Relativistic effects on apparent angular size

[nutgeb]

Oh my, when I have DrGreg riled up I know its time to reconsider!

I figured out my mistake. In the earth-moon rocket scenario, the Earth observer and the rocket observer agree on the simultaneity of the event when the rocket passes the moon; however, these two observers disagree about how far the moon is from the earth, and the time at which the moon-passing occurs! If the rocket's velocity is ~ 8.65 c, which is relativistic $$\gamma = 2$$, the rocket observer perceives the earth-moon distance to be half of what the Earth observer perceives. Because of this effect, the Lorentz contraction of the distance to the rocket (in the Earth frame) and to the Earth (in the rocket's frame) does not affect the simultaneity of the moon-passing event.

Let's switch back to my disk and detector scenario. I have attached a JPEG showing the scenario first from the stationary detector's frame, and then from the moving disk's (Disk M's) frame. These are not Minkowski diagrams, they are simple plots of the x and time axes at 3 separate snapshots in time. The moving disk's velocity away from the detector is again at $$\gamma = 2$$. At t = t' = 0, the moving disk is exactly passing the detector's location at the origin x = x' = 0.

A stationary ruler extends from the detector in the positive-x direction. Three stationary Disks A, B and C are positioned at the x = 5, 10 and 20 marks along the detector's ruler. The moving disk drags another long ruler behind it. The detector perceives the moving disk's ruler to be Lorentz contracted by half. Symmetrically, the moving disk perceives the detector's ruler to also be Lorentz contracted by half. (The contraction of the rulers can be seen from the fact that the markings on them are half as far apart).

The detector and the moving disk observer agree on the simultaneity of the events when the moving disk passes Disks A, B and C sequentially. However, they disagree on how far each of the 3 Disks is from the detector, and they also disagree on how much time elapses before each event occurs. Relative to the detector's perception, the moving disk observer perceives himself to pass Disk C after half the elapsed time and at half the distance.

This diagram indicates that when the moving disk passes Disk C, the detector sees the moving disk to have an apparent angular size corresponding a distance of 20. However, at the same event, the moving disk sees the detector to have an apparent angular size corresponding to a distance of 10, that is, twice the angular size that the detector simultaneously perceives the moving disk to have.

This suggests a conclusion that an observer does not perceive any increase in the apparent angular size of a radially moving object as compared to other objects which are stationary in that observer's frame, despite the fact that the distance to the moving object is Lorentz contracted in the observer's frame.

P.S.: Note that the time measured by the moving disk observer is proper time because he uses his own wristwatch to make all time measurements as he passes each object. The distances measured by the detector to Disks A, B and C are proper distances because they are measured while the detector is at rest compared to those 3 disks. All other time and distance measurements in this scenario are not 'proper' measurements.

 

[sylas]

despite the fact that the distance to the moving object is Lorentz contracted in the observer's frame.

The distance to a disk is not Lorentz contracted by motion of what you observe. A moving disk and a stationary disk at the same location are at the same distance.

(I misread your diagrams at first sight, and have deleted an incorrect comment I made at first sight.)

Added in edit. In your disagram, disk M is moving with gamma = 2, and so v = 0.866 c

This does not appear on your distance axis, or else you are using very odd distance units.

Cheers -- sylas

 

[A.T.]

Of course a moving object and a stationary object that are at the same location are at the same distance in the stationary observer's frame.

Good

That does not mean that the intervening distance traveled by the moving object was not Lorentz contracted.

Depends in which frame:

- In the observer's frame the moving object travels the same distance, as is measured between observer and the stationary object with a stationary ruler.

- In the moving object's frame the distance between the observer and stationary object is contracted.

You are obviously referring to the later while in your first sentence you were talking about the first.

The Lorentz contraction is demonstrated by the contraction of the ruler dragging behind the moving object all the way to the observer.

The observer doesn't measure distances in his frame with moving rulers.

I will rely on Taylor and Wheeler's textbook for my interpretation that distances traveled are Lorentz contracted.

In the traveler's frame. Not in the observer's frame where start and end point are at rest.

If you wish to push a different interpretation, please reference a reliable source.

Your source is great. But your ability to misunderstand things even greater.

 

[nutgeb]

The stationary detector measurers (by its own stationary tape measure) that the moving disk has moved 20 units of distance by the time it passes stationary Disk C.

The stationary detector can visually confirm that the 40 unit mark on the tape measure dragged behind the moving disk passes by the detector just as the moving disk passes Disk C.

The detector knows that, before it was set in motion, the tape measure dragged behind the moving disk showed the same separation between markings as the detector's own tape measure. Therefore it is reasonable for the stationary detector to conclude that the distance traveled by the moving disk has been Lorentz contracted in the detector's own frame.

The fact that distances traveled by a moving object are Lorentz contracted in the stationary origin's rest frame is frequently mentioned in the context of the Milne cosmology model. See, e.g. Matt McIrvin's Milne Cosmology http://world.std.com/~mmcirvin/milne.html" :

"Well, an effect of special relativity is that an object that moves relative to some reference frame has a reduced length, as calculated in that reference frame; it becomes squashed in the direction of motion. This is called the Lorentz contraction. The length converges to zero as the speed approaches the speed of light. Distances between moving objects are similarly contracted."

He also references Ned Wright's http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH"which shows distances to receding comovers to be Lorentz contracted in the Milne model, in the origin frame. Ned says: "We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c * t_o / 2." In other words, the maximum distance to the most distant galaxy (in the origin frame) is the maximum light travel distance, at a velocity of c, divided by 2. I have attached another diagram which shows how this is indeed the maximum distance that will be measured in the origin frame to any comoving galaxy at any time. My chart shows that the maximum distance any object can move away from the origin in 1 year is .5 lightyear, and the distance maximizes at a velocity ~ .7c.

 

[A.T.]

He also references Ned Wright's http://www.astro.ucla.edu/~wright/cosmo_02.htm#DH"which shows distances to receding comovers to be Lorentz contracted in the Milne model, in the origin frame. Ned says: "We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c * t_o / 2." In other words, the maximum distance to the most distant galaxy (in the origin frame) is the maximum light travel distance, at a velocity of c, divided by 2.

That has nothing to do with Lorentz contraction of distances. It is a consequence signal delay due to finite light speed. Lorentz contraction in not an effect of signal delay. The formula above has nothing to do with Lorentz contraction, because the contraction factor for a galaxy receding the almost c is nearly 0, and not 0.5.

 

[nutgeb]

OK, I give up. Travel distance is not Lorentz contracted in the stationary origin frame. Therefore radial distances from the origin are not contracted in the Milne Cosmology. Matt McIrvin's page is wrong about that.

The correct reason why the origin observer in a Milne model sees an inhomogeneous distribution of comoving particles is because the Milne methodology necessarily (and rather artificially) packs an infinite number of comoving particles into the velocity ranges asymptotically closest to c. It does this so that the distances and velocities of all comoving particles will be related to each other by Lorentz transformations.

Milne is useless to model a single uniform kinematic/ballistic Big Bang explosion of an initial central object made up of a finite number of particles. Milne models a highly non-uniform explosion of an initial central object made up of an infinite number of particles.

 

[sylas]

OK, I give up. Travel distance is not Lorentz contracted in the stationary origin frame. Therefore radial distances from the origin are not contracted in the Milne Cosmology. Matt McIrvin's page is wrong about that.

No... you continue to misunderstand what you are reading. THAT is the problem.

Distances ARE contracted in precisely the way Matt describes. Not in the way you have described.

Cheers -- sylas

 

[nutgeb]

Distances ARE contracted in precisely the way Matt describes.

Matt has one sentence on the subject: "Distances between moving objects are similarly contracted." There is nothing precise about Matt's statement, so your reference to it is nonsense.

Here is Matt's sentence in full context:

"Well, an effect of special relativity is that an object that moves relative to some reference frame has a reduced length, as calculated in that reference frame; it becomes squashed in the direction of motion. This is called the Lorentz contraction. The length converges to zero as the speed approaches the speed of light. Distances between moving objects are similarly contracted. So there could be a whole infinity of galaxies in that finite expanding bubble, packed in cleverly! They'd get closer and closer together, approaching infinite density, in the limit of galaxies near the surface. We can always pack in more galaxies by stuffing faster-moving and therefore flatter ones in the space remaining to the surface of the bubble."

Sylas, please explain CLEARLY in your own words how you understand the distances between comoving particles to be Lorentz contracted in the Milne model, in the frame of the stationary origin.

Since the comoving particles in the Milne model can be of arbitrary size, including being infinitesimally small, there is no significance at all to the Lorentz contraction of the individual individual particles themselves ("becoming squashed", in Matt's precise terminology). The only relevant consideration here is the contraction of distances between particles.

 

[sylas]

Matt has one sentence on the subject: "Distances between moving objects are similarly contracted." There is nothing precise about Matt's statement, so your reference to it is nonsense.

Matt has a whole page, and I read the whole page before posting. I stand by my comment without hesitation; it is not nonsense. What Matt means is clear, in the context of the page, and Matt is correct.

Here is Matt's sentence in full context:

"Well, an effect of special relativity is that an object that moves relative to some reference frame has a reduced length, as calculated in that reference frame; it becomes squashed in the direction of motion. This is called the Lorentz contraction. The length converges to zero as the speed approaches the speed of light. Distances between moving objects are similarly contracted. So there could be a whole infinity of galaxies in that finite expanding bubble, packed in cleverly! They'd get closer and closer together, approaching infinite density, in the limit of galaxies near the surface. We can always pack in more galaxies by stuffing faster-moving and therefore flatter ones in the space remaining to the surface of the bubble."

Sylas, please explain CLEARLY in your own words how you understand the distances between comoving particles to be Lorentz contracted in the Milne model, in the frame of the stationary origin.

Nutgeb, I have no confidence in my ability to explain this in words you will understand clearly. Matt's words seem perfectly clear to me.

Matt does NOT say that the distance from us to a remote galaxy is "Lorentz contracted". But that is what you were saying about the disks. I have no idea what it will take for you to see this difference, or to understand that there's no implication that the distance to a remote object as WE measure it is "Lorentz contracted" by the velocity of the object being observed. What is contracted is the size of the moving object, in the direct of motion, by comparison with the size it has in its own rest frame. In the same way, the separation of two objects moving at similar velocities is contracted by comprison to the separation as seen in the rest frame of those moving objects.

Hences: what is contracted is the distance BETWEEN two remote galaxies that are a long way away and receding from us at high velocity. THAT is what Matt is saying in the above extract. It's a comparison between the distance between those galaxies in our frame, and the distance between those galaxies in their own frame.

We see a certain distance between those galaxies, which is LESS than the distance as it would be seen by those galaxies themselves, because of the effects of Lorentz contraction. So we see receding galaxies crowded together in the distance, but an observer in those galaxies will see them further apart.

In the Milne model, you have objects moving away from you at constant velocities that increase with their distance. That is, objects that are further away are receding more rapidly.

Furthermore, in the frame of any of these objects, they observe the same expansions as we do.

Furthermore, the recession velocities are such that there is a finite time t₀ so that all galaxies we see are receding at velocities so that we were all right next to one another a time t₀ ago.

Since the comoving particles in the Milne model can be of arbitrary size, including being infinitesimally small, there is no significance at all to the Lorentz contraction of the individual individual particles themselves ("becoming squashed", in Matt's precise terminology). The only relevant consideration here is the contraction of distances between particles.

The remote particles are "squashed" by precisely the same factor as their separation distances. That's explicit in the passage you quoted and you even bolded it.

Now you say that there's no significance in this.

That's just weird. You've got Matt's own explicit comments, and you simply turn them on their head and assert the direct opposite for some reason.

Please don't demand me to explain this to you "clearly". I do not claim any great ability to explain this where so many others have evidently failed to express things in terms you can understand. If you still think my previous answer was "wrong" then so be it. I don't mind.

Cheers -- sylas

 

[nutgeb]

Sylas, I will do my best to focus on the science and not respond to your relentless expressions of irritation. I appreciate your explanation.

I agree that in the Milne model, the distance between the origin and various comoving particles is not Lorentz contracted.

I also agree that, as viewed from the origin, the separation between comoving particles is Lorentz contracted at larger distances and velocities.

I also agree that, at the location of, and in the comoving frame of, a particle that is distant from the origin, the distances between nearby particles are not Lorentz contracted.

Beyond those points, I will give more thought to what it all means.

The remote particles are "squashed" by precisely the same factor as their separation distances.

You missed my point. I'll let it drop rather than repeating it.

 

[sylas]

Sylas, I will do my best to focus on the science and not respond to your relentless expressions of irritation. I appreciate your explanation.

Thanks. I prefer to think of myself as frustrated rather than irritated. I'd love to help better, but I don't think I can. So I'll leave it at that and wish you the best. I share your view that DrGreg is someone to listen to with special attention; his expertise and lack of apparent irritation is legendary.

If I can think of something useful to say, I will, but taking a break from writing more posts seems sensible.

All the best -- sylas