Fundamentals of Time and Relativity

In the Minkowski space of spacetime coordinates $\{x^0,x^1,x^2,x^3\}=\{t,x,y,z\}$, the motion of a mass point in an inertial frame is described by a timelike worldline $x^{\mu }(\tau ),\mu =0,1,2,3$ with the proper time $\tau$ as time parameter; the velocity of light is set equal to one.

To simplify the notation, it is convenient to introduce a set of base vectors ${\text{{e}}_{\mu }\text{}}$ that satisfy the orthonormality relation ${\text{e}}_{\mu }\cdot {\text{e}}_{\nu }=g_{\mu \nu }$. The dot indicates the Minkowski inner product. Relative to the chosen origin of the inertial frame, the world line then has the coordinate- independent vector representation

The four-velocity $u(\tau )$ and the four-acceleration $a(\tau )$ are defined as the derivatives of $x(\tau )$ with respect to the proper time:

The norm of the four-velocity is calculated as the Minkowski product

Here the Lorentz factor is the function

 of the three-velocity $v=dx/dt$.

Because the norm of the four-velocity is a constant, the velocity four-vector is orthogonal to the four-acceleration: $d{u}^2/d\tau =2u(\tau )\cdot a(\tau )=0$. In an instantaneously co-moving inertial reference frame, we have $u=0$, $\gamma =1$. It follows that $a(\tau )\cdot a(\tau )=-a^2$ where $a$ is the acceleration measured in the instantaneous rest frame; that is, the acceleration felt by an observer moving with the particle.