🚀 go-pugleaf

john@jlc.mv.com (John Leslie) writes:

>   Could someone have mercy on those of us whose college physics is
>twenty or more years in the past?  What is "four-vectors"?

Be careful what you wish for... people like me are listening!
What follows is a lengthy primer on four-vectors in special
relativity.  In fact, much of basic special relativity is
implicit in it.

(Be patient: the most mathematically difficult part of
this post is right at the beginning, and the physics is
all at the end, in part 2!)

                 I. Lorentz transformations

A four-vector, in this context, is a vector in four-dimensional
Minkowski spacetime, a space in which time functions as a fourth
dimension.  A four-vector transforms in the usual way under
rotations, and in a well-defined way under velocity boosts, 
similar to rotations but involving hyperbolic rather than 
trigonometric functions of the appropriate parameter, and 
some different minus signs.  

Imagine a rotation in the xy plane in ordinary three-space.  A
vector in the xy plane transforms as follows under a rotation
by the angle theta:

x -> x' = x cos theta - y sin theta
y -> y' = x sin theta + y cos theta

Likewise, a four-vector in the xt plane (t = time!) transforms
like this under a velocity boost with "rapidity parameter" phi:

x -> x' = x cosh phi + t sinh phi
t -> t' = x sinh phi + t cosh phi 

where by x and t, I mean the corresponding vector components.

Cosh phi is the famous time dilation factor, gamma, which
is 1/sqrt(1-v^2/c^2), with v the relative velocity.  Sinh phi
is gamma times v/c, or just gamma*v if c = 1.  Quantities
such as velocity, energy, and momentum can be expressed in terms
of four-vectors, making notation more compact and easy to
handle, and making it obvious when something is a relativistic
invariant and when it isn't.  

          II. Invariant length and inner products

Consider an ordinary vector V in three-dimensional space.  The
squared length of this vector,

     2   2   2
    V + V + V  
     x   y   z  ,

is independent of rotations in three-dimensional space.  There
is a similar quantity associated with a four-vector A,

    2   2   2   2
 + A - A - A - A
    t   x   y   z  ,

which is *invariant* under both rotations and velocity boosts
(or, in the language usually used, "arbitrary Lorentz
transformations.")  Notice the different sign on the t 
(time) component!  The *overall* sign is just a convention, but
the *difference* in sign between the time component and the
spatial components is what makes this quantity invariant
under boosts.  If you like, you can prove it for a four-vector
in the xt plane, with the y and z components set to zero;
it's easy if you know that cosh^2 phi - sinh^2 phi = 1.

(The overall sign convention varies greatly among authors!
Many like to set the first term negative and the
others positive.  I'll continue to use this one, though.)
 
This is made more general by redefining the dot or inner
product of two vectors.  For three-vectors, the dot product
of vectors P and Q is defined as

  P Q  +  P Q  +  P Q     , abbreviated as    P Q
   x x     y y     z z                         i i

where the sum over repeated indices is understood.  For
four-vectors, the inner product of R and S is
 
  R S  -  R S  -  R S  -  R S               (1)
   t t     x x     y y     z z  .

Just as the dot product of two vectors is invariant under
rotations, this inner product is invariant under Lorentz
transformations, both rotations and boosts.

                   III.  The metric

A shorter way of writing this, using the summation convention,
is to define the "metric tensor", g, which is a matrix

            1  0  0  0
    g   =   0 -1  0  0
     ab     0  0 -1  0
            0  0  0 -1

Then the inner product (1) can be abbreviated 
                                           T
   R  g   S         or, if you like, just R gS.
    a  ab  b 

In the index version of the expression, the repeated
indices imply sums over all four possible values of both
a and b.

(It is sometimes customary to use Greek indices when the
index can vary over all four dimensions.  ASCII doesn't
include Greek, so I'm using a and b instead.)

To keep track of when we need to use the metric tensor, we can
use upper indices (which look like exponents, but aren't) and
lower indices, and use the rule that you do sums over products
where an upper index is the same as a lower index:  the inner
product of R and S is then

    a                                   b
   R  S        where        S   =  g   S
       a                     a      ab      ,

and we use the metric tensor g to "lower indices."
But this is really just another compact abbreviation for (1).

-- 
Matt McIrvin, grad student, Dept. of Physics, Harvard University
mcirvin@husc.harvard.edu      mumble mumble mumble mumble mumble