7 Astrophysics and Cosmology
Based on the black hole theorem, theorems on the structure and origin of the Universe are
proved in Section7.5.
Finally, in Section7.6, we have derived 1) the PID cosmological model of our Universe,
2) the nature of dark energy and dark matter, and 3) the gravitational force formulas.
7.1 Astrophysical Fluid Dynamics
The main objective of this section is to establish fluid dynamical models for astrophysical and
cosmological objects such as stars, galaxies, and clustersof galaxies, based on the Principle
of Symmetry-Breaking2.14.
7.1.1 Fluid dynamic equations on Riemannian manifolds
To consider astrophysical fluid dynamics, we first need to discuss the Navier-Stokes equations
on Riemannian manifolds.
Let(M,gij)be ann-dimensional Riemannian manifold. The fluid motion onMare
governed by the Navier-Stokes equations given by
(7.1.1)
∂u
∂t+ (u·∇)u=ν∆u−1
ρ∇p+f forx∈M,divu= 0 ,whereu= (u^1 ,···,un)is the velocity field,pis the pressure,fis the external force,ρis
the mass density,νis the dynamic viscosity, and the differential operator∆is the Laplace-
Beltrami operator defined as∆u= (∆u^1 ,···,∆un)with
(7.1.2) ∆ui=div(∇ui)+gijRjkuk,
div(∇ui) =gkl[
∂
∂xl(
∂ui
∂xk+Γik juj)
+Γil j(
∂uj
∂xk+Γksjus)
(7.1.3)
−Γklj(
∂ui
∂xj+Γijsus)]
.
HereRijis the Ricci curvature tensor andΓik jthe Levi-Civita connection:
Rij=1
2
gkl(
∂^2 gkl
∂xi∂xj+
∂^2 gij
∂xk∂xl−
∂^2 gk j
∂xi∂xl−
∂^2 gli
∂xj∂xk)
(7.1.4)
+gklgrs(
ΓrklΓsij−ΓrilΓsk j)
,
Γik j=1
2
gil(
∂gkl
∂xj+
∂gjl
∂xk−
∂gk j
∂xl)
(7.1.5).
The nonlinear convection term(u·∇)uin (7.1.1) is defined by
(7.1.6)
(u·∇)u= (uiDiu^1 ,···,uiDiun),uiDiuk=ui∂uk
∂xi+Γkijuiuj