Invariant intervals
Let’s begin with the space-time invariant interval r´² − ct´² = r² − ct². Then let us solve the equations: r´ = Ar + Bt and t´ = Cr + Dt. ⇒ r´ = 0 = Ar + Bt → r = −tB/A = vt where v = −B/A {or} B = −Av ⇒ r´ […]
2D light clock
The famous Michelson-Morley experiment used what could be described as a 2D light clock since their interferometer combined two light clocks at right angles. Their hypothesis was that this would show the Earth moving through the aether, but they failed to detect any motion. Einstein explained this failure as a feature of relativity. In other
Harmonic operations
This revised post follows up on harmonic (or reciprocal or parallel) addition mentioned in a previous post here. See Grossman & Katz, Non-Newtonian Calculus (Lee Press, Pigeon Cove, MA: 1972) p.59f; and Kent E. Erickson, “A New Operation for Analyzing Series-Paralled Networks,” IRE Trans. on Circuit Theory, March 1959, pp.124-126. See also here and here.
Lorentz for spacetime and timespace
This post builds on the previous posts here and here, and follows the approach of J.-M. Levy here. First, consider the Lorentz transformation for spacetime along the x axis per Levy’s section III: Let us now envision two frames in ‘standard configuration’ with K´ having velocity v with respect to K and let x, t
Light clocks with multidimensional time
A previous post on this subject is here. One reference for this post is V. A. Ugarov’s Special Theory of Relativity (Mir, 1979). A light clock is a device with an emission-reflection-reception cycle of light that registers the current timeline point and placeline point in units of cycle length and duration. Consider two identical light
Clock-rods
A clock-rod is a linear or planar clock with a parallel rod attached to it. A mechanical or electronic clock-rod might look like this: Three clock-rods mutually perpendicular would measure length and duration in all directions. A light clock-rod is conceptually like this: The clock and rod are parallel to each other so that parallel
Kinematic derivations
Displacement with distime: displacement s, duration (time interval) t, velocities v1 and v, acceleration a: To prove: v = v1 + at a = (v – v1) / t by definition ⇒ at = (v – v1) multiply by t ⇒ v = v1 + at To prove: s = v1t + ½ at2 vavg
Space as time and time as space
Galileo used the distance of uniform motion as a measure of the distime, i.e., time interval (Dialogues Concerning Two New Sciences Tr. by Henry Crew and Alfonso de Salvio, 1914): Accordingly we see that while the body moves from b to c with uniform speed, it also falls perpendicularly through the distance ci, and at
Metric postulates for time geometry
Geometry was developed by the ancient Greeks in the language of length, but it is an abstraction that may be applied to anything that conforms to its definitions and axioms. Here we apply it to duration. We will use Brossard’s “Metric Postulates for Space Geometry” [American Mathematical Monthly, Vol. 74, No. 7, Aug.-Sep., 1967, pp.
Length and duration for time and space
Length and duration are defined by their measurement. Length is that which is measured by a rigid rod or its equivalent. Length is “spatial extension” (cf. Dict. of Physics). Duration is that which is measured by a clock or its equivalent. Duration is “time measured by a clock or comparable mechanism” (Dict. of Physics). Time