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

82 3+1 equations for matter and electromagnetic field
The Newtonian limit of this equation is [cf. Eqs. (5.15) and (5.59)]
∂Ui
∂t + Uj DjUi = − 1
DiP − DiΦ, (5.69)
ρ0
i.e. the standard Euler equation in presence of a gravitational field of potential Φ.
5.3.6 Further developments
For further developments in 3+1 relativistic hydrodynamics, we refer to the review article by
Font [124]. Let us also point out that the 3+1 decomposition presented above is not very
convenient for discussing conservation laws, such as the relativistic generalizations of Bernoulli’s
theorem or Kelvin’s circulation theorem. For this purpose the Carter-Lichnerowicz approach,
which is based on exterior calculus, is much more powerfull, as discussed in Ref. [143].
5.4 Electromagnetic field
not written up yet; see Ref. [258].
5.5 3+1 magnetohydrodynamics
not written up yet; see Refs. [45, 235, 22].