Two and Four Clock Algorithms for Determination of Absolute Velocity

The process of determining absolute velocity begins with observations relative to an inertial frame of reference and relies on clocks so constructed that they will remain synchronized regardless of what motions they are subjected to between synchronization verifications made by bringing the clocks together, not by light signals over distance.

Two Clock Algorithm:

A two clock algorithm for determining absolute velocity is quite simple but holds little promise of high precision. We synchronize two co-located clocks *A* and *B*, deploy them to a fixed-length separation and equip them with means to exchange light signals through a vacuum and measure the times *T _{AB}* light takes from

The fixed-length separation will be *l* = *c*⋅σ/2 . The clocks should be brought back together at the end of the procedure to verify maintenance of synchronization.

Four Clock Algorithm:

The four clock algorithm requires four synchronized clocks to be deployed to vertices of a tetrahedron. Because it can be scaled to any size, it has the potential for high precision. Data used are the twelve timings of light signals in each direction between pairs of vertices. The edge lengths of the tetrahedron are measured relative to the inertial frame by the round trip light time σ. Coordinates of the vertices are computed from edge lengths. The solution to be found is the magnitude of **β** and the cosines of the angles between **β** and the respective edges of the tetrahedron; this will be independent of the orientation and handedness of the coordinate system.

In order to calculate **β** (vector with magnitude β = ||**β**||) we select three edges, (optionally, to minimize rounding error, the edges having the ratio ρ = δ/σ furthest from zero.) We form a 3 by 3 matrix *M* with each row being a unit vector parallel to a chosen edge. We also form a column vector * R* with elements equal to ρ for each chosen edge taken in the same order. Multiplying the vector by the inverse of the matrix,

The remainder of this document gives (1) the algorithm for finding coordinates of vertices and unit direction vectors of edges from edge lengths, and (2) the derivation of the calculation.

Algorithm for Finding Coordinates of Vertices and Unit Direction Vectors of Edges from Tetrahedron Edge Lengths

We number vertices [1,2,3,4], (optionally, so that ρ will be non-negative when the forward direction is from lower numbered vertex to higher numbered vertex.)

We use pairs of vertices to name edge lengths, σ_{1,2} = σ_{2,1}, and dihedral angles, ω_{1,2} = ω_{2,1}.

We use ordered pairs of vertices to name edge vectors *U*_{1,2} = -*U*_{2,1}, unit direction vectors *u*_{1,2} = *U*_{1,2}/σ_{1,2}, and angles at the first named vertex with rays to the un-named vertices, thus α_{1,2} denotes ∠314=∠413.

To find a vertex angle cosine for a triangular face of the tetrahedron use the law of cosines. Example: for the angle α_{1,2} at vertex 1 of the triangle formed without vertex 2, (that is, ∠314), cos(α_{1,2}) = (σ_{1,3}^{2} + σ_{1,4}^{2} − σ_{3,4}^{2})/(2⋅σ_{1,3}⋅σ_{1,4})

To find cosine of the dihedral angle ω_{1,2} at edge σ_{1,2} we use the first spherical law of cosines (for sides), bearing in mind that the measure of a side of a spherical triangle equals the angle measure of the angle at the center of sphere with rays to the ends of the side. So, for example, taking vertex 1 as center of the sphere, cos(ω_{1,2}) = (cos(α_{1,2}) −cos(α_{1,3})⋅cos(α_{1,4}))/(sin(α_{1,3})⋅sin(α_{1,4}))

Position vectors ** U** for the vertices are direction vectors from vertex 1 which is assigned origin of coordinates,

Derivation of the Calculation of **β**

Figure 1 represents the travel, at light-speed *c* relative to the absolutely 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 inertial movement of a tetrahedral vertex at velocity ** v** through point

The cosine law and quadratic formula yield solutions for *t*_{1} and *t*_{2} in the triangles ** ADB** and

(1) |

(2) |

(3) |

Then, *t*_{2} from triangle ** CDB** in equations (4) through (6):

(4) |

(5) |

(6) |

From these values of *t*_{1} and *t*_{2}, with the substitution of β for *v*/*c*, we find ρ for the tetrahedron edge.

(7) |

Applying the following transformations of sines and cosines to the inertial frame values,

(7) |

(7) |

we obtain ρ = β⋅cos(φ_{β}) = **β**•* u*. The matrix equation

- Physics Fixes Home
- The Greatest Haberdasher of All Time, (A Fable)
- What We Know
- PDF: Refutation of Lorentz-Einstein Special Relativity
- PDF: Introduction to FitzGerald Relativity [DRAFT]
- Graphical Portrayal of Electromagnetic Radiation
- Cosmology: The Genesis of Spiral Galaxies
- Reconciling Olbers' Dark Sky Paradox, Dark Matter and Cosmic Background Radiation
- Simultaneity: An Improved Definition of Simultaneous Events
- Updating Century Old Relativity Theory
- A Revealing Test of the Compatibility of Special Relativity Postulates
- Appendix: An Algorithm for Determination of Absolute Velocity
- Introduction
- De Sitter's Astronomical Proof of the Constancy of Light Speed
- The Michelson-Morley Experiment
- Definitions and Notation
- Idealized Observation of Clocks in the Rest Frame Coordinate System
- Idealized Determination of Absolute Velocity of an Inertial Frame Coordinate System
- Two and Four Clock Algorithms for Determination of Absolute Velocity
- Basic Transformation of Coordinates and Measures, Corresponding Axes Parallel, Co-Linear x-Axes Parallel to β, and t = 0 When Origins Coincide.

- A Fresh Exploration of Relativity
- Suggested Reading

Copyright © 2014 by David Bryan Wallace, Cape Coral, Florida, USA