# Special relativity as a uniquely flat solution

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SPECIAL RELATIVITY

#### Special relativity as a uniquely flat solution

1905 –

The relativistic equations of special relativity are the only set that allow a flat-spacetime solution.

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## Defining a range

In the relativistic ellipse exercise, the principle of relativity generates a spectrum of potential relativistic equations, given by the relationship $CT × {(1 - \frac{v^2}{c^2})}^x$, where the exponent, $x$ has a value between $0$ and $1$

## Logical possibilities

#### x equals 0.5

If x=0.5 exactly, the relativistic ellipse has the same width for any value of $v$, and simply elongates by the Lorentz factor – its internal wavelength distances can be fitted back into the original spherical outline by a simple Lorentz contraction without introducing any intrinsic curvature. Although this contraction is arguably a form of distortion, it is a '"flat"' distortion – angles certainly change, but every line that is “straight” before the contraction is still straight afterwards.
All other values of $x$ require more complex distortions that involve curvature:

#### x is greater than 0.5

If x is any greater than 0.5, then wavelengths and wavelength-distances inside the ellipse diagram are correspondingly longer, and a simple uniform contraction is not sufficient to cram everything back inside a circle of the original radius – a normalised map of the internal distances has to curve out of the plane. For values of $0.5 \lt x \lt= 1$, the region around a moving body appears to be associated with an effective tilted gravity-well, or the deepening and tilting of the body’s existing gravity-well. Solutions in this range are gravitoelectromagnetic, and associate the relative motion of masses with non-Euclidean (non-flat) distortions of the lightbeam grid.
– In other words, they associate positive recoverable kinetic energy with positive curvature.

#### x is less than 0.5

By contrast, if x is any less than 0.5, the wavelengths at any given nominal positive velocity will all be shorter than the $x=0.5$ solution. If $x=0.5$ represents zero curvature with relative velocity, then $0 =\lt x \lt 0.5$ represents solutions that associate positive recoverable kinetic energy with negative curvature – a situation that doesn’t seem credible in a physical model.

## Results

This exercise suggest five main conclusions:

SR has the unique relativistic solution for flat spacetime
The assumptions of the principle of relativity and perfectly flat spacetime are enough to let us derive the equations of special relativity as the only possible solution. Pages and pages of unnecessary calculations and overly-complicated proofs are not required.
In SR, it's relativity and flat spacetime that are important
Although Einstein quoted his two postulates as "relativity" and "constant lightspeed", we can have relativity and locally constant lightspeed in a curved model, without getting the SR equations. For SR, Einstein took c-constancy to mean global c-constancy, which is another way of saying that the lightbeam geometry of the region is flat (SR's implicit third postulate).
Newtonian mechanics was never really a flat-spacetime theory
Since C19th Newtonian optics generates the shift equations of x=1, this tells us that Newtonian physics does not "fit" flat spacetime. To be geometrically consistent, a Newtonian system has to involve velocity-dependent curvature, and any geometrical implementation of NM has to be more sophisticated than special relativity,'s flat Minkowski spacetime.
Any relativistic alternative to SR must involve curved spacetime
If SR "owns" the flat-spacetime solution, any competing relativistic model must occupy part of the (positive) curved-spacetime range, $x\gt0.5$.
Much of the C20th testing was pretty badly thought-out
If we want to test whether special relativity is the correct theory of relativity, we need to test where the shift equations fall in the range $0.5\lt=x\lt=1$. We need to test SR against redder predictions, but C20th testing typically only compared SR against bluer predictions, which were never particularly credible in the first place.

## Notes

It was typical in the C20th for texts to give the impression that SR's were the only possible set of relativistic equations. While technically wrong, this worldview was encouraged by using the word "relativistic Doppler" for the SR Doppler predictions (implying that there were only one set of relativistic Doppler equations), and using the word "relativistic" more generally as meaning "SR-compatible". For instance we see Newtonian relationships $x=1$ referred to as "non-relativistic", when in fact they are relativistic ... just not compatible with flat spacetime.