4-momentum (idea)

4-momentum is the generalization of momentum to the 4-vector formalism used in relativistic mechanics. The 4-momentum includes both the relativistic momentum and the mechanical energy in a single, mathematically-convenient form.

Definition and Rationale

Ordinary non-relativistic momentum is defined as p=mv, where m is the mass of the particle and v is its velocity. The velocity is simply the time derivative of position, dx/dt. Momentum by this definition is both simple and useful and appears in numerous places in classical mechanics. Things become more complicated once relativity is included. In a relativistic theory, time does not flow the same in different reference frames, complicating things considerably. It is possible to derive relatively complicated formulae that relate ordinary velocities in different frames of reference, but people hoped for a better way.

The solution is to define the velocity in terms of a specific timescale, that of the particle's rest frame, referred to as its proper time. This is something of an odd concept, since it involves considering the position in one frame relative to the time in another frame. In fact, in the frame in which we consider the time, the velocity is by definition zero. Unfortunately, this is the price that must be paid for an invariant defintion of time. Fortunately, the proper time is related to the time in another frame fairly simply by the time dilation formula; our time t above is related to the proper time τ by t=τ/γ where γ is a velocity-dependent factor common in relativistic mechanics. So we have a new definition of velocity, u = dx/dτ = γ(dx/dt), called the relativistic velocity, and thus we can relativistically define momentum as p=mu.

The astute observer will note that there's nothing 4-ish about this quantity; there are three spatial dimensions so these vectors are three-dimensional objects. In relativity, space and time are linked, so we really should be considering time as a coordinate as well. We can't just replace x=(x,y,z) with x^μ=(t,x,y,z), because time has different units: seconds rather than meters. To make everything into a length, we can simply multiply the time by the speed of light c, making the 4-vector position x^μ=(ct,x,y,z).

We can now define a 4-velocity in terms of this; u^μ=dx^μ/dτ, and thus the 4-momentum as p^μ=mu^μ. It so happens that this works out to be p^μ=γm(c,v_x,v_y,v_z), where v_i is the component of the ordinary non-relativistic velocity in the i-direction (where i stands for any of x, y, or z). The latter three components are the relativistic momentum defined two paragraphs above, and the first (time) component is the relativistic energy, divided by the speed of light to match units.

Interpretation and Utility

4-vector quantities have three especially convenient features. The first is that they all transform identically, through the ordinary Lorentz transform used for position, 4-position being the archetype of all 4-vectors. The second property is that an equation involving only 4-vectors and invariant quantities is automatically true in all reference frames dealt with by special relativity. The third property, and the one that is most noteworthy, is that the product of any two 4-vectors is an invariant, so it has the same value regardless of the reference frame used. In particular, the product of a 4-vector with itself, its "length", remains constant regardless of the frame of reference.

In the case of 4-momentum, the product p^μp_μ is m²c². This is very useful, as the mass is a particularly simple property of a body. In general, in calculations using the conservation of 4-momentum, multiplying by one of the 4-momenta simplifies calculations by allowing each term of the equation to be evaluated in whatever reference frame is most convenient (often the center of mass frame, or the rest frame of one of the bodies). This allows us to avoid the cumbersome machinery of the Lorentz transform, while simplifying large systems of equations.

Conclusion

While 4-momentum may not be the most intuitive concept in physics, its utility and compatibility with the concepts of special relativity make it appear relatively frequently. In modern, fully-relativistic theories such as quantum electrodynamics and quantum field theory, it is the only definition of momentum that appears, and even string theory uses a similar definition, extended to its 10, 11, or 26-dimensional spacetime.