3+1 formalism and bases of numerical relativity - LUTh ...

78 3+1 equations for matter and electromagnetic field
Figure 5.2: Coordinate velocity V of the fluid defined as the ratio of the fluid displacement with respect to the
line of constant spatial coordinates to the coordinate time increment dt.
From the very definition of the shift vector (cf. Sec. 4.2.2), the drift of the coordinate line
x i = const from the Eulerian observer worldline between t and t + dt is the vector dt β. Hence
we have (cf. Fig. 5.2)
dℓ = dt β + dx. (5.40)
Dividing this relation by dτ, using Eqs. (5.33), (3.15) and (5.37) yields
U = 1
(V + β) . (5.41)
N
On this expression, it is clear that at the Newtonian limit as given by (5.15), U = V .
5.3.2 Baryon number conservation
In addition to ∇ · T = 0, the perfect fluid must obey to the fundamental law of baryon number
conservation:
∇ · jB = 0 , (5.42)
where jB is the baryon number 4-current, expressible in terms of the fluid 4-velocity and
the fluid proper baryon number density nB as
jB = nBu . (5.43)
The baryon number density measured by the Eulerian observer is
NB := −jB · n. (5.44)