Observing Christmas Lights in a Black Hole: What You See

[D.S.Beyer]

TL;DR Summary

If a string of blinking Christmas lights extends from the center of a black hole out to a large radius r. What do I see, if I am perpendicular to the line of lights, at radius r?

If a string of blinking Christmas lights extends from the center of a black hole out to a large radius r.
What do I see, if I am perpendicular to the line of lights, at radius r?

Experiment specifics

• 3 solar mass, non-charged, non spinning black hole.
• Observer is 1,000 Au from the center of the black hole. ( r = 1,000au )
• Lights are each individually battery powered. (to avoid signal complications)
• Lights blink in perfect unison, 1 blink per second. (as measured in an inertial frame).
• Lights are spaced 1 per meter (as measured in an inertial frame).
• Lights emit perfectly white light (as measured in an inertial frame).
• The ‘cable’ connecting each light to each other is indestructible and massless. (sigh)
• The lights are bright enough to be seen at greater than 10,000au. (The energy required for such brightness is not a factor in the experiment.)

I’d like to discuss the

1. apparent visual spacing of the lights as they approach the event horizon,
2. the apparent flashing timing,
3. the color of the flashes, and
4. the total length (proper distance) of the line of Christmas Lights, vs the observed length from my frame of reference

Thoughts?

 

[PeterDonis]

If a string of blinking Christmas lights extends from the center of a black hole out to a large radius r.
What do I see, if I am perpendicular to the line of lights, at radius r?

Your specification of the experiment leaves out the most important point: are the lights freely falling into the hole, or are they stationary (i.e., each light held at a fixed altitude by a rocket or some other such method)?

There are also some other issues with your formulation. The most important is that there is no such thing as "the center of a black hole" as a point in space. The "center" is a moment of time, inside the horizon, to the future of all events inside the horizon. So the inner endpoint of the string of lights can't be at the center, and you need to decide where it is--it must be somewhere outside the horizon. Where?

In addition:

• Lights blink in perfect unison, 1 blink per second. (as measured in an inertial frame)
• Lights are spaced 1 per meter (as measured in an inertial frame).
• Lights emit perfectly white light (as measured in an inertial frame).

First, the "as measured in an inertial frame" here must mean "as measured in a local inertial frame centered on the given light", since there are no global inertial frames in a curved spacetime. Is that what you meant?

Second, it is impossible for each light to blink once per second as measured in its local inertial frame, and for the lights to blink "in perfect unison", as in their blinks being globally synchronized. This is true whether the lights are free-falling into the hole or are stationary. So your stated scenario is impossible and cannot be analyzed. You need to make it consistent somehow: either drop the "in unison", or drop the "1 blink per second as measured in a local inertial frame". Which do you want to drop?

 

[PeroK]

I think that you should plan to celebrate Christmas some other way!

 

[D.S.Beyer]

Your specification of the experiment leaves out the most important point: are the lights freely falling into the hole, or are they stationary (i.e., each light held at a fixed altitude by a rocket or some other such method)?

There are also some other issues with your formulation. The most important is that there is no such thing as "the center of a black hole" as a point in space. The "center" is a moment of time, inside the horizon, to the future of all events inside the horizon. So the inner endpoint of the string of lights can't be at the center, and you need to decide where it is--it must be somewhere outside the horizon. Where?

In addition:

First, the "as measured in an inertial frame" here must mean "as measured in a local inertial frame centered on the given light", since there are no global inertial frames in a curved spacetime. Is that what you meant?

Second, it is impossible for each light to blink once per second as measured in its local inertial frame, and for the lights to blink "in perfect unison", as in their blinks being globally synchronized. This is true whether the lights are free-falling into the hole or are stationary. So your stated scenario is impossible and cannot be analyzed. You need to make it consistent somehow: either drop the "in unison", or drop the "1 blink per second as measured in a local inertial frame". Which do you want to drop?

Awesome. I'll try to nail these details down.
1. I would like the lights to be held at a fixed position. I attempted to describe this by including an indestructible massless cable, on which each light is affixed at 1 per meter. (I am also interested in the case of free-falling lights, but since I've started down this path let's try and flush it out)

2. The center of the black hole problem. This is tricky. I guess I sort of assumed that the event horizon is a sphere (in this case) and the center of that sphere is the center of the black hole. But that does beg the question of what coordinate system are we measure these things with.
...I have a terrible feeling that this point will spiral into it's own problem, so I guess let's just change the problem to have the lights end at "1 meter from the event horizon"

3. Light spacing, blinking ect. What I was hoping to conve here is that if you gathered up all the lights into your reference frame, they would measure 1 per meter, blink 1 per sec (within reason of your 'small lab'), and have perfectly white light. So, the all start synchronized in the reference frame of the observer, then are lowered into position. aaarg... but that is going to cause it's own problems, as during the lowering things will happen that affect the timing. Can we theoretically teleport the lights into position? After we have measured and synchronized them?

 

[D.S.Beyer]

I think that you should plan to celebrate Christmas some other way!

I am open to suggestions. Maybe we calculate how fast Santa needs to travel to get toys to everyone?

 

[PeterDonis]

1. I would like the lights to be held at a fixed position.

Ok.

I attempted to describe this by including an indestructible massless cable, on which each light is affixed at 1 per meter.

To be clear, I am assuming this means "1 per meter" as measured in the local inertial frames at each light's location along the cable.

I guess I sort of assumed that the event horizon is a sphere (in this case) and the center of that sphere is the center of the black hole.

And this is not correct. First, the horizon is not a single 2-sphere; it is a null 3-surface composed of an infinite family of 2-spheres connected along a null line. Second, none of those 2-spheres have a "center" at all in the ordinary sense. A black hole's event horizon is not just a 2-sphere in ordinary Euclidean 3-space. It is something quite different.

that does beg the question of what coordinate system are we measure these things with.

Everything I have said above or in my previous post is independent of any choice of coordinates. You simply have an incorrect understanding of the geometry involved. The best way to avoid having to deal with that here is to keep everything outside the horizon, as it looks like you have decided to do; see next comment.

let's just change the problem to have the lights end at "1 meter from the event horizon"

Yes, this is fine. Then we can just use standard Schwarzschild coordinates, since those are the easiest to work with as long as everything is outside the horizon.

if you gathered up all the lights into your reference frame

Which would mean moving them, which would mean whatever you learned would tell you nothing about what they were like before they were moved.

the all start synchronized in the reference frame of the observer, then are lowered into position. aaarg... but that is going to cause it's own problems, as during the lowering things will happen that affect the timing.

Exactly.

Can we theoretically teleport the lights into position? After we have measured and synchronized them?

No. The fundamental physical fact that you are struggling unsuccessfully to avoid here is that gravitational time dilation prevents the lights from blinking in unison if they each blink once per second according to their own local clocks. There is no way to avoid that.

 

[PeterDonis]

@D.S.Beyer it might help to take a step back here. What is your overall objective in analyzing this scenario? What are you trying to learn from it?

 

[D.S.Beyer]

@D.S.Beyer it might help to take a step back here. What is your overall objective in analyzing this scenario? What are you trying to learn from it?

@PeterDonis thanks for bearing with me on this.
I am essentially trying to setup a scenario where a visual ruler (of time ie blinking lights and space ie length) extends from a relativity low spacetime curvature (Minkowski) to a relatively high spacetime curvature (Schwarzschild black hole).

I am interested in the observed length contraction and time dilation of the ruler.
I'd like to learn more about on where the change in the ruler takes place, from an observers perspective, and hopefully mesh this with the terrible and abundant space/proper length (rubber sheet) diagrams.

An long term goal is to create new visualizations of gravity with animations.

 

[DaveC426913]

I'd like to learn more about on where the change in the ruler takes place...

Along its entire length from infinity down to the EH - to a greater or lesser degree.

 

[PeterDonis]

I am interested in the observed length contraction and time dilation of the ruler.

Lights closer to the horizon will appear to the observer to be blinking slower. That part is easy, since gravitational time dilation is all that is involved.

How the spacing between the lights will appear is more problematic. There are at least two issues involved:

(1) "Length contraction" is not really a good term for what is going on with respect to the spatial geometry. The spatial geometry around the black hole is non-Euclidean; it's not that the length between two adjacent lights "contracts" as you get closer to the horizon, it's that the non-Euclidean geometry simply has more room for lights with a constant 1-meter spacing than a Euclidean calculation based on the circumferences of circles that get gradually closer to the horizon would suggest.

(2) There is also the complication of light bending due to the hole's gravity, which makes the direction of a given light signal coming to your observer appear to be slightly further away from the hole than it would be if the hole weren't there. This effect gets larger as you get closer to the horizon, so the lights closer to the horizon will appear further from the horizon than they actually are, by a larger amount than lights further away from the horizon. The net effect is to make the lights appear more "squeezed together" to the observer, the closer to the horizon they are.

Perhaps the best way to convey these issues is to imagine a second scenario, where spacetime is flat, and we pick out a 2-sphere with the same surface area as the hole's event horizon, put it in the center, and string a cable of lights from 1 meter outside the 2-sphere all the way out to 1,000 AU, at the same orientation relative to the observer. In other words, what we are holding constant are the surface areas of two 2-spheres: the "horizon" 2-sphere, and the "1,000 AU" 2-sphere (the one the observer is located on).

Then what the observer will see in the two scenarios can be compared, qualitatively, as follows:

Flat spacetime: The number of lights will be exactly what you would calculate by dividing (1,000 AU minus the radius of a 2-sphere with surface area equal to the horizon sphere) by 1 meter. The lights will all be blinking in unison, and they will all appear equally spaced. (Technically, there will be a slight decrease in the apparent visual angle between two lights as you go further out from the "horizon" 2-sphere, but I think this can be neglected here.)

Black hole spacetime: There will be a larger total number of lights; they will not blink in unison (the ones closer in will blink slower); and they will not appear equally spaced (the lights closer to the horizon 2-sphere will appear squeezed together more than the lights further out). Also, the innermost light will appear slightly further out (again because of light bending).

 

[D.S.Beyer]

Lights closer to the horizon will appear to the observer to be blinking slower. That part is easy, since gravitational time dilation is all that is involved.

Awesome.

(1) "Length contraction" is not really a good term for what is going on with respect to the spatial geometry. The spatial geometry around the black hole is non-Euclidean; it's not that the length between two adjacent lights "contracts" as you get closer to the horizon, it's that the non-Euclidean geometry simply has more room for lights with a constant 1-meter spacing than a Euclidean calculation based on the circumferences of circles that get gradually closer to the horizon would suggest.

I am a little shaky on the difference between length contraction in SR vs length contraction in GR.
I am, most likely mistakingly, working under the assumption that the gravitational field in GR acts like the acceleration of an object in SR, to produce the length contracted effects in the direction of acceleration.
(...begins to open large can of worms...)

(2) There is also the complication of light bending due to the hole's gravity, which makes the direction of a given light signal coming to your observer appear to be slightly further away from the hole than it would be if the hole weren't there. This effect gets larger as you get closer to the horizon, so the lights closer to the horizon will appear further from the horizon than they actually are, by a larger amount than lights further away from the horizon. The net effect is to make the lights appear more "squeezed together" to the observer, the closer to the horizon they are.

Wow. This makes sense, but makes things wicked complicated. Are we talking about null-geodesics here?

Perhaps the best way to convey these issues is to imagine a second scenario, where spacetime is flat, and we pick out a 2-sphere with the same surface area as the hole's event horizon, put it in the center, and string a cable of lights from 1 meter outside the 2-sphere all the way out to 1,000 AU, at the same orientation relative to the observer. In other words, what we are holding constant are the surface areas of two 2-spheres: the "horizon" 2-sphere, and the "1,000 AU" 2-sphere (the one the observer is located on).

Then what the observer will see in the two scenarios can be compared, qualitatively, as follows:

Flat spacetime: The number of lights will be exactly what you would calculate by dividing (1,000 AU minus the radius of a 2-sphere with surface area equal to the horizon sphere) by 1 meter. The lights will all be blinking in unison, and they will all appear equally spaced. (Technically, there will be a slight decrease in the apparent visual angle between two lights as you go further out from the "horizon" 2-sphere, but I think this can be neglected here.)

Thank you for restating the experiment in solid terms.

Black hole spacetime: There will be a larger total number of lights; they will not blink in unison (the ones closer in will blink slower); and they will not appear equally spaced (the lights closer to the horizon 2-sphere will appear squeezed together more than the lights further out). Also, the innermost light will appear slightly further out (again because of light bending).

Perfect.
So if I were to try and show this in an animation... I'm getting the vibe that the majority of the lights as viewed from the observer at 1,000 au, look pretty 'normal'. But, zoomed in on, say, the last 1000 meters, we start to see visual effects of time dilation and space contraction that would appease the short attention spans of a normal person.

As DaveC426913 notes, the effects occur across the entire length of the light rope. But the effects occurrence is connected with some parabolic or exponential increase. If this was all SR I would guess that the effects were driven by the Lorentz Factor (Gamma).

Any links to equations that I can fiddle with in Wolfram would be fun at this point.

 

[PeterDonis]

I am, most likely mistakingly, working under the assumption that the gravitational field in GR acts like the acceleration of an object in SR

This is indeed mistaken. Acceleration is not what produces length contraction in SR; length contraction in SR is just an effect of looking at the same object from the standpoint of two different inertial frames.

As I said before, "length contraction" is not a good term for what is happening in the curved spacetime in this scenario. You have illustrated why, by showing that using such a term leads to a mistaken attempt to find some kind of analogy with SR that just isn't there.

Are we talking about null-geodesics here?

Yes.

I'm getting the vibe that the majority of the lights as viewed from the observer at 1,000 au, look pretty 'normal'. But, zoomed in on, say, the last 1000 meters, we start to see visual effects of time dilation and space contraction that would appease the short attention spans of a normal person.

It would be more than just the last 1000 meters (since a 3 solar mass black hole has a Schwarzschild radius of about 9 km or 9000 meters, and visual effects will be significant out to some significant multiple of the Schwarzschild radius), but yes, the portion in which there would be significant effects would be only a small fraction of the whole 1,000 AU, the part closest to the hole.

As @DaveC426913 notes, the effects occur across the entire length of the light rope.

But, as he said, "to a greater or lesser degree". Technically, the effects are not zero even at 1,000 AU for a 3 solar mass black hole. But they are so tiny as to be way below our ability to detect with our current technology. At what point the effects become negligible as you move out from the inner end of the string of lights depends on how accurate you are assuming the measurements are.

the effects occurrence is connected with some parabolic or exponential increase.

No, it's neither of these.

If this was all SR I would guess that the effects were driven by the Lorentz Factor (Gamma).

But it isn't all SR, and it's best not to even try to find analogies with SR (as I noted above).

Any links to equations

The equation for gravitational time dilation is simple, but it's the only one relevant to this problem that is. The tick rate of a clock at Schwarzschild radial coordinate $r$, as compared with that of a clock at infinity (which, for practical purposes, means "the clock at 1,000 AU") is $\sqrt{1 - 2 G M / c^2 r}$, where $G$ is Newton's gravitational constant, $M$ is the mass of the hole, and $c$ is the speed of light.

Note that $r$ is not the same as "distance from the horizon"; the Schwarzschild radial coordinate labels 2-spheres by their surface area, i.e., the radial coordinate $r$ labels a 2-sphere with surface area $4 \pi r^2$. So, for example, "1 meter above the horizon" does not correspond to an $r$ of the Schwarzschild radius plus 1 meter. The relationship between $r$ and actual radial distance, close to the horizon, is a more complicated equation:

$$
d = \sqrt{r \left( r - \frac{2 G M}{c^2} \right)} + \frac{2 G M}{c^2} \ln \left( \sqrt{\frac{c^2 r}{2 G M}} + \sqrt{\frac{c^2 r}{2 G M} -1} \right)
$$

 

[Nugatory]

I am a little shaky on the difference between length contraction in SR vs length contraction in GR.

If you are going to move from SR to GR, an often effective first step is to try to forget that you ever heard the terms "length contraction" and "time dilation" - these SR descriptions of what's going on are not wrong but they are applicable only to particular special cases (that's why it's called "special" relativity) and will get in the way once you move past these special cases. Instead, work on understanding the relativity of simultaneity (we have many good threads here) and use that as a starting point for understanding how the phenomena that SR calls length contraction and time dilation can appear in the special case of observing a moving clock and a moving ruler. That's not sufficient to explain the GR effects, but it is sufficient to bring out the subtleties involved in discussing distant clocks and rulers.

 

[PAllen]

Just a comment on "how can you have a sphere without a center?". A simple analogy is an example of a circle without a center. Consider that the 'universe' (really manifold) is a cylinder. A circle around the axis (as viewed with the cylinder embedded in 3-space) has no center within the 'universe'. On the other hand, many small circles embedded in the cylinder do have centers within the cylinder. Quite analogous to this, the geometry and topology of a black hole region has this same feature regarding 2-spheres, rather than circles. Some have centers, some don't. The horizon 2-spheres lack a center in precisely the sense of the 'big circles' of a cylinder.