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Units, Constants and Useful Formulas
Energy and Momentum of a Particle
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The four-vector formalism is both powerful and elegant and is used
to derive the important equation that relates the physical quantities
energy, mass and momentum in Special Relativity.
The 4-velocity is the rate of change of both time and space coordinates
with respect to the proper time of the object. The 4-velocity is a
tangent vector to the world line. Thus,
4-vector of velocity, u^{μ} = dx^{μ}/dτ
Time component:
x^{0} = ct = cγτ
dx^{0}/dτ = cγ
Spacial components:
dτ^{2} = dt^{2} - dx^{2}/c^{2}
Rewrite
dτ = √{(dt^{2} - dx^{2}/c^{2})/dt^{2}}dt
= √(1 - v^{2}/c^{2})dt
dt/dτ = 1/√(1 - v^{2}/c^{2})
dx/dτ = (dx/dt)(dt/dτ) = v_{x}/√(1 - v^{2}/c^{2}) same for y and z
The complete 4-vector is written notationally in terms of the time
and space components as:
dx^{μ}/dτ = {c/√(1 - v^{2}/c^{2}), v_{x}/√(1 - v^{2}/c^{2})
Alternatively, and generalizing to all directions
u^{μ} = {γc, γv}
Likewise we can identify the 4-momentum as
p^{μ} = {γmc, γmv}
Note: The relativistic mass is defined as γm where m is the invariant/rest/proper
mass. Some physicists reject this as not really meaningful, however.
Einstein recognized p^{0} when multiplied by c as the KE. Thus
E = mc^{2}/√(1 - v^{2}/c^{2})
For small v/c expand as binomial.
mc^{2}(1 - v^{2}/c^{2})^{-1/2} = (1 + v^{2}/2c^{2})mc^{2}
E = mc^{2} + mv^{2}/2
Which is agreement with Newton (rest energy + KE).
Relationship between Energy and Momentum
----------------------------------------
E^{2} = m^{2}c^{4}/(1 - v^{2}/c^{2})
Put c = 1 to simplify math.
E^{2} = m^{2}/(1 - v^{2})
p^{2} = m^{2}v^{2}/(1 - v^{2})
E^{2} - p^{2} = m^{2}
Add back in c to make it dimensionally correct.
E^{2} - p^{2}c^{2} = m^{2}c^{4}
where m (the rest or invariant mass) is the mass of a body that is isolated (free) and
at rest relative to the observer.
When p = 0, E = mc^{2}
For massless particle m = 0
E = √p^{2}c^{2} = pc