Question

If we intercept an electron having total energy of 1533 MeV that came from Vega, which is 26 ly from us, how far in light-years was the trip in the rest frame of the electron?

Short answer

In the rest frame of the electron, the distance travelled by the electron from Vega to earth is 77896 light-years.

Step-by-step solution

Lorentz factor.

The result of 2^nd postulate of the special theory of relativity is that the clocks run slower for a moving object when measured from a rest frame. The factor by which the clock is running differently is called the Lorentz factor.

The relativistic kinetic energy relation is given by

$K=m_e{c}^2\left(\gamma -1\right)$

Where $m_e$is the electron’s rest mass, and ,$\gamma$is the Lorentz factor. Using this relation to determine the Lorentz factor,

$$\begin{gathered} \gamma =\frac{K}{m_e{c}^2}+1 \\ =\frac{1533\times {10}^6\times 1.6\times {10}^{-19}J}{\left(9.1\times {10}^{-31}kg\right){(3\times {10}^{8}m/s)}^2}+1 \\ =2995+1 \\ =2996 \end{gathered}$$

We know that the Lorentz factor is expressed in terms of speed parameter $(\beta =v/c)$as,

$$\begin{gathered} \gamma =\frac{1}{\sqrt{1-{\beta }^2}} \\ \beta =\sqrt{1-\frac{1}{{\gamma }^2}} \\ =\sqrt{1-\frac{1}{{\left(2996\right)}^2}} \\ =0.9999 \end{gathered}$$

Hence the electron is traveling at a speed of 0.9999c.

Length contraction

The length of an object measured in the object’s inertial reference frame is called proper length. When the length of this object is measured in any other inertial frame moving at relative speed is always shorter than the proper length. This is known as length contraction.

$L=L_o\sqrt{1-{\beta }^2}=\frac{L_o}{\gamma }$

Here, the length measured from the earth’s frame is 26 ly, and the Lorentz factor is found to have a value of 2996.The proper distance, i.e., the distance measured from the electron’s reference frame between vegan and earth, is

$$\begin{gathered} L=\frac{L_o}{\gamma } \\ {L}_o=\gamma L \\ =2996\left(26ly\right) \\ =77896ly \end{gathered}$$

In the rest frame of the electron, the distance travelled by the electron from Vega to earth is 77896 light-years.