Relativistic Energy, Momentum, and Mass Shells

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Rest mass is the invariant length of a future timelike four-momentum, whereas energy and three-momentum are components measured by a chosen observer. The mass shell ties those quantities together. For a multiparticle system, invariant mass comes from the total four-momentum rather than from adding constituent rest masses.

Required background. Calculus and Linear Algebra provides derivatives, contractions, and series expansions; Variational Mechanics and Symmetries supplies canonical momentum and the symmetry origin of conserved charges.

The single-particle mass shell

For a massive particle, the four-velocity and four-momentum are

U^\mu=\frac{dx^\mu}{d\tau} =\gamma(1,\mathbf v), \qquad p^\mu=mU^\mu=(E,\mathbf p).

Here $\mathbf v=d\mathbf x/dt$ , $|\mathbf v|\lt1$ , and

\gamma=(1-|\mathbf v|^2)^{-1/2}.

The normalization $U^2=-1$ gives

p^\mu p_\mu=-m^2, \qquad E^2=|\mathbf p|^2+m^2,

with $E=\gamma m$ and $\mathbf p=\gamma m\mathbf v$ . The physical mass shell is the future sheet $p^0=E\gt0$ ; note that lowering the index gives $p_0=-E$ . Tong derives four-momentum and its invariant norm in Tong 2013, §7.4.3, printed pp. 136–137 (PDF), using the opposite metric signature.

At low speed,

E=m+\frac12m|\mathbf v|^2+O(mv^4) =m+\frac{|\mathbf p|^2}{2m}+O(p^4/m^3).

Thus the Newtonian kinetic energy is the first correction to the rest energy. Restoring $c$ gives $E^2=|\mathbf p|^2c^2+m^2c^4$ .

For a massless particle, a nonzero future-null momentum obeys $p^2=0$ and $E=|\mathbf p|\gt0$ . It has no rest frame: no finite Lorentz boost turns a nonzero null vector into $(E,\mathbf0)$ . The null mass shell and this absence of a rest frame are treated in Tong 2013, §7.4.4, printed pp. 138–140 (PDF).

From particles to a system

For an asymptotically free incoming or outgoing particle state—well before or after a localized interaction, with any emitted quanta included—the total four-momentum is

P^\mu=\sum_a p_a^\mu.

This particle sum is not the general finite-time definition for an interacting system: matter and fields can carry four-momentum between identifiable particles. For a general isolated interacting system, choose a spacelike Cauchy surface $\Sigma$ with future-directed unit normal $n^\mu$ and define

P^\nu =\int_\Sigma T_{\rm total}^{\mu\nu}\,d\Sigma_\mu, \qquad d\Sigma_\mu\equiv-n_\mu\,d\Sigma,

where $T_{\rm total}^{\mu\nu}$ is the complete matter-plus-field stress-energy tensor. On an inertial $t={\rm constant}$ slice, this convention gives $P^\nu=\int T_{\rm total}^{0\nu}\,d^3x$ . Local conservation $\partial_\mu T_{\rm total}^{\mu\nu}=0$ and vanishing flux at the outer boundary make the charge independent of the chosen surface. Tong obtains energy-momentum as the charge of spacetime translations in Tong, Quantum Field Theory, §1.3.2, printed pp. 14–16 (PDF); the hypersurface integral is its covariant form.

When this total $P^\mu$ is future causal, define the system invariant mass by

M_{\rm sys}^2=-P^\mu P_\mu.

Thus $M_{\rm sys}^2\ge0$ . When $P$ is timelike, $M_{\rm sys}\gt0$ is the total energy in the center-of-momentum frame. When $P$ is nonzero null, $M_{\rm sys}=0$ and no such rest frame exists. The degenerate vector $P^\mu=0$ is invariant under every Lorentz transformation and is not a nonzero massless system momentum. In neither physical case is system mass generally the sum of constituent rest masses, because internal kinetic, binding, and field energies contribute.

For any chosen inertial observer $u^\mu$ , the total measured energy is $E_{(u)}=-u\cdot P$ , but $M_{\rm sys}$ is observer-independent. If $P$ is future timelike, a boost reaches the center-of-momentum frame:

P^\mu=(M_{\rm sys},\mathbf0).

Consequently, system mass includes every form of energy remaining in that frame. A hot box is more massive than the same box after it radiates and loses the radiation; a bound system’s mass includes its negative binding energy. An open subsystem can have a time-dependent invariant mass because four-momentum crosses its boundary.

Transfer check: two photons

Two photons each have energy $\mathcal E$ . Compute the system invariant mass when they move in opposite directions and when they move in the same direction.

Solution

For opposite directions along $x$ ,

p_1^\mu=(\mathcal E,\mathcal E,0,0), \qquad p_2^\mu=(\mathcal E,-\mathcal E,0,0).

Each photon is null:

p_1^2=p_2^2=-\mathcal E^2+\mathcal E^2=0.

Their total is

P^\mu=(2\mathcal E,0,0,0),

M_{\rm sys}^2=-P^2=4\mathcal E^2, \qquad M_{\rm sys}=2\mathcal E.

The center-of-momentum frame is already the displayed frame. Although each constituent has zero rest mass, the two-photon system has nonzero invariant mass.

If both photons move in the $+x$ direction, then

P^\mu=(2\mathcal E,2\mathcal E,0,0), \qquad P^2=0,

and $M_{\rm sys}=0$ . The system has no center-of-momentum rest frame. More generally, photons with energies $\mathcal E_1,\mathcal E_2$ and angle $\theta$ between their directions satisfy

M_{\rm sys}^2 =2\mathcal E_1\mathcal E_2(1-\cos\theta),

which reproduces both limits.

It is best to use “mass” for invariant mass and avoid “relativistic mass” for the frame-dependent energy. Other common failures are adding rest masses instead of four-momenta, choosing the past sheet $p^0\lt0$ for an ordinary incoming particle, or assuming every collection has a rest frame. A sum of future-directed physical particle momenta is future causal, but a selected open or effective subsystem requires its own consistency check.

The conservation of $P^\mu$ is not merely a kinematic rule. Poincaré Symmetry and Relativistic Conservation Laws derives its symmetry origin and distinguishes point-particle expressions from field charges.

References

Tong, David. Dynamics and Relativity. §§7.4.2–7.4.4, printed pp. 134–140, and §7.5, printed pp. 145–147 (PDF), University of Cambridge, Lent Term 2013. Tong uses $(+---)$ , so the mass-shell square has the opposite sign.
Tong, David. Quantum Field Theory. §1.3.2, “Spacetime Translations,” printed pp. 14–16 (PDF), University of Cambridge, Michaelmas Terms 2006 and 2007. The stress-energy formulas here are translated to the site’s $(-+++)$ convention.