Note well that we are not YET introducing proper notation for co- and contravariant tensors as we don't know what that means. Actually the notation for ordinary coordinates should be and we will have to determine whether any given 4-vector quantity that is a function of the coordinates transforms like the coordinate or like a differential of the coordinate in order to determine if it is co- or contravariant. Similarly, we have not yet discussed how to form the various dyadic products of co- and contravariant vectors - some will form scalars, some vectors, some second rank tensors. In other words, the results below are all correct, but the notation sucks and this suckiness will make certain parts of doing the algebra more difficult than it needs to be.
I may rewrite this whole combined multichapter stretch, as I'm not certain of the pegagogical value of presenting things incorrectly or in an elementary form and then correctly in an elegant form as Jackson does. In the meantime, please bear with the notation below allowing for the fact that much of it is just wrong.
Coordinate 4-vectors are .
Arbitrary 4-vectors are .
If the ``arbitrary'' vector transforms like the coordinates, then
A_0' & = & &gamma#gamma;(A_0 - &beta#beta; ·A)
A_&par#parallel;' & = & &gamma#gamma;(A_&par#parallel;- &beta#beta;A_0)
A_&perp#perp;' & = & A_&perp#perp;
and
&Delta#Delta;A^2 & = & A_0^2 - (A_1^2 + A_2^2 + A_3^2)
& = & A_0^2 - A ·A
is an invariant of the transformation. Note: whenever I boldface a
vector quantity, I mean the 3D euclidean (cartesian) vector in ordinary space.
In that case I will write the time (0) component explicitly. When I want to
refer to a 4-vector generically, I will not boldface it (e. g. -- A
vs
).
Kids! Amaze your friends! Astound your neighbors! Show that A_0' B_0' - A' ·B' = A_0 B_0 - A ·B is an invariant of the Lorentz transformation for arbitrary 4-vectors . This is (or will be) homework.
Now, we have a few definitions of ``new words'' to learn. Most of you probably already know them from one context or another, but we all need to agree at this point to talk a common language, so we will review the definitions carefully and avoid confusion.
Electromagnetic signals (and anything else travelling at speed ) travel on the light cone. An event is a coordinate . We are usually interested in causally connected events on a world line. This might be, for example, the trajectory of a massive particle (like one on the tip of your nose) with . Causally connected world line trajectories must live inside the light cone of each event that lies upon them.
Consider two events. If we define the invariant interval S_12^2 = c^2 (t_1 - t_2)^2 - x_1 -x_2^2 then we have a
Both events are inside each other's light cone. These events can be ``causally connected'', because a light signal given off by one can reach the other from the ``inside''. In this case, a suitable Lorentz transformation can make , but always.
Both events are outside each other's light cone. These events are ``causally disconnected'', because a light signal given off by one can not reach the other. If nothing goes faster than light, then those particular events did not speak to one another. Note that this does not mean that earlier (and later) events on each world line to not connect. The events are disconnected, not the world lines themselves.
In this case, a suitable Lorentz transformation can make , but always.
Both events are on each other's light cone. These events are ``causally connected'' by electromagnetic radiation. The field produced by charges at one event are directly interacting with charges at the other event, and vice versa.
Note well that the event pairs considered above can be made spatially coincident, temporally coincident, or both, by suitably boosting the frame. Events with a timelike separation can be made spatially coincident. Events with a spacelike separation can be made to occur at the same time, or in either order. Events with a lightlike separation will always have a lightlike separation in all frames.
We are about to run into a profound philosophical difficulty. Physics is dedicated to dynamics - typically solving initial value problems and hence predicting the dynamical evolution of systems in time. Unfortunately, we just eliminated time as an independent variable. By making it a part of our geometry, it is no longer available as an independent parameter that we can use to write traditional equations of motion.
There are likely to other significant consequences of this decision, as many of the quantities studied in physics are tensor forms defined with respect to spatial geometry. That is, when I compute ``charge'' or ``momentum'' or ``electric field'' or a ``rotation matrix'', I'm computing 0th, 1st or 2nd rank tensors that inherit their directional character (or lack of it) from the underlying spatial coordinate system. Well, we've just made that underlying coordinate system four dimensional and so quantities like ``momentum'' and ``electric field'' will have to be reconsidered. We may need to find new ``timelike'' coordinates to associate with some of these, and perhaps reclassify others as different sorts of tensors.
Finally, we need to recover a ``time'' that can be used to write down some sort of equations of motion or we can't make a ``physics''. This will prove to be very difficult. For one thing, we can no longer expect to be able to solve initial value problems, as time is now a symmetric coordinate. The trajectories of particles are determined by their relativistic interaction connections and differential ``equations of motion'' with boundary conditions on a closed four dimensional hypersurface at four-infinity! That means that it is impossible in principle to predict future trajectories from only a knowledge of those trajectories in the past. It is amazing how few people in physics are willing to internally acknowledge that fact. Accept it. It is true. You will be happier for it.
Anyway, there are at least two ways around this (mathematical) difficulty. One is to introduce a ``hypertime'' - yet another dimension containing a parameter that can serve us as time has served in the past^{17.3}. This, however, introduces a fifth dimension which we need (currently) like a fifth wheel. Maybe God lives in hypertime, but there are infinite difficulties associated with our trying to implement it in the complete absence of physical probes. Say hello to Plane Joe from Flatland. Leave it to masochistic theorists to play games with 10, 26, or even 4096 dimensional projective manifolds at least until you are ready to become one of them.
The second way is to introduce the proper time. This is the time measured in the ``rest frame'' of a particle as it moves along its world line. As such, it is still not an ``absolute'' time like we are used to but it is the closest that we can come to it.
Note well that proper time does not really solve our philosophical problems, because one must still ask how the ``particle'' measures time. If it carries with it a little ``clock'', that clock must have moving parts and some sort of associated period, and those parts have in turn their own proper time. If it is a point particle, its clock must either be in internal degrees of freedom - you begin to see why those theorists mentioned above work their way up to higher dimensional spaces - or else the particle infers the passage of time from what it ``sees'' of the rest of the Universe via its interaction connections and doesn't really have a proper time at all because it cannot have its own proper clock.
It does, however, solve our immediate mathematical problem (that of finding a suitable parameter in terms of which to describe the evolution of a system) so we'll go with it anyway.