def BrillLindquist(ComputeADMGlobalsOnly=False):
global Cartxyz, gammaCartDD, KCartDD, alphaCart, betaCartU, BCartU
# Step 2: Setting up Brill-Lindquist initial data
# Step 2.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
global Cartxyz, gammaCartDD, KCartDD, alphaCart, betaCartU, BCartU
Cartxyz = ixp.declarerank1("Cartxyz")
# Step 2.b: Set psi, the conformal factor:
psi = sp.sympify(1)
psi += BH1_mass / (2 * sp.sqrt((Cartxyz[0] - BH1_posn_x)**2 +
(Cartxyz[1] - BH1_posn_y)**2 +
(Cartxyz[2] - BH1_posn_z)**2))
psi += BH2_mass / (2 * sp.sqrt((Cartxyz[0] - BH2_posn_x)**2 +
(Cartxyz[1] - BH2_posn_y)**2 +
(Cartxyz[2] - BH2_posn_z)**2))
# Step 2.c: Set all needed ADM variables in Cartesian coordinates
gammaCartDD = ixp.zerorank2()
KCartDD = ixp.zerorank2() # K_{ij} = 0 for these initial data
for i in range(DIM):
gammaCartDD[i][i] = psi**4
alphaCart = 1 / psi**2
betaCartU = ixp.zerorank1(
) # We generally choose \beta^i = 0 for these initial data
BCartU = ixp.zerorank1(
) # We generally choose B^i = 0 for these initial data
if ComputeADMGlobalsOnly == True:
return
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Cartesian",Cartxyz,
gammaCartDD,KCartDD,alphaCart,betaCartU,BCartU)
import BSSN.BSSN_ID_function_string as bIDf
global returnfunction
returnfunction = bIDf.BSSN_ID_function_string(cf, hDD, lambdaU, aDD, trK,
alpha, vetU, betU)
def StaticTrumpet(ComputeADMGlobalsOnly = False):
global Sph_r_th_ph,r,th,ph, gammaSphDD, KSphDD, alphaSph, betaSphU, BSphU
# All gridfunctions will be written in terms of spherical coordinates (r, th, ph):
r,th,ph = sp.symbols('r th ph', real=True)
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM",DIM)
# Step 1: Set psi, the conformal factor:
# Auxiliary variables:
psi0 = sp.symbols('psi0', real=True)
# *** The StaticTrumpet conformal factor ***
# Dennison and Baumgarte (2014) Eq. 13
# https://arxiv.org/pdf/1403.5484.pdf
# psi = sqrt{1 + M/r }
psi0 = sp.sqrt(1 + M/r)
# *** The physical spatial metric in spherical basis ***
# Set the upper-triangle of the matrix...
# Eq. 15
# gamma_{ij} = psi^4 * eta_{ij}
# eta_00 = 1, eta_11 = r^2, eta_22 = r^2 * sin^2 (theta)
gammaSphDD = ixp.zerorank2()
gammaSphDD[0][0] = psi0**4
gammaSphDD[1][1] = psi0**4 * r**2
gammaSphDD[2][2] = psi0**4 * r**2*sp.sin(th)**2
# ... then apply symmetries to get the other components
# *** The physical trace-free extrinsic curvature in spherical basis ***
# Set the upper-triangle of the matrix...
# Eq.19 and 20
KSphDD = ixp.zerorank2()
# K_{rr} = M / r^2
KSphDD[0][0] = -M / r**2
# K_{theta theta} = K_{phi phi} / sin^2 theta = M
KSphDD[1][1] = M
KSphDD[2][2] = M * sp.sin(th)**2
# ... then apply symmetries to get the other components
# Lapse function and shift vector
# Eq. 15
# alpha = r / (r+M)
alphaSph = r / (r + M)
betaSphU = ixp.zerorank1()
# beta^r = Mr / (r + M)^2
betaSphU[0] = M*r / (r + M)**2
BSphU = ixp.zerorank1()
if ComputeADMGlobalsOnly == True:
return
# Validated against original SENR:
#print(sp.mathematica_code(gammaSphDD[1][1]))
Sph_r_th_ph = [r,th,ph]
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Spherical", Sph_r_th_ph,
gammaSphDD,KSphDD,alphaSph,betaSphU,BSphU)
import BSSN.BSSN_ID_function_string as bIDf
global returnfunction
returnfunction = bIDf.BSSN_ID_function_string(cf, hDD, lambdaU, aDD, trK, alpha, vetU, betU)
def ShiftedKerrSchild(ComputeADMGlobalsOnly = False):
global Sph_r_th_ph,r,th,ph, rho2, gammaSphDD, KSphDD, alphaSph, betaSphU, BSphU
# All gridfunctions will be written in terms of spherical coordinates (r, th, ph):
r,th,ph = sp.symbols('r th ph', real=True)
thismodule = "ShiftedKerrSchild"
DIM = 3
par.set_parval_from_str("grid::DIM",DIM)
# Input parameters:
M, a, r0 = par.Cparameters("REAL", thismodule, ["M", "a", "r0"])
# Auxiliary variables:
rho2 = sp.symbols('rho2', real=True)
# r_{KS} = r + r0
rKS = r+r0
# rho^2 = rKS^2 + a^2*cos^2(theta)
rho2 = rKS*rKS + a*a*sp.cos(th)**2
# alpha = 1/sqrt{1 + M*rKS/rho^2}
alphaSph = 1/(sp.sqrt(1 + 2*M*rKS/rho2))
# Initialize the shift vector, \beta^i, to zero.
betaSphU = ixp.zerorank1()
# beta^r = alpha^2*2Mr/rho^2
betaSphU[0] = alphaSph*alphaSph*2*M*rKS/rho2
# Time derivative of shift vector beta^i, B^i, is zero.
BSphU = ixp.zerorank1()
# Initialize \gamma_{ij} to zero.
gammaSphDD = ixp.zerorank2()
# gammaDD{rKS rKS} = 1 +2M*rKS/rho^2
gammaSphDD[0][0] = 1 + 2*M*rKS/rho2
# gammaDD{rKS phi} = -a*gammaDD{r r}*sin^2(theta)
gammaSphDD[0][2] = gammaSphDD[2][0] = -a*gammaSphDD[0][0]*sp.sin(th)**2
# gammaDD{theta theta} = rho^2
gammaSphDD[1][1] = rho2
# gammaDD{phi phi} = (rKS^2 + a^2 + 2Mr/rho^2*a^2*sin^2(theta))*sin^2(theta)
gammaSphDD[2][2] = (rKS*rKS + a*a + 2*M*rKS*a*a*sp.sin(th)**2/rho2)*sp.sin(th)**2
# *** Define Useful Quantities A, B, D ***
# A = (a^2*cos^2(2theta) + a^2 + 2r^2)
A = (a*a*sp.cos(2*th) + a*a + 2*rKS*rKS)
# B = A + 4M*rKS
B = A + 4*M*rKS
# D = \sqrt(2M*rKS/(a^2cos^2(theta) + rKS^2) + 1)
D = sp.sqrt(2*M*rKS/(a*a*sp.cos(th)**2 + rKS*rKS) + 1)
# *** The extrinsic curvature in spherical polar coordinates ***
# Establish the 3x3 zero-matrix
KSphDD = ixp.zerorank2()
# *** Fill in the nonzero components ***
# *** This will create an upper-triangular matrix ***
# K_{r r} = D(A+2Mr)/(A^2*B)[4M(a^2*cos(2theta) + a^2 - 2r^2)]
KSphDD[0][0] = D*(A+2*M*rKS)/(A*A*B)*(4*M*(a*a*sp.cos(2*th)+a*a-2*rKS*rKS))
# K_{r theta} = D/(AB)[8a^2*Mr*sin(theta)cos(theta)]
KSphDD[0][1] = KSphDD[1][0] = D/(A*B)*(8*a*a*M*rKS*sp.sin(th)*sp.cos(th))
# K_{r phi} = D/A^2[-2aMsin^2(theta)(a^2cos(2theta)+a^2-2r^2)]
KSphDD[0][2] = KSphDD[2][0] = D/(A*A)*(-2*a*M*sp.sin(th)**2*(a*a*sp.cos(2*th)+a*a-2*rKS*rKS))
# K_{theta theta} = D/B[4Mr^2]
KSphDD[1][1] = D/B*(4*M*rKS*rKS)
# K_{theta phi} = D/(AB)*(-8*a^3*Mr*sin^3(theta)cos(theta))
KSphDD[1][2] = KSphDD[2][1] = D/(A*B)*(-8*a**3*M*rKS*sp.sin(th)**3*sp.cos(th))
# K_{phi phi} = D/(A^2*B)[2Mr*sin^2(theta)(a^4(M+3r)
# +4a^2r^2(2r-M)+4a^2r*cos(2theta)(a^2+r(M+2r))+8r^5)]
KSphDD[2][2] = D/(A*A*B)*(2*M*rKS*sp.sin(th)**2*(a**4*(rKS-M)*sp.cos(4*th)\
+ a**4*(M+3*rKS)+4*a*a*rKS*rKS*(2*rKS-M)\
+ 4*a*a*rKS*sp.cos(2*th)*(a*a + rKS*(M + 2*rKS)) + 8*rKS**5))
if ComputeADMGlobalsOnly == True:
return
# Validated against original SENR:
#print(sp.mathematica_code(gammaSphDD[1][1]))
Sph_r_th_ph = [r,th,ph]
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Spherical", Sph_r_th_ph,
gammaSphDD,KSphDD,alphaSph,betaSphU,BSphU)
global returnfunction
returnfunction = bIDf.BSSN_ID_function_string(cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU)
def UIUCBlackHole(ComputeADMGlobalsOnly=False,
include_NRPy_basic_defines_and_pickle=False):
global Sph_r_th_ph, r, th, ph, gammaSphDD, KSphDD, alphaSph, betaSphU, BSphU
# All gridfunctions will be written in terms of spherical coordinates (r, th, ph):
r, th, ph = sp.symbols('r th ph', real=True)
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
# Step 1: Set psi, the conformal factor:
# Spin per unit mass
a = M * chi
# Defined under equation 1 in Liu, Etienne, & Shapiro (2009) https://arxiv.org/pdf/1001.4077.pdf
# Boyer - Lindquist outer horizon
rp = M + sp.sqrt(M**2 - a**2)
# Boyer - Lindquist inner horizon
rm = M - sp.sqrt(M**2 - a**2)
# Boyer - Lindquist radius in terms of UIUC radius
# Eq. 11
# r_{BL} = r * ( 1 + r_+ / 4r )^2
rBL = r * (1 + rp / (4 * r))**2
# Expressions found below Eq. 2
# Sigma = r_{BL}^2 + a^2 cos^2 theta
SIG = rBL**2 + a**2 * sp.cos(th)**2
# Delta = r_{BL}^2 - 2Mr_{BL} + a^2
DEL = rBL**2 - 2 * M * rBL + a**2
# A = (r_{BL}^2 + a^2)^2 - Delta a^2 sin^2 theta
AA = (rBL**2 + a**2)**2 - DEL * a**2 * sp.sin(th)**2
# *** The ADM 3-metric in spherical basis ***
gammaSphDD = ixp.zerorank2()
# Declare the nonzero components of the 3-metric
# (Eq. 13 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf):
# ds^2 = Sigma (r + r_+/4)^2 / ( r^3 (r_{BL} - r_- ) * dr^2 +
# Sigma d theta^2 + (A sin^2 theta) / Sigma * d\phi^2
gammaSphDD[0][0] = ((SIG * (r + rp / 4)**2) / (r**3 * (rBL - rm)))
gammaSphDD[1][1] = SIG
gammaSphDD[2][2] = AA / SIG * sp.sin(th)**2
# *** The physical trace-free extrinsic curvature in spherical basis ***
# Nonzero components of the extrinsic curvature K, given by
# Eq. 14 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf:
KSphDD = ixp.zerorank2()
# K_{r phi} = K_{phi r} = (Ma sin^2 theta) / (Sigma sqrt{A Sigma}) *
# [3r^4_{BL} + 2a^2 r^2_{BL} - a^4 - a^2 (r^2_{BL} - a^2) sin^2 theta] *
# (1 + r_+ / 4r) (1 / sqrt{r(r_{BL} - r_-)})
KSphDD[0][2] = KSphDD[2][0] = (M*a*sp.sin(th)**2)/(SIG*sp.sqrt(AA*SIG))*\
(3*rBL**4 + 2*a**2*rBL**2 - a**4- a**2*(rBL**2 - a**2)*\
sp.sin(th)**2)*(1 + rp/(4*r))*1/sp.sqrt(r*(rBL - rm))
# Components of the extrinsic curvature K, given by
# Eq. 15 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf:
# K_{theta phi} = K_{phi theta} = -(2a^3 Mr_{BL} cos theta sin^3 theta) /
# (Sigma sqrt{A Sigma}) x (r - r_+ / 4) sqrt{(r_{BL} - r_-) / r }
KSphDD[1][2] = KSphDD[2][1] = -((2*a**3*M*rBL*sp.cos(th)*sp.sin(th)**3)/ \
(SIG*sp.sqrt(AA*SIG)))*(r - rp/4)*sp.sqrt((rBL - rm)/r)
alphaSph = sp.sympify(
1
) # We generally choose alpha = 1/psi**2 (psi = BSSN conformal factor) for these initial data
betaSphU = ixp.zerorank1(
) # We generally choose \beta^i = 0 for these initial data
BSphU = ixp.zerorank1(
) # We generally choose B^i = 0 for these initial data
if ComputeADMGlobalsOnly:
return
# Validated against original SENR: KSphDD[0][2], KSphDD[1][2], gammaSphDD[2][2], gammaSphDD[0][0], gammaSphDD[1][1]
#print(sp.mathematica_code(gammaSphDD[1][1]))
Sph_r_th_ph = [r, th, ph]
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Spherical", Sph_r_th_ph,
gammaSphDD,KSphDD,alphaSph,betaSphU,BSphU)
# Let's choose alpha = 1/psi**2 (psi = BSSN conformal factor) for these initial data,
# where psi = exp(phi); chi = 1/psi**4; W = 1/psi**2
if par.parval_from_str("EvolvedConformalFactor_cf") == "phi":
alpha = sp.exp(-2 * cf)
elif par.parval_from_str("EvolvedConformalFactor_cf") == "chi":
alpha = sp.sqrt(cf)
elif par.parval_from_str("EvolvedConformalFactor_cf") == "W":
alpha = cf
else:
print("Error EvolvedConformalFactor_cf type = \"" +
par.parval_from_str("EvolvedConformalFactor_cf") + "\" unknown.")
sys.exit(1)
import BSSN.BSSN_ID_function_string as bIDf
# Generates initial_data() C function & stores to outC_function_dict["initial_data"]
bIDf.BSSN_ID_function_string(
cf,
hDD,
lambdaU,
aDD,
trK,
alpha,
vetU,
betU,
include_NRPy_basic_defines=include_NRPy_basic_defines_and_pickle)
if include_NRPy_basic_defines_and_pickle:
return pickle_NRPy_env()
def UIUCBlackHole(ComputeADMGlobalsOnly=False):
global Sph_r_th_ph, r, th, ph, gammaSphDD, KSphDD, alphaSph, betaSphU, BSphU
# All gridfunctions will be written in terms of spherical coordinates (r, th, ph):
r, th, ph = sp.symbols('r th ph', real=True)
thismodule = "UIUCBlackHole"
# Step 0: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
# Step 1: Set psi, the conformal factor:
# The UIUC initial data represent a Kerr black hole with mass M
# and dimensionless spin chi in UIUC quasi-isotropic coordinates,
# see https://arxiv.org/abs/1001.4077
# Input parameters:
M, chi = par.Cparameters("REAL", thismodule, ["M", "chi"])
# Auxiliary variables:
a, rp, rm, rBL, SIG, DEL, AA = sp.symbols('a rp rm rBL SIG DEL AA',
real=True)
# Spin per unit mass
a = M * chi
# Defined under equation 1 in Liu, Etienne, & Shapiro (2009) https://arxiv.org/pdf/1001.4077.pdf
# Boyer - Lindquist outer horizon
rp = M + sp.sqrt(M**2 - a**2)
# Boyer - Lindquist inner horizon
rm = M - sp.sqrt(M**2 - a**2)
# Boyer - Lindquist radius in terms of UIUC radius
# Eq. 11
# r_{BL} = r * ( 1 + r_+ / 4r )^2
rBL = r * (1 + rp / (4 * r))**2
# Expressions found below Eq. 2
# Sigma = r_{BL}^2 + a^2 cos^2 theta
SIG = rBL**2 + a**2 * sp.cos(th)**2
# Delta = r_{BL}^2 - 2Mr_{BL} + a^2
DEL = rBL**2 - 2 * M * rBL + a**2
# A = (r_{BL}^2 + a^2)^2 - Delta a^2 sin^2 theta
AA = (rBL**2 + a**2)**2 - DEL * a**2 * sp.sin(th)**2
# *** The ADM 3-metric in spherical basis ***
gammaSphDD = ixp.zerorank2()
# Declare the nonzero components of the 3-metric
# (Eq. 13 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf):
# ds^2 = Sigma (r + r_+/4)^2 / ( r^3 (r_{BL} - r_- ) * dr^2 +
# Sigma d theta^2 + (A sin^2 theta) / Sigma * d\phi^2
gammaSphDD[0][0] = ((SIG * (r + rp / 4)**2) / (r**3 * (rBL - rm)))
gammaSphDD[1][1] = SIG
gammaSphDD[2][2] = AA / SIG * sp.sin(th)**2
# *** The physical trace-free extrinsic curvature in spherical basis ***
# Nonzero components of the extrinsic curvature K, given by
# Eq. 14 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf:
KSphDD = ixp.zerorank2()
# K_{r phi} = K_{phi r} = (Ma sin^2 theta) / (Sigma sqrt{A Sigma}) *
# [3r^4_{BL} + 2a^2 r^2_{BL} - a^4 - a^2 (r^2_{BL} - a^2) sin^2 theta] *
# (1 + r_+ / 4r) (1 / sqrt{r(r_{BL} - r_-)})
KSphDD[0][2] = KSphDD[2][0] = (M*a*sp.sin(th)**2)/(SIG*sp.sqrt(AA*SIG))*\
(3*rBL**4 + 2*a**2*rBL**2 - a**4- a**2*(rBL**2 - a**2)*\
sp.sin(th)**2)*(1 + rp/(4*r))*1/sp.sqrt(r*(rBL - rm))
# Components of the extrinsic curvature K, given by
# Eq. 15 of Liu, Etienne, & Shapiro, https://arxiv.org/pdf/1001.4077.pdf:
# K_{theta phi} = K_{phi theta} = -(2a^3 Mr_{BL} cos theta sin^3 theta) /
# (Sigma sqrt{A Sigma}) x (r - r_+ / 4) sqrt{(r_{BL} - r_-) / r }
KSphDD[1][2] = KSphDD[2][1] = -((2*a**3*M*rBL*sp.cos(th)*sp.sin(th)**3)/ \
(SIG*sp.sqrt(AA*SIG)))*(r - rp/4)*sp.sqrt((rBL - rm)/r)
alphaSph = sp.sympify(
1
) # We generally choose alpha = 1/psi**2 (psi = BSSN conformal factor) for these initial data
betaSphU = ixp.zerorank1(
) # We generally choose \beta^i = 0 for these initial data
BSphU = ixp.zerorank1(
) # We generally choose B^i = 0 for these initial data
if ComputeADMGlobalsOnly == True:
return
# Validated against original SENR: KSphDD[0][2], KSphDD[1][2], gammaSphDD[2][2], gammaSphDD[0][0], gammaSphDD[1][1]
#print(sp.mathematica_code(gammaSphDD[1][1]))
Sph_r_th_ph = [r, th, ph]
cf,hDD,lambdaU,aDD,trK,alpha,vetU,betU = \
AtoB.Convert_Spherical_or_Cartesian_ADM_to_BSSN_curvilinear("Spherical", Sph_r_th_ph,
gammaSphDD,KSphDD,alphaSph,betaSphU,BSphU)
global returnfunction
returnfunction = bIDf.BSSN_ID_function_string(cf, hDD, lambdaU, aDD, trK,
alpha, vetU, betU)