E=mc2

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,

                                                $c^2dm=(mv)dv+v^2dm$

substituting, the value of in the differential of the kinectic energy, we have as follows
                                                      $dE_k=c^2dm$

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, $E=mc^2$