Could a universe with something like special relativity based on position instead of velocity be logically consistent?

Both time dilation and length contraction depend on something called the Lorentz factor, commonly denoted by $\gamma$. It is defined in terms of the velocity, $v$, and speed of light, $c$, as $$\gamma=\frac{1}{\sqrt{1-v^2/c^2}}$$ Some like to simplify this a bit by writing $\beta=v/c$, and substituting accordingly.

However, if we were to substitute in $x$ for $v$, we have some issues:

• Units. $x$ has units of distance, while $c$ has units of speed. I suppose you can throw in a constant there, but that seems a bit ad hoc. Speaking of which, that brings us to a different problem.

• The derivation. A rather elegant derivation is given in this answer, as well as some applications to energy and momentum. Basically, if we start with the assumption that the line element $ds^2$ is given by $$ds^2=c^2dt^2-dx^2-dy^2-dx^2$$ then we can find $\gamma$ with only a small bit of work, including introducing a quantity called $d\tau$, the proper time. The expression for the line element, by the way, assumes we use the $(+,-,-,-)$ metric signature.

  The thing is, this equation - the Minkowski metric - is the cornerstone of Minkowski spacetime, the foundation for special relativity. In order to change this so that you get your desired result, then you have to create an entirely new structure for this "flat" spacetime.

  An excellent geometric derivation of the Lorentz factor is given in Elements of Astrophysics (beware - large file!), on page 20.

posted almost 9 years ago

HDE 226868‭

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