mu-meson experiment

Question

for the experiment that moving mu-mesons live longer, we used as an example their straight-line motion in the atmosphere. But if make mu-mesons in a laboratory and cause them to go in a curve with a magnet, will they last exactly as much longer as they do when they are moving in a straight line?if so, doesn't it violates relativity principle?

Answer 1

It is possible to calculate the time dilation for a particle moving in a curve, though the maths involved is a bit more complicated than for motion in a straight line. If you're interested I describe how to do the calculation for the special case of circular motion in my answer to Is gravitational time dilation different from other forms of time dilation?.

The end result is that the time dilation in circular motion is related to the particle speed in exactly the same way as it is for linear motion. So if the velocity stays constant using a magnet to bend the particle trajectory into a circle would not change the lifetime.

Answer 2

"moving mu-mesons live longer"

They live longer than muons "at rest", but how are muons "at rest" defined, and how does one measure their lifetime? Here is the answer:

http://cosmic.lbl.gov/more/SeanFottrell.pdf "The lifetime of muons at rest [...] Some of these muons are stopped within the plastic of the detector and the electronics are designed to measure the time between their arrival and their subsequent decay. The amount of time that a muon existed before it reached the detector had no effect on how long it continued to live once it entered the detector. Therefore, the decay times measured by the detector gave an accurate value of the muon's lifetime. After two kinds of noise were subtracted from the data, the results from three data sets yielded an average lifetime of 2.07x 10^(-6)s, in good agreement with the accepted value of 2.20x 10^(-6)s."

http://www.physics.rutgers.edu/ugrad/389/muon/muon-rutgers.pdf "In order to measure the decay constant for a muon at rest (or the corresponding mean-life) one must stop and detect a muon, wait for and detect its decay products, and measure the time interval between capture and decay. Since muons decaying at rest are selected, it is the proper lifetime that is measured. Lifetimes of muons in flight are time-dilated (velocity dependent), and can be much longer..."

Clearly the muons "at rest" are not at rest actually - they are undergoing a catastrophe. For that reason their lifetime is shorter than the lifetime of moving muons (which are not undergoing a catastrophe). There is no time dilation.