===========================
Special Relativity
===========================
.. warning:: This is a rough work in progress!! Likely to be factual errors,
poor grammar, etc.
.. note:: Most of this content is based on a 2002 Caltech course taught by
Kip Thorn [PH237]_
*See also `physics/general relativity`__*
As opposed to general relativity, special relativity takes place in a *flat*
Minkowski space time: a 4-space with three spatial dimensions and one time
dimension.
+----------------+--------------------+
| Index notation | Variable | Type |
+----------------+--------------------+
| `$x^0$`:m: | `$t$`:m: | Time |
| `$x^1$`:m: | `$x$`:m: | Spatial |
| `$x^2$`:m: | `$y$`:m: | Spatial |
| `$x^3$`:m: | `$z$`:m: | Spatial |
+----------------+--------------------+
Separations
-------------
The separation `$(\Delta s)^2$`:m: between two events in space time, in a given
Lorentzian/inertial frame, is defined
as:
:m:`$$ (\Delta s)^2 \equiv -(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 $$`
or
:m:`$$ (\Delta s)^2 \equiv -(\Delta x^0)^2 + \sum_{i,j} \delta_{ij} \Delta x^i \Delta x^j$$`
where :m:`$\delta_{ij}$` is the Kronecker delta (unity or 1 when
:m:`$i=j$`; zero otherwise), and the indices i and j are over the spatial
dimensions 1,2,3 (corresponding to x,y,z). It can be shown that this separation
is Lorentz-invariant; the scalar value of separation between two events does
not depend on the inertial frame chosen.
Note the negative sign in front of the time dimension. The are three types of
separations: **space-like** when :m:`$(\Delta s)^2 > 0$`, **null-** or
**light-like** when :m:`$(\Delta s)^2 = 0$`, and **time-like** when
:m:`$(\Delta s)^2 < 0$`. When dealing with time-like separations, ignore the
implication of an imaginary number. The difference in time :m:`$\Delta \Tau$`
is always real: :m:`($\Delta \Tau)^2= -(\Delta s)^2$`.
References
----------------
.. [PH237] `Gravitational Waves`:title: (aka ph237), a course taught by Kip Thorne at Caltech in 2002. See http://elmer.tapir.caltech.edu/ph237/ for notes and lecture videos.