The Physics
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Opus in profectus

# Mass-Energy

## Discussion

### the basic idea

These ideas are completely disorganized. Relativity has a different equation for (almost) everything. It's like classical physics just isn't good enough. There's a different one for time (time dilation) and a different one for space (length contraction) and now there's a different one for momentum and another different one for energy.

The relativistic equation for momentum looks like this…

 p = mv √(1 − v2/c2)

When v is small (as it is for the kinds of speeds we deal with in everyday life) the denominator is approximately equal to one and the equation reduces to its classical version…

 v ≪ c ⇒ p ≈ mv

which is as it should be. Relativity doesn't replace classical physics, it supplements it. All equations in special relativity should reduce to classical equations at low velocities. This is known as the correspondence principle.

The relativistic equation for energy looks like this…

 E = mc2 √(1 − v2/c2)

Applying the correspondence principal is not so easy here.

 v ≪ c ⇒ E ≈ mc2

This equation is like nothing in classical mechanics. It's very famous, but it has no relation to classical physics. Let's try a more sophisticated approach using the binomial series (sometimes called the Taylor series).

(a + b)n =
 ∞ ∑ k = 0

n
an − kbk
k

If we apply this to the denominator of the energy formula…

√(1 − v2/c2)

where…

 a = 1 b = −v2/c2 n = −½

we get…

 E = mc2 ⎛⎝ 1 + 1 v2 + 3 v4 + 5 v6 + 35 v8 + 63 v10 +… ⎞⎠ 2 c2 8 c4 16 c6 128 c8 256 c10

Multiply all the terms in the expansion by mc2. The first term is the rest energy.

E0 = mc2

The second term is the classical equation for kinetic energy.

 K = 1 mv2 2

the higher order terms are corrections that become more and more noticeable as the speed approaches the speed of light

relativistic mass

 m' = m √(1 − v2/c2)

for moving massed particles

E2 = p2c2 + m02c4

for massed particles at rest

E = m0c2

for massless particles

E = pc