AST 204 13 February 2008 Lecture 4
A. More Spacetime and Geometry
I. Like Time or Distance? Timelike and Spacelike Vectors
The square of the interval (or modulus, as we shall call it for 4vectors not connecting two
events) associated with a 4vector
vectoru
is, as we have seen,
s
2
=
u
2
t
−
u
2
x
−
u
2
y
−
u
2
z
=
u
2
t
−
u
2
,
where
u
is an ordinary 3vector, the
space part
of
vectoru
. We should emphasize here that this
separation is
not
a real geometric notion–it is associated with a particular reference frame
(coordinate system), and is no more ‘natural’ than artificially separating a 3vector into
a zcomponent and a 2vector in the
x
−
y
plane.
It is nevertheless often useful to do,
just as similar artificial constructs in 3space are. It is clear that
s
2
can have either sign,
depending on the relative sizes of the time component and the length of the space part,
and can, of course, also be
zero
. In the last case the 4vector is called
null
, and is, as we
have seen, parallel to the path of a photon in spacetime. If the sign of
s
2
is
positive
, the
vector is called
timelike
; the simplest example is a vector with
no
space part at all: the
vector connecting two events occurring at the same place in some reference frame. If, on
the other hand,
s
2
is
negative
, the vector is called
spacelike
; again the simplest example is
a 3vector with no time component–this is a vector connecting two simultaneous events in
some frame, for example. In this case the length of the 3vector is (
−
s
2
)
1
/
2
.
The path of a particle in spacetime is called its
world line
. Now in a frame in which it is
instantaneously at rest, however complex its motion, its velocity is zero, and so the 4vector
tangent to its world line has only a time component, and is hence
timelike
. Furthermore,
if we let
τ
be the time kept by a clock carried by this particle (the particle’s
proper time
,
then the 4vector
vectoru
=
dvectorx
dτ
=
parenleftbigg
dt
dτ
,
d
x
dτ
parenrightbigg
clearly has modulus unity, because again in the frame in which the particle is instan
taneously at rest the time component is 1 and the space part vanishes.
vectoru
is called the
4velocity
of the particle; for very slowly moving particles it is simply (1
,
v
).
Newtonian
mechanics is that regime in which all the particles satisfy the approximation that the veloc
ity is much, much less than unity (c), so their 4velocities have time components negligibly
different from 1
.
Notice that if a particle starts from rest in some frame, the 4velocity has modulus unity
and always does so, which means that it remains timelike, which means that

dx

< dt
, or

dx
/dt

<
1—
Particles accelerating from rest in any frame can NEVER exceed the velocity
of light.
We will see physically what the reason for this is shortly.
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 Spring '08
 Knapp
 Astronomy, Dot Product, Space, General Relativity, Special Relativity, time component

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