Four Causal Classes of Newtonian Frames

Abstract

The causal characters (spacelike, lightlike, timelike) of the coordinate lines, coordinate surfaces and coordinate hypersurfaces of a coordinate system in Relativity define what is called its causal class. It is known that, in any relativistic space-time, there exist one hundred and ninety nine such causal classes. But in Newtonian physics (where only spacelike and timelike characters exist) the corresponding causal classes have not been discussed until recently. Here it is shown that, in sharp contrast with the relativistic case, in Newtonian space-time the different causal classes of coordinate systems are drastically reduced to four. These causal classes admit simple geometric descriptions and physical interpretations. For example, it is shown that one can generate coordinate systems of the four causal classes by means of the sole linear synchronization group, i.e. by coordinate transformations that only change the origin of time linearly. The relativistic analogs of these examples are also considered.

Relativistic versus Newtonian Frames

“…As a consequence, the number of causally different Newtonian classes of frames is equal to the dimension of the space-time. Hence, see [7] …”

“…In summary, it can be shown (see [7]) that one has the following implications valid only for Newtonian frames:…”

“…In fact, see [7], in any space-time every coordinate x  plays two extreme roles: that of a hypersurface for every constant value , of gradient con Only when both causal orientations coincide, it is possible to extend to the coordinate x  itself the character of the common causal orientation of its two mentioned variations.…”

Emission and null coordinates: geometrical properties and physical construction

“…The ordered set of these causal characters is called the causal class of the coordinate system. It is already a well known result that the number of different causal classes of coordinate systems in Relativity is 199 [11] while in Newtonian physics there exists only four causal classes [12].…”

Spherical symmetric parabolic dust collapse: 𝒞1 matching metric with zero intrinsic energy

“…Moreover, these 3-spaces are the level surfaces of the A function of this solution because it is a function of the sole proper time τ , A(τ ). This means that the gradient dA is everywhere timelike, g D (dA, dA) < 0, on the whole metric domain (which is a T -region,30 i. e, the Datt function A defines a timelike gradient coordinate according to the parlance used in Refs 32,33)…”