Of Particular Significance

# Chapter 8, Endnote 9

• Quote: Ironically, most particle physicists don’t use either of these two simple interpretations of the relativity formula. We use a third one!

• Endnote: Taking E to be total energy and m to be rest mass, we view E = m[c2] as true only for stationary objects. Otherwise E > m[c2]; in words, the total energy of a moving object always exceeds its internal energy (by an amount that can be easily expressed in terms of the quantity called momentum.)

We particle physicists write

• $E^2=(pc)^2 + (mc^2)^2 \ \ \ (1)$

where p is momentum. In terms of the object’s speed v, we don’t use Newton’s formula for momentum, p = mv, but instead we use an Einsteinian version,

• $p = (E/c^2) v \ . \ \ \ (2)$

Note: this is not the same as Newton’s formula p = mv, because we particle physicists use equation (1), which says that for a moving object E is NOT equal to mc2 ! [Got to keep on your toes, here!]

Instead, combining equations (1) and (2), we find

• $E^2=(Ev/c)^2 + (mc^2)^2 \ .$

Bringing the second term to the left-hand side and factoring.

• $E^2 (1- v^2/c^2) = (mc^2)^2 \ .$

Solving for E gives

• $E = m c^2/ \sqrt{1 - v^2/c^2} \ . \ \ \ (3)$

a formula which appears in endnote 7 of this chapter and the ensuing discussion.

Let’s return now to equation (1). For a moving object, v is non-zero, so p is non-zero too, and p2 is positive. Therefore

• $E^2=(pc)^2 + (mc^2)^2 > (mc^2)^2 \ ,$

and thus, taking the square root of both sides (and remembering both motion energy and mass are always positive), we find that

• $E > mc^2$

for a moving object. This can also be seen from equation (3); if v is non-zero, then (v/c)2 is positive but less than 1, so the square root is less than 1, and one over the square root is greater than 1.

Meanwhile for a stationary object, v and p are zero, so returning to equation (1) and taking the square root, or simply using equation (3), we find

• $E = mc^2 \ .$

[See also this article in which I discuss and give intuition as to how Einstein’s formulas were a sensible generalization of Newton’s formulas.]

Search