Length measures space and duration measures time. Length is a scalar, which combined with direction describes 3D space. Duration is a scalar, which combined with direction describes 3D time.
In relativity this might be considered trivial; length and duration may be converted into each other by multiplying or dividing by the speed of light. However, the actual measurement will either use rods or clocks, and so be qualitatively different. The act of measurement determines the type of measure more than the units.
Space and time both have three-dimensional geometries. For most purposes one needs to focus on either 3D space or 3D time and ‘scalarize’ the other. Space is scalarized by replacing each point with its distance from a reference point (usually called the origin). Such a scalarized point of 3D space is called a station.
Time is scalarized by replacing each timepoint with its distime from a reference timepoint. Such a scalarized timepoint of 3D time is called a time (as expected).
Space-time means 3D space measured by distance with scalar time. Time-space means 3D time measured by distime with scalar station.
Space-time and time-space are represented by geometries formed from 3D Euclidean geometry plus the additional structural properties below.
Tim Maudlin in his Philosophy of Physics: Space and Time (Princeton, 2012) introduces Newton’s Laws without invoking Newton’s notions of absolute space, time, and motion. This is done by employing the geometry of Galilean space-time.
Let’s follow his account but substitute time-space for space-time. That includes time-space versions of all units of measure, including velocity, acceleration, mass, force, etc., as specified in the parallel glossary above.
A time-space diagram represents three dimensions of distime and one dimension of distance, as station. This is like a space-time diagram as pictured below but with only one dimension of distance (as station) and multiple dimensions of distime:
Maudlin expresses Newton’s First Law in terms of space-time diagrams (p.59):
First Law (Space-Time Diagram Version): The trajectory of a body is represented by a straight line in a space-time diagram, except insofar as it is compelled to change its state by impressed forces.
Here is the time-space analogue:
First Law (Time-Space Diagram Version): The trajectory of a body is represented by a straight line in a time-space diagram, except insofar as it is compelled to change its state by released releases.
Maudlin states that inertial motion is “a motion that a body with no external forces on can have”; and “inertial motions are specified as exactly the motions represented by straight lines in a space-time diagram.”
Facilial motion is the motion that a body can have with no releases in it, which is specified as exactly the motion represented by a straight line in a time-space diagram.
The Second Law, in this language, would say that the effect of a release is represented by a curvature or bending of the world-line with the time-space diagram. The world-line will bend in the direction of the release, and the amount of curvature will be proportional to the release and inversely proportional to the vass of the body.
Maudlin writes: “In essence, the inertial mass of an object is a measure of how much it resists having its world-line bent by forces; the bigger the mass, the less the trajectory bends.”
In essence, the facilial vass of a body is a measure of how much it resists having its world-line bent by releases: the larger the vass, the less the trajectory bends. So the dynamics could be stated as
D = n Bend,
where D is the release, n is the vass, and Bend is a measure of the curvature of the world-line.
(Pg. 60) A time-space point or event is essentially a place with time-space at a station, for example, the firing of a starter pistol to begin a race. This is idealized as a point with no duration in any direction and no length.
Continuing with page 61:
A time-space event corresponds to a single point in a time-space diagram. Several different sorts of structure are postulated for time-space:
In contrast with space-time, in which there is “an absolute, objective lapse of time between any pair of events” such that a “complete set of events forms a simultaneity slice”, time-space has the following:
First, there is an absolute station difference between any pair of events. There is a metric for the series of stations, so it is well defined.
Given this absolute station structure, time-space can be partitioned into sets of events that all happen at the same station, that is, events between which the station interval is zero. Each such complete set of events forms a co-location slice with time-space, which is a foliation of the time-space.
Time-space postulates a particular temporal geometry on each co-location slice. In particular, it postulates that the events on each slice have the temporal structure of 3D Euclidean geometry.
Continuing with page 62:
The Laws require an additional bit of trans-station geometry: geometrical structure among events at different stations. Time-space diagrams have a different geometrical structure than the time-space they are meant to represent, so some aspect of the picture must be ignored when interpreting the diagrams. The topology of the time-space corresponds to the topology of the time-space diagram, so continuous trajectories will be represented by continuous curves in the picture.
Time-space contains an affine structure, so that world-lines can be characterized as either straight or curved. This affine structure is called the facilial structure of the time-space. A body subject to no internal release is in facilial motion and so has a facilial trajectory through time-space, which are just the straight trajectories.
With this time-space there are no absolute lenticities but there are absolute relentations: the bending of a world-line with time-space.
Continuing with page 65:
We now have the resources to say, in general, what a facilial coordinate system is. With time-space a coordinate system will assign four numbers to each event: three temporal coordinates and one station coordinate. The differences in values of the station coordinate should be proportional to the station interval between events. So all the events that take place on a single co-located slice should be assigned the same station coordinate. And on each station slice, the three temporal coordinates should form a Cartesian coordinate system. Distime and other purely temporal geometry only obtains among events that occur at the same station. In order to be a facilial coordinate system, something more is required: the coordinate curves associated with the station variable must be facilial trajectories.
The critical point to notice is that the distinction between facilial and non facilial coordinate systems is parasitic on the prior distinction between facilial and non facilial trajectories with time-space itself. It is only because the physical time-space has the right kind of affine structure that a meaningful distinction among coordinate systems is possible.
(Pg. 66) Time-space structure is not directly observable, but it nonetheless plays an essential role in the formulation of physical theory.