Definitions and Notation
To accommodate various platforms with differing requirements for parsing mathematical expressions, I may use suffixes, example_0, as an alternative notation for subscripts, example0. At times, subscripting may be reserved for indexes with numeric significance, then we may write, for example, v_forward instead of vforward. Circumstances may arise that require both suffix and subscript, v_01.
Some Greek letters will be employed:
beta β for velocity as a fraction of light speed, (β as a suffix designates relative to the frame with velocity β)
for difference, sum and the ratio difference/sum of one way timings of light in opposite directions, respectively,
delta δ = T_forward − T_reverse
sigma σ = T_forward + T_reverse
rho ρ = δ/σ,
for angles, alpha α, phi φ, psi ψ, and theta θ,
phi φ designates angles measured from β
psi ψ designates dihedral angles at an edge parallel to β
theta θ designates the angle between distinct β vectors
alpha α will serve for all other angles.
Where an explicit separator is desired in a suffix, iota ι or lowercase o will be employed interchangeably,
T1,2 = T_12 = T_1ι2 = T_1o2.
Conforming to current SI definitions, time and light speed are primary standards from which the standard for distance is derived. Length shall be distinguished from distance. Length of an object shall be measured in its proper frame by an interferometric standard, that is to say, by time of round trip travel for light. The standard for speed in the rest frame is c the free space speed of light. For practical reasons, speed and velocity may also be expressed as distance or length per unit time. It will be important to distinguish between relative and absolute speed. Absolute speed (relative to the rest frame) is computed as absolute distance per unit time. One can speak of a difference of absolute speed, but that will not be the same as relative speed. Relative speed will be computed as length per unit time and will depend on frame of reference.
The SI standard of time measure is oscillations emanating from a cesium atom at rest. Notice the “at rest” requirement added in 1997, http://www.bipm.org/en/si/si_brochure/chapter2/2-1/second.html. Chronological time t in all frames of reference is the same as in the rest frame. Numeric subscripts (or suffixes) will usually indicate sequence, typically with t0 = 0 when origins of coordinate systems coincide. Time intervals may be denoted with uppercase,
T1,2 = t2 − t1.
In an effort to avoid ambiguity I shall use the word “length” and variable names beginning with l to denote length of an object measured in its proper frame. Lengths used to calculate relative speed (relative to a given inertial frame) shall be the lengths at rest in that frame. By convention, proper coordinates of an inertial frame of reference will be given in terms of length measured at rest relative to that frame of reference. I shall use the word “distance” or “span” and I will use variable names starting with d or s to represent measure by a length standard in a different frame, with the modifier “absolute” or a subscript or suffix zero, span0 or span_0, for measure in the rest frame and the modifier “relative” or a subscript or suffix that designates the frame of reference otherwise. Generally multi-character literal variable names subscripted or suffixed with zero represent absolute or rest frame values; otherwise a subscript or suffix may designate a specific inertial frame. If velocity measure were computed from length and time, it would be frame of reference dependent, because moving solids and the absolute span of the moving length standard are subject to contraction. Bearing this in mind, I express velocity of a moving frame relative to the rest frame coordinates using the light-speed-fraction vector β = v0/c, where v0 is the rest frame measure of the inertially moving frame's velocity. Angle measure will also be frame of reference dependent, and the axes of a moving coordinate system will not be orthogonal in the stationary system unless one axis of the moving system is aligned with the direction of its movement. A subscript or suffix denoting frame of reference may be used. Note: Because length is always measured in its proper frame, a length name like lβ is never used. The length of a solid material object is independent of its absolute velocity; the absolute span of the object diminishes as the object contracts with increasing absolute velocity. For a given object, l ≥ s0.
Synchronization of clocks absolutely at rest may be accomplished by light signals (as in Einstein's relativity), but β must be taken into account for synchronizing moving clocks at a distance.
In discussion of angles I shall use names beginning φ to denote an angle measured from the direction of β, ψ_12 to denote the dihedral angle between two angles φ_1 and φ_2, θ_12 to denote the angle between the vectors β_1 and β_2 of two inertial frames of reference, and α to denote other angles. It will be expedient to name and use the cosines and sines of certain angles rather than using the angle itself in calculations, and this will be done by prefixing "cos" or "sin" to the angle name, e.g. cos(φ0) = cos_φ_0. Prefixes and suffixes are part of the variable name, so exponents are placed accordingly, cos2(φ0) = cos_φ_02, not cos2_φ_0.
Copyright © 2014 by David Bryan Wallace, Cape Coral, Florida, USA