Deriving e=mc^2

We begin with the definition of relativistic mass:

.

We can now define relativistic momentum as:

.

Since force is defined as



we can define force in the relativistic context analogously:

.

In classical physics work is defined as

.

We can reduce this to a problem in a single dimension for simplicity, and recall that the kinetic energy added to a body is equal to the work done on the body by an external force:

.

From here we can say:

.

Applying the definition of relativistic mass, the last expression becomes:

[

.

Integrating through, we have



The term on the left, , is the total energy of the body after an external force is applied, the term on the right, is the rest energy, and the final expression is the kinetic energy of the body.

The kinetic energy can be expressed more clearly, in my opinion, as

.

And there you have it. Mass is energy!