ABSTRACT
The 2-momentum of an object in (two-dimensional) Minkowski space mov-
ing at speed v = tanhβ is given by
p = mu =
( E p
) =
( m coshβ m sinhβ
) . (8.1)
Consider N equally spaced identical particles, as shown in Figure 8.1, and
in a spacetime diagram in Figure 8.2. If the separation between particles
in their rest frame is , then the number density of particles in their rest
frame is
n = N
. (8.2)
As shown in Figure 8.3, an observer at rest watching the particles go past
will measure
¯ =
coshβ (8.3)
due to length contraction, and will therefore observe a number density
n¯ = n coshβ. (8.4)
How many such particles move past an observer at rest per unit time? The
distance traveled by each particle is v · 1 = tanhβ, so the number flux of particles is given by
n¯ tanhβ = n sinhβ. (8.5)
This leads us to define the number flux 2-vector
N = nu =
( n coshβ n sinhβ
) , (8.6)
where n is the number density of the particles in their rest frame.
