FOUR-VECTORS AND PROPER TIME

Section §02 left a nagging question. Time dilation: $\Delta t' = \gamma \Delta t$. Length contraction: $L' = L/\gamma$. Relativistic velocity-addition: a fraction with $\gamma$ hiding inside the denominator. Doppler shift: $\sqrt{(1+\beta)/(1-\beta)}$, which is $e^\eta$ in rapidity notation where $\eta = \mathrm{atanh}(\beta)$. Energy: $E = \gamma mc^2$. Momentum: $p = \gamma mv$. The same factor $\gamma$ haunts every formula. Why?

The answer is not a coincidence. It is a theorem about geometry. 's Cologne lecture of 1908 — three years after 's two postulates — stated the conclusion flat: space and time are not separately conserved quantities. They are coordinates in a 4-dimensional pseudo-Euclidean manifold, and $\gamma$ is not a kinematic correction factor. It is the cosine of a hyperbolic rotation angle in that manifold. Every §02 formula is a different projection of the same rotation onto a different axis.

§03.4 makes that precise. By the end of this topic you will have a single geometric object — the Four-vector — that carries all six §02 formulas as its components or contractions. The §02 algebra was the §03 geometry written out slowly, piece by piece, in coordinates.

In Newtonian mechanics the velocity of a particle is $\mathbf{v} = d\mathbf{r}/dt$: the rate of change of position with respect to lab time. This fails relativistically because $t$ is frame-dependent — different observers disagree on how much lab time has elapsed. The fix is to differentiate with respect to something every observer agrees on: Proper time $\tau$, the time elapsed on a clock carried by the particle itself.

The result is the Four-velocity:

where $dx^\mu = (c\,dt, dx, dy, dz)$ is an infinitesimal displacement in spacetime and $d\tau = dt/\gamma$ converts from lab time to proper time. The four components are: a time-like piece $u^0 = \gamma c$ and three spatial pieces $u^i = \gamma v_i$. At rest ($\beta = 0$, $\gamma = 1$) this reduces to $u^\mu = (c, 0, 0, 0)$: a particle at rest is still "moving through time" at speed $c$. liked to put it bluntly: everything moves through spacetime at $c$; what changes between observers is only the split between the time direction and the space directions.

The four-velocity is a Lorentz covariant object: every component transforms under a Lorentz boost by the same rule as $(ct, x, y, z)$. That covariance is why $\gamma$ appears in every relativistic formula — any quantity built from four-vectors inherits the same transformation law, and $\gamma$ is the boost's natural parameter.

The time coordinate $t$ in the lab is observer-dependent. The Proper time $\tau$ is not. It is the arc length along a World-line in the Minkowski metric, and every observer computes the same number when they integrate it along the same worldline.

The relation is:

For a particle moving at constant $\beta$, integrating over a lab time interval $\Delta t$ gives $\Delta\tau = \Delta t / \gamma$ — exactly the §02.1 time-dilation formula, re-derived as a geometric statement: the clock carried by the particle measures a shorter arc length along its kinked worldline than the straight coordinate time axis measures. There is no mechanism or force making the moving clock tick slow. The Minkowski metric simply assigns less arc length to that path. The §02 "moving clocks run slow" is the same sentence as "a straight line is the longest geodesic between two events in Minkowski spacetime."

This is the §03 reformulation of §02.1: not "moving clocks run slow because of something physics does to them" but "the Minkowski metric assigns less proper time to a non-geodesic path." The same integral, taken over an arbitrarily curved worldline, gives the §03.5 twin paradox result without any additional assumptions.

The Minkowski norm-squared of the four-velocity is:

The Minkowski norm of the four-velocity is $c^2$, identically, for every particle in every inertial frame at every velocity. It is a Lorentz Invariant interval: a scalar that every observer in every frame computes to be the same value. The four-velocity vector is not an ordinary 3D arrow that gets length-contracted or rotated; it lives on the upper sheet of the hyperboloid $u^\mu u_\mu = c^2$ in four-dimensional velocity space, and a Lorentz boost slides the vector along that hyperboloid without changing its Minkowski length.

This is the geometric statement of "everyone moves through spacetime at $c$." A particle at rest has all of its $c$-worth of spacetime velocity directed along the time axis: $u^\mu = (c, 0, 0, 0)$. A particle moving at $\beta c$ redirects some of that velocity into the spatial direction: $u^\mu = (\gamma c, \gamma \beta c, 0, 0)$. The total Minkowski speed stays $c$. The §02 $\gamma$ factor is the geometry of that redistribution.

The scene above makes this literal. The rapidity $\eta = \mathrm{atanh}(\beta)$ is the single parameter that controls the hyperbolic rotation. As $\eta$ increases from $0$ to $\infty$, the four-velocity vector sweeps the hyperboloid $u^\mu u_\mu = c^2$. Each §02 formula is a monotone function of the same $\eta$. There is one rotation. There are six projections. The §02 chapter was a guided tour of six windows onto the same geometric room.

The electromagnetic analogue lives in §11.5 of the electromagnetism module. The Four-potential $A^\mu = (\phi/c, \mathbf{A})$ is exactly the same construction applied to the EM field: a four-component object that transforms under Lorentz boosts by the same rule as $(ct, x, y, z)$, with a Lorentz-invariant norm related to the gauge freedom. See the EM application for the canonical analogue of the four-velocity machinery built here.

Two consequences follow immediately from what §03.4 established.

The first is the twin paradox. If proper time is the Minkowski arc length along a worldline, then two worldlines connecting the same pair of events accumulate, in general, different amounts of proper time. The straight worldline (the stay-at-home twin) is the longest geodesic; the kinked worldline (the traveling twin) is shorter. §03.5 works this out explicitly: the kinked worldline has less proper time not because of any force or physiological effect on the traveler, but because the Minkowski metric measures less arc length along the broken path.

The second is the four-momentum. If Four-velocity $u^\mu = \gamma(c, \mathbf{v})$ is the geometric generalization of 3-velocity, then the obvious relativistic generalization of momentum is $p^\mu = m u^\mu = \gamma m(c, \mathbf{v})$. The time component is $p^0 = \gamma mc$ — and $p^0 c = \gamma mc^2$ is the total energy, which in the rest frame reduces to $E_0 = mc^2$. The mass-energy equivalence of §04.2 is the $\mu = 0$ component of the four-momentum norm. §04.1 builds this in full. The same four-vector machinery that gave us proper time gives us $E = mc^2$ as a component identity, not a separate postulate.