Changes

The result makes the matter more of a mystery.

The result makes the matter more of a mystery.

::Let <math>1 - a = b < 0</math>. Then <math>\gamma = \frac{1}{\sqrt{b}} = \frac{1}{z}</math> where <math>z = x + iy</math> is an arbitrary complex number, with <math>i = \sqrt{-1}</math>, i.e. the "imaginary number."

::Let <math>1 - a = b < 0 </math>. Then <math>\gamma = \frac{1}{\sqrt{b}} = \frac{1}{z}</math> where <math>z = x + iy</math> is an arbitrary complex number, with <math>i = \sqrt{-1}</math>, i.e. the "imaginary number."

In other words, the Lorentz factor would involve an imaginary number.

In other words, the Lorentz factor would involve an imaginary number.

::<math>t = \gamma (t' - \delta x \frac{v}{c^2}) = \gamma t'(1 - \frac{\delta x}{t'} \frac{v}{c^2}))</math>.

::<math>t = \gamma (t' - \delta x \frac{v}{c^2}) = \gamma t'(1 - \frac{\delta x}{t'} \frac{v}{c^2}))</math>.

Plugging in values of <math>\gamma = 2</math>, <math> \frac{v}{c} = .866<//math>, <math>t' = 1</math>, and <math>\frac{\delta x}{t'} = 2c</math> (or twice the speed of light) give

Plugging in values of <math>\gamma = 2</math>, <math> \frac{v}{c} = .866<//math>, <math>t' = 1</math>, and <math>\frac{\delta x}{t'} = 2c</math> (or twice the speed of light) gives

::<math>t = 2(1 - \frac{2c}{1} \frac{.866}{c}) = 2 - 4(.866) = -1.464

::<math>t = 2(1 - \frac{2c}{1} \frac{.866}{c}) = 2 - 4(.866) = -1.464</math>

so <i>negative</i> time has passed for the external observer; the tachyon has moved back in time according to the rest of the universe. This creates obvious potential for paradox.  

so <i>negative</i> time has passed for the external observer; the tachyon has moved back in time according to the rest of the universe. This creates obvious potential for paradox.