A general expression is obtained for the space-time interval between neighboring events in a one-dimensional space in which it is possible to set up a rigid reference frame. Particular expressions are then obtained for the interval for the special cases of a rigid frame at rest in a uniform gravitational field and a rigid frame uniformly accelerating in field-free space. The two expressions are not equivalent and are used to show why, how, and to what extent observations made in a rigid enclosure at rest in a gravitational field are not equivalent to observations made in a rigid enclosure that is uniformly accelerating in field-free space. Two facts of particular interest that are demonstrated in the course of the analysis are the following: (i) Two spatially separated particles that are simultaneously released from rest and allowed to fall freely in a uniform gravitational field will not remain at rest with respect to one another. (ii) Uniformly accelerating reference frames and inertial frames are the only possible one-dimensional rigid frames in flat space-time.
A uniformly accelerated reference frame S is defined as a set of observers who remain at rest with respect to a given observer A who is accelerating at a constant rate with respect to the instantaneously comoving inertial frames. The one-dimensional uniformly accelerated reference frame S is considered. The world lines of A and the other observers making up S are determined. Coordinates useful for describing events in S are carefully defined and the transformation equations between different sets of them are derived. The variation with position in S of the speed and frequency of light waves is determined. The motion of a free-particle in S is determined. Various phenomena in S, ordinarily associated with general relativity, are considered, in particular the asymmetric aging of twins at rest at different positions and the existence of horizons.
The meaning of spatial geometry in a reference frame is carefully analyzed. It is shown that in a uniformly accelerating reference frame spatial geometry is Euclidean if distance is measured with measuring rods and non-Euclidean if distance is measured with light signals. The distance function and the square of the line element associated with each mode of measurement are obtained.
A simple statement of Noether’s theorem as it applies to classical mechanics is given and proved. It is then used to generate constants of the motion associated with Lagrangians possessing certain transformation properties.
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