fixup: specify if integrands need to be integrated with conformal/fla…

…t elements.

src/Elliptic/Systems/Xcts/Events/ObserveAdmIntegrals.cpp

@@ -104,6 +104,9 @@ void local_adm_integrals(
     const auto& face_mesh = mesh.slice_away(boundary_direction.dimension());
     const auto& conformal_face_normal =
         conformal_face_normals.at(boundary_direction);
+    const auto face_normal_magnitude = magnitude(conformal_face_normal);
+    const auto flat_face_normal = tenex::evaluate<ti::i>(
+        conformal_face_normal(ti::i) / face_normal_magnitude());
 
     // Compute curved and flat area elements
     const auto face_sqrt_det_conformal_metric =
@@ -131,17 +134,16 @@ void local_adm_integrals(
     const auto contracted_mass_integrand =
         tenex::evaluate(mass_integrand(ti::I) * conformal_face_normal(ti::i));
     const auto contracted_linear_momentum_integrand = tenex::evaluate<ti::I>(
-        linear_momentum_integrand(ti::I, ti::J) * conformal_face_normal(ti::j));
+        linear_momentum_integrand(ti::I, ti::J) * flat_face_normal(ti::j));
 
     // Take integrals
     adm_mass->get() += definite_integral(
         get(contracted_mass_integrand) * get(conformal_area_element),
         face_mesh);
     for (int I = 0; I < 3; I++) {
-      adm_linear_momentum->get(I) +=
-          definite_integral(contracted_linear_momentum_integrand.get(I) *
-                                get(conformal_area_element),
-                            face_mesh);
+      adm_linear_momentum->get(I) += definite_integral(
+          contracted_linear_momentum_integrand.get(I) * get(flat_area_element),
+          face_mesh);
       center_of_mass->get(I) += definite_integral(
           center_of_mass_integrand.get(I) * get(flat_area_element), face_mesh);
     }

src/PointwiseFunctions/Xcts/AdmLinearMomentum.hpp

@@ -23,6 +23,10 @@ namespace Xcts {
  *                    \Big) \, dS_j.
  * \end{equation}
  *
+ * \note For consistency with `adm_linear_momentum_volume_integrand`, this
+ * integrand needs to be contracted with the Euclidean face normal and
+ * integrated with the Euclidean area element.
+ *
  * \param result output pointer
  * \param conformal_factor the conformal factor $\psi$
  * \param inv_spatial_metric the inverse spatial metric $\gamma^{ij}$
@@ -62,6 +66,9 @@ tnsr::II<DataVector, 3> adm_linear_momentum_surface_integrand(
  * where $1/(8\pi) P^{jk}$ is the result from
  * `adm_linear_momentum_surface_integrand`.
  *
+ * \note For consistency with `adm_linear_momentum_surface_integrand`, this
+ * integrand needs to be integrated with the Euclidean volume element.
+ *
  * \param result output pointer
  * \param surface_integrand the quantity $1/(8\pi) P^{ij}$ (result of
  * `adm_linear_momentum_surface_integrand`)

src/PointwiseFunctions/Xcts/AdmMass.hpp

@@ -21,13 +21,17 @@ namespace Xcts {
  *                     \bar\gamma^{jk} \bar\Gamma^i_{jk}
  *                     - \bar\gamma^{ij} \bar\Gamma_{j}
  *                     - 8 \bar\gamma^{ij} \partial_j \psi
- *                  \Big) dS_i.
+ *                  \Big) d\bar{S}_i.
  * \end{equation}
  *
  * \note We don't use the other versions presented in \cite BaumgarteShapiro of
  * this integral because they make assumptions like $\bar\gamma = 1$,
  * $\bar\Gamma^i_{ij} = 0$ and faster fall-off of the conformal metric.
  *
+ * \note For consistency with `adm_mass_volume_integrand`, this integrand needs
+ * to be contracted with the conformal face normal and integrated with the
+ * conformal area element.
+ *
  * \param result output pointer
  * \param deriv_conformal_factor the gradient of the conformal factor
  * $\partial_i \psi$
@@ -69,7 +73,7 @@ tnsr::I<DataVector, 3> adm_mass_surface_integrand(
  *                    - \bar\gamma^{ij} \partial_i \bar\Gamma_j
  *                    - \bar\Gamma_l \bar\gamma^{lj} \bar\Gamma_j
  *                    - 8 \bar D^2 \psi
- *                  \Big) dV,
+ *                  \Big) d\bar{V},
  * \end{equation}
  *
  * where we can use the Hamiltonian constraint (Eq. 3.37 in
@@ -86,6 +90,9 @@ tnsr::I<DataVector, 3> adm_mass_surface_integrand(
  * \note This is similar to Eq. 3.149 in \cite BaumgarteShapiro, except that
  * here we don't assume $\bar\gamma = 1$.
  *
+ * \note For consistency with `adm_mass_surface_integrand`, this integrand needs
+ * to be integrated with the conformal volume element.
+ *
  * \param result output pointer
  * \param conformal_factor the conformal factor
  * \param conformal_ricci_scalar the conformal Ricci scalar $\bar R$

src/PointwiseFunctions/Xcts/CenterOfMass.hpp

@@ -26,6 +26,9 @@ namespace Xcts {
  * \note We don't include the ADM mass $M_{ADM}$ in this integrand. After
  * integrating the result of this function, you have to divide by $M_{ADM}$.
  *
+ * \note For consistency with `center_of_mass_volume_integrand`, this
+ * integrand needs to be integrated with the Euclidean area element.
+ *
  * \see `Xcts::adm_mass_surface_integrand`
  *
  * \warning This integral assumes that the conformal metric falls off to
@@ -71,6 +74,9 @@ tnsr::I<DataVector, 3> center_of_mass_surface_integrand(
  *
  * where $n^i = x^i / r$ and $r = \sqrt{x^2 + y^2 + z^2}$.
  *
+ * \note For consistency with `center_of_mass_surface_integrand`, this
+ * integrand needs to be integrated with the Euclidean volume element.
+ *
  * \see `center_of_mass_surface_integrand`
  *
  * \param result output pointer