91Ó°ÊÓ

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

$K=m_e{c}^2\left(\mathrm{γ}-1\right)$

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

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

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

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

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.

$L=L_o\sqrt{1-{\mathrm{β}}^2}=\frac{L_o}{\mathrm{γ}}$

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

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

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