Fundamentals of Time and Relativity

Let us consider the four-momentum $p^{\mu }=m(u^0,u)$ of a relativistic particle. The temporal component is the energy of the particle. Thus, we can write $p^{\mu }=(E,p)$ which is usually referred to as the energy-momentum vector. Since the invariant mass $m$ is positive, the energy-momentum vector is time-like and satisfies the mass-shell equation:

This implies the relativistic energy-momentum relation

The four-momentum $p(\tau )$ is a vector in Minkowski space tangent to the worldline $x(\tau )$ under consideration. We can picture this four-momentum as a vector in the tangent space, i.e., a Minkowski space with its origin at the spacetime point $x(\tau )$. The orientation of the orthonormal basis ${\text{{e}}_{\mu }\text{}}$ in this tangent space can be freely chosen, but two choices are rather natural: (i) alignment with the base vectors of the Minkowski space of spacetime points (observer frame), (ii) alignment of ${\text{e}}_0(\tau )$ with the tangent to the worldline at $x(\tau )$ (co-moving frame). This is illustrated in the Minkowski diagram below.

Particle refers here to an idealized model of a quantity of energy that is localized and stable (within some relevant time and energy scale). A composite body, like a proton, a grain of solid or a celestial body, can also be conceptualized as a particle, as long as localization and stability are principal features. The rest mass is then a complex sum of energies of the moving constituent particles, binding energies, energies from internal reaction processes, etc.