Four-vectors
REFERENCE: Hartle
Perhaps you've begun to think about vectors in
(ct)
four-dimensional space - after all, vectors are a
major part of the language of physics. We will
b
call these vectors four-vectors to distinguish
a + b
them from the old three-dimensional vectors of
Newtonian physics and Euclidean space.
We will
a
also need to distinguish between the old three2a
vectors and the new four vectors.
We will
-a
reserve bold type from now on for four vectors,
(x)
and italicized normal type with a small arrow
will be reserved for three-vectors. Thus V is a
[Fig. 1]
four-vector and V is a three-vector (and V is a
scalar).
Since it is hard to write in "bold"
type, we will designate a hand-written four-vector by a squiggly
underline, or an underline arrow.
Thus V and V
both represent the

hand-written four-vector V.
We define a four-vector as a directed line segment in four-dimensional
Minkowski space just as we defined a three-vector as a directed line
segment in Euclidean space. Four vectors can be graphically added and
subtracted in the usual way, and they can be multiplied by a scalar to
increase or decrease their length, or reverse their direction (see Fig.
1).
Vectors can be of three types illustrated in Fig
2. Timelike vectors have a timelike separation
between tail and tip, null vectors have null
separation between tail and tip, and spacelike
vectors
have
spacelike
(ct)
separation
between
tail
and tip. The null vectors
are so-named because, as
at
a
et
you
may
recall,
the
separation
between
two
ey ex ay (x)
points on a light line is
zero (null). See "The new
ax
(y)
spacetime geometry" if you
[Fig. 3]
have forgotten.
(ct)
Null
Timelike
Spacelike
(x)
(y)
[Fig. 2]



Just as we have defined the "basis three-vectors" i, j, and k of unit
length in Euclidean space, we will define the basis four-vectors of
unit length in Minkowski space:
[Eq. 1]
{et, ex, ey, ez} = {e0, e1, e2, e3}  e.
Note
that
basis
vectors
use
subscripts,
and
components
use
superscripts. Any four-vector a can be specified by its components
along the axes,
a = (at, ax, ay, az) = (a0, a1, a2, a3)  a,
[Eq. 2]
or in term of the basis four-vectors
a = atet + axex + ayey + azez  3=0ae = ae.
[Eq. 3]
The last equality makes use of a special convention - namely that
repeated upper and lower indices are understood to be summed.
Greek
indices (, , , , etc.) vary from 0 to 3 (through all four
dimensions) and the Roman indices (i, j, k, etc.) vary from 1 to 3.
Thus we can write the four-vector a as
a = ae = a0e0 + a1e1 + a2e2 + a3e3 = atet + axex + ayey + azez.
[Eq. 4]

We can write the three-vector a as








a = aiei = a1e1 + a2e2 + a3e3 = axex + ayey + azez.
[Eq. 5]

And we can write the four-vector a in terms of the three-vector a as

a = a = (at, a ) = (at, ai ).
Consider two events A and B located at coordinates
(ctA,xA,yA,zA) and (ctB,xB,yB,zB), respectively (see Fig. 4).
We then can define the displacement four-vector from A to
B as
x = (ctB - ctA, xB - xA, yB - yA, zB - zA),
or
[Eq. 7]
x = xB - xA.
[Eq. 6]
(ct)
B
x
A
(x)
[Fig. 4]
Take note of the new use of the letter x. It is used as a
generalized coordinate in Eq. 7. Thus xB2  yB, xA0  ctA,
and so on.
One must look at context to determine its usage. Eq. 7 is
really four equations, since  cycles through the values 0 through 3.
Thus Eq. 7 is equivalent to
x0 = xB0 - xA0, x1 = xB1 - xA1, x2 = xB2 - xA2, x3 = xB3 - xA3,
or xt = xBt - xAt, xx = xBx - xAx, xy = xBy - xAy, xz = xBz - xAz,
or ct = ctB - ctA, x = xB - xA, y = yB - yA, z = zB - zA.
We call  a dummy index. We could have expressed Eq. 7 as x = xB xA, or used any other Greek index. Indices that show up on both sides
of an equality are cycled, producing one equation for each value for a
total of four equations).
Four-vectors can be transformed from one IRF to another moving at speed
v relative to the positive x-axis via the Lorentz boost:
ct' = (ct - xv/c), x' = (x - vt), y' = y, z' = z.
For example, suppose we wish to transform the four vector given in (S)
by a = (cat, ax, ay, az), into the corresponding four-vector in (S') by
a' = (cat',ax',ay',az').
We simply match up the components with the
correct coordinate. Thus
cat  ct, ax  x, ay  y, az  z,
and
cat'  ct', ax'  x', ay'  y', az'  z',
so that
cat' = (cat - axv/c),
ax' = (ax - vat),
ay' = ay,
az' = az.
Finally,
a' = (cat', ax', ay', az') = [(cat - axv/c) , (ax - vat), ay, az ].