This is where I began,
Simple Nature: http://www.lightandmatter.com/html_books/0sn/ I've also found
Spacetime Physics by Taylor and Wheeler very helpful. The first chapter of Blandford and Thorne:
Applications of Classical Physics ( http://www.pma.caltech.edu/Courses/ph136/yr2008/ ) is very nicely set out, with alternating sections on Newtonian physics (Euclidean space) and special relativity (flat spacetime).
A Lorentz coordinate system, also called an intertial frame of reference, has three constant spatial unit basis vectors at right angles to each other, just like a Cartesian coordinate system, and one constant temporal unit basis vector "at right angles" to all of the spatial basis vectors. A "right angle" here is defined by the formula for finding the magnitude. How can we compare lengths of space and time? The existence of a constant speed of light,
c, gives us a natural way to do this. We pick units in which
c = 1, for example years (of time) and light years (of distance), or seconds and light seconds (the distance light travels in a second). Or if we want units of space instead, we could use metres and the time it takes light to travel one metre through a vacuum (about 3.3 nanoseconds).
Traditionally the relationship between time and space is represented by drawing one axis of space horizontally, and the time axis vertically. Then the
worldline (trajectory through spacetime) of a particle at rest in one Lorentz coordinate system makes a vertical line in that coordinate system because its position doesn't change. All Lorentz coordinate systems are moving at constant velocity relative to each other, so the worldline of a particle at rest in one is also a straight line in any other Lorentz coordinate system, although it will have a different slope if the other is moving at some nonzero velocity relative to the first.
You can change from one Lorentz coordinate system to another by translations (linear movements in space), rotations (i.e. circular rotations by any angle around any axis in space) or by boosts (also called "hyperbolic rotations"). This last kind of transformation represents a switch to a system moving with some nonzero velocity relative to the first. Analogous to the way a circular rotation "mixes" two of the space components of a vector with each other, hyperbolic rotations "mix" the time component with one of the space components, hence the phenomena called time dilation and length contraction and relativity of simultaneity. Translations, rotations, reflections and boosts are called Poincaré transformtions. Rotations, reflections and boosts are called Lorentz transformations. Rotations and boosts alone are called the proper Lorentz transformations. But often in elementary texts, the expression Lorentz transformation is synonymous with boost.
Unlike Euclidean space, in which one vector can be rotated and scaled into any other vector, there are certain distinct regions of spacetime from which it isn't possible to rotate or boost a vector into one of the other regions. For each event (point in spacetime), there is its timelike future, its timelike past, its lightlike future, its lightlike past, and its spacelike "elsewhere". Future lightlike vectors point along a trajectory that a massless particle, such as a photon, could take through spacetime. Although they have a nonzero spatial length, their spacetime magnitude is always zero. Think of a vector representing a lightlike displacement of one light year. Its time component is, by the definition of lightlike, the same: one year. One year (of time) minus one year (of space) = 0. In this case,
0=\sqrt{ \Delta t^2-\left ( \frac{\Delta x}{c}\right )^2- \left ( \frac{\Delta y}{c}\right )^2 - \left ( \frac{\Delta z}{c}\right )^2.
Future timelike vectors point in the direction of the possible spacetime trajectory of a particle with mass. The magnitude of a timelike displacement vector is called its
proper time and often denoted with the Greek letter tau:
\Delta \tau =\sqrt{ \Delta t^2-\left ( \frac{\Delta x}{c}\right )^2- \left ( \frac{\Delta y}{c}\right )^2 - \left ( \frac{\Delta z}{c}\right )^2}.
More generally, proper time is the arc length of the possible spacetime trajectory of a particle with mass:
\Delta \tau = \int \sqrt{1-\frac{v^2}{c^2}} \enspace dt.
Past timelike and lightlike vectors are the negative of future ones; they point in the spacetime direction a particle could have come from. Lightlike vectors all lie in the surface of what's called a
light cone. The past part of the surface of the light cone of an event is the set of past events from which light is reaching that event (think of all the events you can see in at once in the night sky: events taking place 4 years ago at Alpha Centauri, 9 years ago at Sirius, the Andromeda Galaxy 2.5 million years ago...); the future part of the surface of the light cone are all of the events in the future which something traveling at the speed of light (in a vacuum) can reach (think of a supernova exploding, and all of the times and places when/where that event will be seen).
That just leaves spacelike vectors, which point in (I suppose you could call them) "superluminal" directions. No particle can travel along a spacelike displacement, but it's still useful to define vectors representing such displacements. They represent the geometrical relationship between pairs of events (different points in spacetime) whose relative location means that neither can causally affect the other. There is no natural, coordinate-independent way of ordering such events in time. If they're simultaneous according to one Lorentz coordinate system, you can always boost to another, traveling at a different velocity, in which one such event will happen first, or to another, traveling in a different direction, in which the other event happens first!
There's a mathematical awkwardness here, because if we define the magnitude of a spacelike vector in the same way as I did for causal vectors (lightlike and timelike), the magnitude of a spacelike vector is an imaginary number. Some authors don't mind that. Others avoid imaginary numbers by defining their spacelike magnitudes slightly differently. For example, that of a spacelike displacement vector:
\Delta \sigma = \sqrt{-\Delta t^2 + \left ( \frac{\Delta x}{c}\right )^2 + \left ( \frac{\Delta y}{c}\right )^2 + \left ( \frac{\Delta z}{c}\right )^2}
or, equivalently, some define the magnitude of all spacetime in a Lorentz coordinate system vectors as
\Delta s = \sqrt{\left| \Delta t^2 - \left ( \frac{\Delta x}{c}\right )^2 - \left ( \frac{\Delta y}{c}\right )^2 - \left ( \frac{\Delta z}{c}\right )^2 \right|}.
Others simply define the magnitude as the square of the above:
\Delta t^2 - \left ( \frac{\Delta x}{c}\right )^2 - \left ( \frac{\Delta y}{c}\right )^2 - \left ( \frac{\Delta z}{c}\right )^2
This has the advantage of avoiding imaginary magnitudes and square roots, and let's you use the same definition for all types of vector, although it doesn't make such a good analogy with Euclidean magnitude.
But don't get bogged down trying to remember all these different definitions! I'm just mentioning them as something to be aware of because it's something I found confusing at first. Best to find a good book that explains things clearly and makes sense to you, and get used to whichever method they use, then look up the others as you meet them.
Finally, I should say that there are two "sign conventions" (
http://en.wikipedia.org/wiki/Sign_convention ), both very common. Some authors use + and - as I've done here, while others swap them over and use a + wherever I've used a -, and a - wherever I've used a plus. It doesn't make any difference to the fundamental principles; it's a bit pesky for beginners, but you soon get used to it. -+++ has the advantage that the space part looks exactly the same as Pythagoras. +--- has the advantage that you can treat proper time as the magnitude of a displacement vector.
So the different conventions to watch out for are:
(1) Time units or space units? If you're reading a book that divides x,y and z by c in the above formulas, it's using time units, e.g. years and light years, or seconds and light seconds. If instead it multiplies t by c, it's using space units: one unit of time is the amount of time it takes light to travel, e.g. a metre. If in doubt, a little dimensional analysis should clear things up. For example, if the formula involves mass, energy and momentum, think where you'd need to put something with dimensions of distance divided by time, c, for the units to balance. If your book has already set c=1, then it doesn't matter; the equations are much simpler and work either way.
(2) Sign convention. +--- or -+++.
(3) How the magnitide of vectors is defined: imaginary magnitudes for either causal or spacelike vectors depending on sign convention; different formulas for causal and spacelike; one formula with absolute value under square root; magnitude defined as the square.
