Can light be reflected in a black hole?

[happyparticle]

TL;DR

Can light be reflected in a black hole to reach an observer's eyes?

Hi,
I have a question about the behavior of light inside a black hole (inside the event horizon).

Is it true that light coming from outside the black hole can be reflected once reaching a mirror inside the black hole?

For instance, if an observer inside the black hole hold a mirror would he see something, knowing that even the light can't move outward?

Is it only a question of frame? The observer is in a free falling frame. If so, why exactly we say that light can't move outward.

 

[Dale]

Assume that the black hole is supermassive and that you are inside the event horizon but far enough out that tidal forces are negligible. Then a mirror will work normally for you. You can look in the mirror and shave or apply makeup as you are used to.

If so, why exactly we say that light can't move outward.

Because it can’t. Light cannot move outward, but you are moving inward too. It can move inward more slowly than you.

 

[Orodruin]

It should also be noted that ”outwards” does not have a well-defined meaning inside the black hole. If we assume standard Schwarzschild coordinates, increasing r for a non-spacelike pathe inside the black hole would be moving in a direction backwards in time.

 

[Dale]

”outwards” does not have a well-defined meaning inside the black hole

Is that correct? You can still foliate the spacetime in a set of nested spheres. And any direction in which those spheres get larger is outward. Those directions are all timelike, but well defined.

Or perhaps you are meaning to define the word “direction” to only only refer to spacelike directions. If so then I think we are not disagreeing, just using different meanings.

 

[PeroK]

Is that correct? You can still foliate the spacetime in a set of nested spheres. And any direction in which those spheres get larger is outward. Those directions are all timelike, but well defined.

Or perhaps you are meaning to define the word “direction” to only only refer to spacelike directions. If so then I think we are not disagreeing, just using different meanings.

I'm convinced that by "outward" the OP meant in the direction of increasing $r$. See this thread for some background reading:

https://www.physicsforums.com/threads/volume-of-a-black-hole-using-the-schwarzschild-metric.1078772/

 

[Ibix]

Or perhaps you are meaning to define the word “direction” to only only refer to spacelike directions. If so then I think we are not disagreeing, just using different meanings.

That would be my reading of what @Orodruin wrote. But then the question is what you meant by "[light] can move inward more slowly than you".

I think the correct way of stating it is that there's no invariant sense of "moving inward" or "moving outward", so there's no problem with light moving in any direction. Fundamentally, light can't escape because the horizon is an outgoing null surface so cannot be reached from the inside, not because there's a spatial direction it can't move.

 

[Ibix]

I'm convinced that by "outward" the OP meant in the direction of increasing $r$. See this thread for some background reading:

https://www.physicsforums.com/threads/volume-of-a-black-hole-using-the-schwarzschild-metric.1078772/

I'd forgotten this was that poster. Let's see if he's still reading viXra or if he's wised up.

 

[Orodruin]

Is that correct? You can still foliate the spacetime in a set of nested spheres. And any direction in which those spheres get larger is outward. Those directions are all timelike, but well defined.

Or perhaps you are meaning to define the word “direction” to only only refer to spacelike directions. If so then I think we are not disagreeing, just using different meanings.

My point is that you cannot "move outwards". That would require a future-directed time-like curve going towards larger r values.

 

[PeterDonis]

You can still foliate the spacetime in a set of nested spheres. And any direction in which those spheres get larger is outward.

Yes.

Those directions are all timelike.

No. There are spacelike and null directions inside the horizon that are outward by your definition (an equivalent definition would be that the areal radius $r$ increases with increasing affine parameter along the curve) as well. The null directions will be past-directed (like the outward timelike ones), but they're there.

One way to see this is to look at a Kruskal diagram that has the hyperbolas of constant $r$ drawn on it. Inside the horizon, as the diagram makes clear, those hyperbolas all have spacelike tangent vectors everywhere, which means that it is always possible to find both spacelike and null vectors that point outward as well as timelike ones.

 

[happyparticle]

Because it can’t. Light cannot move outward, but you are moving inward too. It can move inward more slowly than you.

I'm not sure to understand how an observer can move faster than light?

Also, if the light from behind reach the mirror this means that the light is faster than the observer.

you are inside the event horizon but far enough out that tidal forces are negligible

But there are still tidal forces. To reach the eyes of an observer, the light should move in the opposite direction, no?

If we assume standard Schwarzschild coordinates, increasing r for a non-spacelike pathe inside the black hole would be moving in a direction backwards in time.

Doesn't that mean the same thing? We can't "move" in past nor we can't move "outward".

 

[Ibix]

I'm not sure to understand how an observer can move faster than light?

Here's a sketch of a Kruskal diagram with two infalling mirrors marked in red. They start exchanging light pulses (yellow) when the second one crosses the horizon. Notice that they have no trouble exchanging light pulses, just limited time to do so.

 

[Dale]

My point is that you cannot "move outwards". That would require a future-directed time-like curve going towards larger r values.

I agree with that. Outwards is meaningful and both you and light cannot go that direction because it is the past.

The light doesn’t go outwards and mirrors function for the same reason that light doesn’t go to the past and mirrors function.

 

[PeterDonis]

Notice that they have no trouble exchanging light pulses, just limited time to do so.

To add one point that might help to clarify the fact that the light pulses are "moving inward", even though they (at least in one direction) are moving inward slower than the infalling observer: if you draw curves of constant areal radius $r$ on the Kruskal diagram, you will see that all of the light pulses--in both directions--always go in the direction of decreasing $r$ (i.e., the direction of constant $r$ along the hyperbolas is always more horizontal than the worldlines of the light pulses), even though, relative to the observer, they are going in opposite directions. In other words, "decreasing $r$" is a global description, not a local one. It tells you how things are moving relative to the global spacetime geometry, not how things are moving locally relative to each other--which is what's important for mirrors to work.

 

[Dale]

I'm not sure to understand how an observer can move faster than light?

As @Ibix said, that probably wasn’t a good description. However, it is important to remember that in non-inertial coordinates the restriction is not to go slower than light, but rather to have a timelike worldline. In non-inertial coordinates you can easily have massive objects go faster than light while still satisfying the timelike worldline criterion.

But there are still tidal forces. To reach the eyes of an observer, the light should move in the opposite direction, no?

Tidal forces are a needless complication.

When you look in your bathroom mirror, it works even though the light doesn’t go to the past. Inside the horizon outwards is in the past. Light doesn’t have to go outwards for a mirror to work for the same reason that light doesn’t have to go to the past for a mirror to work

 

[Ibix]

In non-inertial coordinates you can easily have massive objects go faster than light while still satisfying the timelike worldline criterion.

Point of clarification: they can move faster than $c$. They will never overtake a light pulse.

 

[PeterDonis]

In non-inertial coordinates you can easily have massive objects go faster than light while still satisfying the timelike worldline criterion.

Note that in this context, "faster than light" means "a coordinate speed greater than $c$". It does not mean "faster than an actual light beam going in the same direction". The coordinate speed of light itself in non-inertial coordinates can also be greater than $c$.

 

[Dale]

Point of clarification: they can move faster than $c$. They will never overtake a light pulse.

Correct. They can move faster than $c$. They can even have a coordinate speed that is greater than the coordinate speed of a pulse of light. But they can never overtake a light pulse.

 

[PeterDonis]

They can even have a coordinate speed that is greater than the coordinate speed of light.

Can you clarify the circumstances under which this can occur?

 

[Dale]

Can you clarify the circumstances under which this can occur?

Sure.

Consider an inertial frame and a rotating frame that shares the same spatial origin. Consider an object at rest in the inertial frame at a distance $r\omega=0.6 c$.

In the rotating frame the object has a speed of $0.6c$. The prograde light has a speed of $1.6c$. The retrograde light has a speed of $0.4c$. So the object with mass has a higher speed than the speed of the retrograde light.

 

[Orodruin]

Can you clarify the circumstances under which this can occur?

Define affine coordinates where light is moving at infinite coordinate speed in one direction and c/2 in the opposite. An observer moving in the direction where the light's coordinate speed is infinite can have any coordinate speed smaller than infinity, which is larger than the coordinate speed of light moving in the opposite direction.

 

[PeterDonis]

In the rotating frame the object has a speed of $0.6c$.

And this speed is retrograde, correct?

The prograde light has a speed of $1.6c$. The retrograde light has a speed of $0.4c$. So the object with mass has a higher speed than the speed of the retrograde light.

I think you have this backwards. The object's motion in this frame is retrograde, so its coordinate speed cannot be higher than the coordinate speed of retrograde light. If it were, the object would be moving faster than light in the same direction, which is not possible.

Viewed from the rotating frame, prograde light is moving in the same direction as a timelike object at rest in that frame, so its coordinate speed in that frame will be $c$ minus the frame's speed of $0.6 c$. Retrograde light is moving in the opposite direction from a timelike object at rest in the rotating frame, so its coordinate speed in that frame will be $c$ plus the frame's speed of $0.6c$.

 

[Dale]

I think you have this backwards.

Oops. Right. I got my labels mixed up.

its coordinate speed in that frame will be c minus the frame's speed of 0.6c.

The absolute value of that, yes.

 

[happyparticle]

Thank you all for your help.