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

5.3 Perfect fluid 81
Finally, by applying formula (4.7), we get the fluid stress tensor with respect to the Eulerian
observer:
or, taking into account Eq. (5.52),
5.3.4 Energy conservation law
S = γ ∗ T = (ρ + P) γ ∗ u
=ΓU
⊗ γ ∗ u
=ΓU
+P γ ∗ g
=γ
= P γ + Γ 2 (ρ + P)U ⊗ U, (5.62)
S = P γ + (E + P)U ⊗ U . (5.63)
By means of Eqs. (5.61) and (5.63), the energy conservation law (5.12) becomes
∂
− Lβ
∂t
E +N D · [(E + P)U] − (E + P)(K + KijU i U j ) +2(E +P)U ·DN = 0 (5.64)
To take the Newtonian limit, we may combine the Newtonian limit of the baryon number
conservation law (5.50) with Eq. (5.18) to get
∂E ′
∂t + D · [(E′ + P)U] = −U · (ρ0DΦ), (5.65)
where E ′ := E − E0 = Ekin + Eint and we clearly recognize in the right-hand side the power
provided to a unit volume fluid element by the gravitational force.
5.3.5 Relativistic Euler equation
Injecting the expressions (5.61) and (5.63) into the momentum conservation law (5.23), we get
∂
− Lβ [(E + P)Ui] + NDj Pδ
∂t j
i + (E + P)Uj
Ui + [Pγij + (E + P)UiUj]D j N
−NK(E + P)Ui + EDiN = 0. (5.66)
Expanding and making use of Eq. (5.64) yields
∂
− Lβ Ui + NU
∂t j DjUi − U j DjN Ui + DiN + NKklU k U l Ui
+ 1
∂
NDiP + Ui − Lβ P = 0. (5.67)
E + P
∂t
Now, from Eq. (5.41), NU j DjUi = V j DjUi + β j DjUi, so that −Lβ Ui + NU j DjUi = V j DjUi −
UjDiβ j [cf. Eq. (A.7)]. Hence the above equation can be written
∂Ui
∂t + V j DjUi + NKklU k U l Ui − UjDiβ j = − 1
∂P
NDiP + Ui
E + P
∂t
−DiN + UiU j DjN.
∂P
− βj
∂xj
(5.68)