Muon Time Dilation in Accelerating Frames

[bksree]

Hi
In the book, "Why does E= mc2" by Cox and Forshaw, while discussing time dilation, the example of a muon is given. The authors explain that muons when circulated in the 14 m diameter AGS facility at Brookhaven at 99.94% of the speed of light, its lifetime is increased from the value of 2.2 microseconds (when it is at rest) to 60 microseconds i.e 29 times longer which he explains is equal to gamma = sqrt(1 - v^2 / C^2).

My question is : The formula, gamma = sqrt(1 - v^2 / C^2), is for an inertial system while the muons while moving in the circular orbit are in an accelerating frame. So is it correct to use this expression for time dilation ?

Thank you

 

[Ibix]

Yes. The correct general formula is that the elapsed time for a particle is$$\begin{eqnarray*}
\Delta \tau&=&\int_{t=0}^{\Delta t}d\tau\\
&=&\frac{1}{c}\int_{t=0}^{\Delta t}\sqrt{c^2dt^2-dx^2-dy^2-dz^2}\\
&=&\frac 1{c}\int_0^{\Delta t}\sqrt{c^2-\left(\frac{\partial x}{\partial t}\right)^2-\left(\frac{\partial y}{\partial t}\right)^2-\left(\frac{\partial z}{\partial t}\right)^2}dt
\end{eqnarray*}$$But the sum of the squares of the partial derivatives is just the velocity squared, and hence$$\begin{eqnarray*}
\Delta \tau&=&\frac 1c\int_0^{\Delta t}\sqrt{c^2-v^2}dt\\
&=&\sqrt{1-\frac{v^2}{c^2}}\Delta t
\end{eqnarray*}$$Note that I've said nothing about the path beyond assuming that $v^2$ is independent of $t$ in the last step, but have assumed that the $t,x,y,z$ coordinates are inertial coordinates.

Note that this case is not reciprocal. An observer circling with the muon would not conclude that lab clocks were ticking slow, but that they were ticking fast - the muon observer's frame is not inertial. So, while some results for inertial motion do hold for non-inertial motion, not all of them do.

 

[A.T.]

n is : The formula, gamma = sqrt(1 - v^2 / C^2), is for an inertial system while the muons while moving in the circular orbit are in an accelerating frame. So is it correct to use this expression for time dilation ?

The reference frame in which v is measured is an inertial reference frame.