These three dimensions in momentum space and three dimensions in real space constitute the 6D phase space. The distribution of neutrinos in this phase space is given by f(r, θ, ϕ, ϵ, μ, Φ). The four-current of particles in the phase space cell dΓcom is given by
(22)$$\begin{eqnarray*}
j^\mu = \int f u^\mu \, \mathrm{d}\Gamma _\mathrm{com}.
\end{eqnarray*}$$The number ΔN of particles in dΓcom within some three volume is
(23)$$\begin{eqnarray*}
\Delta N &amp;=&amp; \int \sqrt{-g}\, \langle \boldsymbol{j}, \mathrm{d}x^\mu \rangle \wedge \mathrm{d}x^\nu \wedge \mathrm{d}x^\xi \wedge \mathrm{d}x^\sigma \nonumber \\
&amp;=&amp; \int \sqrt{-g} j^\mu \, \mathrm{d}x^\nu \, \mathrm{d}x^\xi \, \mathrm{d}x^\sigma ,
\end{eqnarray*}$$where $\langle \boldsymbol {j},dx^\mu \rangle$ denotes the contraction of the four-vector $\boldsymbol {j}$ with the volume form. This general form can be used to obtain both the number of particles within a space-like three-volume cell d3x as well as the number of particles flowing through cell interfaces per coordinate time dt (two space and one time dimensions). The number of particles within the phase space volume dΓcom in a cell $\mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3$ is
(24)$$\begin{eqnarray*}
N = \iiiint \sqrt{-g} f u^0 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3,
\end{eqnarray*}$$and the flux across a cell interface with x1 = const. is
(25)$$\begin{eqnarray*}
F = \iiiint \sqrt{-g} f u^1 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^2 \, \mathrm{d}x^3 \, \mathrm{d}t.
\end{eqnarray*}$$For a conformally flat metric (Isenberg 2008,used in coconut), with
(26)$$\begin{eqnarray*}
g_{\mu \nu }= \begin{pmatrix}-\alpha ^2+\beta _i \beta ^i &amp; \quad \beta _x &amp; \quad \beta _y &amp; \quad \beta _z \\
\beta _x &amp; \quad \phi ^4 &amp; \quad 0 &amp; \quad 0\\
\beta _y &amp; \quad 0 &amp; \quad \phi ^4 &amp; \quad 0\\
\beta _z &amp; \quad 0 &amp; \quad 0 &amp; \quad \phi ^4 \\
\end{pmatrix},
\end{eqnarray*}$$in terms of the lapse function α, the conformal factor ϕ, the shift vector βi, and βi = ϕ4βi, these can be simplified to
(27)$$\begin{eqnarray*}
N = \iiiint \phi ^6 f \hat{u}^0 \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^1 \, \mathrm{d}x^2 \, \mathrm{d}x^3,
\end{eqnarray*}$$(28)$$\begin{eqnarray*}
F = \iiiint \phi ^6 f \left(\frac{\alpha }{\phi ^2}\hat{u}^1+\beta ^r \hat{u}^0\right) \, \mathrm{d}\Gamma _\mathrm{com}\, \mathrm{d}x^2 \, \mathrm{d}x^3 \, \mathrm{d}t.
\end{eqnarray*}$$in terms of the Eulerian velocity components $\hat{u}^\mu$. It is critical to recognize that because of the invariance of the distribution function we can directly use f in the comoving frame to compute the number of neutrinos within phase space cell dΓcomv that cross a zone boundary during the time interval dt. Only the neutrino four-velocity in equation (22) must be transformed into the coordinate frame. The problem, however, is that the current jμ only obeys a conservation law ∇μjμ = 0 if we define dΓcom away from a given reference point xμ as the phase space volume filled by particles from dΓcom at xμ (or in technical terms if we define dΓcom away from xμ by an exponential map). In general, the exponential map of dΓcom(xμ) from xμ to another point x′μ will differ from the momentum space volume dΓcom(x′μ) defined by the same comoving-frame momentum space coordinates, which necessitates the remapping step that we outlined in Section 2.1 and will specify in detail in the next section.
Citation: Isenberg 2008
Citation function: Uses; Uses