Basics of Special Relativity
**************************************

The Postulates, Spacetime Diagram, and Metric
=====================================================================


Special relativity was developed out of two postulates [Schutz2009]_

1. Princple of relativity (Galileo),
2. Universality of speed of light (Einstein).

Using these two postulates, where the first key definition is interval of events squared

.. math::
   \Delta s^2,

we can derive basically all the relations we need. Some other intuitions will also be applied to the derivations.


Using a spacetime diagram, we can prove that this is invariant under transformation of frames [Schutz2009]_.

.. admonition:: Hyperbolic Space
   :class: note

   If anyone realizes that spacetime is in fact hyperbolic space by looking at the expression of intervals :math:`\Delta s^2`, the transformation is determined.


As we know the invariant quantity of the physical laws, the transformation of vectors can be found out of it, which is basically a rotation in hyperbolic space.



Metric Conventions
==============================


The metric in Eq. :eq:`eq-sr-calculate-interval` is 'derived' from the interval.

To write it down, there are different convention. We choose the signature :math:`+2` metric in special relativity

.. math::
   \eta_{\mu\nu}=\begin{pmatrix}
	-1 & 0 & 0 & 0\\
	0 & 1 & 0 & 0\\
	0 & 0 & 1 & 0\\
	0 & 0 & 0 & 1\\
   \end{pmatrix}.


In most cases, we use natural unit :math:`c=1`.


.. admonition:: d'Alembert operator
   :class: toggle

   d'Alembert operator, or wave operator, is the Lapace operator in Minkowski space. [1]_

   .. math::
      \Box \equiv \partial _ \mu\partial^\nu = \eta _{\mu\nu}\partial^\mu \partial^\nu


   In the usual {t,x,y,z} natural orthonormal basis,

   .. math::
      \Box & = -\partial_t^2+\partial_x^2+\partial_y^2+\partial_z^2 \\
      & = -\partial_t^2+\Delta^2 \\
      & = -\partial_t^2+\nabla



   On wiki [2]_ , they give some applications to it.
   	* klein-Gordon equation
   	  :math:`(\Box+m^2)\phi=0`
   	* wave equation for electromagnetic field in vacuum:
   	  For the electromagnetic four-potential $\Box A^\mu=0$\footnote{Gauge}
   	* wave equation for small vibrations
   	  :math:`\Box_c u(t,x)=0\rightarrow u_{tt}-c^2 u_{xx}=0`





Hyperbolic Geometric Description
==================================

.. admonition:: A Coincidence?
   :class: note

   Let's start from this coincidence.

   .. _special-relativity-velocity-addition:

   .. figure:: assets/special-relativity/specialRelativityVelocityAddition.png
      :align: center

      Addition of velocities

   Recall that in special relativity, velocity addition is

   .. math::
      v_S = \frac{u+v_O}{1+ \beta v/c},
      :label: eqn-velocity-transformation-1d

   where :math:`v_S` is the velocity measured in moving frame S, :math:`v_O` is the velocity measured in frame O. This :math:`\beta` is the factor :math:`u/c` where u is the velocity of the moving frame measure in frame O.

   At the same time, we have the following hyper trig relation.

   .. math::
      \tanh (\alpha + \beta) = \frac{\tanh \alpha + \tanh \beta}{1 + \tanh \alpha \tanh \beta}.

   Isn't this addition of angles the same as the velocity addition?


The algebra of relativity is mostly based on invariance of a new distance under a new rotation. Here we are not going to repeat the derivation of these transformations from the beginning, instead we would like to have a look at the really amazing part of this mathematical theory.


As shown in :numref:`special-relativity-velocity-addition`, we define quantities in two different frames, the frame O and frame S. The velocity of frame S measured in frame O is :math:`u`. Out of this velocity we define a quantity

.. math::
   \tanh \alpha_u = \frac{u}{c},

In fact, any velocity divided by speed of light should be a hyperbolic tangent,

.. math::
   \tanh \alpha_{v_x} = \frac{v_x}{c}.

With this definition of hyperbolic tangent, we notice that

.. math::
   \gamma = \frac{1}{\sqrt{1 - u^2/c^2}} = \cosh\alpha_u.

Suppose we have an object moving with velocity :math:`v_S` in frame S. The velocity measured in frame O is the addition of the velocity of frame S itself and the velocity :math:`v_S`, except the addition rule is not the usual plus but the rule stated in Eq. (:eq:`eqn-velocity-transformation-1d`). We apply the definitions of the hyperbolic trig function,

.. math::
   \frac{v_{S}}{c} = \tanh(\alpha_u + \alpha_{v_O}) = \frac{\tanh \alpha_{u} + \tanh \alpha_{v_0}}{1 + \tanh \alpha_{U} \tanh \alpha_{v_O}} = \frac{u/c + v_O/c}{1+ \frac{u}{c} \frac{v_{O}}{c}}.

We could imagine the algebra of velocities would be simply summations if we define 'velocity' as :math:`\arctan \frac{v_x}{c}`.

Addition of velocities is not that fundamental. What's more important is :highlight-text:`the transformation of coordinate`, as we have always been talking about. In the old school language, the coordinate transformation is

.. math::
   \begin{pmatrix}
   t_O\\
   x_O
   \end{pmatrix} = \gamma \begin{pmatrix}
   1 &  u/c^2 \\
   u & 1
   \end{pmatrix}\begin{pmatrix}
   t_S\\
   x_S
   \end{pmatrix},

where

.. math::
   \gamma = \frac{1}{\sqrt{1 - u^2/c^2}} = \cosh\alpha_u.

If we use the language of hyperbolic trig functions, this transformation becomes

.. math::
   \begin{pmatrix}
   t_O\\
   x_O
   \end{pmatrix} = &\cosh\alpha_u \begin{pmatrix}
   1 &  (\tanh \alpha_u)/c \\
   c(\tanh\alpha_u) & 1
   \end{pmatrix}\begin{pmatrix}
   t_S\\
   x_S
   \end{pmatrix}\\
   =& \begin{pmatrix}
   \cosh\alpha_u &  (\sin \alpha_u)/c \\
   c(\sin\alpha_u) & \cosh\alpha_u
   \end{pmatrix}\begin{pmatrix}
   t_S\\
   x_S
   \end{pmatrix}.

To make the transformation symmetric, we consider

.. math::
   \begin{pmatrix}
   c t_O\\
   x_O
   \end{pmatrix} = \begin{pmatrix}
   \cosh\alpha_u &  \sin \alpha_u \\
   \sin\alpha_u & \cosh\alpha_u
   \end{pmatrix}\begin{pmatrix}
   ct_S\\
   x_S
   \end{pmatrix}.

.. admonition:: Natural Unit
   :class: hint

   Look at these tedious steps. Why not just use natural units and set :math:`c=1`. We should.

This is basically the rotation matrix in hyperbolic spacetime.

.. admonition:: Rotation in Euclidean Space
   :class: toggle

   The rotations in Euclidean space is described as

   .. math::
      \begin{pmatrix}
      x'\\
      y'
      \end{pmatrix} = \begin{pmatrix}
      \cos\theta &  -\sin \theta \\
      \sin\theta & \cos\theta
      \end{pmatrix}\begin{pmatrix}
      x\\
      y
      \end{pmatrix}.

It is quite different from the rotations in Euclidean space.


Since we are talking about geometry, space-time diagram will be extremely important. The length contraction, time dilation, and even doppler shift can be explained and calculated using the hyperbolic trig functions. Triangles on the space-time diagram are described in :ref:`visualizations-of-hyperbolic-space`.


.. index:: time-dilation

Time Dilation
=========================

Use a spacetime diagram.


Length Contraction
=========================

Use a spacetime diagram.


Footnotes
==========

1. *The Geometry of Special Relativity* by Tevian Dray.


.. [1] Actually, there are more general definations for Lapacian, which includes this d'Alembertian of course.
.. [2] wiki:D'Alembert\_operator
.. [Schutz2009] *A First Course in General Relativity*