From 8c62b232273c58cf9cd6a939ad61b05a9725ebce Mon Sep 17 00:00:00 2001 From: bryan newbold Date: Sun, 13 Jul 2008 20:56:44 -0400 Subject: start of SR --- physics/special relativity | 53 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 53 insertions(+) create mode 100644 physics/special relativity (limited to 'physics') diff --git a/physics/special relativity b/physics/special relativity new file mode 100644 index 0000000..41bf4b8 --- /dev/null +++ b/physics/special relativity @@ -0,0 +1,53 @@ +=========================== +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. -- cgit v1.2.3