7.1 Astrophysical Fluid Dynamics
the pressure term is
(7.1.7) ∇p=
(
g^1 k
∂p
∂xk,···,gnk
∂p
∂xk)
,
the divergence ofuis
(7.1.8) divu=
∂uk
∂xk
+Γkjkuj=1
√
g∂(
√
guk)
∂xk,
andg=det(gij).
By (7.1.2) and (7.1.6)-(7.1.8), the Navier-Stokes equations (7.1.1) can be equivalently
written as
(7.1.9)
∂ui
∂t+uk∂ui
∂xk+Γik jukuj=ν[
div(∇ui)+gijRjkuk]
−
1
ρgij∂p
∂xj+fi,∂uk
∂xk
+Γkjkuj= 0.Remark 7.1.In the Navier-Stokes equations (7.1.1), the Laplace operator∆can be taken in
two forms:
(7.1.10) ∆=dδ+δd the Laplace-Beltrami operator,
(7.1.11) ∆=div·∇ the Laplace operator.
Here we choose (7.1.10) instead of (7.1.11) to represent the viscous term in (7.1.1). The
reason is that the Laplace-Beltrami operator
(dδ+δd)ui=div·∇ui+gijRjkukgives rise to an additional termgijRjkuk. In fluid dynamics, the termμdiv·∇urepresents
the viscous (frictional) force, and the termgijRjkukis the force generated by space curvature
and gravitational interaction. Hence physically, it is more natural to take (7.1.10) instead of
(7.1.11).
Remark 7.2. In the fluid dynamic equations (7.1.9), the symmetry of general relativity
breaks, and the space and time are treated independently.
7.1.2 Schwarzschild and Tolman-Oppenheimer-Volkoff (TOV) metrics
We recall in this section the classical Schwarzschild and TOV metrics for centrally symmetric
gravitational fields.
Schwarzschild metric
Many stars in the Universe are spherically-shaped, generating centrally symmetric gravi-
tational fields. It is known that the Riemannian metric of a spherically symmetric gravitation
field takes the following form:
(7.1.12) ds^2 =−euc^2 dt^2 +evdr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),