates to the first derivative of the potential, which, again, can only be defined in a stationary spacetime.

Since the region at and inside the horizon of a black hole is not stationary, none of the above even makes sense there.

gravitational potential can be related to the coordinate speed of light

Only in a stationary region of spacetime, where "potential" has meaning. And even there, the coordinate speed of light has no physical meaning; it's not something anyone can actually measure. But that's irrelevant for this discussion, because...

Hence light emitted at the horizon stays at the horizon forever. It doesn't go up, stop, and then fall back.

No, this is not the reason light stays at the horizon--because the horizon, and the region inside it, is not stationary, so none of the concepts above are meaningful there.

As for matter falling into the hole, I suspect it doesn't even make it as far as the event horizon, but that's one for another day.

This is wrong too, as many, many previous threads here on PF have discussed. If you continue to post these incorrect claims, you will receive a warning.

Here's what Don Koks said

I like the Usenet Physics FAQ, and it's often a good source, but that doesn't mean it's always right, or that its authors always are. The coordinate speed of light, which is what "speeds up" as a light ray moves upward in the gravitational potential field in a stationary spacetime, and "slows down" as the light moves downward, has no physical meaning. Don Koks knows this, because he adds the qualification that nobody actually measures it; anyone actually measuring, locally, the speed of light will find it to be $c$. He attributes this to the measuring device "speeding up" or "slowing down" exactly in sync with the light itself; but that is attributing a physical meaning to the coordinate speed of light that it simply doesn't have.

This is a good illustration of why, when push comes to shove, we don't use pop science articles, even good ones like the Usenet Physics FAQ, as references here on PF; we only use peer-reviewed scientific papers or textbooks, and even then we only use those in which the physics is set out rigorously in math, not heuristically in ordinary language. Ordinary language is simply too imprecise to rely on if you really want to understand the physics.

 

[PeterDonis]

Here's what Don Koks said

To expand on this issue a bit more, I'll bring up something which I'm surprised you didn't bring up, since it is a better argument for your case than what you quoted. This is from the updated article in the Usenet Physics FAQ by Koks that you linked to:

"That the speed of light depends on position when measured by a non-inertial observer is a fact routinely used by laser gyroscopes that form the core of some inertial navigation systems. These gyroscopes send light around a closed loop, and if the loop rotates, an observer riding on the loop will measure light to travel more slowly when it traverses the loop in one direction than when it traverses the loop in the opposite direction."

What Koks is referring to here is the Sagnac Effect, and it is indeed real. However, the quote above subtly misdescribes what is actually observed. What is actually observed is that, if light beams are sent around a rotating ring in opposite directions, the counter-rotating beam will arrive back at the source in a shorter time than the co-rotating beam, and the difference in times is related to the angular velocity of rotation.

In other words, what is directly measured is not that "light travels more slowly" in the co-rotating beam than in the counter-rotating beam. If light speed detectors were placed at various points in the ring, to measure the speed of the light beams passing them, they would measure the beams to be moving at $c$. What is directly measure is the difference in travel times of the two beams; and that cannot be due to a change in the speed of light, because we can measure that to be the same locally. Instead, it is due to the way the geometry of spacetime works. A heuristic way of describing this is to say that, relative to an inertial frame, the light source is moving towards the counter-rotating beam and away from the co-rotating beam--in other words, it's because the rotation of the ring breaks the symmetry between the two different directions in space around the ring.

Since the difference in travel times is an invariant, it must still be present if we switch to a non-inertial frame in which the ring is at rest. In this frame, the coordinate speed of light is indeed different in the two directions; but once again, that coordinate speed has no physical meaning, as we can show by putting light speed detectors at various points around the ring and measuring the speeds of both beams to be $c$. So the difference in travel times has to be due to something else--once again, to a breaking of the symmetry between the two opposite spatial directions around the ring.

How else could we test this? Here's one way: put a sensor on the ring that detects the frequency of incoming light from some distant object at a fixed position (relative to an inertial frame). If the ring is not rotating, it can be placed so that the sensor continuously detects that light, and reports a constant frequency, the same as the (known) frequency of emission. If we now start the ring rotating, the sensor will, first of all, only detect the light intermittently, once per rotation; and each time it detects the light, it will measure it to have a different frequency--either blueshifted or redshifted, depending on the direction of the incoming light beam relative to the ring. The frequency shift will be directly related to the angular velocity of rotation of the ring, and will give a direct measurement of the breaking of the symmetry of space in the frame in which the ring is at rest. We could make this even stronger by having two incoming beams, coming in from opposite directions, and measuring that one beam's frequency shift is exactly equal in magnitude and opposite in direction to the other beam's frequency shift when the ring is rotating.

The point of all this is that, instead of blindly saying that "light travels faster or slower in a gravitational potential" based on one measurement, we should be looking at all the measurements we can make, and coming up with a single scheme of description that covers all of them. That's how scientific theories work. And "the speed of light changes" is not a workable single scheme that accounts for all of the experiments. The geometry of spacetime, with the speed of light always locally measured to be $c$, is a workable single scheme. That's why we prefer it here on PF.

 

[Max Hofbauer]

Nothing shot vertically upwards has a chance to escape a g-field... in the long run.
To illustrate this we need another Gedanken Experiment:

In a universe holding only the Gedanken-Planet, but no other celestial
bodies, you would never be able to shoot so far away that gravity could
be overridden by dark energy...

In the case of a black hole, it seems perfectly possible for a beam of light
to escape, only to fall back later. In order for us to see that beam, it would
have to reach all the way to our retinas, but we are too far away.

Maybe anyone, bound to be gobbled up by a black hole, can witness a bright
dot in the very center of it.

 

[PeterDonis]

In a universe holding only the Gedanken-Planet, but no other celestial
bodies, you would never be able to shoot so far away that gravity could
be overridden by dark energy...

For an object shot upward to escape the Gedanken-Planet, it doesn't need to have the Gedanken-Planet's gravity overridden by dark energy. It just needs to be shot upward fast enough that the Gedanken-Planet's gravity will never slow it down to zero speed and make it fall back again.

In the case of a black hole, it seems perfectly possible for a beam of light
to escape, only to fall back later

It may seem that way to you, but it's not correct. Light emitted radially outward at a black hole's horizon stays at the horizon forever; it doesn't move upward and then fall back. (Light emitted radially outward inside a black hole's horizon falls inward continuously.)

In order for us to see that beam, it would
have to reach all the way to our retinas, but we are too far away.

Incorrect. You would not be able to see light from a black hole's horizon even if you were "hovering" just a very small distance above the horizon. The only way to see a black hole's horizon or what's inside it is to fall in, in which case you yourself can never escape back out.

Maybe anyone, bound to be gobbled up by a black hole, can witness a bright
dot in the very center of it.

Even if you fall inside the hole, you will never see what's at the "center", because there is no "center" in the usual sense. The singularity at $r = 0$ is in your future if you fall in; it's not a place in space, it's a moment of time. The only way to reach it is to move into the future, which you can't help doing; and once you reach it, you cease to exist.

 

[Grinkle]

once you reach it, you cease to exist

Or at least we can say we don't know / have no model for what happens "after time ends" (sic) for particles. Playing on words here, hope I don't come off as just trying to sound clever, but I think the logic is sound - if everything at the singularity does not exist, then the singularity is either a consequential phenomena somehow made of nothing or it also does not exist.

I appreciate the insight that reaching the singularity is really describing a particular tick of the clock more than a particular increment forward in distance.

 

[PeterDonis]

Or at least we can say we don't know / have no model for what happens "after time ends" (sic) for particles.

More precisely, we don't know that the classical model I described, which says that things that reach the singularity cease to exist, is correct; in fact most physicists think it isn't. But we don't have a good model to replace it; we don't have a good theory of quantum gravity (yet), and that's what we would need to know what sort of model would replace the classical model in situations where the classical model predicts a singularity.

if everything at the singularity does not exist

That's not what the classical model says. It says that things that reach the singularity cease to exist. Everything up until that point exists just fine.

As I said above, most physicists believe the classical model is not correct in this situation, but that's not really because of the "cease to exist" part. It's because, even before anything reaches the singularity, according to the classical model, it will encounter spacetime curvatures that are so large that we expect quantum gravity effects to become important and the classical theory to break down. And, as I said above, we don't know what those effects are.

 

[greswd]

Farsight is that you?