Matters relating to length and duration in physics and transportation
Dual Euclidean transformations are required to transform six dimensions of length and duration: one Euclidean transformation for length space with time and one Euclidean transformation for duration space with distance. The two Euclidean transformations are: x′ = x − vt and z′ = z − ws where x and x′ are length space vectors, t …
The Euclidean geometry is a category with point positions as the objects and Euclidean transformations as the morphisms. In kinematics there are two Euclidean geometries: that of length and that of duration. They are in turn categories: Length space is a category with point locations as the objects and Euclidean transformations of points as the …
The traditional Aristotelian square of opposition is like that of first-order logic apart from existential import: Or in words: Outer negation is the contradictory, i.e., affirm/deny, and inner negation is the contrary, i.e., all/none. For quantifiers (or other operators) there is a duality square: Outer negation is negation of the whole quantifier; inner negation is …
Attempted derivations of the Lorentz transformation in the previous post here, which is similar to the light wavefronts approach here, do not work. The reason is that independent and dependent variables are treated alike, but they are not. I suspect this applies to all derivations of the Lorentz transformation. Let us look at the first …
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. …
