### Question

I read somewhere that Einstein's formula is mostly quoted in the abbreviated version E=mc^{2} but it should actually be written E^{2}=m^{2}c^{4}. Is that true?

Asked by: Bret W.

### Answer

The expression E = mc^{2} 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 = [(p^{2}c^{2}) + m^{2}c^{4}]^{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:

E^{2} = (p^{2}c^{2}) + (m^{2}c^{4}).

Notice that you get E^{2} = m^{2}c^{4} 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=mc^{2}

b. E^{2} = m^{2}c^{4}

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