Relativistic effects on apparent angular size

[A.T.]

Therefore, in AT's scenario, if the rocket passes next to the moon and then passes next to the earth, the rocket will perceive the Earth and moon clocks to be unsynchronized from each other, despite the fact that those two clocks are synchronized in the earth-moon inertial frame. Thus in the rocket's frame,...

Why do you keep talking about the rocket's frame and rocket passing the earth, while completely ignoring my very simple question about the observer in the Earth's frame? Let me reiterate it:

So when a rocket moving at 0.9c relative to Earth passes by the moon, an observer on earth views the distance to moon & rocket Lorentz contracted?

Note that "rocket passing by the moon" is a single event in spacetime. And I'm interested in the length measured in a single frame of reference: the Earth's frame. So please stop bringing up "failure of simultaneity".

 

[sylas]

Thanks for the straight answer. Unfortunately it's wrong. Please read section 4.7 of the Taylor & Wheeler's textbook which I linked to in an earlier post.

I think you do not understand that basic text.

Let me make the situation of your original post precise, using actual numbers. We have a disk, at a distance of 6 light years from an observer. It is at rest with respect to the observer. The disk is 1 meter thick, and 10 meters in radius, in the rest frame of the observer, and face on to an observer.

We accelerate the disk, in a very short period of time (1 nanosecond), up to 60% of light speed, directly away from the observer.

In the frame of the observer, the disk is now 80 centimeters thick. It is Lorentz contracted, along the direction of motion. During the acceleration, it can have moved no more than 18 centimeters, which is 60% of a light-nanosecond. Hence it is still 6 light years distant, and it is still 10 meters in radius.

Of course, it takes six years from the acceleration for the observer see the disk undergo that acceleration.

There is no Lorentz contraction of the distance to the disk. There is a Lorentz contraction of the disk itself.

The angular size of the disk depends on its radius, not its thickness. That is unaltered.

We know that the distance to the disk is unaltered by the velocity of the disk, because a twin disk which was not accelerated is also seen six years later. The light travel time from the disk to the observer, as measured by the observer, is independent of the velocity of the disk.

Why you think that the act of accelerating the disk results in a different distance, I am unsure. You apparently think Taylor and Wheeler imply such a thing. But they don’t. It is true that distances to things can be Lorentz contracted… for example, if it was the observer that was accelerated rather than the disk.

But in the case you describe, there is no significant effect on the distance to the disk over the brief interval of acceleration, because the disk hasn’t had time to move to a new location. There’s no Lorentz contraction involved.

OK I apologize. I would appreciate if instead of telling me to "go work the problem", you show me how you work it to obtain a different result from mine -- or point me to a reliable source that shows it. Especially when the question relates to the heuristics rather than to the math itself. It's easy to generate an answer that is mathematically correct but does not correctly reflect the workings of a particular scenario.

As I have said before, I find it very hard indeed to follow your reasoning. It might just be me; there are other folks who seem to follow rather better. So I don't know where you are going wrong, and I often find it hard to tell what you are saying. In the initial posts of this thread, however, the situation seems clear enough, and corresponds to what I have tried to express above in numbers. If this is what you did intend to describe, then you are making inferences that do not follow from Taylor and Wheeler.

My advice -- for whatever it is worth -- remains precisely the same as given previously. Specify clearly the events you mean, in some inertial frame. Use Lorentz transformations to get the time and distance in another inertial frame.

The Lorentz transformations are independent of the velocity of objects being observed. They just depend on the frame of the observer, and let you map co-ordinates of an event between different frames.

Cheers -- sylas

 

[DrGreg]

nutgeb

Lorentz contraction applies when different inertial observers each measure the distance between the same two objects that are a constant distance apart. Because each measures the distance to be constant, it doesn't matter exactly when each observer makes their measurement.

When two objects are moving relative to each other, it's not so simple. As each observer now measures a varying distance, it matters how the two observers' measurements are synchronised to each other. There is no simple yes/no answer as to whether length contraction occurs, it depends on the details of a specific problem and when each observer makes their measurement.

So simply claiming that Lorentz contraction occurs is meaningless unless you spell out exactly which two measurements you are comparing and exactly when each measurement is taken. The advice that others have given you, to forget Lorentz contraction and apply the full Lorentz transformation, is good advice.

Another misconception that seems to arise on this forum from time to time is that Lorentz contraction applies to two different measurements made by the same observer. There's no automatic reason for that to be true, unless there are additional reasons such that the conditions I spelled out in my opening sentence can be applied.

Now, considering the specific example in this thread of the rocket flying past the moon. Both the rocket and the moon are (momentarily) at the same place at the same time. This is something all observers must agree on. Let's say the rocket actually crashed into the moon with an explosion. Any observer watching this would see the moon and the rocket approach each other and the explosion occur just as they met. And any observer is forced to agree that the distance to the moon, the distance to the rocket and the distance to the explosion are all equal at that event (and very nearly equal just before the explosion). It would be nonsense to say that the distance to the rocket was half the distance to the moon milliseconds before the explosion (both distances being measured by the same observer).

 

[Austin0]

#1 This ought to be easy. The angular size of an object, in special relativity, depends on the distance from where light is emitted to where it is received, as given by the frame of the observer.

#2 You can derive the same result for angular size in the frame of the observer, or any other inertial frame by considering Lorentz contraction on a pinhole camera of a moving viewer.

I've described it, with some diagrams, in [post=2217788]msg #21[/post] of "most basic of thought experiments in special relativity".

Cheers -- sylas

#1 Would seem to mean that angular distortion doesn't occur because the distance as measured in the observer frame is by definition, not contracted, and the image must arrive from exactly where it was located .

#2 The pinhole camera illustration is very clear .
Is the interpretation of this, that angular size distortion as observed from a moving frame is purely an optical effect occurring in a camera or or human eye?
In light of #1 this would seem a necessary conclusion.

Thanks

 

[sylas]

#1 Would seem to mean that angular distortion doesn't occur because the distance as measured in the observer frame is by definition, not contracted, and the image must arrive from exactly where it was located .

I'm not sure what you mean.

The angular size of an object of a known size can be calculated directly in the frame where the observer is at rest. In this frame, you simply need to know the distance from the object being viewed to the observer. Put together with the actual size of the object, and you can calculate the subtended angle.

There's no "contraction" involved in this; it's the usual way of calculating subtended angles.

#2 The pinhole camera illustration is very clear .
Is the interpretation of this, that angular size distortion as observed from a moving frame is purely an optical effect occurring in a camera or or human eye?
In light of #1 this would seem a necessary conclusion.

The second calculation is in a frame where you infer the angular size that is apparent to an observer, represented as the camera. This is calculated using numbers for a different frame. The camera is "Lorentz contracted", and the distance to the event of emission of photons used to view the image is different also. Plus there's the effect of moving towards or away from the photon stream.

But put it together, and you get the same answer as calculated entirely in the observer's frame.

Cheers -- sylas

 

[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