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The Muon Experiment and Spacetime Geometry

#1
confused2 Offline
Special Relativity follows..

The purpose of this post isn't really to discuss the muon experiment - I'm really trying to clarify the nature of spacetime as a geometry. In the following analysis c appears as a constant in the same manner as there is a constant used to convert litres to gallons. It would have been surprising if the French had randomly chosen the natural unit for measuring liquids (the gallon) and so we have a conversion factor between one and the other. In the same way we have c as a conversion factor between time and distance.  

Basically we have muons created in the upper atmosphere heading down at close to the speed of light (about 0.98 of the speed of light). They are counted at two points one 10km above the other. Muons decay very rapidly so the difference between the two counts is a good indication of the elapsed time between the two counting stations.

In the simplest notation - in one dimension of length and one of time, where s is the spacetime interval, c is a constant (found by experiment), x is a distance, t is time and v is the muon velocity in the Earth frame
s²=x²-c²t²      .... we can derive this later if there is any interest
Compare this with Pythagoras' Theorum
r²=p²+q² <- the spacetime interval is the same geometryish thing but with distance and time (and a conversion factor).

In the nature of special relativity the time in the Earth frame isn't going to be the same as the time in the muon frame so I'll put a ' after times and distances in the Earth frame. So t' is Earth time and t is muon time.

In the muon frame elapsed time =t we have
s²=-c²t²    < note no distance here*, the first and second counts are in the same place as far as the muon is concerned.

In the Earth frame t' and x' (where x' ~10km) we have
s²=x'²-c²t'²
since s² is the same in both frames
c²t²=x'²-c²t'²
we know x' = vt' from Newton and the definition of velocity
so
-c²t²=(vt')²-c²t'²
c²t²=c²t'²-(vt')²
t= t'√(1-v²/c²)
so t=t_muon is less than t'=t_Earthclock

The t= t'√(1-v²/c²) should be the same as wiki or Einstein or anyone else gets.

There may be mistakes and/or typos or I may simply be wrong.
For more details and another analysis see:-
http://hyperphysics.phy-astr.gsu.edu/hba.../muon.html

I'd like to emphasise that I've used the speed of light (as a constant) to predict the result of an experiment that doesn't involve light. Just time, distance, velocity and muons.
The s²=x²-c²t²  might seem assplucked - deriving it takes a few more lines of high school algebra.

* - it took me years to work out why.
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#2
Syne Offline
No, c is not a conversion factor. For one, converting between time and space is nonsense (as one contracts and the other dilates in SR), and for another, the Lorentz transformation is how you convert between different frames of reference...where the speed of light is invariant (the only thing that does not change). Your analogy to converting between different units of measurement does the opposite; it holds the measuring units invariant while changing the factor needed to make each specific conversion.
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#3
confused2 Offline
Syne Wrote:Your analogy to converting between different units of measurement does the opposite; it holds the measuring units invariant while changing the factor needed to make each specific conversion.
If we choose to measure distance in light years and time in years then the conversion factor © becomes 1. Where s is the invariant spacetime interval, we'd have  
s²=x²-t²  
which is often done,  depending on the forums and individuals you encounter.

Syne Wrote:..as one contracts and the other dilates in SR)
This is implicit in s²=x²-t², s² is invariant (betwen frames) so if x² increases then t² must decrease - this is where your contraction and dilation actually come from.

The OP derived the the standard (accepted) equation for time dilation of one frame seen from another using s²=x²-c²t² as a starting point.

The muon experiment involves just two events located in spacetime (first and second counters), the geometry is entirely linear between those events so I don't think there is any need for the complexity of the full Lorenz Transform - it will (inevitably) give the same result (try it!).
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#4
Syne Offline
(Oct 13, 2018 12:47 AM)confused2 Wrote:
Syne Wrote:Your analogy to converting between different units of measurement does the opposite; it holds the measuring units invariant while changing the factor needed to make each specific conversion.
If we choose to measure distance in light years and time in years then the conversion factor  © becomes 1. Where s is the invariant spacetime interval, we'd have  
s²=x²-t²  
which is often done,  depending on the forums and individuals you encounter.
That's a choice of convenient units; it still leaves c invariant.
Quote:
Syne Wrote:..as one contracts and the other dilates in SR)
This is implicit in s²=x²-t², s² is invariant (betwen frames) so if x² increases then t² must decrease - this is where your contraction and dilation actually come from.

The OP derived the the standard (accepted) equation for time dilation of one frame seen from another using s²=x²-c²t² as a starting point.

The muon experiment involves just two events located in spacetime (first and second counters), the geometry is entirely linear between those events so I don't think there is any need for the complexity of the full Lorenz Transform - it will (inevitably) give the same result (try it!).

Contraction and dilation mean space and time are dependent upon relative velocity, and the spacetime interval is wholly mediated by c. So again, you've given no reason to assume spacetime more fundamental than causality.
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#5
Secular Sanity Offline
(Oct 13, 2018 04:20 AM)Syne Wrote: Contraction and dilation mean space and time are dependent upon relative velocity, and the spacetime interval is wholly mediated by c. So again, you've given no reason to assume spacetime more fundamental than causality.

He’s right, you know. Read the speed of light and causality and then the speed of light as a limit.

"These considerations show that the speed of light as a limit is a consequence of the properties of spacetime, and not of the properties of objects such as technologically imperfect space ships. The prohibition of faster-than-light motion, therefore, has nothing in particular to do with electromagnetic waves or light, but comes as a consequence of the structure of spacetime."

https://en.wikipedia.org/wiki/Minkowski_..._causality
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#6
Syne Offline
(Oct 13, 2018 03:12 PM)Secular Sanity Wrote:
(Oct 13, 2018 04:20 AM)Syne Wrote: Contraction and dilation mean space and time are dependent upon relative velocity, and the spacetime interval is wholly mediated by c. So again, you've given no reason to assume spacetime more fundamental than causality.

He’s right, you know. Read the speed of light and causality and then the speed of light as a limit.

"These considerations show that the speed of light as a limit is a consequence of the properties of spacetime, and not of the properties of objects such as technologically imperfect space ships. The prohibition of faster-than-light motion, therefore, has nothing in particular to do with electromagnetic waves or light, but comes as a consequence of the structure of spacetime."

https://en.wikipedia.org/wiki/Minkowski_..._causality

No one claimed that electromagnetic waves of light are fundamental, deary. That's why we've been talking about whether spacetime or causality is more fundamental, where causality relates all frames via spacetime intervals and is what limits the speed of light...not equivalent to light. Try to keep up. Rolleyes
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#7
Secular Sanity Offline
(Oct 13, 2018 08:47 PM)Syne Wrote: No one claimed that electromagnetic waves of light are fundamental, deary. That's why we've been talking about whether spacetime or causality is more fundamental, where causality relates all frames via spacetime intervals and is what limits the speed of light...not equivalent to light. Try to keep up.  Rolleyes

The principle of causality just states that there is no way to send a message faster than light.

See previous link.

"Any general technical means of sending signals faster than light would permit information to be sent into the originator's own past. In the diagram, an observer at O in the x-ct system sends a message moving faster than light to A. At A, it is received by another observer, moving so as to be in the x′-ct′ system, who sends it back, again faster than light, arriving at B. But B is in the past relative to O. The absurdity of this process becomes obvious when both observers subsequently confirm that they received no message at all, but all messages were directed towards the other observer as can be seen graphically in the Minkowski diagram.

If it were possible to accelerate an observer to the speed of light, their space and time axes would coincide with their angle bisector.
The coordinate system would collapse, in concordance with the fact that due to time dilation, time would effectively stop passing for them.

These considerations show that the speed of light as a limit is a consequence of the properties of spacetime. The prohibition of faster-than-light motion, therefore, has nothing in particular to do with electromagnetic waves or light, but comes as a consequence of the structure of spacetime."
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#8
Syne Offline
And as your citation says, causality also guarantees that the order of events cannot change, even while the timing of events viewed from different frames does. So no cause can ever be observed to occur after its effect.

No, these show that both the speed of light and spacetime are a consequence of causality. Again, how would a spacetime that can, itself, exceed the speed of light be the primary enforcer of such a limit?

It is popularly imagined that Special Relativity forbids travel faster than the speed of light or the propagation of signals faster than the speed of light. However, the actual theory does not contain this assumption. The original theory, framed by Einstein in 1905, states that the speed of light in free space is constant in all inertial frames of reference so how did people in general come to believe that this implies a speed limit? The idea of a speed limit comes from two predictions of the theory, that inertia increases towards infinite as velocity approaches light speed and that causality, the succession of cause and effect, is violated if we could signal at speeds above the speed of light.

The inertial constraint does not apply to particles without a rest mass, such as the photon, or to particles that might oscillate between massless and massive forms. The possibility that causality would be violated if signals could travel faster than the speed of light is a more interesting problem however. The relationship between Faster than Light signal speeds and causality will be considered and it will be shown that if a Faster than Light signal were ever discovered then either Special Relativity or Causality will be false.
- https://en.wikibooks.org/wiki/Special_Re...Relativity


The Lorentz transformations have a mathematical property called linearity, since x' and t' are obtained as linear combinations of x and t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that we tacitly assumed while performing the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere. All inertial observers will agree on what constitutes accelerating and non-accelerating motion. Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.
- https://en.wikipedia.org/wiki/Spacetime#...formations


So you're claiming that a faster than light signal would violate causality but not SR? That's what saying spacetime is more fundamental than causality means. That a crucial principle to all science, cause and effect, is dependent upon a relatively recent theory about how differently moving observers relate to each other.
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#9
Secular Sanity Offline
I don't know. I'll have to think about. Do you have any resources?
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#10
Syne Offline
(Oct 13, 2018 11:33 PM)Secular Sanity Wrote: I don't know. I'll have to think about. Do you have any resources?

I'm afraid I can't teach rigorous thinking.
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