Explorations of multidimensional space and time with linear and angular motion.
Velocity is defined as: where x is the displacement and t = ‖t‖ is the independent time interval, the distime of a parallel reference motion. The inverse of v is the function defined by the reciprocal of this derivative: The converse of v is w, the lenticity, which is defined as: where t is the …
There is a symmetry principle for length and duration: Measures of length and duration are interchangeable and the forms of equations remain the same if all measures of length are interchanged with their corresponding measures of duration and vice versa. The three-dimensionality of length is fully reflected in the three-dimensionality of duration. This is a …
Following the previous post here, we use Jacobian matrices to transform location and chronation vectors between inertial observers. As before, let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their scalar magnitude without it. Velocity = V, lenticity = W, displacement = x, distimement = z, independent distance = s, independent distime …
In order to combine the three dimensions of length space and three dimensions of duration space in definitions for motion in six dimensions (3+3), it is necessary to use Jacobian matrices. The 3+1 and 1+3 dimensional definitions are simplifications of these. Let matrices be written with upper case boldface. Let vectors be written in lowercase boldface …
The following is generalized from the explanation of Galilean invariance here. Chorocosm (inertial frames) Among the axioms from Newton’s theory are: (1) There exists an original inertial frame in which Newton’s laws are true. An inertial frame is a reference frame in uniform motion relative to the original inertial frame. (2) All inertial frames share …
Kinecosm is the world of motion, which is the subject of kinematics. Since the extent of motion has two measures: length and duration, the kinecosm has two subworlds: Length space is the three-dimensional world of length, which is commonly called space. Duration space is the three-dimensional world of duration. Chorocosm is length space with time. …
The length part of the Galilean transformation is: with the relative velocity v. The time part of the Galilean transformation is: so that time is the same for all observers. Einstein made time relative and symmetric with length (at least in one dimension) by assuming an absolute speed of light, c. With β = v/c …
Consider an analogue clock with a dial and a hand, i.e., a pointer, reimagined as two circular bands: The Dial band is at rest relative to the observer and the Hand band turns clockwise at a fixed rate. The pointer on the Hand band points to the marking on the Dial band that indicates the …
This post is similar to the post on light clocks here. Simple harmonic motion (SHM) is like the spring below: The equation for describing the period is where T is the period, m is the mass, and k is the spring constant. The displacement for each cycle is zero since it returns to its starting point. …
Space is the three-dimensional domain in which motion occurs. Time is a one-dimensional domain in which a reference uniform motion occurs. The extent of a motion is measured either as a length or a duration by a rod-clock. A rod-clock is a linear rod combined with a linear clock, like this: The pointer moves in uniform …
