Is There a Discrepancy in Distance Measurements?

[Hernik]

Hi! Love this forum. Let me see if I understand this right:

It's a clear and beautiful night. I look at the sky. In the direction of the moon I spot a proton heading straight for me. It's a visitor in our galaxy. So it brought a camera like most tourists do. A technologically advanced camera with autofocus with a memory function. The proton is traveling at relativistic velocity. When it passes the moon I know it must be approximately 384.000 kilometers away... soon after that it reaches the earth.

It hits a molecule in the atmosphere and is scattered in the form of other particles. But the camera is tough! It lands undamaged next to me. I look at the pictures. The proton took a snapshot of me exactly when it passed the moon. I check the memory of the autofocus and compare it to my measurement: The protons distance from the moon to me was much smaller than my distance from me to the moon.

Can this be right - there is no agreement between two bodies that move relative to each other on the distance that separates them? The same distance has two different "lenghts"?

- Henrik

 

[JesseM]

It's correct to say that the distance in your frame is larger than the distance in the proton's frame, but neither distance is what would be seen visually if either of you were using a camera, the apparent visual distance is distorted due to something called the http://th.physik.uni-frankfurt.de/~scherer/qmd/mpegs/lampa_terrell_penrose_info.html, I gave some explanation and links in [post=2984821]this post[/post].

 

[DaveC426913]

The protons distance from the moon to me was much smaller than my distance from me to the moon.

Yes. From the proton's frame of reference, the Moon and Earth are rushing toward it at relativistic velocities. Because of this, the proton sees what it expects to see; objects flattened into discs along their line of travel, with a commensurate shortening of distance between them.

 

[JesseM]

Yes. From the proton's frame of reference, the Moon and Earth are rushing toward it at relativistic velocities. Because of this, the proton sees what it expects to see; objects flattened into discs along their line of travel, with a commensurate shortening of distance between them.

"See" is potentially misleading as I suggested in my comment above, optically the proton will actually see the length of oncoming objects stretched rather than shrunk (though of course their length is still shrunk in the proton's rest frame), as shown in the animations on this page.

 

[yuiop]

Yes. From the proton's frame of reference, the Moon and Earth are rushing toward it at relativistic velocities. Because of this, the proton sees what it expects to see; objects flattened into discs along their line of travel, with a commensurate shortening of distance between them.

On the other hand the proton also sees something it might not expect (unless it has read this forum). The Earth appears much smaller (subtends a smaller angle) than in a similar photo taken by an identical camera and lens that is stationary on the Moon. Paradoxically, if you suddenly accelerate towards something it suddenly shrinks in size and appears visually further away even though the measured distance is shorter than when you were at rest with the object.

 

[DaveC426913]

On the other hand the proton also sees something it might not expect (unless it has read this forum). The Earth appears much smaller (subtends a smaller angle) than in a similar photo taken by an identical camera and lens that is stationary on the Moon. Paradoxically, if you suddenly accelerate towards something it suddenly shrinks in size and appears visually further away even though the measured distance is shorter than when you were at rest with the object.

Why is that?

 

[PAllen]

Why is that?

The effect is called relativistic aberration. It is actualy covered ok in wikipedia:

http://en.wikipedia.org/wiki/Relativistic_aberration

 

[yuiop]

The effect is called relativistic aberration. It is actualy covered ok in wikipedia:

http://en.wikipedia.org/wiki/Relativistic_aberration

Actually it looks like we are all missing the point of the OP. We are talking about the visual or optical effects due to light travel times, while Hernik is asking about good old measured regular length contraction.

I check the memory of the autofocus and compare it to my measurement: The protons distance from the moon to me was much smaller than my distance from me to the moon.

Can this be right - there is no agreement between two bodies that move relative to each other on the distance that separates them? The same distance has two different "lenghts"?

- Henrik

The bolded part of the quote above shows he is talking about length contraction which makes distances appear shorter, while aberration effect due to light travel times causes approaching objects to appear optically further away.

To answer Henrik's original question, yes, measured lengths (not visual) are relative and depend on the relative motion of the observer.

 

[JesseM]

Actually it looks like we are all missing the point of the OP. We are talking about the visual or optical effects due to light travel times, while Hernik is asking about good old measured regular length contraction.

I think his question suggested he wasn't aware of the distinction between the two, since he also talked about what would be seen in a photograph. And I did say that his comment would be correct with regard to the length in each frame at the beginning of my first post.

 

[yuiop]

If I have done the calculation correctly, the de-magnification effect is given by:

h' = h \sqrt\frac{1-v/c}{1+v/c}}

where h is the height of the image on the film backplate when the camera is at rest with the photographed object and h' is the height of the image when the camera is moving towards the object with relative velocity v. Note the similarity to the relativistic Doppler shift.

 

[yuiop]

Why is that?

Why is it paradoxical, or why is that objects appear smaller when you are moving towards them?

Assuming the latter, take the case of the proton moving towards the Earth as seen by an observer in the Earth rest frame. The distance between the front of the camera and the back of the camera is length contracted. (Assume a pin hole camera for simplicity.) When a photon enters the lens the backplate is moving towards the photon so the effective distance between the front plate and the backplate is reduced by a factor of \sqrt{(1-v^2)/(1+v^2)}. This shorter effective distance means rays have less time to diverge between the lens and the backplate and the image subtends a smaller angle making the image look smaller and further away.

In the rest frame of the proton camera, the light rays entering the camera when the camera is alongside the Moon left the Earth at an earlier time than its current instantaneous position suggests and so the image is smaller than when the camera is at rest with the Earth and Moon.

Alternatively using the aberration effect, if light rays appear to be coming towards a camera at a given angle from an object, when the camera and object are at rest wrt each other, then the light rays will appear to be coming from a smaller angle when the camera and object are moving towards each other and from a wider angle when they are moving away from each other. I think these articles covers it slightly better than Wikipedia:

http://physics.sharif.edu/~astronomy/images/UsefulMaterial/aberration%20essay.pdf
http://demonstrations.wolfram.com/RelativisticAberrationAndDopplerShift/
http://arxiv.org/ftp/astro-ph/papers/0505/0505206.pdf

Interestingly if two identical in size bright stars are orbiting each other in a plane that we share, then the star approaching us should look smaller and the star receding from us should look larger. I wonder if that has ever actually been observed?

 

[yuiop]

Can this be right - there is no agreement between two bodies that move relative to each other on the distance that separates them? The same distance has two different "lenghts"?

A similar scenario has been observed in muons. The half life of mouns is such that they should ordinarily expire before they hit the ground and yet they are observed in significant quantities at sea level. The relativistic explanation is that due to time dilation they live longer in the Earth frame due to their relativistic motion (>0.9c) In the rest frame of the muons, the muons half life is unchanged, but to them the height of the Earth's atmosphere is length contracted and that is why they live long enough to have a short stay at the seaside .

 

[Hernik]

Thank you for all the very thorough answers. Very interesting how the images change for cameras that move relative to their subjects. I enjoyed the links very much.

- henrik

 

[yuiop]

Thank you for all the very thorough answers. Very interesting how the images change for cameras that move relative to their subjects. I enjoyed the links very much.

- henrik

You're welcome

You might like this animation of relativistic abberation too The relevant part starts after about 1 minute. It can be seen that the clouds at the distant horizon shrink towards the vanishing point, giving the impression you are moving backwards if you concentrate on the sky only.

 

[yuiop]

When a photon enters the lens the backplate is moving towards the photon so the effective distance between the front plate and the backplate is reduced by a factor of \sqrt{(1-v^2)/(1+v^2)}.

Correction, the factor should have read \sqrt{(1-v/c)/(1+v/c)} in agreement with post #10.