Re: The Cosmic Muons Paradox

On Aug 27, 8:22 am, mluttgens <mluttg...@xxxxxxxxx> wrote:
On 27 août, 14:48, PD <thedraperfam...@xxxxxxxxx> wrote:



On Aug 27, 6:46 am, mluttgens <mluttg...@xxxxxxxxx> wrote:

On 27 août, 03:52, PD <thedraperfam...@xxxxxxxxx> wrote:

On Aug 26, 8:07 pm, mluttgens <mluttg...@xxxxxxxxx> wrote:

On 26 août, 23:16, PD <thedraperfam...@xxxxxxxxx> wrote:

On Aug 26, 10:08 am, mluttgens <mluttg...@xxxxxxxxx> wrote:

The cosmic muons paradox
_______________________

In special relativity, the time dilation effect is reciprocal:
as observed from the point of view of either of two clocks
which are in motion with respect to each other, it will be
the other clock that is time dilated.
(cf.http://en.wikipedia.org/wiki/Time_dilation)

Fromhttp://www.mth.uct.ac.za/omei/gr/chap1/node9.html:

"A very interesting example of the slowing down of time
is that of the cosmic ray muons . These are particles that
disintegrate spontaneously after an average lifetime of about
2.2 microseconds. It is clear that in its short lifetime a muon
cannot, even at the speed of light, travel more than 600 m.
But although the muons are created at the top of the atmosphere,
some 10 km up, we can detect them down here on earth. How
can that be!!? From the muon's point of view (i.e.from their frame
of reference) they only live about 2 microseconds. However from
our point of view they live considerably longer, indeed long enough
to reach the surface of the earth (by a factor f = 1/sqrt(1-v^2/c^2)."

According to SR, the lifetime of the muons moving at v relative
to the Earth's surface is dilated by f according to the Earth
observer. Experimentally, this is a real effect.

Such effect should be reciprocal, according to SR. But if it were,
an Earth observer should not be able to detect muons at Earth's
level. Conclusively, the time dilation effect is *not* reciprocal
in the case of the muons.

How do relativists explain such paradox?

Let's note that the twin paradox is explained by a "turnaround"
in which the ship undergoes non-inertial motion, while the Earth
has no such turnaround.
But the muons don't present a turnaround, hence another
explanation than a breaking of symmetry must be found.

Marcel Luttgens

Marcel, the reciprocity happens under a very special set of
circumstances.

I would say that it normally happens. Only in very special set
of circumstances, it doesn't (like for instance the "turnaround"
in the twin paradox.

Let's take two frames A and B in relative motion.

1. We'll consider two events E1 and E2 that satisfy a special
criterion: they occupy the same spatial location in A. There is a time
T_A that is the time interval between these two events E1 and E2. Now,
in B, these two events will not occupy the same spatial location, of
course, but we can still measure a time interval T_B between E1 and
E2. We will find that T_B > T_A. This is what time dilation means.
2. Now what if we asked the same thing in reverse? Now we'll consider
two events E3 and E4 that satisfy the same special criterion: they
occupy the same spatial location in B. NOTE!! E3 and E4 cannot
possibly be the same two events as E1 and E2, because E1 and E2 do not
satisfy the criterion of being in the same spatial location in B.
There is a time T'_B that is the time interval between these two
events E3 and E4. Now, in A, these two events will not occupy the same
spatial location, of course, but we can still measure a time interval
T'_A between E3 and E4. We will find that T'_A > T'_B. This is what
the reciprocity of time dilation means.

Now you understand a little bit better, I hope, what the careful
statement about the reciprocity of time dilation means and why it is
so important to make note of the actual events. Your referencing the
muon case only includes two events: E1 = muon creation at the top of
the atmosphere, E2 =  muon landing at the surface of the earth.
Nowhere do you have E3 and E4 to apply the reciprocity statement.

You seem to be close to a solution, but which are events E3
and E4 in the case of the cosmic muons?

Just to clarify something, Marcel, this is not the solution to some
difficult conundrum or soft spot in relativity. It is just explaining
basics to you. It would be like explaining to you why 2 * 0 = 0 does
not imply that 0/0 = 2.

There aren't any explicit E3 and E4, which is why the reciprocity is
not obvious.
You can choose E3 to be the same as E2, if you like. Now what you need
is an E4 which occurs at the same place but at a different time in the
Earth frame. Then in the muon frame, the time between E3 and E4 will
be longer than the time measured in the Earth frame. Voila,
reciprocity.

Should I conclude that in no real experiment (as opposed to
idealized thought experiments), the time dilation effect is
reciprocal in SR?

No, you should not conclude that. The time dilation effect is
reciprocal in every real experiment where there are two distinct pairs
of events that meet the criteria I mentioned. If you have a real
experiment where you are considering only two events, then of course
time dilation is not reciprocal between those two events (see my very
simple rendering of your problem where it is OBVIOUS that this is the
case), and special relativity has NEVER said that time dilation is
reciprocal between one pair of events. If you ever thought it did,
then this was a result of a basic and very simple misunderstanding of
relativity on your part, which I hope has been corrected.

Thank you for your explanation.
But could you refer to *one* real experiment where time dilation
is reciprocal?

I'm not aware of any experiment specifically designed to test
reciprocity of time dilation. There are a number of predictions made
by relativity that still are open to testing, which is one of the
reasons there is still plenty opportunity for experimental work. As a
guiding principle, theories don't become accepted when *all* of the
conceivable applications have been checked out.

However, the g-2 experiment does provide a good example where the
effects of mutual time dilation can be seen, even though the
experiment was not specifically designed to measure that.


Marcel Luttgens

Marcel Luttgens

Marcel Luttgens

I'm sure you're frustrated with the quality of the wiki articles that
gave you too-shallow information to avoid this confusion. This is the
point of having more in-depth materials when it comes time to really
understand relativity, and it is also the point of puzzles like this
to eradicate oversimplified understanding. You get what you pay for,
Marcel

Do you understand my point about trying to learn relativity from
wikipedia now?



.

Relevant Pages

  • Re: The Cosmic Muons Paradox
    ... But although the muons are created at the top of the atmosphere, ... to the Earth's surface is dilated by f according to the Earth ... the time dilation effect is *not* reciprocal ... statement about the reciprocity of time dilation means and why it is ...
    (sci.physics.relativity)
  • Re: The Cosmic Muons Paradox
    ... But although the muons are created at the top of the atmosphere, ... to the Earth's surface is dilated by f according to the Earth ... the time dilation effect is *not* reciprocal ... statement about the reciprocity of time dilation means and why it is ...
    (sci.physics.relativity)
  • Re: The Cosmic Muons Paradox
    ... But although the muons are created at the top of the atmosphere, ... to the Earth's surface is dilated by f according to the Earth ... the time dilation effect is *not* reciprocal ... statement about the reciprocity of time dilation means and why it is ...
    (sci.physics.relativity)
  • Re: The Cosmic Muons Paradox
    ... But although the muons are created at the top of the ... we can detect them down here on earth. ... the time dilation effect is *not* ... the reciprocity of time dilation means. ...
    (sci.physics.relativity)
  • Re: The Cosmic Muons Paradox
    ... In special relativity, the time dilation effect is reciprocal: ... But although the muons are created at the top of the atmosphere, ... the reciprocity of time dilation means. ...
    (sci.physics.relativity)
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