Four-vectors and Minkowski contractions

Use Cartesian inertial coordinates $x^\mu=(ct,x,y,z)$ and the metric $\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$. Greek indices run from zero through three; spatial indices run from one through three. Time is multiplied by the vacuum light speed so every coordinate has units of metres. A four-vector is defined by its Lorentz transformation law, rather than by the fact that somebody has arranged four numbers in a column. This distinction will matter when electric and magnetic components are assembled into a tensor.

Lowering an index means contracting with the metric. Thus the spatial components change sign, while the time component does not. A repeated index is summed when it appears once up and once down. Two upper indices do not define a scalar contraction without an additional metric. These rules prevent an accidental Euclidean sum from being mistaken for an invariant. They also tell us that the derivative has lower components $\partial_\mu=(c^{-1}\partial_t,\nabla)$; raising its index changes the spatial derivative signs.

Let the primed observer move at speed $V$ along positive x relative to the unprimed observer. Our passive boost has $\beta=V/c$ and negative off-diagonal entries. The inverse boost replaces beta by its negative. Stating which observer moves avoids the common mistake of using a correct boost matrix for the opposite transformation. Expanding the two transformed squares shows that the cross terms cancel and the remaining coefficient is $\gamma^2(1-\beta^2)=1$. This algebra explains interval preservation directly.

With this signature a positive squared interval is timelike, a negative one spacelike, and zero null. Along an infinitesimal timelike worldline displacement, $d\tau=\sqrt{dx^\mu dx_\mu}/c$. For a finite endpoint separation, its norm divided by c equals elapsed clock time only on the connecting inertial worldline. An accelerated clock requires integrating the proper-time increments along its actual path. A spacelike separation cannot be turned into a timelike one by boosting. Its coordinate time may change sign, but its causal classification cannot. Null separation does not mean the events coincide: a light signal can connect different events with zero interval.

For a massive particle define $u^\mu=dx^\mu/d\tau=\gamma(c,\mathbf v)$, where tau is its own proper time. Its norm is always $c^2$, even when its ordinary velocity changes. Four-momentum is $p^\mu=mu^\mu=(\mathcal E/c,\mathbf p)$ for fixed rest mass m. The invariant mass shell follows by contraction. Differentiating the fixed four-velocity norm also gives $u_\mu a^\mu=0$. Four-acceleration is orthogonal to four-velocity in Minkowski geometry; it need not be perpendicular to ordinary velocity in a spatial diagram.

An observer with normalized four-velocity $w^\mu$ measures particle energy $p_\mu w^\mu$. In that observer's rest coordinates this reduces to c times the particle's temporal momentum component. Energy is therefore an observer-dependent contraction involving two physical objects, whereas rest mass belongs to the particle alone. Massless particles still have meaningful four-momentum and zero momentum norm, but cannot be assigned a rest-frame proper-time four-velocity. Applying the massive definition to a photon would divide by a vanishing clock increment.