Length Contraction in the Muon Experiment

[Pete]

Agreed.

[Aer]

Agreed.

Well then, lets define the length of the known universe as 30b ly in the Earth frame (15b ly in each direction). This makes the length of the known universe 6b ly (3b ly in each direction) in the Muon frame as defined from the Muon experiment.

So light from the edge of the universe takes 15b yrs to arrive to our point in space according to the Earth and this same light takes 3b yrs to arrive to our point in space according to the Muon.

[Pete]

And what do you conclude?

[Aer]

And what do you conclude?

According to our Muon frame, how much time has elapsed on Earth if 3b yrs have elapsed in our (Muon) frame?

[James R]

Aer:

[quote=Aer]

You must know that I am going to bring up the aburdity of this statement. You might as well forget trying to preach your point of view to me.

I am quite serious. How is a distance meaningful if it is not a distance between two objects? Can you give me an example of a distance which is NOT between two objects, other than an empty statement like "3 kilometres" (3 kilometres where? What for? According to whom?)

[Aer]

I am quite serious. How is a distance meaningful if it is not a distance between two objects?

Take a distance from a slice of spacetime. That is, freeze time - pick two arbitrary points in one frame, get the distance between them, go to the other frame, the arbitrary points are no longer arbitrary as they have 1-to-1 correspondence in actual reality in this frame. That is if we place objects at these two points, both frame can see the two objects. Just because two objects don't exist there doesn't mean there isn't a distance between these two points.

[superluminal]

You must know that I am going to bring up the aburdity of this statement. You might as well forget trying to preach your point of view to me.

I am quite serious. How is a distance meaningful if it is not a distance between two objects? Can you give me an example of a distance which is NOT between two objects, other than an empty statement like "3 kilometres" (3 kilometres where? What for? According to whom?)

It's a fascinating point that one rarely, if ever, consciously examines. What is a distance without two objects? The only example I can think of is points in a geometric plane or space - an abstraction. Fascinating.

[Pete]

Analysis time:

Events of interest (coordinates given for Earth frame):
O - Earth-muon collision ( I'll aribtrarily set this to be the Origin: x = 0, t = 0)
A - Earth at the beginning of time (x = 0, t = -15b yr)
B - light emitted at the beginning of time at the -x end of the known Universe (x = -15b ly, t = -15b yr)
C - light emitted at the beginning of time at the +x end of the known Universe (x = 15b ly, t = -15b yr)
D - muon at the beginning of time (x = -14.7b ly, t = -15b yr)

Note that I'm assuming that time began across the known Universe 15 billion years ago in Earth's frame (you see where this is going already, don't you?)

Now I'm going to transform these events to the Muon frame. I'm just plugging numbers into the Lorentz transform, with v = 0.98c.

Muon frame:
O: x' = 0, t' = 0
A: x' = 73.5b ly, t' = -75b yr
B: x' = -1.5b ly, t' = -1.5b yr
C: x' = 148.5b ly, t' = 148.5by
D: x' = 0, t' = -3b yr

You might like to plot these points on a graph.


Conclusion of the analysis:
According to the muon:
The known universe is 150b ly across.**
Time began 3b yrs ago for the muon.
Time began 75b yrs ago on Earth.
Time began 1.5b yrs ago at the -x end of the known Universe.
Time began 148.5b yrs ago at the +x end of the known Universe.



**I didn't expect this! This is why I rely on the Lorentz transform, rather than trying to use length contraction, time dilation, and relative simultaneity piecemeal.

Things to cogitate on:
What is the "end of the known universe", really? It is a place now, or a past event?

[Pete]

It's a fascinating point that one rarely, if ever, consciously examines. What is a distance without two objects? The only example I can think of is points in a geometric plane or space - an abstraction. Fascinating.

I know what you mean!

I posted my thoughts on this topic a bit earlier, deep in the heart of the highly stimulating discussion with Aer:

What is length, under the umbrella of SR?
It's the magnitude of the spacetime interval between two simultaneous events.

According to this view, length isn't directly transferrable between frames - To transfer a length from one frame to another, you have to change one or both events so that they are simultaneous in the new frame. This is OK, as long as you make sure that the new event(s) have the same spatial coordinates in the original frame as the old events. THis is easy and natural if you have an object at rest in the first frame to mark the spatial coordinates.

[Aer]

Analysis time:

B - light emitted at the beginning of time at the -x end of the known Universe (x = -15b ly, t = -15b yr)
C - light emitted at the beginning of time at the +x end of the known Universe (x = 15b ly, t = -15b yr)

B: x' = -1.5b ly, t' = -1.5b yr
C: x' = 148.5b ly, t' = 148.5by

Assuming your assumptions leading to your lorentz transformations and calculations are correct (I checked neither yet) then we have a very interesting result above. The same light hitting the Earth can be presumed to be hitting the muon when the Earth and muon collide. Yet it took different distances and different time intervals for the light to reach the muon/earth from different directions.

And I am much too tired to be doing any analysis of mathematical equations

[James R]

Aer:

Take a distance from a slice of spacetime. That is, freeze time - pick two arbitrary points in one frame, get the distance between them, go to the other frame, the arbitrary points are no longer arbitrary as they have 1-to-1 correspondence in actual reality in this frame.

Frames, spacetime. Sounds very abstract to me.

So, you define two points in spacetime, and you denote the distance between them to be the spatial separation at one instant of time. Right? Then you switch frames. How do you now define the distance between the same two points? Are the points even the same any more? How do you know where your points are? By switching frames, didn't you exchange one spacetime for another? At the very least, it seems to me you swapped a bit of space for a bit of time and vice versa.

What I am saying is that while coordinates in spacetimes can change, all observers agree on the existence or non-existence of objects in spacetime.

That is if we place objects at these two points, both frame can see the two objects. Just because two objects don't exist there doesn't mean there isn't a distance between these two points.

But by specifying two points in the first place, you are presupposing an observer. And when it comes down to it, all real observers are also objects.

[superluminal]

Yes Pete. I remember that post.

It's the magnitude of the spacetime interval between two simultaneous events.

I committed your phrase to memory - I really like it!

[Pete]

Thanks, super - that's high praise!

[Aer]

At the very least, it seems to me you swapped a bit of space for a bit of time and vice versa.

Yes, maybe this is true. However, if we send a signal to a moving object to record its observation and can calculate when the moving object will recieve this signal to "record" at that split time interval, then can we not just record what we see at that calculated time that the moving object recieves the signal? Of course I see the relativitity of simultaneity rearing its ugly ass again

What I am saying is that while coordinates in spacetimes can change, all observers agree on the existence or non-existence of objects in spacetime.

Let's leave the discussion to where it exists with Pete and I right now. We all moved on from this discussion long ago - and to be honest, I keep forgetting what I am replying to in this thread

[Aer]

Does anyone wish to verify Pete's results:

Analysis time:

B - light emitted at the beginning of time at the -x end of the known Universe (x = -15b ly, t = -15b yr)
C - light emitted at the beginning of time at the +x end of the known Universe (x = 15b ly, t = -15b yr)

B - light emitted at the beginning of time at the -x end of the known Universe (x = -15b ly, t = -15b yr)
C - light emitted at the beginning of time at the +x end of the known Universe (x = 15b ly, t = -15b yr)

B: x' = -1.5b ly, t' = -1.5b yr
C: x' = 148.5b ly, t' = 148.5by

[Pete]

Assuming your assumptions leading to your lorentz transformations and calculations are correct (I checked neither yet) then we have a very interesting result above. The same light hitting the Earth can be presumed to be hitting the muon when the Earth and muon collide. Yet it took different distances and different time intervals for the light to reach the muon/earth from different directions.

Yep - the "beginning of time" was simultaneous in the Earth frame, but not the muon frame.

I think this means that assuming a universal instant (the beginning of time) defines a universally observable reference frame.

An observer on Earth can tell that they are at rest in the Universe's "creation rest frame", while an observer with the muon can tell that they are moving at 0.98c relative to that frame.

This sounds kind of like the CMBR rest frame - we can tell by looking at the CMBR that we're moving at 600km/s relative to some Universally observable reference frame.

[Pete]

to be honest, I keep forgetting what I am replying to in this thread

I hear you... it's been entertaining, but exhausting

[Aer]

Yep - the "beginning of time" was simultaneous in the Earth frame, but not the muon frame.

It seems time could have never have began as no one can agree on when it happened.

I think this means that assuming a universal instant (the beginning of time) defines a universally observable reference frame.

Come again? I rather like my Earth reference frame - Einstein told me it was perfectly OK to use.

An observer on Earth can tell that they are at rest in the Universe's "creation rest frame", while an observer with the muon can tell that they are moving at 0.98c relative to that frame.

This sounds kind of like the CMBR rest frame - we can tell by looking at the CMBR that we're moving at 600km/s relative to some Universally observable reference frame.

If all you said above is true, we should have some light coming from one direction that took longer to get here than light from the opposite direction, no?

[Aer]

And how did the length of the known universe expand in the muon's frame when we agreed that the muon's frame was length contracted?

Are you sure your calculations are correct? I certainly wouldn't be disappointed as this is nonsense, but

[Pete]

If all you said above is true, we should have some light coming from one direction that took longer to get here than light from the opposite direction, no?

Yes! (unless GR says something different about it) But not by much. The relativistic gamma factor for 600km/s is only 1.000002

And how did the length of the known universe expand in the muon's frame when we agreed that the muon's frame was length contracted?

Apparently we were wrong.
I'm not completely sure how this pans out as far as length contraction goes... length contraction is hard to follow without objects with length to track.

Are you sure your calculations are correct?

I checked them twice... I've been looking for an online Lorentz transform calculator, but no joy.

But, it's not hard to do.
In our case with the units we're using, v = 0.98, gamma = 5, c = 1, so:
x' = 5(x - 0.98 t)
t' = 5(t - 0.98 x)

I certainly wouldn't be disappointed as this is nonsense

Are you sure?