Add boost to KerrSchild #5610

guilara · 2023-11-01T14:02:12Z

Proposed changes

Add a boost to the KerrSchild solution. This is done by boosting the coordinates, null form and derivatives of the H function defining the KerrSchild solution. (For the case when there is a boost the solution depends on time as well, but we don't code up the time dependence of the coordinates for simplicity.)

Evolving this solution will require time dependent translation maps to keep the excision surface inside the apparent horizon (#5576)

This reuses items from develop...nilsvu:spectre:kerr_boost

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Further comments

nikwit

nice work overall. Two points:

1.) I think we discussed that with a boost, KerrSchild is technically time dependent but you have only implemented it for t=0. It is therefore technically no longer a AnalyticSolution. Can you make sure to document this?

2.) As we discussed, could you write a test that computes tensors like the extrinsic curvature and the spacetime metric once boosted and once unboosted and make sure they correspond to a manual transformation?

nikwit · 2023-11-21T14:00:03Z

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.hpp

+ * ## Boost of the Kerr-Schild solution
+ *
+ * We add initial momentum to the solution by applying a Lorentz boost to the
+ * metric. Given the form of the Kerr-Schild metric in terms of a covariant


Can you rephrase this to something more readable? Perhaps "The Kerr-Schild metric can be expressed covariantly in terms of (...). We construct the boosted metric by boosting these object individually."

nikwit · 2023-11-21T14:02:38Z

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.hpp

        internal_tags::x_minus_center<DataType, Frame> /*meta*/) const;

-    void operator()(const gsl::not_null<Scalar<DataType>*> a_dot_x,
-                    const gsl::not_null<CachedBuffer*> cache,
+    void operator()(gsl::not_null<Scalar<DataType>*> a_dot_x,


why did you delete all these const specifiers?

clang-tidy was complaining. Said only has effect in the cpp

nikwit · 2023-11-21T14:19:21Z

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.cpp

@@ -391,8 +450,7 @@ void KerrSchild::IntermediateComputer<DataType, Frame>::operator()(
  const auto& lapse = get(cache->get_var(*this, gr::Tags::Lapse<DataType>{}));
  const auto& lapse_squared =
      get(cache->get_var(*this, internal_tags::lapse_squared<DataType>{}));
-  get(*deriv_lapse_multiplier) =
-      -square(null_vector_0_) * lapse * lapse_squared;
+  get(*deriv_lapse_multiplier) = -lapse * lapse_squared;


why can you set square(null_vector_0_) to 1 here but not other places?

I don't think its set to one, just that it seems not in the formula for the spatial derivative of the lapse.

nikwit · 2023-11-21T14:25:51Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.cpp

+template <size_t SpatialDim>
+tnsr::Ab<double, SpatialDim, Frame::NoFrame> lorentz_boost_matrix(
+    const std::array<double, SpatialDim>& velocity) {
+  tnsr::I<double, SpatialDim, Frame::NoFrame> velocity_as_tensor{};


tnsr::I<double, SpatialDim, Frame::NoFrame> velocity_as_tensor(velocity) should work I think

Done. Seems to work

nikwit · 2023-11-21T14:30:07Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.hpp

+/*!
+ * \ingroup SpecialRelativityGroup
+ * \brief Apply a Lorentz boost to the spatial part of a one form.
+ * \details The zero component of the one form is also specified.


Maybe "This requires passing the 0th component of the one form as an additional argument.

nikwit · 2023-11-21T14:43:06Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.cpp

@@ -58,6 +59,80 @@ void lorentz_boost_matrix(
    (*boost_matrix).get(i + 1, i + 1) += 1.0;
  }
 }
+
+template <size_t SpatialDim>
+tnsr::Ab<double, SpatialDim, Frame::NoFrame> lorentz_boost_matrix(


can you make this a symmetric tensor?

Can I? This has one upper and one lower index

Not currently supported.

nikwit · 2023-11-21T14:54:05Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.cpp

+  }
+  const auto boost_matrix_1 = lorentz_boost_matrix(velocity_first_index);
+  const auto boost_matrix_2 = lorentz_boost_matrix(velocity_second_index);
+  for (size_t i = 0; i < SpatialDim + 1; ++i) {


[optional] tenex::evaluate<ti::a, ti::b>(result, boost_matrix_1(ti::C, ti::a) * boost_matrix_2(ti::D, ti::b) * tensor(ti::c, ti::d) is quite concise.

nikwit · 2023-11-21T15:02:28Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.hpp

+ * Note that the Lorentz matrix is symmetric.
+ */
+template <typename DataType, size_t SpatialDim, typename Frame>
+void lorentz_boost(


Ideally, there would be one implementation that works for a generic tensor type so do not have all of these overloads. If you think that would be overkill for now, maybe just add a comment that this can be done in the future.

I think I wouldn't want to delay this anymore. Added comment.

nikwit · 2023-11-21T15:06:31Z

tests/Unit/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Test_WrappedGr.cpp

@@ -44,7 +44,7 @@ void compare_different_wrapped_solutions(const double mass,
 template <typename SolutionType>
 void test_copy_and_move(const SolutionType& solution) {
  test_copy_semantics(solution);
-  auto solution_copy = solution;
+  const auto& solution_copy = solution;


this is just a reference now. Can you just pass in solution directly?

nikwit · 2023-11-21T15:10:35Z

tests/Unit/Helpers/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/VerifyGrSolution.hpp

@@ -426,5 +426,55 @@ void verify_consistency(const Solution& solution, const double time,
                                              derivative_delta),
      derivative_approx);
 }
+
+/// Check the consistency of dependent quantities returned by a


dependent on what?

On other metric quantities. Changed doc.

guilara · 2024-02-21T16:55:53Z

@nikwit I've added one more test to manually boost the spacetime metric and check that I get the same as the implementation. Not entirely sure how to do it for the extrinsic curvature. I think I need to upgrade it to 4d tensor (not sure about this) then boost it and compare a 4d tensor obtained from the implementation.

guilara · 2024-02-21T17:01:49Z

@nikwit was suggesting moving the KerrBoost to AnalyticData instead of AnalyticSolutions while I suggest leaving it here and add a warning in the doc.

The reasoning is mainly because the Kerr boost as it is currently implemented (and as it is implemented in SpEC) is only usable for initial data and some assumptions are being made. For instance, we are not coding up the time dependence at all (the solution now is both function of position and time since its moving, but we set t = 0), and assumptions on the time derivative of the lapse and shift (which will be rewritten anyway when evolved).

@nilsvu any thoughts on this?

nilsvu · 2024-02-21T19:47:19Z

@geoffrey4444 Could you review this one too, since you wrote the Lorentz transformations here and boosts in SpEC?

nilsvu

Yes please keep this class in AnalyticSolution with the warning in the docs.

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.hpp

nilsvu · 2024-02-21T21:57:05Z

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.hpp

+ * metric. Since the Kerr-Schild metric can be expressed covariantly in terms of
+ * the Minkowski metric, a scalar function and a one form, we constructing the
+ * metric by boosting the these objects individually. Notice that we also need
+ * to appropriately boost the coordinates to the boosted frame.


I believe we're evaluating the boosted Kerr-Schild metric in the boosted frame, meaning that in these coordinates the black hole does not move, correct? So we inverse-boost the input coordinates to get to the unboosted frame, evaluate Kerr-Schild quantities in the unboosted frame, then boost these quantities. Please clarify the docs here.

Will clarify. I got confused about it myself at some point. We start with coordinates in the frame where the black hole is moving. Then boost them to the frame where the black hole is at rest, and where we can construct the stationary solution. Then we boost all covariant quantities back to the frame where it is moving.

Hm, for initial data I thought we're solving in a frame where the BHs are at rest.. @geoffrey4444 can you clarify how this works in SpEC?

Talked about this in the meeting today. I think you're correct. For initial data we go into a rotating frame that gets rid of the translation but keeps the Lorentz contraction.

nilsvu · 2024-02-21T22:02:36Z

src/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/KerrSchild.hpp

+ * \warning While technically the boosted Kerr-Schild metric is dependent on
+ * both the time and space coordinates, we have implemented it only at $t = 0$
+ * as in SpEC. Therefore it is technically not an analytic solution and should
+ * not be used to compute errors with respect to it.


The solution should be stationary in the boosted frame; so shouldn't all time derivatives be zero and this be a solution at any $t$? Am I misunderstanding something?

I believe it is stationary in the rest frame, which is not the frame we want. In the frame we want, the black hole is moving and so the metric will be dependent on t and x. I didn't code this t dependence since I didn't see this in SpEC --it seems to me it should only enter in the function that computes the radius to the center.

I also didn't code the time derivatives of the lapse and shift in the frame where the black hole is moving (which are zero in the rest frame of course) but left them as zero. This should be fine because I think they are replaced by the gauge functions we use to evolve this.

Ok I think you're correct. Let me look at the SpEC implementation too.

nilsvu · 2024-02-21T22:03:48Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.cpp

@@ -58,6 +59,80 @@ void lorentz_boost_matrix(
    (*boost_matrix).get(i + 1, i + 1) += 1.0;
  }
 }
+
+template <size_t SpatialDim>
+tnsr::Ab<double, SpatialDim, Frame::NoFrame> lorentz_boost_matrix(


Not currently supported.

nilsvu · 2024-02-21T22:14:32Z

src/PointwiseFunctions/SpecialRelativity/LorentzBoostMatrix.hpp

+/// @{
+/*!
+ * \ingroup SpecialRelativityGroup
+ * \brief Apply a Lorentz boost to the spatial part of a one form.


The argument type is a vector (tnsr::I), so I also think this documentation should read vector. Refer to the docs of lorentz_boost_matrix above for details and math. I also find those docs confusing though. I believe they say that $\Lambda^a_\bar{a}$ is computed, which boosts a vector from barred to unbarred or a one-form from unbarred to barred (consistent with Blandford/Thorne Sec. 2.7). So my suggestion is:

The docs and argument for this function should read vector (they boost a vector)

The function for the one-form below should apply the inverse Lorentz transformation (by inverting the sign of the velocity). This means you add a minus sign to the function below and remove the minus signs from where the function is used in Kerr-Schild. Make sure to clarify this in the docs of the function.

@geoffrey4444 can you confirm or clarify this?

tests/Unit/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Test_KerrSchild.cpp

nilsvu · 2024-02-21T22:19:41Z

tests/Unit/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Test_KerrSchild.cpp

@@ -366,13 +380,116 @@ void test_zero_spin_optimization(const DataType& used_for_size) {
      });
 }

+template <typename Frame, typename DataType>
+void test_boosted_for_zero_velocity(const DataType& used_for_size) {


nilsvu · 2024-02-21T22:22:06Z

tests/Unit/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Test_KerrSchild.cpp

+  // Need to evaluate at the boosted coordinates
+  // Boost coordinates


These are the unboosted coords, no? (related to comments above)

Maybe the terminology ''boosted/unboosted'' is a bit confusing. Rest frame (of the BH) coords I would say.

nilsvu · 2024-02-21T22:23:33Z

tests/Unit/PointwiseFunctions/AnalyticSolutions/GeneralRelativity/Test_KerrSchild.cpp

+
+template <typename Frame, typename DataType>
+void test_boosted_spacetime_metric(const DataType& used_for_size) {
+  // Test that the extrinsic curvature of the boosted solution indeed


extrinsic curvature -> spacetime metric? Also test extrinsic curvature?

Not yet sure how to do it. I think I would need to elevate the extrinsic curvature to a 4d tensor in both frames.

nilsvu · 2024-02-21T22:26:03Z

tests/Unit/PointwiseFunctions/SpecialRelativity/Test_LorentzBoostMatrix.cpp

@@ -86,3 +167,9 @@ SPECTRE_TEST_CASE(
  d = large_velocity_squared;
  CHECK_FOR_DOUBLES(test_lorentz_boost_matrix_analytic, (1, 2, 3));
 }
+
+SPECTRE_TEST_CASE("Unit.PointwiseFunctions.SpecialRelativity.LorentzBoost",


combine with test case above

geoffrey4444 · 2024-02-22T01:36:07Z

@geoffrey4444 Could you review this one too, since you wrote the Lorentz transformations here and boosts in SpEC?

I'll try but really swamped this week.

guilara · 2024-06-26T13:57:45Z

@nilsvu I have rebased this and added an assert that would throw an error if we call a tag I can't compute for the boost.

geoffrey4444

@nilsvu does this look ready to go to you?

nilsvu

Looks good. Thanks for fixup this up. Pleas squash your commits. Also to keep the git blame record straight, could you add this to your commit message:

# Two empty lines


Co-authored-by: Nils Vu <nilsvu@caltech.edu>

nilsvu · 2024-06-29T10:58:08Z

It's two empty lines, no > symbols

Add more Lorentz boost functions Add instantiation of time deriv of spatial metric Add boosted Kerr initial data Expand doc Fix initialization of lorentz boost matrix Remove unused reference Add test of boosted spacetime metric Throw error for unsupported tag Co-authored-by: Nils Vu <nilsvu@caltech.edu>

guilara · 2024-06-29T11:03:22Z

It's two empty lines, no > symbols

Oh, sure

nilsvu · 2024-06-29T16:08:27Z

Thank you for getting this done @guilara !

guilara · 2024-07-01T12:57:28Z

Thank you for getting this done @guilara !

Thank you for all the detailed help with it

guilara changed the title ~~Add Boosted Kerr Initial Data~~ Add Boosted KerrSchild Initial Data Nov 1, 2023

guilara force-pushed the PRs-KerrBoost branch 9 times, most recently from 624a213 to 1366cf9 Compare November 3, 2023 14:12

guilara marked this pull request as ready for review November 3, 2023 14:17

guilara changed the title ~~Add Boosted KerrSchild Initial Data~~ Add boost to KerrSchild Nov 3, 2023

guilara requested a review from nikwit November 10, 2023 10:24

nikwit suggested changes Nov 21, 2023

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guilara force-pushed the PRs-KerrBoost branch from 1366cf9 to b40a32d Compare February 15, 2024 16:22

guilara requested a review from nikwit February 15, 2024 16:57

guilara force-pushed the PRs-KerrBoost branch 2 times, most recently from eafc3ec to b2532c8 Compare February 21, 2024 16:52

nilsvu requested a review from geoffrey4444 February 21, 2024 19:46

nilsvu requested changes Feb 21, 2024

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guilara force-pushed the PRs-KerrBoost branch from 06b64c9 to 3d24e12 Compare June 25, 2024 10:23

guilara requested a review from nilsvu June 26, 2024 13:56

geoffrey4444 approved these changes Jun 27, 2024

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nilsvu approved these changes Jun 28, 2024

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guilara force-pushed the PRs-KerrBoost branch from 7e44d28 to 390d9fe Compare June 29, 2024 10:53

guilara force-pushed the PRs-KerrBoost branch from 390d9fe to e4a5d54 Compare June 29, 2024 11:02

nilsvu merged commit a18184a into sxs-collaboration:develop Jun 29, 2024
22 checks passed

ctrandrsn pushed a commit to ctrandrsn/spectre-ctr that referenced this pull request Jul 8, 2024

Merge pull request sxs-collaboration#5610 from guilara/PRs-KerrBoost

96fe826

		// Need to evaluate at the boosted coordinates
		// Boost coordinates

Add boost to KerrSchild #5610

Add boost to KerrSchild #5610

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guilara commented Nov 1, 2023 • edited Loading

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guilara commented Feb 21, 2024

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