is the
metric tensor of
special relativity with
metric signature for definiteness chosen to be (–1, 1, 1, 1). The negativity of the norm reflects that the momentum is a
timelike four-vector for massive particles. The other choice of signature would flip signs in certain formulas (like for the norm here). This choice is not important, but once made it must for consistency be kept throughout.
The Minkowski norm is Lorentz invariant, meaning its value is not changed by Lorentz transformations/boosting into different frames of reference. More generally, for any two four-momenta p and q, the quantity p ⋅ q is invariant.
Relation to four-velocity
For a massive particle, the four-momentum is given by the particle's
invariant massm multiplied by the particle's
four-velocity,
where the four-velocity u is
and
is the Lorentz factor (associated with the speed ), c is the
speed of light.
Derivation
There are several ways to arrive at the correct expression for four-momentum. One way is to first define the four-velocity u = dx/dτ and simply define p = mu, being content that it is a four-vector with the correct units and correct behavior. Another, more satisfactory, approach is to begin with the
principle of least action and use the
Lagrangian framework to derive the four-momentum, including the expression for the energy.[2] One may at once, using the observations detailed below, define four-momentum from the
actionS. Given that in general for a closed system with
generalized coordinatesqi and
canonical momentapi,[3]
it is immediate (recalling x0 = ct, x1 = x, x2 = y, x3 = z and x0 = −x0, x1 = x1, x2 = x2, x3 = x3 in the present metric convention) that
is a covariant four-vector with the three-vector part being the (negative of) canonical momentum.
Observations
Consider initially a system of one degree of freedom q. In the derivation of the
equations of motion from the action using
Hamilton's principle, one finds (generally) in an intermediate stage for the
variation of the action,
The assumption is then that the varied paths satisfy δq(t1) = δq(t2) = 0, from which
Lagrange's equations follow at once. When the equations of motion are known (or simply assumed to be satisfied), one may let go of the requirement δq(t2) = 0. In this case the path is assumed to satisfy the equations of motion, and the action is a function of the upper integration limit δq(t2), but t2 is still fixed. The above equation becomes with S = S(q), and defining δq(t2) = δq, and letting in more degrees of freedom,
Observing that
one concludes
In a similar fashion, keep endpoints fixed, but let t2 = t vary. This time, the system is allowed to move through configuration space at "arbitrary speed" or with "more or less energy", the field equations still assumed to hold and variation can be carried out on the integral, but instead observe
where L is the relativistic
Lagrangian for a free particle. From this,
glossing over these details,
The variation of the action is
To calculate δds, observe first that δds2 = 2dsδds and that
So
or
and thus
which is just
where the second step employs the field equations duμ/ds = 0, (δxμ)t1 = 0, and (δxμ)t2 ≡ δxμ as in the observations above. Now compare the last three expressions to find
with norm −m2c2, and the famed result for the relativistic energy,
where mr is the now unfashionable
relativistic mass, follows. By comparing the expressions for momentum and energy directly, one has
that holds for massless particles as well. Squaring the expressions for energy and three-momentum and relating them gives the
energy–momentum relation,
It is also possible to derive the results from the Lagrangian directly. By definition,[5]
which constitute the standard formulae for canonical momentum and energy of a closed (time-independent Lagrangian) system. With this approach it is less clear that the energy and momentum are parts of a four-vector.
The energy and the three-momentum are separately conserved quantities for isolated systems in the Lagrangian framework. Hence four-momentum is conserved as well. More on this below.
More pedestrian approaches include expected behavior in electrodynamics.[6] In this approach, the starting point is application of
Lorentz force law and
Newton's second law in the rest frame of the particle. The transformation properties of the electromagnetic field tensor, including invariance of
electric charge, are then used to transform to the lab frame, and the resulting expression (again Lorentz force law) is interpreted in the spirit of Newton's second law, leading to the correct expression for the relativistic three- momentum. The disadvantage, of course, is that it isn't immediately clear that the result applies to all particles, whether charged or not, and that it doesn't yield the complete four-vector.
It is also possible to avoid electromagnetism and use well tuned experiments of thought involving well-trained physicists throwing billiard balls, utilizing knowledge of the
velocity addition formula and assuming conservation of momentum.[7][8] This too gives only the three-vector part.
Conservation of four-momentum
As shown above, there are three conservation laws (not independent, the last two imply the first and vice versa):
The four-
momentump (either covariant or contravariant) is conserved.
The
3-spacemomentum is conserved (not to be confused with the classic non-relativistic momentum ).
Note that the invariant mass of a system of particles may be more than the sum of the particles' rest masses, since
kinetic energy in the system center-of-mass frame and
potential energy from forces between the particles contribute to the invariant mass. As an example, two particles with four-momenta (5 GeV/c, 4 GeV/c, 0, 0) and (5 GeV/c, −4 GeV/c, 0, 0) each have (rest) mass 3GeV/c2 separately, but their total mass (the system mass) is 10GeV/c2. If these particles were to collide and stick, the mass of the composite object would be 10GeV/c2.
One practical application from
particle physics of the conservation of the
invariant mass involves combining the four-momenta pA and pB of two daughter particles produced in the decay of a heavier particle with four-momentum pC to find the mass of the heavier particle. Conservation of four-momentum gives pCμ = pAμ + pBμ, while the mass M of the heavier particle is given by −PC ⋅ PC = M2c2. By measuring the energies and three-momenta of the daughter particles, one can reconstruct the invariant mass of the two-particle system, which must be equal to M. This technique is used, e.g., in experimental searches for
Z′ bosons at high-energy particle
colliders, where the Z′ boson would show up as a bump in the invariant mass spectrum of
electron–
positron or
muon–antimuon pairs.
If the mass of an object does not change, the Minkowski inner product of its four-momentum and corresponding
four-accelerationAμ is simply zero. The four-acceleration is proportional to the proper time derivative of the four-momentum divided by the particle's mass, so
Canonical momentum in the presence of an electromagnetic potential
This, in turn, allows the potential energy from the charged particle in an electrostatic potential and the
Lorentz force on the charged particle moving in a magnetic field to be incorporated in a compact way, in
relativistic quantum mechanics.
^Taylor, Edwin; Wheeler, John (1992). Spacetime physics introduction to special relativity. New York: W. H. Freeman and Company. p. 191.
ISBN978-0-7167-2327-1.
Landau, L.D.; Lifshitz, E.M. (2000). The classical theory of fields. 4th rev. English edition, reprinted with corrections; translated from the Russian by Morton Hamermesh. Oxford: Butterworth Heinemann.
ISBN9780750627689.