## Special Relativity

Waves need a medium to travel through – don’t they? James Clerk Maxwell showed that light was a wave, an electromagnetic wave. So how did it cross the empty vacuum of space? And so the Victorians decided that empty space must be filled with something you couldn’t see, called it the luminiferous aether, and set out to find it.

(P.S. A photon of light ‘crosses’ space? It moves? It travels? I wonder what the photon would think? I wonder what I would see if I sat on the photon as it crossed space? I wonder why it is whenever I think of the Universe, I imagine an egg falling to the floor, breaking open, and all its innards slowly spilling out? 🙂 Anyway… )

A couple of guys (Michelson and Morley) set out to become famous and find the aether. They failed. It couldn’t be found. It didn’t exist. Light crossed empty space without a medium and even freakier all observers measure the speed of light with the same value! You’d have thought that if one light beam met another light beam coming in the opposite direction, they’d meet at a closing speed of twice the speed of light. They didn’t. Once times the speed of light was the maximum speed that anything can attain, even photons.

A **Reference Frame **is a set of coordinates that describes a stage upon which events can occur: A z-axis, a y-axis, an x-axis, a t-axis, enough to describe any motion. **Inertial Reference Frames** are ones that are in Uniform Motion. Like you on a train, not speeding up, not slowing down, and not changing direction. You can pour a cup of coffee on the train, as easily as you could in the station cafe.

In the old Galilean physics, the two frames are related to one another by:

*Δx′ = Δx – vΔt* (to keep it simple, I’ve set the motion to be along the x-axis only)

*Δy′ = Δy
*

*Δz′ = Δz
*

*Δt′ = Δt
*

However, far beyond and unnoticed in our everyday experience, the consequence of this fixed speed of light is that the true translation from one frame to the other is;

*dx′ = γ (dx-dt)
*

*dy′ = dy
*

*dz′ = dz
*

*dt′ = γ (dt-(v/c ^{2})dx)
*

Where *γ = 1 / (1-(v ^{2}/c^{2}))^{-1/2}* (The Lorentz Factor)

So the time coordinate is dependent on the relative motion of the frames (Moving clocks run slow!). Time and Space are related and become Space-Time.

The spatial coordinate in the direction of motion is also affected (it contracts!). So as you approach the speed of light, your time relative to those you left behind, slows to a halt, and the distance you need to travel vanishes! Intuitive or what!? 😉

In Special Relativity it is best not to think in distance (you’ll only get yourself confused when they shrink!), best to think in a space-time way,** Space-Time intervals **between events.

So it turns out that* E ≠ mc ^{2}* after all. In reality it is

*E*= γ

*mc*which with a little algebraic manipulation shows that,

^{2}E_{tot} = (m^{2}c^{4}+p^{2}c^{2})^{-1/2}

P = (E_{tot}^{2}/c^{2}-p^{2}c^{2})^{-1/2}

mc^{2} = (E_{tot}^{2} –p^{2}c^{2})^{-1/2}

And for mass-less photons,

E_{phot} = pc

P_{phot} = h/λ

Should we be surprised that the world doesn’t behave like the billiard ball world of our limited experience?