Mathematical Principles of Theoretical Physics

7.1. ASTROPHYSICAL FLUID DYNAMICS 407

whereAμis the electromagnetic potential, the time components ofgμ νare as

g 00 =−

(

1 +

2

c^2

ψ

)

, g 0 k=gk 0 = 0 for 1≤k≤ 3 ,

andψis the gravitational potential.

2.Fluid dynamic equations.The Newton Second Law can be expressed as

(7.1.63)

dP

dτ

=Force,

whereτis the proper time given by

(7.1.64) dτ=

√

−g 00 dt,

Pis the momentum density field, formally defined by

dx

dτ

=

1

ρ

P,

withρbeing the energy density,

dP

dτ

=

∂P

∂ τ

+

∂P

∂xk

dxk

dτ

=

∂P

∂ τ

+

1

ρ

(P·∇)P,

and

ν∆P+μ∇(divP) the frictional force,

−∇p the pressure gradient,

c^2

2

ρ( 1 −βT)∇g 00 =−ρ( 1 −βT)∇ψ the gravitational force.

Hence, the momentum form of the fluid dynamic equations (7.1.63) is written as

(7.1.65)

∂P

∂ τ

+

1

ρ

(P·∇)P=ν∆P+μ∇(divP)−∇p−ρ( 1 −βT)∇ψ,

where the differential operators∆,∇and(P·∇)are with respect to the space metricgij( 1 ≤
i,j≤ 3 )determined by (7.1.62), as defined in (7.1.2)-(7.1.8).

3.Heat conduction equation:

(7.1.66)

∂T

∂ τ

+

1

ρ

(P·∇)T=κ∆ ̃T+Q,

where ̃∆is defined as
̃∆T=−√^1
g

∂

∂xi

(

√

ggij

∂T

∂xj

),

andg=det(gij), 1 ≤i,j≤3.

4.Energy-momentum conservation:

(7.1.67)

∂ ρ

∂ τ

+divP= 0 ,

whereρis the energy density: