A Fresh Exploration of Relativity Theory
by
David Bryan Wallace
[This paper is a reproduction of my earliest essay on relativity theory, written in 1997. The digital original was lost, but a recently found printout enabled me to recreate it, with a few typo corrections and clarity improvements.]
Certain customary features of twentieth century relativity are not required by the physical theory but are simply precedent following conventions. Moreover, these conventions frequently obscured the physics and led to error. Some of the errors gained greater popular acceptance than correct theory. Here presented is a careful derivation of relativity without the troublesome conventions.
Section 1: Implications of Michelson and Morley's observations.
Let us first, without introducing any new concepts, recapitulate the calculation of Michelson and Morley that guided their expectation. Remember that Michelson and Morley assumed dimensional invariance, i. e. neither length contraction nor time dilation.
Figure 1 represents the passage, at lightspeed c relative to the fixed points A, B, C, of a signal from the point A to the point B and thence, being reflected, to point C; and the passage of an observer moving inertially at velocity v through point A at the moment of the origin of the signal there, to the point C at the moment when the signal reaches C. Point D is the fixed point through which the moving observer passes simultaneous in the fixed system with the reflection of the light signal at point B.
The angle between the interferometer's velocity v and the interferometer's orientation is labeled φ. Let us derive a generalized equation for φ. Then, we can write the two special cases that Michelson and Morley mentioned, φ = 0° and φ = 90°.
The cosine law and quadratic formula yield solutions for t_{1} and t_{2} in the triangles ADB and CDB respectively. First, t_{1} from triangle ADB in equations (1) through (3):
(1) cosine law for ΔADB 
(2) eq1 in standard quadratic form 
(3) quadratic formula solution 
Then, t_{2} from triangle CDB in equations (4) through (6):
(4) cosine law for ΔCDB 
(5) eq4 in standard quadratic form 
(6) quadratic formula solution 
If the moving observer agreed with the fixed observer's measurements of time and distance as Michelson and Morley expected, then the moving observer would observe the speed of light relative to himself to be twice the distance d divided by the total time as in equation (7):
(7) hypothetical apparent light speed 
However, the empirical outcome of the Michelson Morley experiment suggests that the moving observer will measure the speed of light relative to himself as c, not as indicated in equation (7). Therefore the moving observer does not agree with the fixed observer's measurements of time and distance. Before launching into speculation to explain this unexpected outcome, note that the experiment of Michelson and Morley did not utilize any clocks or time measurement, nor did it involve any notions of synchronization at a distance. It yields evidence regarding interferometric measurement of length, but nothing about measurement of time. The measurement of length by a solid measuring stick apparently yields identical measures as the measurement of length by interferometry, independent of motion. This, of course, must be true for the interferometric standardization of length measure to work as practiced by national and international standards organizations.
The reason interferometric length and measuring stick length did not differ as expected by Michelson and Morley was rightly conjectured by FitzGerald: "... the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocity to that of light." If the electromagnetic forces that constitute atomic bonds within solid matter, which give it its solid qualities including dimensional stability, so act as to keep length dimensions stable as determined by interferometry, is it not then obvious that the physics of the bond is interferometerlike or, equivalently, a standing wave resonance. Bonds being electromagnetic in nature, it would be odd were it otherwise. This is sufficient to account for Michelson and Morley's observations. This postulate is the cornerstone of this exploration into relativity theory. Unlike FitzGerald, Lorentz (whose precedent we shall not follow) assumed invariance of dimensions perpendicular to the velocity.
Section 2: Transformation of lengths and angles from the fixed viewpoint.
In this exploration of relativity we do not follow Lorentz's precedent; we devise transformations to use instead of Lorentz transformations. Based on the empirical result of Michelson and Morley, we assume that moving observers will find the round trip speed of light in their frame of reference to be c because they rely on their local measuring sticks or interferometers to standardize their metrics.
Because it is specified that light moves at speed c relative to our fixed points, we define simultaneity in the fixed system as proposed by Einstein, by the process of reflecting light signals between pairs of points with the stipulation that time required for transmission is the same in each direction. Note that we shall not use this procedure to define simultaneity in a moving system, but shall make the fixed system the standard for time measurements and synchronization determinations, thus we conform to the SI standard of time based on a cesium atom at rest. We shall not assume that clocks are immune to rate changes due to motion, but we shall not say that the clock's rate change is a change in the rate of time. The moving observer will need to know his absolute velocity and the transformations between his local frame and the rest frame. His contracted measuring stick will yield a larger length measure than the stationary measure. [The subscript β indicates the moving observer's value.]
(8) length transformation 
When φ=π/2, sin(φ)=1, we have the relativistic transformation for lengths and velocities perpendicular to the relative motion.
(9) perpendicular length transformation 
When φ=0, sin(φ)=0, we have the relativistic transformation for lengths and velocities parallel to the relative motion.
(10) parallel length transformation 
Therefore, the relativistic transformation for the relative motion itself cannot be other than
(11) relative velocity transformation 
Because the tangent of the angle φ is a perpendicular length divided by a parallel length the transformation for tan(φ) is
(12) transformation of tan(φ) 
Similarly, the sine of φ is a perpendicular length divided by a length at angle φ, so the transformation of sin(φ) is
(13) transformation of sin(φ) 
Section 3: Introducing two convenient changes of variables.
Equation 11 can be written in standard quadratic form:
(14) 
Two changes of variables will facilitate derivation of inverses. Let
(15) 
and let
(16) 
Then equation (14) becomes
(17) 
The quadratic formula yields
(18) 
Also from equation (17) are other useful identities,
(19) 
(20) 
(21) 
(22) 
Now we can easily represent R and B in either frame of reference:
(23) 
(24) 
and their positive integral powers:
(25) 
(26) 
(27) 
(28) 
(29) 
(30) 
and so on.
Section 4: Inverse transformation for relative velocity and for parallel and perpendicular length.
Each of the relationships in section 2 has an inverse that is the transformation of the moving observer's measurement to the fixed viewpoint. From equation (15) we have:
(31) 
(32) 
Similarly, the inverse for length when φ = 0 [from equations (10) and (20)]:
(33) 
Solve for d and substitute for B and R [from equations (23) and (24)]:
(34) 
This is the inverse for length parallel to the velocity of the moving observer, φ = 0.
For the perpendicular case, when φ = π/2:
(35) 
Solve for d and substitute for B and R [from equations (23) and (24)]:
(36) 
This is the inverse for length perpendicular to the velocityof the moving observer, φ = π/2.
Section 5: Inverse transformation for angles.
Beginning with the relationship for tangent established in section 2:
(37) 
Solve for tan(#phi;) and substitute for B and R [from equations (23) and (24)]:
(38) 
For sine, square both sides of the relationship established in equation (13) and substituting 1BR for R^{2} from equation (19)
(39) 
Solve for sin^{2}(φ).
(40) 
(41) 
Take the square root of each side and substitute for B and R [from equations (23) and (24)]:
(42) 
(43) 
Section 6: Inverse transformation for length in general.
From equation (8)
(44) 
Solve for d and substitute for sin^{2}(φ) from equation (40)
(45) 
(46) 
(47) 
(48) 
(49) 
Substitute for B and R from equations (23), (24), (25) and (26)
(50) 
Section 7: Table of example values
The maximum difference between φ and φ_{β} occurs when
(51) 
R  φ in degrees where max φφ_{β} occurs 

0.999  78.060794  499.74987 
0.998  75.887004 
249.74975 
0.996  73.35737  124.7495 
0.992  70.43989  62.248996 
0.984  67.115389  30.997984 
0.968  63.391453  15.370935 
0.936  59.319519  7.5542355 
0.872  55.021558  3.639156 
0.744  50.736662  1.6664278 
0.618034  48.438143  1 
0.5  47.058597  0.6666667 
0.25  45.462203  0.2666667 
0.125  45.112789  0.1269841 
0.0625  45.028031  0.0627451 
0.03125  45.006998  0.0312805 
0.015625  45.001749  0.0156288 
Copyright © 2016 by David Bryan Wallace, Cape Coral, Florida, USA