To be brief, I will skip the garrulous formalities of stipulating the conditions (perfect vacuum, no curvature from gravity, near-infinite acceleration without mortal or energy constraints to ignore acceleration time, etc.) for pure cases of special relativity.

A very powerful tool in relative mechanics is the simple coordinate transform from one frame of reference to another.  What is it?  It is a simple tool to provide a linear transformation of coordinates from one non-accelerating frame of reference to another.  What consequence does it have?  Without this tool to transform coordinate systems, it would be difficult to describe one moving system relative to another when speeds are very great.  On the scale of things on Earth today, there is little need for this coordination, but on the cosmological scale, and in terms of a future world where speeds may easily approach ever-increasing fractions of light, this is of great significance.  So what’s the “problem”?  The problem is that there have been serious technical errors caused by implied assumptions of rotation symmetry since the transform was first introduced nearly a hundred years ago.  See for example, the figures of the “View from the Bridge” in the article Negative Energy, Wormholes and Warp Drive on page 89 in Scientific American’s special edition “the Edge of Physics” Volume 13 Number 1.  The problem may initially seem trivial, but has a substantial impact in clarity of the symmetry, as you will see.

I would like to propose an off-shoot of the familiar “Twin Paradox” we have all grown so accustomed to.  Of course, any good book on Special Relativity will explain how there really is no paradox at all, and that an ether theory is not necessary to explain the logic which seems to grate against common sense.  Even Einstein, in his original postulate of Relativity, addressed this quandary.  Almost all of the explanations correctly address the assumptions, but many explanations are indigent.  On that note, let me introduce “Feenburg’s Twin Paradox”, after E. Feenburg from his description of this scenario in a paper in the Journal of Physics: Doppler Effect and Time Dilation. 27:190, March 1959.  The setup defined is simple, and can be very easily tested with technology of today.  I am only using the basic set-up proposed by Feenburg, however, and for his interpretation of correctly addressing the matter – I leave you the option to research his attempt at an explanation.

Case-1: there are two twin observers, A and B, on Earth.  Next, there is a distant periodic variable star we may assume at rest with Earth’s frame of reference and along a designated coordinate axis of y.  Twin A and B both observe the star’s frequency as f0 when at rest on the Earth.  B leaves the Earth at a velocity V along the x-coordinate (perpendicular to the direction of the star from the frame of A).  B must observe a different frequency of the star, where A obviously observes the same frequency f0 to remain constant.  After a set length of proper time, ½*tb, twin B stops and begins the return to Earth maintaining the same speed (for simplicity) as during the departure.  So the transverse Doppler effect must warrant that the frequency of the variable with respect to B’s clocks and rulers to be fb=f0/√(1-v2/c2) where c is the speed of light in the vacuum of space.  The number of cycles of the star observed by both A and B, must be identical, and B must age less than A due to simple time dilation.  The time of B’s round-trip measured by A must be ta = tb/√(1-v2/c2).  So now where’s the paradox?

If we are to maintain the axiom that outward and inward perpendicular coordinates are to remain perpendicular in the case of the above scenario, the coordinate y-axis remains unchanged as twin B accelerates into the x-direction so that if the