The easiest way in which to answer this question is to consider the acceleration 4-vector [see Equation (1726)]

(1836) |

Using the standard transformation, (1694)-(1697), for 4-vectors, we obtain

(1837) | ||

(1838) |

Equation (1839) can be written

where is the constant force (in the -direction) acting on the particle in .

Equation (1841) is equivalent to

where

Thus, we can account for the ever decreasing acceleration of a particle subject to a constant force [see Equation (1839)] by supposing that the inertial mass of the particle increases with its velocity according to the rule (1843). Henceforth, is termed the

The rate of increase of the particle's energy satisfies

(1842) |

This equation can be written

(1843) |

which can be integrated to yield Einstein's famous formula

(1844) |

The 3-momentum of a particle is defined

(1845) |

where is its 3-velocity. Thus, by analogy with Equation (1842), Newton's law of motion can be written

(1846) |

where is the 3-force acting on the particle.

The 4-momentum of a particle is defined

(1847) |

where is its 4-velocity. The 4-force acting on the particle obeys

(1848) |

where is its 4-acceleration. It is easily demonstrated that

(1849) |

because

(1850) |