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Opus in profectus

Mass-Energy

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momentum and energy separately

These ideas are completely disorganized. Keep that in mind when reading this.

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 (relativistic momentum) and another different one for energy (relativistic energy).

The equation for relativistic 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 equation for relativistic energy looks like this…

E =  mc2
√(1 − v2/c2)

Applying the correspondence principal to give us the classical equations is not so easy here. Once again, at low speeds the denominator is one, but the numerator we're left with is something new. Something with no classical counterpart. Something famous.

v ≪ c  ⇒  E ≈ mc2

This equation says that an object at rest has energy, which is why it is sometimes called the rest energy equation. It also says that the reason an object at rest has any energy at all is because it has mass, which is why this equation is also known as the mass-energy equivalence.

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 zeroeth term is the rest energy.

E0 = mc2

The first term is the classical equation for kinetic energy.

E1 =  1  mv2
2

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

momentum and energy together

The momentum equation…

p =  mv
√(1 − v2/c2)

and the energy equation…

E =  mc2
√(1 − v2/c2)

have a common thing — the Lorentz factor, a.k.a. the relativistic gamma…

γ =  1
√(1 − v2/c2)

which means they can be written in a more compact form like this…

p = γmv

E = γmc2

For no immediately apparent reason, start with this expression…

E2 − p2c2

Replace energy and momentum with their gamma versions like this…

γ2m2c4 − γ2m2v2c2

The identity rule allows us to multiply the second term by 1 in the form of c2/c2.

γ2m2c4 − γ2m2v2c2(c2/c2)

Using the associative and commutative properties of multiplication, move things around in the second term.

γ2m2c4 − γ2m2(v2/c2)(c2c2)

Simplify a little.

γ2m2c4 − γ2m2(v2/c2)c4

And pull out like terms.

γ2m2c4(1 − v2/c2)

Notice that the stuff in parentheses is the reciprocal of γ2, which means everything cancels out except the stuff in the middle…

m2c4

Thus…

E2 − p2c2 = m2c4

or…

E2 = p2c2 + m2c4

This is the relativistic energy-momentum relation. For massed particles at rest we get the famous mass-energy relationship or the rest energy equation…

v = 0  ⇒  E = mc2

For massless particles like photons…

m = 0  ⇒  E = pc
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