Changes

Imagine someone else is watching the clock from elsewhere, but not moving with the clock. Label the time they see passing as <math>t'</math>. This may or may not be equal to <math>t</math>, the time seen by the observer moving with the clock.

Imagine someone else is watching the clock from elsewhere, but not moving with the clock. Label the time they see passing as <math>t'</math>. This may or may not be equal to <math>t</math>, the time seen by the observer moving with the clock.

Now, suppose the clock - and its attached observer - begin to move with a velocity <math>v</math> relative to the external observer. In other words, the external observer thinks they are stationary, and sees the the clock and attached observer as moving with velocity <math>v</math>. The movement of the clock is <i>perpendicular</i> to the beam of light bouncing between mirrors. The clock starts moving the instant that the light bounces off the "bottom" mirror.

Now, suppose the clock - and its attached observer - begin to move with a velocity <math>v</math> relative to the external observer. In other words, the external observer thinks they are stationary, and sees the clock and attached observer as moving with velocity <math>v</math>. The movement of the clock is ''perpendicular'' to the beam of light bouncing between mirrors. The clock starts moving the instant that the light bounces off the "bottom" mirror.

The distance the clock travels to the right <i>before</i> the light reaches the "top" mirror and bounces back to the "bottom" is given by the usual equation of motion

The distance the clock travels to the right ''before'' the light reaches the "top" mirror and bounces back to the "bottom" is given by the usual equation of motion

::<math>Distance = vt'</math>  

::<math>Distance = vt'</math>  

::<math>\gamma = \frac{1}{\sqrt{1 - \frac{c^2}{c^2}}} = \frac{1}{\sqrt{1 - 1}} = \frac{1}{0} = \infty</math>

::<math>\gamma = \frac{1}{\sqrt{1 - \frac{c^2}{c^2}}} = \frac{1}{\sqrt{1 - 1}} = \frac{1}{0} = \infty</math>

Using that in the energy-mass equation, one finds there would be infinite energy in a particle of any mass going the speed of light. Since energy is conserved, the energy must come from <i>somewhere</i>, so an infinite amount of energy would have to be put into an observer to move them to the speed of light. (This is a key example of light speed being an insurmountable barrier in space travel.) Appropriately, since a photon is known to be without mass, it is the only particle capable of avoiding this limitation and thus travel at the speed of light.

Using that in the energy-mass equation, one finds there would be infinite energy in a particle of any mass going the speed of light. Since energy is conserved, the energy must come from ''somewhere'', so an infinite amount of energy would have to be put into an observer to move them to the speed of light. (This is a key example of light speed being an insurmountable barrier in space travel.) Appropriately, since a photon is known to be without mass, it is the only particle capable of avoiding this limitation and thus travel at the speed of light.

Similarly, if one were to plug larger and larger Lorentz factors into the time dilation equation

Similarly, if one were to plug larger and larger Lorentz factors into the time dilation equation

::<math>l' = \frac{l}{\gamma} = l\sqrt{1 - \frac{v^2}{c^2}}</math> where <math>l</math> is the "proper length" of the body according to someone moving with it and <math>l'</math> is the length of the body as seen by an external, stationary observer

::<math>l' = \frac{l}{\gamma} = l\sqrt{1 - \frac{v^2}{c^2}}</math> where <math>l</math> is the "proper length" of the body according to someone moving with it and <math>l'</math> is the length of the body as seen by an external, stationary observer

Since we are dividing by the Lorentz factor, the perceived length will always be <i>less</i> than that of the undilated, proper length. For a body moving at the speed of light, subtlety requires one to directly reexamine the Lorentz contraction

Since we are dividing by the Lorentz factor, the perceived length will always be ''less'' than that of the undilated, proper length. For a body moving at the speed of light, subtlety requires one to directly reexamine the Lorentz contraction

::<math>l' = l\sqrt{1 - \frac{c^2}{c^2}} = 1\sqrt{1 - 1} = 0</math>

::<math>l' = l\sqrt{1 - \frac{c^2}{c^2}} = 1\sqrt{1 - 1} = 0</math>

In other words, a body moving towards the speed of light appears shorter and shorter, until at the speed of light it seems to disappear entirely.

In other words, a body moving towards the speed of light appears shorter and shorter, until at the speed of light it seems to disappear entirely.

Note that, for physically tangible, sublight situations, <math>\gamma > 1</math> always. So <i>multiplying</i> something by the Lorentz factor will <i>increase</i> the quantity while <i>dividing</i> by it will <i>decrease</i> the quantity.

Note that, for physically tangible, sublight situations, <math>\gamma > 1</math> always. So ''multiplying'' something by the Lorentz factor will ''increase'' the quantity while ''dividing'' by it will ''decrease'' the quantity.

===Faster Than Light===

===Faster Than Light===

Physically, the resulting relativistic equations - for time, length, energy, etc. - would have to either have something imaginary about them with which to cancel the imaginary number, or the resulting relativistic phenomenon would simply be "imaginary," whatever that may mean. (The label "imaginary" is per se just a name given to <math>\sqrt{-1}</math>.)

Physically, the resulting relativistic equations - for time, length, energy, etc. - would have to either have something imaginary about them with which to cancel the imaginary number, or the resulting relativistic phenomenon would simply be "imaginary," whatever that may mean. (The label "imaginary" is per se just a name given to <math>\sqrt{-1}</math>.)

A common interpetation using time dilation is that a tachyon would travel backwards in time. The basis for this logic is in the kinematic equations called the Lorentz transformations. There are several; the relevant one is

A common interpretation using time dilation is that a tachyon would travel backwards in time. The basis for this logic is in the kinematic equations called the Lorentz transformations. There are several; the relevant one is

::<math>t = \gamma (t' - x \frac{v}{c^2}) = \gamma t'(1 - \frac{x}{t'} \frac{v}{c^2})</math> where <math>x</math> is the change in position.

::<math>t = \gamma (t' - x \frac{v}{c^2}) = \gamma t'(1 - \frac{x}{t'} \frac{v}{c^2})</math> where <math>x</math> is the change in position.

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

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

so <i>negative</i> time has passed; the tachyon seems to have arrived before it left, and so has moved back in time according to one of the available reference frames. This creates obvious potential for paradox.  

so ''negative'' time has passed; the tachyon seems to have arrived before it left, and so has moved back in time according to one of the available reference frames. This creates obvious potential for paradox.  

This possibility of effect precluding cause indicates that, if tachyons truly existed, then there may be no free will, and actions would always be pre-determined. Otherwise, one observer may have an "early warning" of something that will happen in another observer's frame and somehow change it; but then special relativity has it that observers in all reference frames must agree on whether a spacetime event occurs at all or not: no in-between.  

This possibility of effect precluding cause indicates that, if tachyons truly existed, then there may be no free will, and actions would always be pre-determined. Otherwise, one observer may have an "early warning" of something that will happen in another observer's frame and somehow change it; but then special relativity has it that observers in all reference frames must agree on whether a spacetime event occurs at all or not: no in-between.