Before moving on further in the study of spacetime, for the purpose of note we look onto the derivation of the formula $E=mc^2$.

We know that, Force is defined as the rate of change of momentum i.e., $F=\frac {d}{dt}(mv)$

According to the theory of relativity, both mass and velocity are a variable quantity. We therefore, can rewrite the above as follows,

                                          $F=\frac {d}{dt}(mv)$ = $m\frac {dv}{dt}$+$v\frac {dm}{dt}$

Let the force $F$ displace the body through a distance $dx$. Then, the increase in the kinectic energy $dE_k$ of the body is equal to the work done $F.dx$

                          Hence, $dE_k= Fdx= (m\frac {dv}{dt}+v\frac {dm}{dt})dx$
                                                        $=(mv)dv + v^2 dm$

Now, according to the law of mass variation with velocity, we have;

                      $m=\gamma m_0$ , where $\gamma$ represent the factor $1/\sqrt{1-\frac{v^2}{c^2}}$
taking the differential of the mass and the velocity in the equation we have the following relation,


substituting, the value of in the differential of the kinectic energy, we have as follows

Integrating the above differential from $ m_0$ to $m $ i.e, $\int_{m_0}^{m}c^2dm$, we have

                                                   $E_k=mc^2-m_0c^2$ or,

                              rearranging,   $E_k+m_0c^2=E_t=mc^2$

This represent the total energy in the system, $E=mc^2$

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