Cat Chasing Red Dot Faster Than c: Results & Analysis

[Halc]

TL;DR Summary

Exploration of how something moving faster than light speed would appear.

This topic assumes special relativity only, so no gravity is involved. I wanted to know how a superluminal thing would appear to an observer, and two of the 'things' I thought of were a moiré pattern and the red dot projected by a laser pointer. The former can be discussed, but this post will focus on the dot.

Suppose we have a spherical projection screen of radius 3 million km. Our laser pointer is at the center, and it projects a dot somewhere on the screen, which gets there 10 seconds after being emitted. Furthermore, if you look close at the dot, it is encoded with a 32 bit integer that takes a minute to loop. The dot can be made to move at arbitrarily high speeds around the screen. There is a cat that cat move very fast, up to 0.9c. The cat is the observer, and is named Felinestein.

First thing to note is that relative to the inertial frame of the sphere, the dot is in exactly one location at any time and the 'clock' it projects runs at normal rate. Relative to different inertial frames, this is not the case, and the dot might appear in multiple places at once, and some of them might count faster or slower or even backwards.

First scenario is a stationary cat. Felinestein is nearsighted up to only 100 km and cannot see the dot move all the way across the sky, so observation is local and very close to being flat. Let's say we put the laser at the end of a second hand and have it go around once a minute so the dot moves at just over c. Felinestein, who is right at the screen where the dot passes, will not see the dot coming just like you don't hear a supersonic jet coming. But he notices it immediately upon it passing, when it coincidentally projects time t0. Which way does it go? Turns out Felinestein sees two dots, both receding. Looking 'upstream', the dot appears to move away very quickly (over 21 c), is blue shifted, and counts backwards to ~228000 very quickly. Looking in the other direction the dot moves away at just over 0.5c, is red shifted, and counts upward to about 228000 at about half speed.
If the laser rotation rate is increased to say 10 seconds, the dot will be moving at 6.3c and the stationary cat will see the upstream dot move away slower. The downstream dot will appear to move away at around .85c and both will count slower to +/- ~38000.

The redshift I am basing on the fact that the numbers observed count faster or slower than the normal rate (of a stationary dot). In the upstream case, the dot appear to be receding, but also counting backwards, so the double negative results in a blue shift, not a red shift. That's the first unintuitive thing, that an approaching superluminal object would still be blueshifted despite appearing to recede from the observer.

I don't thing there will by any equivalent of a sonic boom. Felinestein is right at the screen and there's no particular bright flash as the dot moves by, but I suspect that if he's off to the side by say a km, there would be one, and the points at which the dot first appears and subsequently splits are not directly opposite him.

What about the moiré pattern? Is there any redshift at all to such a phenomenon? I'd think not since it is effectively a pattern of holes in a screen with a stationary (everywhere uniform) light source behind. There's no moving light to shift.

I'll have Felinestein start to chase the dot in a subsequent post. This seems to have been enough for an intro so far.

 

[anuttarasammyak]

I am afraid DOTS on the screen are not physical substance to which maximum speed c or red and blue shifts are applicable.

Say wide sea waves come to land and splash along the coast at the same time. If we prepare long boards and set it along the coast but with an angle. The splash dots move on the board and we can make it as fast as we like, much more than c, by decreasing the angle. Nothing extraordinary take place there. Phase velocity would be applicable to speed of dots.

 

[Halc]

I am afraid DOTS on the screen are not physical substance to which maximum speed c or red and blue shifts are applicable. Phase velocity would be applicable to speed of dots.

Well that's why I picked it. I needed something that could move faster than c.
It very much does redshift since the classic beam of light is physical and the path it takes from emitter to cat gets shorter over time while approaching, and longer over time after it has passed.
I can do the same effect (superluminal shift) using a stationary laser and multiple mirrors all moving at sub-lightspeed, thus reducing the path length from emitter to detector at a rate far in excess of c.

Phase velocity is applicable to the moiré pattern, but not the one dot, which isn't a wave.

 

[sysprog]

I can do the same effect (superluminal shift) using a stationary laser and multiple mirrors all moving at sub-lightspeed, thus reducing the path length from emitter to detector at a rate far in excess of c.

? Would you please clarify the underlined part of the claim?

 

[jbriggs444]

? Would you please clarify the underlined part of the claim?

Suppose, for instance, that we have a stationary laser shining at a stationary mirror that is 100 light seconds away. A 200 second round trip.

At 90 light seconds away, there is another mirror with is slowly moving across the line of sight so that less than ten seconds later, it will occlude laser resulting in a 180 second round trip.

At 80 light seconds away, a third mirror does the same thing and so on until a mirror slides in place immediately in front of the laser.

Within less than 100 elapsed seconds the path length has reduced from 200 seconds to 0 seconds.

 

[Vanadium 50]

Well that's why I picked it. I needed something that could move faster than c.

But a dot on a screen is not a "thing". "Something moving faster than c" is a subset of "thing", so it's not what you think it is.

 

[Halc]

Would you please clarify the underlined part of the claim?

You're right. I didn't do the arithmetic and couldn't change the pathlength faster than c using the mirror method. My simple scenario was a co-located emitter/detector set up with a stationary local mirror and a remote one incoming, with the signal bouncing back and forth multiple times before hitting the local detector. In a scenario with the remote mirror incoming at half lightspeed and two round trips for the light, the path decreased at a rate of 8/9 c. Faster mirrors and more reflections will approach but not exceed c.

Now the question is, can similar logic be used for the red dot? The path there (distance between emitter at center and the cat, by way of red dot) does very much decrease and increase at arbitrarily high rates.

Something moving faster than c" is a subset of "thing", so it's not what you think it is.

I'm not making any metaphysical assertions about if the dot qualifies as a thing or not. The dot has a location event at time t, and a different location event at time t+1 (locations/times in frame of sphere) and those events are separated in a space-like manner.
The focus on the discussion was what the cat would see.

Within less than 100 elapsed seconds the path length has reduced from 200 seconds to 0 seconds.

Yes, but only by sawing off bits of the beam that never get to return. No redshift would be observed by this scenario despite it being a sort of discreet version of what the dot is doing. If the laser had numbers, some of the numbers would never be seen by the observer in your scenario.

 

[Ibix]

It very much does redshift since the classic beam of light is physical and the path it takes from emitter to cat gets shorter over time while approaching, and longer over time after it has passed.

I think that depends on your model of reflection. Are you getting coherent reflection from mirrors carefully angled to reflect light to your cat, or are you getting absorption and diffuse re-emission? In the latter case you wouldn't get Doppler effects in the wavelength (although you would still in the timestamps) because there is not necessarily a well-defined phase relationship between light coming from even nearby points. I presume you intended the former, but you didn't say.

 

[Halc]

Are you getting coherent reflection from mirrors carefully angled to reflect light to your cat, or are you getting absorption and diffuse re-emission? In the latter case you wouldn't get Doppler effects in the wavelength (although you would still in the timestamps) because there is not necessarily a well-defined phase relationship between light coming from even nearby points.

I had envisioned something like a cinema screen, or for that matter, a painted wall upon which I typically shine my laser cat toy. So that seems to be diffuse re-emission, and you answered one of the questions I had which is if red/blue shift would actually be observed or not. I mean, laser light is in phase, but not necessarily the light re-emitting from the dot.
I don't see how the carefully aligned mirrors would work since they cannot reflect light tangential to the screen since they'd hit the next mirror. OK, it's curved, but that curvature (100 km disk taken out of a 3m km sphere) is functionally a flat surface, done deliberately so I didn't have to do the trig involved when the curvature departed from flat.

Also, would the carefully aligned tiny mirrors give any redshift even if they didn't get in each other's way? It then seems to reduce to the scenario of @jbriggs444 with the mirrors a lot closer together, which produces no redshift due to the mirrors not moving.

OK then, it seems that our cat isn't going to observer any redshift except as manifested in the rate (positive or negative) that the projected numbers are seen. That was after all the purpose of doing that.

 

[Dale]

I agree, I don’t see how any redshift would arise.

 

[Ibix]

I agree, I don’t see how any redshift would arise.

I see Doppler arising in the case of reflection (rather than re-emission) because successive waves have different path lengths from emission to reception. If the cat is on (or near enough) the shell then there is a point where the beam reaches him having traveled a distance $r$ (the radius of the sphere) but the previous wave crest traveled $r$ to its reflection event and then some extra distance to the cat.

 

[jartsa]

I see Doppler arising in the case of reflection (rather than re-emission) because successive waves have different path lengths from emission to reception. If the cat is on (or near enough) the shell then there is a point where the beam reaches him having traveled a distance $r$ (the radius of the sphere) but the previous wave crest traveled $r$ to its reflection event and then some extra distance to the cat.

So a wave-crest becomes longer, like in Doppler-shift. But the energy of said crest must stay unchanged, unlike in Doppler-shift, because otherwise energy would be lost.

 

[Halc]

Felinestein is nearsighted up to only 100 km and cannot see the dot move all the way across the sky

Let's remove that restriction and consider what the (still stationary) cat sees from other vantage points. He is at the edge and suddenly sees two new dots appear 'here'. The one dot goes forward and the other backwards. Neither can just disappear, so they have to meet. That means that Felinestein must always see an odd number of dots. So when the new part of red dots appear 'here', there's also the old one working its way around the circle each minute. The dot that appears to move backwards goes out to meet it, at which point they both disappear.
If the dots move fast enough, there will be more than three dots with no upper bound.

An observer at the center always sees one dot. In the case of the one-minute circuit, the dot is 120 degrees behind where the laser pointer currently points.

For our edge-located cat, there was no particular 'sonic boom' of light, a particularly bright light that is an artifact of emissions from the dot at multiple locations arriving at once at the observer. But now if we place the cat to the side, we get that effect. If the cat moves too close to the center, it will never see more than one dot. The maximum flash (the light equivalent of the sonic boom) seems to occur right at the point where he'd see multiple dots if he were any closer. This location is closer to the middle the faster the light goes, and as best I can work out is simply
$$d =\frac{r(s - 1)}{s}$$ where d is the distance from edge, s is linear speed >= 1 of the red dot, as a fraction of c, and r is the radius of the screen. Any closer to the edge than that and multiple dots will be seen as it comes around. The pairs of new dots will appear upwind in a direction perpendicular to the line between the cat and the center of the circle.

 

[Halc]

This is a thread in the relativity section, so the question also becomes: What happens to a faster-than-light object in other frames. In the frame of the laser pointer, the red dot is always in one location only, exactly 10 seconds after being emitted by the pointer, regardless of the rate that the laser pointer changes direction. Not so in other inertial frames. Note that there's no observer this time. We're talking about how many dots there are at a given time.

This is a pretty simple concept. The worldline of the red dot (rotating in a plane) is a 3D helix (ignoring the z direction in which it is constantly zero) like a slinky. Like a real slinky in Euclidean space, if you stretch it out enough, tilting it will change its shape but its shadow will not intersect itself. If it is barely extended, even a slight tilt will cause overlaps in its shadow. The Euclidean analogy isn't perfect, but the concept is there.

So if the dot rotates around in the x-y plane, then relative to an inertial where the pointer is moving in the +x direction, the max and min speed of the red dot occurs at the y=max/min positions. So on the max-speed side, consider two nearby events on the dot worldline. In the frame of the laser pointer, all nearby events on the superluminal dot worldline have space-like separation. So consider the inertial frame where those two events are simultaneous, which can be done with any pair of events with spacelike separation. In this frame and in any frame where the pointer moves faster there must be a discontinuity in the worldline of the dot: There will be at some point three dots, with one moving backwards and counting backwards. The faster the laser pointer spins, the less speed is needed for a frame where this occurs.

I do want to do one more post directly relevant to the title. Felinestein needs to chase the dot. So far he's always been just sitting there, very cat like. I encourage others to post their thoughts on what it would look like to the cat when it chases a dot moving at just over light speed. He can't catch it of course. What if he chases the phantom dot that moves backwards?