Rod-clocks in a frame of reference

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

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Lorentz with round-trip light

This builds on the post Lorentz transformation derivations but given the round-trip light postulate (RTLP) here which states: The mean round-trip speed of light in vacant space is a constant, c, which is independent of the motion of the emitting body. From this empirical principle the round-trip Lorentz transformations may be derived, which are of the

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Newtonian light wave front

Consider the standard relativity configuration. Let a spherical light wave be emitted from the coincident coordinate origins at t=0 and t′=0. For the rest frame, the spherical wave front is given by x² + y² + z² = c²t². For the frame moving at velocity v parallel to the x-axis, two-way light must be considered.

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Time-space transformations

This post is based on David Tong’s Newtonian Mechanics, 1.2.1 Galilean Relativity. Given one facilial frame system, S, in which a tempicle has coordinates z(s), we can always construct another facilial frame system, S′, in which a tempical has coordinates z′(s) by any combination of the following transformations: Translations: z′ = z + a, for

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Lagrange’s equations in time-space

This post is based on the article Deriving Lagrange’s equations using elementary calculus by Josef Hanc, Edwin F. Taylow, and Slavomir Tuleja (AJP 72(4) 2004), which provides a derivation of Lagrange’s equations from the principle of least action using elementary calculus. A tempicle moves along the t axis with potential lethargy W(t), which is location-independent.

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Definitions of mass and vass

The conservation of momentum states (see here): For a system of objects, a component of the momentum (p = mv, the mass times the velocity) along a chosen direction is constant, if no net outside force with a component in this chosen direction acts on the system. The corresponding principle for levamentum states: For a system

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Polar plot in space and time

This post builds on others such as this. It’s not unusual to see a map of travel time from a central location, for example, this map of Washington, DC, USA (click to enlarge): Two-dimensions of travel time are represented as in space but could be represented as in time. Here is an example, first of

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Relative velocity and lenticity

Consider a particle P in uniform motion. Suppose two inertial observers observe its motion. Observer K is stationary relative to the ground, and observer L is in uniform motion in the same direction as P but at a different rate. (A) Say the spatial position of P relative to K is x(P, K), the spatial

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