Here we expand 4D Galilean spacetime into 6D Galilean spacetime, based on section 1.3 Galilean spacetime of The Geometry of Relativistic Spacetime: from Euclid’s Geometry to Minkowski’s Spacetime by Jacques Bros (Séminaire Poincaré 1 (2005) 1 – 45).
[p.3] We start with a representation space whose points are interpreted as the “physical events”. Any motion of a particle which is physically possible between two given events A and B is represented by a certain world-line with end-points A and B. There is an absolute orientation of such worldline, which can be called its “time-arrow”: its physical meaning is that one of the end-point events, e.g. B, is in the future of the other one A.
[p.6] From the viewpoint of mathematical physics, the use of geometry in more than three dimensions turns out to be necessary, if one wishes to represent phenomena whose description necessitates more than three independent quantities. A typical example is the six dimensional space Rab6 = Ra3 × Rb3 of the positions (a; b) of pairs of material points (or pointlike particles) in mutual interaction. Trajectories of such pairs are represented by curves in R6, described in terms of a parameter t by equations of the form a = a(t); b = b(t).
[p.7] In the following, we shall be concerned with a very special type of geometry of representation, called spacetime, whose purpose is to provide a visualization of the motion phenomena throughout their whole history. If we consider motions in the Euclidean space R3, we need additional time-coordinates and therefore an affine space R6 for representing geometrically all the events of the world. Such a map is intended to picture in an idealistic way the whole history of the world: the motion of any material point (or of any observer) will be represented as a curve, called a world-line, which describes all its history from the remote past to the far future. The usual notion of trajectory will then appear as the projection of the world-line onto the Euclidean space R3 . The world-line is a geometrical concept which contains all the information on the motion, which is not the case for the trajectory: two different worldlines (i.e. motions) may project onto the same trajectory.
6D Galilean spacetime as a geometry of representation of motion phenomena
In its simplest form, which we shall call 6D Galilean spacetime, the concept of spacetime appears as a geometry of representation for the phenomena of motion, relative to an observer called O0, with the following premise: the distance or distime between two events A and B is an absolute quantity; its value is the same for observers moving in an arbitrary way between A and B, provided they are equipped with identical clocks.
With the previous notation, x = x now denotes a point, or equivalently three coordinates called space coordinates, in the usual Euclidean space R3, while y ≡ t denotes an instant . A point X = (x; t) in R6 represents the event which takes place at instant t at the point x of Euclidean space R3. In particular, the origin O0 represents the event called “here and now” (at a certain instant) by the observer O0, who stands “at rest” at x = 0; by definition, this means that the observer’s worldline the time axes with equation x = 0. For O0, the coordinate hyperplane with equation t = 0 represents the set of all simultaneous events which constitute the “present”. Similarly, every fixed value t0 of t the hyperplane with equation t = t0 is a complete set of simultaneous events, which we call the set of simultaneity and which belongs to the future or to the past according to whether ¦t¦ is positive or negative [¦t¦ = ±|t| depending on the axis of order]. The whole future and the whole past are represented respectively by the open half-spaces ¦t¦ > 0 or ¦t¦ < 0 of R6. In such a representation of the events, one says that the time axes associated with the Euclidean space R3 of “present events” constitute the reference frame of the observer O0, (the choice of the “present time” t = 0 is of course a matter of convention for O0).
Let Ov be an observer in uniform motion with vector velocity v with respect to O0 and passing by O, which means that he shares with O0 the same and unique event that we called “here and now”. The time axes v for this observer is defined by the corresponding worldline, namely the straightline with (vector) equation x = v |t|.
For any such observer, the sets of simultaneity t = t0 are the same as for the observer O0. More precisely, every event M = (x; t) of spacetime is perceived by the observer Ov as having coordinates (x´; t´) such that x´ = x − v |t| and t´ = t. This change of coordinates from O0 to Ov is also called a Galilean transformation; it implies the basic property of additivity of velocities: a uniform motion with worldline x = v |t| is seen by Ov as a uniform motion with equation x´ = v |t|, with velocity vector v´ = v − v0. For example, in a train whose velocity is v0 = 100 kmh, a passenger walking longitudinally with velocity v´ = 5 kmh has a velocity with respect to the earth which is v = 105 kmh or 95 kmh according to whether the forward or backward direction of the train has been chosen by that passenger…
We note that the Galilean changes of coordinates do not preserve the notion of orthogonality in R6. If for convenience we choose to represent the simultaneity sets as “horizontal spaces” (the dimension of space being unfortunately reduced to two in our visual perception…) and the time axes of the observer at rest O0 by vertical space, the reference frame for O0 will associate the [p.8] oblique time axes v with the horizontal space. But the observer at rest enjoys no special physical properties with respect to any other observer in uniform motion (that’s the “Galilean principle of relativity” which follows from the law of inertia). So the verticality of the time axes could have been chosen for representing the worldline of any given uniform motion: there is nothing deep in that choice. One can also say that the Galilean spacetime is defined for O0 up to the arbitrariness in the choice of the time axes or in mathematical terms up to a Galilean transformation: it is the equivalence class of all these representations. But the same representation of spacetime is then also acceptable by any observer Ov in uniform motion, which expresses precisely in geometrical terms the content of the Galilean relativity principle.
Here it is also worthwhile to point out that, in contrast with the “horizontal” Euclidean subspaces R3, the Galilean spacetime R6 is only an affine space; it is not equipped with any physically sensible global notion of orthogonality and distance. But this is consistent with our standard perception: why would space and time strangely mix each other in some supergeometry? Galilean spacetime is just a geometry of representation in a very poor sense: it has no global geometrical structure.