Relativity

#physics

The note introduces us to the concept of transformation of coordinates, and how they behave with near lights-peed movement.

Under Galilean transformations, for a frame $S^{\prime }$ moving relative to $S$ with a velocity $v$ in the $x$ direction, the coordinates transform as $x^{\prime }=x-vt$, $y^{\prime }=y,z^{\prime }=z$, and most importantly, $t^{\prime }=t$
This means that time flows uniformly for everyone, and the fundamental laws of mechanics holds everywhere.

But this also implies that light has different speed in different reference frames, since velocities are additive. $c^{\prime }=c\pm u$

In the 19th century when electromagnetism came around, Maxwell's equations suggested that $$c = \frac{1}{{ \mu_{0}\epsilon_{0} }}$$also implying that it is invariant, or atleast, under galilean transformations, these equations would change their fundamental forms in different reference frames.

To combat without ditching Galilean transforms, scientists proposed the "ether". An invisible, absolute, stationary medium permeating the universe, through which light travels.
If this were true, Earth's orbit should create an "ether wind", causing light to travel at different speeds, depending on it's relative motion to earth.

In 1887, the Michelson-Morley Experiment attempted to measure this "ether wind", using an extremely precise interferometer.

Hence, it was proved that the ether does not exist.

Einstein's Postulates

He came in with an attempt to explain the above stuff, and gave two postulates,

1. The principle of Relativity : The laws of physics are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.
2. The constancy of the speed of light: Observers in all inertial frames measure the same speed of light in vacuum, $3\times {10}^8\mathrm{m}/\mathrm{s}$.

Thus with this, Einstein showed that time and space are not continuous, and that they are closely related as space-time.

A direct consequence of this is the problem of simultaneity. Two events that are simultaneous in one reference frame may not be simultaneous in a different moving frame.

To deal with this new discovery, Galilean transformations had to be replaced with Lorentz Transformations, which enforces the constancy of the speed of light.
If we consider two frames $S$ and $S^{\prime }$, whose axes are aligned, with the latter moving with a speed $v$ along the common $x-x^{\prime }$ direction, where their origins coincided at $t=t^{\prime }=0$

Derivation of Lorentz Transform

We assume that light spreads outwards in a sphere in both frames as governed by the equations $$\begin{align}x^2 + y^2 + z^2 &= c^2t^2 \ x'^2 + y'^2 + z'^2 &= c^2t'^2 \end{align}$$
This is basically describing a light-impulse that spreads outwards from origin at $t=0$
Since we have $x^{\prime }=x-vt$, at $t=0$ in Galilean Transform, meaning that in Lorentz Transform, it must be governed with something like $x^{\prime }=\gamma (x-vt)$, and by relativity, $x=\gamma (x^{\prime }+v{t}^{\prime })$

Since our light pulse travels with the same speed in both frames, if we consider it's advance only along $x$ axis, we have $x=ct,x^{\prime }=c{t}^{\prime }$

Substituting these into our previous relations involving $x,x^{\prime }$, we get $$\begin{align}ct' = \gamma(ct-vt) &= \gamma t(c-v)\ ct &= \gamma t'(c+v) \end{align}$$
We multiply the two equations and solve for $\gamma$ $$\begin{align} c^2tt' &= \gamma^2tt'(c^2-v^2) \ \gamma^2 &=\frac{c^2}{c^2 - v^2} \ &\implies \frac{1}{1-\frac{v^2}{c^2}} \end{align}$$
Hence we have obtained $\gamma$, and we call the ratio of $v$ and $c$ as $\beta$.

Thus the Lorentz Factor is $$\gamma = \frac{1}{\sqrt{ 1-\beta^2 }};\ \ \ \ \beta = \frac{v}{c}$$
To derive the time transform, we substitute the value of $x^{\prime }$ back into the expression for $x$, $$\begin{align} x &= \gamma(\gamma(x-vt)+ vt') \ \ vt' &= \frac{x}{\gamma} - \gamma x + \gamma vt \end{align}$$Dividing by $v$ and simplifying we obtain $$t' = \gamma\left( t- \frac{vx}{c^2} \right)$$

Spacetime vector

To describe an "event" in spacetime, we define the spacetime 4 vector : $$X = \begin{pmatrix} ct \ x \ y \ z\end{pmatrix}$$
Then we can define the Lorentz transform between two frames as a simple matrix multiplication.
For a boost along x-axis, the Transform Matrix is denoted by $\mathrm{\Lambda }$ and therefore the equation becomes, $X^{\prime }=\mathrm{\Lambda }X$ which is $$\begin{pmatrix}ct' \ x' \ y' \ z'\end{pmatrix} = \begin{pmatrix}\gamma & -\beta \gamma & 0 & 0 \ -\beta\gamma & \gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1\end{pmatrix}\begin{pmatrix}ct \ x \ y \ z\end{pmatrix}$$
This matrix preserves spacetime interval, which is the core idea of special relativity.

Essentially, the spacetime interval between two events, is given by $$\Delta s^2 =\Delta x^2 + \Delta y^2 + \Delta z^2 -c^2\Delta t^2$$
If this spacetime interval is transformed to some other inertial frame, then $$\Delta s'^2 = \Delta s^2$$which is the core principle of conservation of spacetime interval

Consequences of relativity

1. Simultaneity of events
2. Time Dilation : Moving clocks run slower. A clock moving at a speed $v$ will experience time governed by $t^{\prime }=\gamma t$ where $t$ is the time elapsed in object frame, and $t^{\prime }$ is how long has elapsed in a stationary frame.
3. Length Contraction : Objects lengths contract along the direction of motion. Observed length $L=L_0^{\prime }/\gamma$ where $L_0$ is the true length in a stationary frame.
4. Invariant spacetime interval.
5. Velocity addition nuances : Instead of adding linearly like before, it is governed by $$u' = \frac{u + v}{1 + \frac{uv}{c^2}}$$where $u$ is the velocity of the object in one frame, $v$ is the velocity of another frame, and $u^{\prime }$ is the observed velocity from that frame.
6. Mass-energy equivalence : An indirect consequence, it results in the formula we all know so well, $$E = mc^2$$
7. Transformation of electric and magnetic fields : Since the two fields are inherently related by Maxwell's equations, they can change form between two frames, ie, a purely electric field may appear as a mix in some other frame.

Time dilation

Any moving object observes time passing at a slower pace, compared to a stationary observer.
If an object moves at a speed $v$ relative to a stationary observer, then the time elapsed for it ($t_0$), is related to the time elapsed for us ($t^{\prime }$) as $$t' = \gamma t_{0}$$where, $\gamma =\frac{1}{\sqrt{1-{\beta }^2}}$, $\beta =v/c$.

Length contraction

Consider a spaceship moving towards a target at relativistic speeds.
An observer on ground will notice that it will take time $t=\mathrm{\Delta }x/v$ in it's frame, and someone on the spaceship will notice that it takes time $t^{\prime }=\mathrm{\Delta }{x}^{\prime }/v$ in their frame, where $v$ is same.

Since we know $$t' = \gamma t$$this must imply that $\mathrm{\Delta }{x}^{\prime }=\mathrm{\Delta }x/\gamma$, which is the concept of length contraction.
Therefore, we have that length of an object in a relativistic frame is related as $$L = \frac{L_{0}}{\gamma}$$

Invariant mass energy

Changing frame of reference will generate an observable difference in momentum and energy, but they are conserved by the equation $$E^2 = (mc^2)^2 + p^2c^2$$
This equation is invariant under all reference frames.

Therefore, for a particle at rest, it implies that $E=m{c}^2$

Velocity addition nuances

Consider boost along x-axis, therefore $$\begin{gather} x' = \gamma(x-vt) \ \ t' = \gamma\left( t-\frac{vx}{c^2} \right)\end{gather}$$
Differentiate both of them, considering constant velocity $$\begin{gather}dx' = \gamma(dx - vdt) \ \ dt' = \gamma\left( dt - \frac{vdx}{c^2} \right)\end{gather}$$
Now, since velocity of an object in a frame is $$\begin{gather}u = \frac{dx}{dt} \ \ u' = \frac{dx'}{dt'}\end{gather}$$
Using this knowledge, and dividing equation $(1)$ by $(2)$ and substituting the necessary values of $u$ and $u^{\prime }$, we obtain

$$\boxed{u' = \frac{u+v}{1 + \frac{uv}{c^2}}}$$