Minkowski generalized Newton’s first law (of inertia) by noticing
that a free particle, which is at (relative) rest or moves by inertia, has a
straight timelike worldline.
The frame-independent distinction between inertial motion and
acceleration is reflected in the straightness and curvature of particle
worldlines in Minkowski space.
A free particle, either at (relative) rest or moving by inertia, has a
straight timelike worldline in Minkowski space. This picture of straight
timelike worldlines representing inertial motion provides the insight
that there is no absolute distinction between a state of rest and a state of
uniform motion, because there is no absolute distinction between two straight
lines. This is the geometric interpretation of Newtons first law (law of
inertia) which remains valid in relativity theory. Indeed, if a particle moves
uniformly in one inertial frame, it does so in all other inertial frames.
Fig.8.1 Two particles in a state of
uniform motion and a state of rest (straight worldlines A and B), in
contrast to the curved worldline C.
In Newtonian mechanics the law of inertia is expressed by the
equation
, with
the mass of the particle. We can equally write
8.5
This defines the relativistic momentum which, in the absence of
force, is a conserved quantity because of the homogeneity of space; see Poincaré
Group.
Given a standard mass, a mass
can be assigned to any other particle by colliding it at
low speed with the standard mass and applying the Newtonian law of
conservation of momentum. Since this becomes exact as the velocities go to
zero, an observer can in principle use this limiting procedure to measure
of a particle at rest.
The mass
(also called rest mass) in equation (7.5) is an
intrinsic property of a particle and an invariant scalar quantity by
definition.
In contrast, the so-called inertial mass
is different in different
inertial coordinate systems. Mass and inertial mass are only equal for a
particle at rest.
