Special Relativity: Dynamics: Energy and Momentum

Energy and Momentum

Note that when we used the term 'energy' we mean γmc², which is a particle's total energy. The particle's 'kinetic energy', however, is the excess energy due to its motion, over and above the energy it has when at rest: KE = γmc² - mc². Thus any particle has an amount of energy mc² when at rest; this is the famous mass-energy relationship which explains the energy release in many nuclear reactions, and explains, for example, why all stable nuclei have a mass that is less than their constituent particles. Because of this kinetic energy is not always conserved it a collision or decay: it is the total energy γmc², as we have seen, that is conserved.

There is also an extremely important relationship between energy and momentum:

E² - \|
= γ²m²c⁴1 -
= m²c⁴

Since m²c⁴ is a constant, independent of reference frame, the quantity E² - |

must also be frame invariant (the same in every inertial frame). Another important relation is that

The above equation suggests that there is a special relationship between energy and momentum. Consider a frame F' moving with speed v with respect to frame F along their mutual x/x'-direction (just like when we derived the Lorentz transformations). There is a particle in F' that has energy E' and momentum p' (and is moving also in the x-direction). What is E and p in the frame F? The answer looks very familiar:

ΔE = γ_v(ΔE' + vΔp')
Δp = γ_v(Δp' + vΔE'/c²)

γ_v is the γ factor associated with the relative speed between the frames (v). Not surprisingly these transformations look precisely like the Lorentz transformations between space and time in different frames. These equations also hold if E and p represent total energy and total momentum of a system of particles. Moreover they make it clear that if E and p are conserved in one frame, then they are conserved in any other inertial frame; this is very important to make the conservation laws we derived above meaningful. This arises just because E and p in one frame must be linear functions of E' and p' in another frame. Since the latter quantities are both conserved any linear function of them must be conserved also. Note that, just as with the spacetime transformations the above applies only to the x-direction (there is nothing special about x, except that we have arbitrarily chosen it to be our direction of motion) and p_y = p_y' and p_z = p_z'.

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