Special Relativity
Origin
In 1905, following the failure osf the Michelson-Morley experiment
to prove the existence of an ether as the carrier medium of light,
Albert Einstein published two seminal papers.
The first introduced the concept of photons to explain the photoelectric
effect. This paper earned him the 1921 Nobel prize in physics.
The second paper, "Zur Elektrodynamik bewegter Körper" established the
theory of special relativity.
Underlying it is the concept of inertial frames of reference. Two objects
moving relative to each other and each having attached a local frame of
reference (coordinte system).
Inertial frames of reference are those
moving with constant velocity. Zhey neither change their speed nor their
direction of motion.
Two Principles of SR
-
No object can move faster than the speed of light in vacuum.
-
The laws of physics are the same in all inertial reference frames.
Gravitation has no place in this theory. That comes later with GR.
The consequences nevertheless are profound. The relation between
two inertial frames is described by the Lorentz transform. Time and
space are both parts of a 4-dimensional spacetime. Effects like time
dilation and length contraction, increase of mass-weight with
increasing speed, the equivalence of mass and energy. These are all
effects that have been experimentally confirmed, most dramatically
$E = m\,c^2$ in nuclear weapons, for instance hydrogen bombs, as the
difference $4\times m_H - m_{He} > 0$ is converted into energy.
The same process takes place inside stars like the the sun, making life
on earth possible.
Spacetime and Lorentz Transform
Space and time do not exist as separate entities, but jointly form a
4-dimension space, named spacetime. Its coordinates are
$$(ct,\, x,\, y,\, z) = (x^0,\, x^1,\, x^2,\, x^3) = \vec{x}$$
Given one inertial frame of reference with coordinates $\vec{x'}$ and
another one with coordinates $\vec{x}$ and constant inter-frame velocity
$\vec{\beta} = c^{-1}\vec{v}$ and with the Lorentz factor defined as
$\gamma = (1 - \vec{\beta}\cdot\vec{\beta})^{-1/2}$ the Lorentz transform
is defined by
$$
\large{
\begin{align}
x'^0 &= \gamma\,(x^0 - \vec{\beta}\cdot\vec{X}) \\
\vec{X'}_{||\,\vec{\beta}} &= \gamma\,(-\vec{\beta}\,x^0
+ \vec{X}_{||\,\vec{\beta}})\\
\vec{X'}_{\perp\,\vec{\beta}} &= \vec{X}_{\perp\,\vec{\beta}}
\end{align}}
$$
In the equations $\vec{X} = (x,\,y,\,z)$ is the spatial component
of $\vec{x}$ In the literature one may find equations, where the x-axis
is chosen parallel to the direction of motion. The component of $\vec{X}$
normal to the direction of motion remains unchanged.
Boost and Rotation
.h4tag.sp1.
The transform shown above is only one part of the full Lorentz transform.
It is called the Lorentz boost and is complemented by a
rotation about an axis which need not conincide with $\vec{\beta}$, but
usually lies in the 3-dimensional subspace formed by the spatial
component. Mathematically nothing would exclude a rotation about the
"time" axis.
.h4tag.sp1.
The inverse transform is easily obtained by simply inverting the direction
of $\beta$, $\vec{\beta'} = -\vec{\beta}$. The two effects following directly
from the transform are
-
time dilation
-
length contraction
The equivalence of mass and energy $E = m_0\,c^2$ follows from the concept
of four-vectors and the expansion of $\gamma$ in a power series in
$\beta = \sqrt{\vec{\beta}^2}$.
$$
\large{
\begin{align}
B(a,\,b) &= \frac{\Gamma(a)\,\Gamma(b)}{\Gamma(a + b)}\\
\gamma &=\frac{1}{\sqrt{1 - \beta^2}}\\
&= 2\,\sum_{k = 0}^\infty
\frac{(-1)^k\,\beta^{2\,k}}
{B(\frac{1}{2} - k,\,k + 1)}\\
&\approx 1 + \frac{1}{2}\beta^2 + \frac{3}{8}\,\beta^4
+ \frac{5}{16}\,\beta^6 + \cdots
\end{align}}
$$
The first term in the expansion $E_0 = \gamma(\beta = 0)\,E = m_0\,c^2$
ist the "rest" energy, that an object with $m > 0$ has, when it is not
moving. The second term $m_0\,c^2\,\beta^2 / 2 = m_0\,v^2 / 2$ is the
classical kinetic energy. Higher-order terms would bes undetectable
in most laboratory settings, but high.energy particle physics with
particle velocities appoaching c do indeed measure an increas of mass
as a function of velocity.
All these efects of course only exist for the oberver at rest relative
to the moving particle. For the particle itself, in its reference frame,
nothing changes. Instead, if it could observe the observer, it would
assign the same effects to her or him. In the paricle's frame the
particle is at rest and the observer is moving.
Minkowski Space
The mathematical structure of spacetime was first explained in some detail
in a talk that Hermann Minkowski gave 1908 in Göttingen.
He described the nature of spacetime and its associated metric. They are
now known as Minkowski space and the Minkowski metric.
Expressed by a a rank-2 mmetric tensor $\eta$ and co- and contravariant
rank 1 tensors the relations are
$$
\large{
\begin{align}
\mathbf{\eta} &= \text{diag}\,(1,\,-1,\,-1,\,-1)\\
\vec{x} &= (\,x^0,\,\,x^1,\,\,x^2,\,\,x^3,\,)\\
&= (\,x^0,\,\vec{X}\,)\\
x_\mu &= \eta_{\mu\nu}\,x^\nu\\
ds^2 &= dx^\mu\,dx_\mu\\
&= dx^\mu\,\eta_{\mu\nu}\,dx^\nu\\
&= (dx^0)^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2
\end{align}}
$$
This describes a flat hyperbolic space. There are two signature
conventions, the one above, sometimes called the East Coast
convention and the West Coast convention. The naming refers to the
prevalence of usage at academic institutions on the east and west coast
of the USA.
$$
\mathbf{\eta} =
\begin{cases}
\text{diag}\,(1,\,-1,\,-1,\,-1) & \text{East Coast / Minkowski convention}\\
\text{diag}\,(-1,\,1,\,1,\,1) & \text{West Coast / Weyl convention}\\
\end{cases}
$$
The west coast convention was introduced by Hermann Weyl in his
1919 book "Raum, Zeit, Materie" about general relativity. He set
$x^0 = i\,c\,t$ as an imaginary number. Calculating a conventional
$L_2$ norm produces the minus sign for the 0-component.
Laplace and d'Alembert Operators
The Laplace operator in $\mathbb{R}^3$ is defined as
$\Delta = \nabla^2 = \sum_{i=1}^3\,\partial_i\,\partial^i$.
It constitutes the spece part of the d'Alembert operator which is its
equivalent for spacetime and was known long before Minkowski
introduced the concept of spaceatime. The d'Alambert operator is
defined as
$$
\square = \partial_\mu\,\eta_{\mu\nu}\,\partial^\nu\quad 0\le\mu,\,\nu\le 3
$$
Klein-Gordon Equation
The quantization $E\to\mathrm{i}\,\hbar\,c\,\partial_0$ and
$\vec{p}\to -\mathrm{i}\,\hbar\,\nabla$, with $E_m = m_0\,c^2$ leads
to the inhomogenous Klein-Gordon equation
$$
(\,\square + \frac{E_m^2}{(\hbar\,c)^2}\,)\,\psi(c\,t,\,\vec{x}) = 0
$$
$$
(\,(\hbar\,c)^2\square + E^2\,)\,\psi(c\,t,\,\vec{x}) = 0
$$
The d'Alembert operator can thus be viewed as the quantization
of the momentum (or energy) four-vector.
Fourier Components
The homogenous Klein-Gordon equation $\square\,\psi = 0$ is hyperbolic,
the four-version of the Laplace equation, and describes a wave with the
solution $A\,e^{\mathrm{i}\,(\omega\,t - \vec{k}\cdot\vec{x})}$amd, with
$a = \ln\,A$ and $z = a + \mathrm{i}\,(\omega\,t - \vec{k}\cdot\vec{x})$,
can be written as $e^z$..
Conventional Fourier components involve frequency $\omega$, time t, wave
vector $\vec{k} = \vec{p}\,/\,\hbar$ and spatial vector $\vec{x}$ in the
combination $\omega\,t - \vec{k}\cdot\vec{x}$.
Defining $E_\omega = \hbar\,\omega$ and
$\vec{E}_p = c\,\vec{p}\ $ one obtains
$$
\begin{align}
\omega\,t - \vec{k}\cdot\vec{x} &=
\frac{E_\omega\,c\,t - \vec{E}_p\cdot\vec{x}}{\hbar\,c}\\
\\
&= \frac{E^\mu\,\eta_{\mu\nu}\,x^\nu}{\hbar\,c}
\end{align}
$$
Here $\{E^\mu\} = \{E_\omega,\,\vec{E}_p\}$ is the four-energy
(equivalent to four-momentum) and $\{x^\nu\} = \{c\,t,\,\vec{x}\}$
is the Minkowski spacetime coordinate. In SI units both numerator
and demominator have dimenion $J\,m$. and the fraction, as it should,
is dimensionless.