Relativistic Energy

The conservation of energy is one of the most predominant laws in physics. We are aware of the fact that energy has many important forms, also each form can be converted to any other. We know that parameters like distance, time, motion, velocity and acceleration are all relativistic in nature, then, we can say that energy must be a relativistic quantity too. In other words, the energy of the object under consideration is depending on the inertial frame of reference that we are in. 

According to classical physics, the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is moderated to include the possibility of mass changing into energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is defined in such a way that it will be conserved in all inertial frames, just like in the case of relativistic momentum. 

Relativistic Energy Formula:

Let us start with the derivation of the relativistic energy formula. When we are calculating the non-relativistic energy we assume that the change in kinetic energy is equal to the work done on the system or the object under consideration. In the relativistic case also assume the same theorem and estimate the relativistic energy. 

To derive the relativistic energy formula we assume that the mass-energy principle holds good under relativity. According to the work-energy theorem, it states that the net work done on a system goes into kinetic energy. In other words, we say the change in kinetic energy can be evaluated by calculating the work done on the system or the object that we were considering. The relativistic energy formula is also known as the energy-momentum relation. 

In classical physics, the kinetic energy is given by the product of mass and square of the velocity, this kinetic energy is valid for the object which has a velocity less than the speed of light. But we know that in relativistic mechanics we assume that the particles are moving with the speed of light. The potential energy of the particle is considered to be almost zero or negligible.

The relativistic energy is also known as the relativistic kinetic energy and it can be derived by a small derivation as given below.  We know that according to the work-energy theorem, it states that the net work done on a system goes into kinetic energy. In other words, we say the change in kinetic energy can be evaluated by calculating the work done on the system or the object that we were considering. Therefore mathematically we write,

\[\Rightarrow E=\int_{0}^{r}dW=\int_{0}^{r}F.dr=\int_{0}^{r}Fdr\]…….(1)

Since both applied force and the displacement are in the same direction we are not considering the scalar product. 

We know that according to Newton’s law of motion force F is equal to,

\[\Rightarrow F=\frac{dp}{dt}=\frac{d(mv)}{dt}\]…….(2)

And the velocity of the particle is given by,

\[\Rightarrow v=\frac{dr}{dt}\]…………(3)

Substituting the values in the equation (1) we get,

\[\Rightarrow E=\int_{0}^{t}\frac{d}{dt}(mv)vdt=m\int_{0}^{v}vdv=\frac{mv^{2}}{2}\]..(4)

Equation (4) resembles the classical kinetic energy formula of the particle moving with velocity v. But, here we are considering the relativistic case and hence substituting the value of mass by rest mass and velocity by Lorentz transformation we get,

\[\Rightarrow m=\gamma m_{0}c^{2}\]

Substituting the value of relativistic mass and velocity we finally calculate the relativistic energy.