I read somewhere that Einstein's formula is mostly quoted in the abbreviated version E=mc² but it should actually be written E²=m²c⁴. Is that true?

Asked by: Bret W.

Answer

The expression E = mc² is the rest energy of an object of rest mass m. In motion, the object's total energy is a sum of its rest energy and its kinetic energy. That is

E = [(p²c²) + m²c⁴]^1/2

where p is the object's momentum. You get this more general equation for energy using something called the momentum-energy 4-vector, which you will learn more about if you study special relativity. The 4 vector contains four components: three spatial components of momentum and one time component of momentum. E is just the magnitude of this vector. If you square both sides of the equation, you get:

E² = (p²c²) + (m²c⁴).

Notice that you get E² = m²c⁴ if the object has zero momentum; that is, if the object is at rest.
Answered by: Philip Zell, Ph.D. Physics, ACT, Inc.

The 2 equations:

a. E=mc²
b. E² = m²c⁴

differ only if there is a physical meaning to negative mass and/or energy values. In a. a negative mass value implies a negative energy value only. In b. the same negative mass value has both negative and positive energy solutions. Also, b. allows for both positive and negative solutions for energy even if mass is positive.

Whether negative mass and energy values have any useful connection to reality is another question.
Answered by: Paul Walorski, A.B.Physics