book:content:gsr - Geometry of General Relativity

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A \textit{spacetime diagram} in special relativity is just a diagram drawn
using hyperbola geometry.  Vertical lines represent the \textit{worldline} of
an observer standing still (in the given reference frame).  Horizontal lines
represent a ``snapshot'' of time, according to that observer.

Lines tilted away from the vertical axis represent objects in motion, whose
speed is given by the (reciprocal) slope of the line.  The basic postulate of
special relativity is that the speed of light is the same for all observers.
Thus, the speed of light is special.  But hyperbola geometry already has a
special slope, namely $\pm1$, corresponding to the asymptotes of all the
hyperbolas.  Equivalently, the two lines with slope $\pm1$ can be thought of
as the degenerate hyperbola $x^2-t^2=0$, all of whose points are at zero
``distance'' from the origin.

Without further ado, we henceforth adopt units such that the speed of light is
1.  In other words, we measure both space and time in the same units,
typically meters.  Some authors indicate this by using coordinates $x$ and
$ct$, with $t$ in seconds and $c=3\times10^8$ meters per second; we measure
time directly in meters, so that $c=1$ (and is dimensionless!).

%% %/*
%% Further details can be found in~\S6.5 of~\cite{GSR}.
%% %*/
%% %*Further details can be found in
%% %*[[gsr>book:gsr:rules|Drawing Spacetime Diagrams]].

A useful thought experiment is to consider two train cars (reference frames),
moving with respect to each other at constant speed.  Suppose a lamp is
turned on at the center of one of the cars when they are even with each other.
(Since the speed of light is constant, it doesn't matter whether the lamp is
moving.)  When does the light reach the front and back of the cars?  Each
observer sees the light travel equal distances at the speed of light, so each
must independently conclude that the light reaches both walls simultaneously.
But each observer sees the other observer moving toward one light beam, and
away from the other, and therefore concludes that the other observer does
\textit{not} see the light reach both walls simultaneously!  Thus, 
observers in relative motion must have \textit{different} notions of
simultaneity.
%/*
A spacetime diagram of this situation is shown in Figure~\ref{doppler}.
%*/
%*A spacetime diagram of this situation is shown in Figure~1.

%/*
See Chapter 6 of~\cite{GSR} for further details.
%*/
%*See [[gsr>book:gsr:ch6|Applications]] for further details.

%/*
\begin{figure}
\centering
%\epsfysize=2in\epsffile{\TOP figures/simult.ps}
\includegraphics[height=2in]{\TOP figures/simult}
\caption[Simultaneity depends on the observer.]
{The heavy lines represent the ends of two train cars, one at rest (vertical
lines), the other moving to the right.  The dashed lines represent beams of
light emitted in the center of both cars.  Each observer sees the light reach
both ends of their car at the same time, corresponding to two
\textit{different} notions of simultaneity (``at the same time''), as denoted
by the thin lines connecting the respective intersection points.}
\label{doppler}
\end{figure}
%*/
%*{{ book:figures:simult.png?300 }}
%* **Figure 1:**
%* The heavy lines represent the ends of two train cars, one at rest (vertical
%* lines), the other moving to the right.  The dashed lines represent beams of
%* light emitted in the center of both cars.  Each observer sees the light
%* reach both ends of their car at the same time, corresponding to two
%* //different// notions of simultaneity, as denoted by the thin lines
%* connecting the respective intersection points.
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