It was thought that light propagated as a wave through the luminiferous aether. This would mean that light travels at different speeds in different directions due to Earth’s motion in space. In 1881 Albert Michelson tried to measure this difference, called the Aether Wind, with no success. In 1886 Michelson and Edward Morley repeated the experiment with higher accuracy, again with no success. It is perhaps the most important failed experiment!
Oliver Heaviside refined Maxwell’s equations into their modern form. He also calculated that the electric and magnetic fields of a moving electric charge would be deformed. This was a problem as the laws of physics have to be the same for every observer.
In 1892 George FitzGerald and Hendrick Lorentz independently hypothesised that moving bodies have their length contracted in the direction of motion. This explained the issue discovered by Heaviside and the absence of the aether.
Henri Poincaré refined the works of Maxwell and Lorentz and discovered what is now known as Special Relativity.
Consider two frames of reference K and K’. Their x axes are aligned and their y and z axes are parallel. Now frame K’ is moving at constant velocity \(v\) in the direction of its x axis. Each frame of reference has an observer stationary within it.
Each frame has a stationary clock consisting of a tube of length \(L\) with a beam of light reflecting back and forth along the tube. The tubes are both parallel to the y axis. Each observer will observe time passing at the same rate. It will take \(\tau = 2L/c\) seconds for the light beam to complete a cycle.
Now the observer in K will see the other observer’s light beam to take a longer time \(t\) to complete the cycle.
The path length of the other light beam can be calculated using Pythagorus’ theorem.
\[l = 2\sqrt{L^2 - \frac{v^2t^2}{4}}\]
The time \(t\) can then be calculated.
\[t = \frac{l}{c} = \frac{2}{c}\sqrt{L^2 - \frac{v^2t^2}{4}}\]
Multiplying by \(c/2\) and squaring gives:
\[\frac{c^2t^2}{4} = L^2 - \frac{v^2t^2}{4}\]
\[t^2(c^2 - v^2) = 4L^2\]
\[t = \frac{2L}{c\sqrt{1 - \frac{v^2}{c^2}}} = \frac{\tau}{\sqrt{1 - \frac{v^2}{c^2}}}\]
The Greek letters \(\beta\) and \(\gamma\) are used as a shorthand notation.
\[\beta = \frac{v}{c}, \gamma = \frac{1}{\sqrt{1 - \beta^2}}\]
\[t = \gamma\tau\]
Thus the observer at K sees the other’s clock running slower by a factor of \(\gamma\).
Now if the clock rod is aligned with the x axis by the observer in frame K’, the observer in frame K will measure the length of the rod in frame K as L’ and the time taken to complete a cycle is \(t' = 2L\gamma/c\).
When the light beam leaves the end of the tube nearest the K frame, the far mirror is moving so light has to travel a further distance \(L + vt'_1\). So the time taken is \(t'_1 = (L +vt'_1)/c\).
\[t'_1 = \frac{L'}{c + v}\]
Likewise the light beam coming back will have a shorter distance to travel as the mirror is moving towards it.
\[t'_2 = \frac{L'}{c - v}\]
Combining the equations gives:
\[t' = \frac{2L\gamma}{c} = t'_1 + t'_2 = \frac{2L'c}{c^2 - v^2}\]
\[L' = L\sqrt{1 - \frac{v^2}{c^2}} = \frac{L}{\gamma}\]
The observer in K sees lengths in the other frame in the direction of motion contracted.
The Lorentz transforms describe the translation from one frame of reference to another when they are moving apart at velocity \(v\) in the x direction.
\[x = \gamma(x' + vt')\]
\[x' = \gamma(x - vt)\]
\[t = \gamma\bigg(t' + \frac{vx'}{c^2}\bigg)\]
\[t' = \gamma\bigg(t - \frac{vx}{c^2}\bigg)\]
This can be expressed in matrix form.
\[ \begin{bmatrix} ct' \\ x' \\ y' \\z' \end{bmatrix} = \begin{bmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} ct \\ x \\ y \\z \end{bmatrix} \]
This is a form of rotation known as a boost. The K’ frame’s \(x'\) and \(ct'\) axes are no longer parallel to the \(x\) and \(ct\) axes in the K frame.
The length \(s\) is invariant under Lorentz transformation.
\[s^2 = x^2 + y^2 + z^2 -c^2t^2 = x'^2 + y'^2 + z'^2 - c^2t'^2\]
If an object has velocity \(\mathbf{u}\) in K and velocity \(\mathbf{u'}\) in K’ then the relationship can be found by differentiation. Again \(v\) is the velocity frame K’ is moving in the x direction relative to frame K.
\[u'_x = \frac{dx'}{dt'} = \frac{dx - vdt}{dt - \frac{vdx}{c^2}} = \frac{u_x - v}{1 - \frac{vu_x}{c^2}}\]
\[u'_y = \frac{dy'}{dt'} = \frac{dy}{\gamma(dt - \frac{vdy}{c^2})} = \frac{u_y}{\gamma(1 - \frac{vu_y}{c^2})}\]
\[u'_z = \frac{dz'}{dt'} = \frac{dz}{\gamma(dt - \frac{vdz}{c^2})} = \frac{u_z}{\gamma(1 - \frac{vu_z}{c^2})}\]
It is important to note that \(u'_x \neq u_x - v\).\
The inverse of the x term is:\
\[u_x = \frac{u'_x + v}{1 + \frac{vu'_x}{c^2}}\]
To give an example. A spacecraft travelling towards a planet at half of the speed of light launches a probe at half of the speed of light towards the planet. How fast is the probe travelling towards the planet? Newton would add the velocities to give the speed of light. Lorentz gives a different answer.
\[u_x = \frac{c/2 + c/2}{1 + \frac{c^2}{4c^2}}=0.8c\]
Now consider that the spacecraft directs a beam of light travelling at the speed of light towards the planet. How fast is the light beam travelling towards the planet? Adding velocities incorrectly gives \(1.5c\). Lorentz gives:
\[u_x = \frac{c + c/2}{1 + \frac{c^2}{2c^2}}=c\]
Light always travels at the speed of light no matter what the speed it was emitted at.
Henri Poincaré and Hendrick Lorentz exchanged letters discussing and refining the Lorentz transformations. Poincaré wrote two papers in 1900 describing what he later call the principle of relativity. This states that no experiment can distinguish between a state of uniform motion and a state of rest.
The principle can be explained by a thought experiment. Alice and Bob are two astronauts in separate spacecraft. Bob is moving away fro Alice at a constant velocity that is a significant percentage of the speed of light.
Special Relativity is only valid if there is no acceleration. So the initial condition has to be that Bob accelerated away from Alice to reach his final velocity.
Alice observes that her spacecraft is its usual length and that her clock is ticking at the usual rate. Bob observes that his spacecraft is its usual length and that his clock is ticking at the usual rate. When Alice observes Bob’s spacecraft, she sees that its length is contracted in the direction of travel and that his clock is running slower than hers.
Poincaré also discovered from the Lorentz transformations that the quantity \(x^2 + y^2 + z^2 - c^2t^2\) is constant. He realised that introducing imaginary time \(ict\) as a fourth spatial coordinate made a Lorentz transformation a simple rotation in four-dimensional space. He thought that it would be too much effort to extend the ideas of four-dimensional geometry.
In 1905 Albert Einstein published a paper on what is now known as Special Relativity. He derived the Lorentz Transformations from the two principles of relativity.
Unusually the paper did not reference any other person’s work. He was aware of most of the papers written by Lorentz and Poincaré.
He also published a paper of mass-energy equivalence \(E=mc^2\). It is now clear that Einstein plagiarised Poincaré’s work.
Poincaré never acknowledged Einstein’s work on relativity - he died in 1912 before the general theory was published.
Einstein acknowledged that Poincaré laid the foundations for relativity decades after Poincaré’s death.
Length contraction and time dilation have been observed and validate the theory of Special Relativity. There is, however, an important restriction on Special Relativity. The effects are only valid if the observers are in inertial frames. If an observer experiences acceleration then Special Relativity is no longer valid during the period of acceleration.
A curious consequence of this is that Special Relativity and General Relativity are incompatible. General Relativity states that if spacetime is curved, there must be an acceleration.
Special Relativity is also compatible with Quantum Theory. The Dirac equation is an important part of Quantum Theory. It is based on Spinors which encapsulate the Lorentz transforms.