Explorations of multidimensional space and time with linear and angular motion.
What would it mean for space to be one-dimensional? It would be similar to 1D time. Imagine a clock, except that instead of marking durations, it marks distances. It’s like an odometer for a vehicle that moves at a constant speed and never stops. Like this with units of distance (metres, kilometres, miles, etc.): […]
In a sense every speed is a local conversion of space and time. But a characteristic speed (or modal speed) has the following properties: (1) The speed is relative to a mode of travel or movement. (2) The speed reflects extreme travel conditions in the mode. (3) The speed serves as a general conversion between
I Multiple dimensions of time are easier to see if we look at transportation. Consider the time table of a railway or subway system. Directions are typically shown by the station at the end of each line. The time table lists arrivals and departures – events in space and time. A railway station some distance
I’ve written about terminology and used new terms before, for example, Movement and dimensions. I want to take another look at coining new terms needed for studying movement and the symmetry of space and time. In what follows, the terms pace, lenticity and relentation have new senses. Consider these parallel terms, defined with respect to
The Lorentz transformation for velocities in all directions is known from the special relativity theory. This may be generalized to lenticities in all directions. Begin with the definition of the vector β from a previous post: β = v/c = k/v if |v| < |c|, with velocity v and speed of light c, v/k =
This is a revision to the derivation of the superluminal Lorentz transformation, as presented previously such as here. The problem with the previous derivation is that it did not distinguish between speed and pace, velocity and lenticity. Consider Case 2 again. This case begins with: r´ = r – tc²/v or t´ = t – r/v.
Movement in its simplest form is a trip from A to B. There are two ways one can look at such a movement: (1) as the line segment from A to B, or (2) as the angle between the lines from A to a reference point and A to B. Each of these may be
The dimensions of an object are measured by movements, whether by moving a measuring device or moving one’s eyes while a measuring device stays in the same position. Movements themselves are measured by comparing them with standard movements, such as a movement with constant velocity. A movement compared with a standard linear movement generates a
Since time is three-dimensional, what is the total time given the time in each dimension? The answer is exactly like the total distance. Consider the times t1, t2, and t3. If these are the coordinates of three successive movements, then the total time is their sum: t = t1 + t2 + t3. But if
The instantaneous movement of a particle may be represented by a velocity vector, which describes an instantaneous motion by its magnitude and direction: the magnitude is the ratio of the differentials of the distance and time in each dimension of the movement (dri/dti); the direction is the direction of the instantaneous tangent in each dimension
