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Maximal Isotropic (Horizon Penetrating) Schwarzschild coordinates for XCTS #6427

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joaorebelo-megum wants to merge 1 commit into sxs-collaboration:develop from joaorebelo-megum:HPMaximalIsotropic

+585 −7

joaorebelo-megum commented Jan 4, 2025

Added to the Schwarzschild AnalyticSolution for the XCTS system.

Proposed changes

Implementation of the Schwarzschild coordinates with a radial transformation such that the radius is isotropic and the coordinates are horizon penetrating. The solution remains maximally sliced.
Implementation of Unit tests.

Code review checklist

The code is documented and the documentation renders correctly. Run
make doc to generate the documentation locally into BUILD_DIR/docs/html.
Then open index.html.
The code follows the stylistic and code quality guidelines listed in the
code review guide.
The PR lists upgrade instructions and is labeled bugfix or
new feature if appropriate.


          Maximal Isotropic (Horizon Penetrating) Schwarzschild coordinates

ba0bcce

Added to the Schwarzschild AnalyticSolution for the XCTS system.

hrueter suggested changes

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Contributor

hrueter left a comment

For the commit messages use imperative mood, i.e. I suggest

Add Maximal Isotropic (Horizon Penetrating) Schwarzschild coordinates

to the Schwarzschild AnalyticSolution for the XCTS system.

Git Commit Message Guidelines

Use the present tense ("Add feature" not "Added feature")
Use the imperative mood ("Move cursor to..." not "Moves cursor to...")
Limit the first line to 72 characters or less
If needed, a blank second line followed by a more complete description

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +143 to +152

+                      return .25 *
+                                 (2. * areal_radius + mass +
+                                  sqrt(4. * square(areal_radius) + 4. * areal_radius * mass +
+. * square(mass))) *
+                                 pow((4. + 3. * sqrt(2.)) * (2. * areal_radius - 3. * mass) /
+                                         (8. * areal_radius + 6. * mass +
+. * sqrt(8. * square(areal_radius) +
+. * areal_radius * mass +
+. * square(mass))),
+. / sqrt(2.)) -

Contributor

hrueter Jan 6, 2025

I think here you should call maximal_isotropic_radius_from_areal instead of reimplementing the formula.

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +164 to +173

+                      return .25 *
+                                 (2. * areal_radius + mass +
+                                  sqrt(4. * square(areal_radius) + 4. * areal_radius * mass +
+. * square(mass))) *
+                                 pow((4. + 3. * sqrt(2.)) * (2. * areal_radius - 3. * mass) /
+                                         (8. * areal_radius + 6. * mass +
+. * sqrt(8. * square(areal_radius) +
+. * areal_radius * mass +
+. * square(mass))),
+. / sqrt(2.)) -

Contributor

hrueter Jan 6, 2025

I think here you should call maximal_isotropic_radius_from_areal instead of reimplementing the formula.

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +454 to +461

+                             (2. * r_areal + mass +
+                              sqrt(4. * square(r_areal) + 4. * r_areal * mass +
+. * square(mass)))) *
+                        pow((8. * r_areal + 6. * mass +
+. * sqrt(8. * square(r_areal) + 8. * r_areal * mass +
+. * square(mass))) /
+                                ((4. + 3. * sqrt(2.)) * (2. * r_areal - 3. * mass)),
+. / (2. * sqrt(2.)));

Contributor

hrueter Jan 6, 2025

I think here you should call (a templated version of) maximal_isotropic_radius_from_areal instead of reimplementing the formula.

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +513 to +520

+                    const DataType S =
+                        sqrt(8. * square(r_areal) + 8. * r_areal * mass + 6. * square(mass));
+                    const double C = 4. + 3. * sqrt(2.);
+                    const DataType D = 8. * r_areal + 6. * mass + 3. * S;
+                    const DataType E = 2. * r_areal - 3. * mass;
+                    const DataType F = 2. * r_areal + mass;
+                    const DataType A = F + S / sqrt(2.);
+                    const DataType B = C * E / D;

Contributor

hrueter Jan 6, 2025

Please add a comment with a formula for r in terms of S, C, D, E, F, A and B

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +700 to +710

+                    const DataType S =
+                        sqrt(8. * square(r_areal) + 8. * r_areal * mass + 6. * square(mass));
+                    const double C = 4. + 3. * sqrt(2.);
+                    const DataType D = 8. * r_areal + 6. * mass + 3. * S;
+                    const DataType E = 2. * r_areal - 3. * mass;
+                    const DataType F = 2. * r_areal + mass;
+                    const DataType A = F + S / sqrt(2.);
+                    const DataType B = C * E / D;
+                    const DataType dAdR = 2. + (4. / sqrt(2.)) * F / S;
+                    const DataType dBdR = C * (2. * D - E * (8. + 12. * F / S)) / square(D);
+                    const DataType drdR = r_iso * (dAdR / A + dBdR / (B * sqrt(2.)));

Contributor

hrueter Jan 6, 2025

Here you repeat the calculation for drdR. I think it would be good to factor that into a new function maximal_isotropic_radius_from_areal_deriv.

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.cpp

Comment on lines +932 to +942

+                    const DataType S =
+                        sqrt(8. * square(r_areal) + 8. * r_areal * mass + 6. * square(mass));
+                    const double C = 4. + 3. * sqrt(2.);
+                    const DataType D = 8. * r_areal + 6. * mass + 3. * S;
+                    const DataType E = 2. * r_areal - 3. * mass;
+                    const DataType F = 2. * r_areal + mass;
+                    const DataType A = F + S / sqrt(2.);
+                    const DataType B = C * E / D;
+                    const DataType dAdR = 2. + (4. / sqrt(2.)) * F / S;
+                    const DataType dBdR = C * (2. * D - E * (8. + 12. * F / S)) / square(D);
+                    const DataType drdR = r_iso * (dAdR / A + dBdR / (B * sqrt(2.)));

Contributor

hrueter Jan 6, 2025

Factor into new function.

hrueter requested a review from nilsvu

January 6, 2025 14:16

nilsvu reviewed

View reviewed changes

src/PointwiseFunctions/AnalyticSolutions/Xcts/Schwarzschild.hpp

+                 *
+                 * The solution remains maximally sliced, i.e. \f$K=0\f$. And the horizon in
+                 * these coordinates is at \f$r\approx 0.7793271080557972 M\f$ due to the
+                 * radial transformation from \f$R=2M\f$.

Member

nilsvu Jan 6, 2025

Are you sure this is correct? My understanding is that $R=2M$ is only the horizon radius for $C=0$ in Eq. (3d) in https://arxiv.org/pdf/gr-qc/0701037. For $C=3\sqrt{3}M^2/4$, as chosen for this coordinate tra'fo, the horizon radius is probably not at $R=2M$, right? The paper mentions R=3M/2 is a limiting surface, which corresponds to $r=0$, but I don't understand if they mean this is the Schwarzschild horizon.

If you agree with this, please also correct the statement above that $R$ is the canonical Schwarzschild radius.

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