Relativity

1830's: Michael Faraday was making charges move with coils and magnets.

He experimentally showed that a changing magnetic field will create a current, and the inverse. These phenomena we now call induction.

1855: Maxwell: "On Faraday's lines of force"

The Maxwell Equations

$$ \begin{align*} \nabla \cdot \mathbf {E} & = 4\pi \rho \\ \nabla \cdot \mathbf {B} & = 0 \\ \nabla \times \mathbf {E} & =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}} \\ \nabla \times \mathbf {B} &={\frac {1}{c}}\left(4\pi \mathbf {J} +{\frac {\partial \mathbf {E} }{\partial t}}\right) \end{align*} $$

These were the laws the described the electromagnet phenomena, just like $F = ma$ described mechanical motions.

Simple Example (in 2d)

Unprimed Frame $$\sqrt{(6-3)^2+(7-2)^2}$$

Primed Frame $$ \sqrt{(1-(-4)^2+(5-2)^2} $$

It was quickly realized that the Maxwell equations did not share the same invariance as Newton's Laws.

This was the first problem.

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And if there is a translational invariance, then one could determine whether or not a frame had a velocity.

The second 'problem' grew out of several experiments that showed that light wasn't acting like a 'regular' wave would.

Normally, a mechanical wave would be affected by the motion of the medium it's traveling in. i.e. a water wave in a moving current will take on the velocity of the current as well as its own wave speed.

Such affects were found to be non-existent through several experiments.

Notably, in response to experiments, Lorentz and others proposed the following transforms, to be used instead of the Galilean transformations.

$$ \begin{align} \label{eq:lorentzx} x & = \gamma \left(x' + \beta c t' \right)\\ \label{eq:lorentzy} y & = y'\\ \label{eq:lorentzz} z & = z'\\ \label{eq:lorentzt} ct & = \gamma \left(ct'+ \beta x' \right) \end{align} $$ where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ and $\beta = \frac{v}{c}$

Einstein's paper took all these pieces and assembled them into a new understanding of our physical world:

On the Electrodynamics of Moving Bodies (pdf)

$\mathbb {R} ^{3}$

Our everday 3d world is known more formally as $$\mathbb {R} ^{3}$$. This is a vector space with 3 real coordinates. We map them onto our human notions of $x,y,z$.

We describe positions using regular vectors: $$\mathbf{r} = \begin{pmatrix} r_x \\ r_y \\ r_z \end{pmatrix} $$

We can express the length of such a vector by the following: \begin{equation} |\mathbf{r}| = \sqrt{r_x^2 + r_y^2 + r_z^2} \end{equation} or, if that was some change in position, then we could just as easily consider: \begin{equation} \mathbf{\Delta r}^2 = \Delta r_x^2 + \Delta r_y^2 + \Delta r_z^2 \end{equation} And using galilan transforms between coordinate systems, everyone would 'agree' that that value of $\Delta \mathbf{r}^2$ remains the same.

Furthermore, we can image the region of space where that quantity is equal to some value: it's a sphere.

In 1d, we can plot ct(x) and see that we get two lines leading away from the origin at 45° These are the world lines of photons.

Let's start with $ds^2 \lt 0$, eg: $$ds^2 = -c^2 dt^2 + dx^2 = -1$$ We can plot these separations using: $$ct = \sqrt{dx^2+1}$$ And all the points along this hyperbola are the same 'spacetime' distance away from us.

These points would all be considered 'time-like' in the separations from the origin.

Consider the 3 events shown in the figure.

1. A ball in your hand $t$ seconds from now.
2. A ball over there $t$ seconds from now.
3. A ball over there $>t$ seconds from now.

Which two have the same spacetime separation?

We can also have $ds^2 \gt 0$, eg: $$ds^2 = -c^2 dt^2 + dx^2 = +1$$ We can plot these separations using: $$ct = \pm \sqrt{dx^2-1}$$ And all the points along this hyperbola are the same 'spacetime' distance away from us.

Space-like separations: these are events we can have no causal influence over.

Time-like separations: if we can travel fast enough, or send signals of light, then we can causally influence these events.

Light-like: the worlds lines of photons.

Shown is a vector $\mathbf{r}$ $$ \mathbf{r} = \begin{pmatrix} r_x \\ r_y \\ r_z \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$$ If we wanted to rotate it, we could transform the vector components: $$ \begin{aligned} x & = x' \cos \theta - y' \sin \theta \\ y & = x' \sin \theta + y' \cos \theta \\ z & = z' \end{aligned} $$

We can express a rotation about an axis as a multiplication of the vector $\mathbf{r}$ with a Rotation Matrix $\mathcal{R}$ $$ \mathcal{R} = \begin{bmatrix}\cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

"Due to the nature of light and energy, it’s impossible to reach the speed of light, nearly 300,000 kilometers per second. It would take an infinite amount of energy. The fastest any human-made object has traveled is only about 0.06 percent of that speed. At that rate, it would take about 6,600 years to reach the nearest exoplanet, Proxima Centauri b, 4.24 light-years away."

"A spacecraft traveling at one-tenth of the speed of light could shave the trip down to a quick 40 years. Future engineers could use nuclear power to achieve that, says Scott Bailey, an engineer at Virginia Tech. But developing that technology could take thousands of years."