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Added spatial and spacetime Weyl tensor computations (e.g. for potent…
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…ial future work on gravitational wave extraction) for both Minkowski and black hole spacetimes, plus updated unit test to confirm that (a) Minkowski space is conformally-flat; and (b) in black hole spacetimes, the Riemann and Weyl tensors are equal.
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@JonathanGorard
JonathanGorard committed May 6, 2024
1 parent 2763af7 commit ee8771b
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Showing 9 changed files with 512 additions and 3 deletions.
156 changes: 154 additions & 2 deletions unit/ctest_gr_spacetime.c
View file
Original file line number Diff line number Diff line change
Expand Up @@ -163,23 +163,28 @@ test_gr_minkowski()
}

double ****spatial_riemann_tensor = gkyl_malloc(sizeof(double***[3]));
double ****spatial_weyl_tensor = gkyl_malloc(sizeof(double***[3]));
double **spatial_ricci_tensor = gkyl_malloc(sizeof(double*[3]));
double spatial_ricci_scalar;

for (int i = 0; i < 3; i++) {
spatial_riemann_tensor[i] = gkyl_malloc(sizeof(double**[3]));
spatial_weyl_tensor[i] = gkyl_malloc(sizeof(double**[3]));
spatial_ricci_tensor[i] = gkyl_malloc(sizeof(double[3]));

for (int j = 0; j < 3; j++) {
spatial_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[3]));
spatial_weyl_tensor[i][j] = gkyl_malloc(sizeof(double*[3]));

for (int k = 0; k < 3; k++) {
spatial_riemann_tensor[i][j][k] = malloc(sizeof(double[3]));
spatial_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[3]));
spatial_weyl_tensor[i][j][k] = gkyl_malloc(sizeof(double[3]));
}
}
}

spacetime->spatial_riemann_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spatial_riemann_tensor);
spacetime->spatial_weyl_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spatial_weyl_tensor);
spacetime->spatial_ricci_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spatial_ricci_tensor);
spacetime->spatial_ricci_scalar_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spatial_ricci_scalar);

Expand All @@ -190,6 +195,7 @@ test_gr_minkowski()
for (int k = 0; k < 3; k++) {
for (int l = 0; l < 3; l++) {
TEST_CHECK( gkyl_compare(spatial_riemann_tensor[i][j][k][l], 0.0, 1e-10) );
TEST_CHECK( gkyl_compare(spatial_weyl_tensor[i][j][k][l], 0.0, 1e-10) );
}
}
}
Expand All @@ -198,23 +204,28 @@ test_gr_minkowski()
TEST_CHECK( gkyl_compare(spatial_ricci_scalar, 0.0, 1e-10) );

double ****spacetime_riemann_tensor = gkyl_malloc(sizeof(double***[4]));
double ****spacetime_weyl_tensor = gkyl_malloc(sizeof(double***[4]));
double **spacetime_ricci_tensor = gkyl_malloc(sizeof(double*[4]));
double spacetime_ricci_scalar;

for (int i = 0; i < 4; i++) {
spacetime_riemann_tensor[i] = gkyl_malloc(sizeof(double**[4]));
spacetime_weyl_tensor[i] = gkyl_malloc(sizeof(double**[4]));
spacetime_ricci_tensor[i] = gkyl_malloc(sizeof(double[4]));

for (int j = 0; j < 4; j++) {
spacetime_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));
spacetime_weyl_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));

for (int k = 0; k < 4; k++) {
spacetime_riemann_tensor[i][j][k] = malloc(sizeof(double[4]));
spacetime_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
spacetime_weyl_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
}
}
}

spacetime->spacetime_riemann_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_riemann_tensor);
spacetime->spacetime_weyl_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_weyl_tensor);
spacetime->spacetime_ricci_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_ricci_tensor);
spacetime->spacetime_ricci_scalar_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_ricci_scalar);

Expand All @@ -225,6 +236,7 @@ test_gr_minkowski()
for (int k = 0; k < 4; k++) {
for (int l = 0; l < 4; l++) {
TEST_CHECK( gkyl_compare(spacetime_riemann_tensor[i][j][k][l], 0.0, 1e-10) );
TEST_CHECK( gkyl_compare(spacetime_weyl_tensor[i][j][k][l], 0.0, 1e-10) );
}
}
}
Expand All @@ -250,12 +262,15 @@ test_gr_minkowski()

for (int k = 0; k < 3; k++) {
gkyl_free(spatial_riemann_tensor[i][j][k]);
gkyl_free(spatial_weyl_tensor[i][j][k]);
}
gkyl_free(spatial_riemann_tensor[i][j]);
gkyl_free(spatial_weyl_tensor[i][j]);
}
gkyl_free(spatial_metric_der[i]);
gkyl_free(spatial_christoffel[i]);
gkyl_free(spatial_riemann_tensor[i]);
gkyl_free(spatial_weyl_tensor[i]);
}
gkyl_free(spatial_metric);
gkyl_free(inv_spatial_metric);
Expand All @@ -265,6 +280,7 @@ test_gr_minkowski()
gkyl_free(shift_vector_der);
gkyl_free(lapse_function_der);
gkyl_free(spatial_riemann_tensor);
gkyl_free(spatial_weyl_tensor);
gkyl_free(spatial_ricci_tensor);

for (int i = 0; i < 4; i++) {
Expand All @@ -278,18 +294,22 @@ test_gr_minkowski()

for (int k = 0; k < 4; k++) {
gkyl_free(spacetime_riemann_tensor[i][j][k]);
gkyl_free(spacetime_weyl_tensor[i][j][k]);
}
gkyl_free(spacetime_riemann_tensor[i][j]);
gkyl_free(spacetime_weyl_tensor[i][j]);
}
gkyl_free(spacetime_metric_der[i]);
gkyl_free(spacetime_christoffel[i]);
gkyl_free(spacetime_riemann_tensor[i]);
gkyl_free(spacetime_weyl_tensor[i]);
}
gkyl_free(spacetime_metric);
gkyl_free(inv_spacetime_metric);
gkyl_free(spacetime_metric_der);
gkyl_free(spacetime_christoffel);
gkyl_free(spacetime_riemann_tensor);
gkyl_free(spacetime_weyl_tensor);
gkyl_free(spacetime_ricci_tensor);
}
}
Expand Down Expand Up @@ -513,6 +533,57 @@ test_gr_schwarzschild()

TEST_CHECK( gkyl_compare(spacetime_ricci_scalar, 0.0, 1e-2) );

double ****spacetime_riemann_tensor = gkyl_malloc(sizeof(double***[4]));
double ****covariant_spacetime_riemann_tensor = gkyl_malloc(sizeof(double***[4]));
double ****spacetime_weyl_tensor = gkyl_malloc(sizeof(double***[4]));

for (int i = 0; i < 4; i++) {
spacetime_riemann_tensor[i] = gkyl_malloc(sizeof(double**[4]));
covariant_spacetime_riemann_tensor[i] = gkyl_malloc(sizeof(double**[4]));
spacetime_weyl_tensor[i] = gkyl_malloc(sizeof(double**[4]));

for (int j = 0; j < 4; j++) {
spacetime_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));
covariant_spacetime_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));
spacetime_weyl_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));

for (int k = 0; k < 4; k++) {
spacetime_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
covariant_spacetime_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
spacetime_weyl_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));

for (int l = 0; l < 4; l++) {
covariant_spacetime_riemann_tensor[i][j][k][l] = 0.0;
}
}
}
}

spacetime->spacetime_riemann_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_riemann_tensor);
spacetime->spacetime_weyl_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_weyl_tensor);

for (int i = 0; i < 4; i++) {
for (int j = 0; j < 4; j++) {
for (int k = 0; k < 4; k++) {
for (int l = 0; l < 4; l++) {
for (int m = 0; m < 4; m++) {
covariant_spacetime_riemann_tensor[i][j][k][l] += spacetime_metric[i][m] * spacetime_riemann_tensor[m][j][k][l];
}
}
}
}
}

for (int i = 0; i < 4; i++) {
for (int j = 0; j < 4; j++) {
for (int k = 0; k < 4; k++) {
for (int l = 0; l < 4; l++) {
TEST_CHECK( gkyl_compare(covariant_spacetime_riemann_tensor[i][j][k][l], spacetime_weyl_tensor[i][j][k][l], 1e-2) );
}
}
}
}

bool in_excision_region;
spacetime->excision_region_func(spacetime, 0.0, x, y, 0.0, &in_excision_region);

Expand Down Expand Up @@ -557,11 +628,23 @@ test_gr_schwarzschild()
gkyl_free(spacetime_metric_der[i][j]);
gkyl_free(spacetime_christoffel[i][j]);
gkyl_free(spacetime_metric_cov_der[i][j]);

for (int k = 0; k < 4; k++) {
gkyl_free(spacetime_riemann_tensor[i][j][k]);
gkyl_free(covariant_spacetime_riemann_tensor[i][j][k]);
gkyl_free(spacetime_weyl_tensor[i][j][k]);
}
gkyl_free(spacetime_riemann_tensor[i][j]);
gkyl_free(covariant_spacetime_riemann_tensor[i][j]);
gkyl_free(spacetime_weyl_tensor[i][j]);
}
gkyl_free(spacetime_metric_der[i]);
gkyl_free(spacetime_christoffel[i]);
gkyl_free(spacetime_metric_cov_der[i]);
gkyl_free(spacetime_ricci_tensor[i]);
gkyl_free(spacetime_riemann_tensor[i]);
gkyl_free(covariant_spacetime_riemann_tensor[i]);
gkyl_free(spacetime_weyl_tensor[i]);
}
gkyl_free(spacetime_metric);
gkyl_free(inv_spacetime_metric);
Expand All @@ -570,6 +653,9 @@ test_gr_schwarzschild()
gkyl_free(spacetime_christoffel);
gkyl_free(spacetime_metric_cov_der);
gkyl_free(spacetime_ricci_tensor);
gkyl_free(spacetime_riemann_tensor);
gkyl_free(covariant_spacetime_riemann_tensor);
gkyl_free(spacetime_weyl_tensor);
}
else {
bool in_excision_region;
Expand Down Expand Up @@ -799,6 +885,57 @@ test_gr_kerr()

TEST_CHECK( gkyl_compare(spacetime_ricci_scalar, 0.0, 1e-2) );

double ****spacetime_riemann_tensor = gkyl_malloc(sizeof(double***[4]));
double ****covariant_spacetime_riemann_tensor = gkyl_malloc(sizeof(double***[4]));
double ****spacetime_weyl_tensor = gkyl_malloc(sizeof(double***[4]));

for (int i = 0; i < 4; i++) {
spacetime_riemann_tensor[i] = gkyl_malloc(sizeof(double**[4]));
covariant_spacetime_riemann_tensor[i] = gkyl_malloc(sizeof(double**[4]));
spacetime_weyl_tensor[i] = gkyl_malloc(sizeof(double**[4]));

for (int j = 0; j < 4; j++) {
spacetime_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));
covariant_spacetime_riemann_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));
spacetime_weyl_tensor[i][j] = gkyl_malloc(sizeof(double*[4]));

for (int k = 0; k < 4; k++) {
spacetime_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
covariant_spacetime_riemann_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));
spacetime_weyl_tensor[i][j][k] = gkyl_malloc(sizeof(double[4]));

for (int l = 0; l < 4; l++) {
covariant_spacetime_riemann_tensor[i][j][k][l] = 0.0;
}
}
}
}

spacetime->spacetime_riemann_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_riemann_tensor);
spacetime->spacetime_weyl_tensor_func(spacetime, 0.0, x, y, 0.0, pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), pow(10.0, -6.0), &spacetime_weyl_tensor);

for (int i = 0; i < 4; i++) {
for (int j = 0; j < 4; j++) {
for (int k = 0; k < 4; k++) {
for (int l = 0; l < 4; l++) {
for (int m = 0; m < 4; m++) {
covariant_spacetime_riemann_tensor[i][j][k][l] += spacetime_metric[i][m] * spacetime_riemann_tensor[m][j][k][l];
}
}
}
}
}

for (int i = 0; i < 4; i++) {
for (int j = 0; j < 4; j++) {
for (int k = 0; k < 4; k++) {
for (int l = 0; l < 4; l++) {
TEST_CHECK( gkyl_compare(covariant_spacetime_riemann_tensor[i][j][k][l], spacetime_weyl_tensor[i][j][k][l], 1e-2) );
}
}
}
}

bool in_excision_region;
spacetime->excision_region_func(spacetime, 0.0, x, y, 0.0, &in_excision_region);

Expand Down Expand Up @@ -843,11 +980,23 @@ test_gr_kerr()
gkyl_free(spacetime_metric_der[i][j]);
gkyl_free(spacetime_christoffel[i][j]);
gkyl_free(spacetime_metric_cov_der[i][j]);

for (int k = 0; k < 4; k++) {
gkyl_free(spacetime_riemann_tensor[i][j][k]);
gkyl_free(covariant_spacetime_riemann_tensor[i][j][k]);
gkyl_free(spacetime_weyl_tensor[i][j][k]);
}
gkyl_free(spacetime_riemann_tensor[i][j]);
gkyl_free(covariant_spacetime_riemann_tensor[i][j]);
gkyl_free(spacetime_weyl_tensor[i][j]);
}
gkyl_free(spacetime_metric_der[i]);
gkyl_free(spacetime_christoffel[i]);
gkyl_free(spacetime_metric_cov_der[i]);
gkyl_free(spacetime_ricci_tensor[i]);
gkyl_free(spacetime_riemann_tensor[i]);
gkyl_free(covariant_spacetime_riemann_tensor[i]);
gkyl_free(spacetime_weyl_tensor[i]);
}
gkyl_free(spacetime_metric);
gkyl_free(inv_spacetime_metric);
Expand All @@ -856,6 +1005,9 @@ test_gr_kerr()
gkyl_free(spacetime_christoffel);
gkyl_free(spacetime_metric_cov_der);
gkyl_free(spacetime_ricci_tensor);
gkyl_free(spacetime_riemann_tensor);
gkyl_free(covariant_spacetime_riemann_tensor);
gkyl_free(spacetime_weyl_tensor);
}
else {
bool in_excision_region;
Expand Down
37 changes: 37 additions & 0 deletions zero/gkyl_gr_blackhole.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -445,6 +445,43 @@ static void
blackhole_spacetime_ricci_scalar(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dt, const double dx, const double dy, const double dz, double* spacetime_ricci_scalar);

/**
* Compute the rank-4 (spatial) Weyl curvature tensor at a given point in a black hole spacetime.
*
* @param spacetime Base spacetime object.
* @param t Time coordinate.
* @param x Spatial coordinate (x-direction).
* @param y Spatial coordinate (y-direction).
* @param z Spatial coordinate (z-direction).
* @param dx Spatial coordinate spacing (x-direction).
* @param dy Spatial coordinate spacing (y-direction).
* @param dz Spatial coordinate spacing (z-direction).
* @param spatial_weyl_tensor Rank-4 spatial Weyl curvature tensor (output).
*/
GKYL_CU_D
static void
blackhole_spatial_weyl_tensor(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dx, const double dy, const double dz, double***** spatial_weyl_tensor);

/**
* Compute the rank-4 (spacetime) Weyl curvature tensor at a given point in a black hole spacetime.
*
* @param spacetime Base spacetime object.
* @param t Time coordinate.
* @param x Spatial coordinate (x-direction).
* @param y Spatial coordinate (y-direction).
* @param z Spatial coordinate (z-direction).
* @param dt Time coordinate spacing.
* @param dx Spatial coordinate spacing (x-direction).
* @param dy Spatial coordinate spacing (y-direction).
* @param dz Spatial coordinate spacing (z-direction).
* @param spacetime_weyl_tensor Rank-4 spacetime Weyl curvature tensor (output).
*/
GKYL_CU_D
static void
blackhole_spacetime_weyl_tensor(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dt, const double dx, const double dy, const double dz, double***** spacetime_weyl_tensor);

/**
* Compute the rank-2 extrinsic curvature tensor at a given point in a black hole spacetime.
*
Expand Down
37 changes: 37 additions & 0 deletions zero/gkyl_gr_minkowski.h
View file
Original file line number Diff line number Diff line change
Expand Up @@ -356,6 +356,43 @@ static void
minkowski_spacetime_ricci_scalar(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dt, const double dx, const double dy, const double dz, double* spacetime_ricci_scalar);

/**
* Compute the rank-4 (spatial) Weyl curvature tensor at a given pont in Minkowski space.
*
* @param spacetime Base spacetime object.
* @param t Time coordinate.
* @param x Spatial coordinate (x-direction).
* @param y Spatial coordinate (y-direction).
* @param z Spatial coordinate (z-direction).
* @param dx Spatial coordinate spacing (x-direction).
* @param dy Spatial coordinate spacing (y-direction).
* @param dz Spatial coordinate spacing (z-direction).
* @param spatial_weyl_tensor Rank-4 spatial Weyl curvature tensor (output).
*/
GKYL_CU_D
static void
minkowski_spatial_weyl_tensor(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dx, const double dy, const double dz, double***** spatial_weyl_tensor);

/**
* Compute the rank-4 (spacetime) Weyl curvature tensor at a given point in Minkowski space.
*
* @param spacetime Base spacetime object.
* @param t Time coordinate.
* @param x Spatial coordinate (x-direction).
* @param y Spatial coordinate (y-direction).
* @param z Spatial coordinate (z-direction).
* @param dt Time coordinate spacing.
* @param dx Spatial coordinate spacing (x-direction).
* @param dy Spatial coordinate spacing (y-direction).
* @param dz Spatial coordinate spacing (z-direction).
* @param spacetime_weyl_tensor Rank-4 spacetime Weyl curvature tensor (output).
*/
GKYL_CU_D
static void
minkowski_spacetime_weyl_tensor(const struct gkyl_gr_spacetime* spacetime, const double t, const double x, const double y, const double z,
const double dt, const double dx, const double dy, const double dz, double***** spacetime_weyl_tensor);

/**
* Compute the rank-2 extrinsic curvature tensor at a given point in Minkowski space.
*
Expand Down
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