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In article <1992Jul21.182314.14031@husc3.harvard.edu>  
mcirvin@husc8.harvard.edu (Mcirvin) writes:

> 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.  

The article was a prety good overview about the subject of four-vectors.  
Yet, two short remarks:

Remark 1:
If you don't set c=1, the coordinates of a point in a Minkowsky-space are  
(if any doubt, consider the units):

 a
x = (ct,x,y,z)  (a is an upper index)

Therefore the transfromation above has to be

   x -> x' = x cosh phi   + ct sinh phi
   t -> t' = x/c sinh phi +  t cosh phi 

Remark 2:
Because of the metric there are two kinds of coordinates neccessary. For  
example the space-vector is given as
 
                    a
   contravariant:  x  = (ct, x, y, z)  (a is an upper index)  

   covariant:      x  = (ct,-x,-y,-z)  (a is a lower index)
                    a

The relationship between a contravariant and a covariant vector is given  
as
                  b
    A    = g   .A      where g    is the described metric tensor
      a      ab                ab

The dot product is aways defined as the product of a covariant and a  
contravariant vector, which explains the minus sign:

              a              a
    A.B :=  A  .B   = A   .B
                  a     a

For example the absolut square of the space-vector is:
      a
    x  .x   = (ct).(ct) + (x).(-x) + (y).(-y) + (z).(-z)
          a

      a        2 2    2    2    2
    x  .x   = c t  - x  - y  - z
          a

I hope this two remarks are comprehensible, too.

--
Harry