def BSSN_source_terms_for_BSSN_RHSs(custom_T4UU=None):
global sourceterm_trK_rhs, sourceterm_a_rhsDD, sourceterm_lambda_rhsU, sourceterm_Lambdabar_rhsU
# Step 3.a: Call BSSN_source_terms_ito_T4UU to get SDD, SD, S, & rho
if custom_T4UU == "unrescaled BSSN source terms already given":
SDD = ixp.declarerank2("SDD", "sym01")
SD = ixp.declarerank1("SD")
S = sp.symbols("S", real=True)
rho = sp.symbols("rho", real=True)
else:
SDD, SD, S, rho = stress_energy_source_terms_ito_T4UU_and_ADM_or_BSSN_metricvars(
"BSSN", custom_T4UU)
PI = par.Cparameters("REAL", thismodule, ["PI"],
"3.14159265358979323846264338327950288")
alpha = sp.symbols("alpha", real=True)
# Step 3.b: trK_rhs
sourceterm_trK_rhs = 4 * PI * alpha * (rho + S)
# Step 3.c: Abar_rhsDD:
# Step 3.c.i: Compute trace-free part of S_{ij}:
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors() # Sets gammabarDD
gammabarUU, dummydet = ixp.symm_matrix_inverter3x3(
Bq.gammabarDD) # Set gammabarUU
tracefree_SDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
tracefree_SDD[i][j] = SDD[i][j]
for i in range(3):
for j in range(3):
for k in range(3):
for m in range(3):
tracefree_SDD[i][j] += -sp.Rational(1, 3) * Bq.gammabarDD[
i][j] * gammabarUU[k][m] * SDD[k][m]
# Step 3.c.ii: Define exp_m4phi = e^{-4 phi}
Bq.phi_and_derivs()
# Step 3.c.iii: Evaluate stress-energy part of AbarDD's RHS
sourceterm_a_rhsDD = ixp.zerorank2()
for i in range(3):
for j in range(3):
Abar_rhsDDij = -8 * PI * alpha * Bq.exp_m4phi * tracefree_SDD[i][j]
sourceterm_a_rhsDD[i][j] = Abar_rhsDDij / rfm.ReDD[i][j]
# Step 3.d: Stress-energy part of Lambdabar_rhsU = stressenergy_Lambdabar_rhsU
sourceterm_Lambdabar_rhsU = ixp.zerorank1()
for i in range(3):
for j in range(3):
sourceterm_Lambdabar_rhsU[
i] += -16 * PI * alpha * gammabarUU[i][j] * SD[j]
sourceterm_lambda_rhsU = ixp.zerorank1()
for i in range(3):
sourceterm_lambda_rhsU[i] = sourceterm_Lambdabar_rhsU[i] / rfm.ReU[i]
def BSSN_RHSs():
# Step 1.c: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
global have_already_called_BSSN_RHSs_function # setting to global enables other modules to see updated value.
have_already_called_BSSN_RHSs_function = True
# Step 1.d: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.e: Import all basic (unrescaled) BSSN scalars & tensors
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors()
gammabarDD = Bq.gammabarDD
AbarDD = Bq.AbarDD
LambdabarU = Bq.LambdabarU
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
# Step 1.f: Import all neeeded rescaled BSSN tensors:
aDD = Bq.aDD
cf = Bq.cf
lambdaU = Bq.lambdaU
# Step 2.a.i: Import derivative expressions for betaU defined in the BSSN.BSSN_quantities module:
Bq.betaU_derivs()
betaU_dD = Bq.betaU_dD
betaU_dDD = Bq.betaU_dDD
# Step 2.a.ii: Import derivative expression for gammabarDD
Bq.gammabar__inverse_and_derivs()
gammabarDD_dupD = Bq.gammabarDD_dupD
# Step 2.a.iii: First term of \partial_t \bar{\gamma}_{i j} right-hand side:
# \beta^k \bar{\gamma}_{ij,k} + \beta^k_{,i} \bar{\gamma}_{kj} + \beta^k_{,j} \bar{\gamma}_{ik}
gammabar_rhsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
gammabar_rhsDD[i][j] += betaU[k] * gammabarDD_dupD[i][j][k] + betaU_dD[k][i] * gammabarDD[k][j] \
+ betaU_dD[k][j] * gammabarDD[i][k]
# Step 2.b.i: First import \bar{A}_{ij} = AbarDD[i][j], and its contraction trAbar = \bar{A}^k_k
# from BSSN.BSSN_quantities
Bq.AbarUU_AbarUD_trAbar_AbarDD_dD()
trAbar = Bq.trAbar
# Step 2.b.ii: Import detgammabar quantities from BSSN.BSSN_quantities:
Bq.detgammabar_and_derivs()
detgammabar = Bq.detgammabar
detgammabar_dD = Bq.detgammabar_dD
# Step 2.b.ii: Compute the contraction \bar{D}_k \beta^k = \beta^k_{,k} + \frac{\beta^k \bar{\gamma}_{,k}}{2 \bar{\gamma}}
Dbarbetacontraction = sp.sympify(0)
for k in range(DIM):
Dbarbetacontraction += betaU_dD[k][
k] + betaU[k] * detgammabar_dD[k] / (2 * detgammabar)
# Step 2.b.iii: Second term of \partial_t \bar{\gamma}_{i j} right-hand side:
# \frac{2}{3} \bar{\gamma}_{i j} \left (\alpha \bar{A}_{k}^{k} - \bar{D}_{k} \beta^{k}\right )
for i in range(DIM):
for j in range(DIM):
gammabar_rhsDD[i][j] += sp.Rational(2, 3) * gammabarDD[i][j] * (
alpha * trAbar - Dbarbetacontraction)
# Step 2.c: Third term of \partial_t \bar{\gamma}_{i j} right-hand side:
# -2 \alpha \bar{A}_{ij}
for i in range(DIM):
for j in range(DIM):
gammabar_rhsDD[i][j] += -2 * alpha * AbarDD[i][j]
# Step 3.a: First term of \partial_t \bar{A}_{i j}:
# \beta^k \partial_k \bar{A}_{ij} + \partial_i \beta^k \bar{A}_{kj} + \partial_j \beta^k \bar{A}_{ik}
# First define AbarDD_dupD:
AbarDD_dupD = Bq.AbarDD_dupD # From Bq.AbarUU_AbarUD_trAbar_AbarDD_dD()
Abar_rhsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Abar_rhsDD[i][j] += betaU[k] * AbarDD_dupD[i][j][k] + betaU_dD[k][i] * AbarDD[k][j] \
+ betaU_dD[k][j] * AbarDD[i][k]
# Step 3.b: Second term of \partial_t \bar{A}_{i j}:
# - (2/3) \bar{A}_{i j} \bar{D}_{k} \beta^{k} - 2 \alpha \bar{A}_{i k} {\bar{A}^{k}}_{j} + \alpha \bar{A}_{i j} K
gammabarUU = Bq.gammabarUU # From Bq.gammabar__inverse_and_derivs()
AbarUD = Bq.AbarUD # From Bq.AbarUU_AbarUD_trAbar()
for i in range(DIM):
for j in range(DIM):
Abar_rhsDD[i][j] += -sp.Rational(2, 3) * AbarDD[i][
j] * Dbarbetacontraction + alpha * AbarDD[i][j] * trK
for k in range(DIM):
Abar_rhsDD[i][j] += -2 * alpha * AbarDD[i][k] * AbarUD[k][j]
# Step 3.c.i: Define partial derivatives of \phi in terms of evolved quantity "cf":
Bq.phi_and_derivs()
phi_dD = Bq.phi_dD
phi_dupD = Bq.phi_dupD
phi_dDD = Bq.phi_dDD
exp_m4phi = Bq.exp_m4phi
phi_dBarD = Bq.phi_dBarD # phi_dBarD = Dbar_i phi = phi_dD (since phi is a scalar)
phi_dBarDD = Bq.phi_dBarDD # phi_dBarDD = Dbar_i Dbar_j phi (covariant derivative)
# Step 3.c.ii: Define RbarDD
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
RbarDD = Bq.RbarDD
# Step 3.c.iii: Define first and second derivatives of \alpha, as well as
# \bar{D}_i \bar{D}_j \alpha, which is defined just like phi
alpha_dD = ixp.declarerank1("alpha_dD")
alpha_dDD = ixp.declarerank2("alpha_dDD", "sym01")
alpha_dBarD = alpha_dD
alpha_dBarDD = ixp.zerorank2()
GammabarUDD = Bq.GammabarUDD # Defined in Bq.gammabar__inverse_and_derivs()
for i in range(DIM):
for j in range(DIM):
alpha_dBarDD[i][j] = alpha_dDD[i][j]
for k in range(DIM):
alpha_dBarDD[i][j] += -GammabarUDD[k][i][j] * alpha_dD[k]
# Step 3.c.iv: Define the terms in curly braces:
curlybrackettermsDD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
curlybrackettermsDD[i][j] = -2 * alpha * phi_dBarDD[i][j] + 4 * alpha * phi_dBarD[i] * phi_dBarD[j] \
+ 2 * alpha_dBarD[i] * phi_dBarD[j] \
+ 2 * alpha_dBarD[j] * phi_dBarD[i] \
- alpha_dBarDD[i][j] + alpha * RbarDD[i][j]
# Step 3.c.v: Compute the trace:
curlybracketterms_trace = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
curlybracketterms_trace += gammabarUU[i][j] * curlybrackettermsDD[
i][j]
# Step 3.c.vi: Third and final term of Abar_rhsDD[i][j]:
for i in range(DIM):
for j in range(DIM):
Abar_rhsDD[i][j] += exp_m4phi * (
curlybrackettermsDD[i][j] -
sp.Rational(1, 3) * gammabarDD[i][j] * curlybracketterms_trace)
# Step 4: Right-hand side of conformal factor variable "cf". Supported
# options include: cf=phi, cf=W=e^(-2*phi) (default), and cf=chi=e^(-4*phi)
# \partial_t phi = \left[\beta^k \partial_k \phi \right] <- TERM 1
# + \frac{1}{6} \left (\bar{D}_{k} \beta^{k} - \alpha K \right ) <- TERM 2
global cf_rhs
cf_rhs = sp.Rational(1, 6) * (Dbarbetacontraction - alpha * trK) # Term 2
for k in range(DIM):
cf_rhs += betaU[k] * phi_dupD[k] # Term 1
# Next multiply to convert phi_rhs to cf_rhs.
if par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "phi":
pass # do nothing; cf_rhs = phi_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "W":
cf_rhs *= -2 * cf # cf_rhs = -2*cf*phi_rhs
elif par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "chi":
cf_rhs *= -4 * cf # cf_rhs = -4*cf*phi_rhs
else:
print("Error: EvolvedConformalFactor_cf == " + par.parval_from_str(
"BSSN.BSSN_quantities::EvolvedConformalFactor_cf") +
" unsupported!")
exit(1)
# Step 5: right-hand side of trK (trace of extrinsic curvature):
# \partial_t K = \beta^k \partial_k K <- TERM 1
# + \frac{1}{3} \alpha K^{2} <- TERM 2
# + \alpha \bar{A}_{i j} \bar{A}^{i j} <- TERM 3
# - - e^{-4 \phi} (\bar{D}_{i} \bar{D}^{i} \alpha + 2 \bar{D}^{i} \alpha \bar{D}_{i} \phi ) <- TERM 4
global trK_rhs
# TERM 2:
trK_rhs = sp.Rational(1, 3) * alpha * trK * trK
trK_dupD = ixp.declarerank1("trK_dupD")
for i in range(DIM):
# TERM 1:
trK_rhs += betaU[i] * trK_dupD[i]
for i in range(DIM):
for j in range(DIM):
# TERM 4:
trK_rhs += -exp_m4phi * gammabarUU[i][j] * (
alpha_dBarDD[i][j] + 2 * alpha_dBarD[j] * phi_dBarD[i])
AbarUU = Bq.AbarUU # From Bq.AbarUU_AbarUD_trAbar()
for i in range(DIM):
for j in range(DIM):
# TERM 3:
trK_rhs += alpha * AbarDD[i][j] * AbarUU[i][j]
# Step 6: right-hand side of \partial_t \bar{\Lambda}^i:
# \partial_t \bar{\Lambda}^i = \beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k <- TERM 1
# + \bar{\gamma}^{j k} \hat{D}_{j} \hat{D}_{k} \beta^{i} <- TERM 2
# + \frac{2}{3} \Delta^{i} \bar{D}_{j} \beta^{j} <- TERM 3
# + \frac{1}{3} \bar{D}^{i} \bar{D}_{j} \beta^{j} <- TERM 4
# - 2 \bar{A}^{i j} (\partial_{j} \alpha - 6 \partial_{j} \phi) <- TERM 5
# + 2 \alpha \bar{A}^{j k} \Delta_{j k}^{i} <- TERM 6
# - \frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_{j} K <- TERM 7
# Step 6.a: Term 1 of \partial_t \bar{\Lambda}^i: \beta^k \partial_k \bar{\Lambda}^i - \partial_k \beta^i \bar{\Lambda}^k
# First we declare \bar{\Lambda}^i and \bar{\Lambda}^i_{,j} in terms of \lambda^i and \lambda^i_{,j}
global LambdabarU_dupD # Used on the RHS of the Gamma-driving shift conditions
LambdabarU_dupD = ixp.zerorank2()
lambdaU_dupD = ixp.declarerank2("lambdaU_dupD", "nosym")
for i in range(DIM):
for j in range(DIM):
LambdabarU_dupD[i][j] = lambdaU_dupD[i][j] * rfm.ReU[i] + lambdaU[
i] * rfm.ReUdD[i][j]
global Lambdabar_rhsU # Used on the RHS of the Gamma-driving shift conditions
Lambdabar_rhsU = ixp.zerorank1()
for i in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[i] += betaU[k] * LambdabarU_dupD[i][k] - betaU_dD[
i][k] * LambdabarU[k] # Term 1
# Step 6.b: Term 2 of \partial_t \bar{\Lambda}^i = \bar{\gamma}^{jk} (Term 2a + Term 2b + Term 2c)
# Term 2a: \bar{\gamma}^{jk} \beta^i_{,kj}
Term2aUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Term2aUDD[i][j][k] += betaU_dDD[i][k][j]
# Term 2b: \hat{\Gamma}^i_{mk,j} \beta^m + \hat{\Gamma}^i_{mk} \beta^m_{,j}
# + \hat{\Gamma}^i_{dj}\beta^d_{,k} - \hat{\Gamma}^d_{kj} \beta^i_{,d}
Term2bUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
Term2bUDD[i][j][k] += rfm.GammahatUDDdD[i][m][k][j] * betaU[m] \
+ rfm.GammahatUDD[i][m][k] * betaU_dD[m][j] \
+ rfm.GammahatUDD[i][m][j] * betaU_dD[m][k] \
- rfm.GammahatUDD[m][k][j] * betaU_dD[i][m]
# Term 2c: \hat{\Gamma}^i_{dj}\hat{\Gamma}^d_{mk} \beta^m - \hat{\Gamma}^d_{kj} \hat{\Gamma}^i_{md} \beta^m
Term2cUDD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for m in range(DIM):
for d in range(DIM):
Term2cUDD[i][j][k] += (rfm.GammahatUDD[i][d][j] * rfm.GammahatUDD[d][m][k] \
- rfm.GammahatUDD[d][k][j] * rfm.GammahatUDD[i][m][d]) * betaU[m]
Lambdabar_rhsUpieceU = ixp.zerorank1()
# Put it all together to get Term 2:
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[i] += gammabarUU[j][k] * (Term2aUDD[i][j][k] +
Term2bUDD[i][j][k] +
Term2cUDD[i][j][k])
Lambdabar_rhsUpieceU[i] += gammabarUU[j][k] * (
Term2aUDD[i][j][k] + Term2bUDD[i][j][k] +
Term2cUDD[i][j][k])
# Step 6.c: Term 3 of \partial_t \bar{\Lambda}^i:
# \frac{2}{3} \Delta^{i} \bar{D}_{j} \beta^{j}
DGammaU = Bq.DGammaU # From Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
for i in range(DIM):
Lambdabar_rhsU[i] += sp.Rational(
2, 3) * DGammaU[i] * Dbarbetacontraction # Term 3
# Step 6.d: Term 4 of \partial_t \bar{\Lambda}^i:
# \frac{1}{3} \bar{D}^{i} \bar{D}_{j} \beta^{j}
detgammabar_dDD = Bq.detgammabar_dDD # From Bq.detgammabar_and_derivs()
Dbarbetacontraction_dBarD = ixp.zerorank1()
for k in range(DIM):
for m in range(DIM):
Dbarbetacontraction_dBarD[m] += betaU_dDD[k][k][m] + \
(betaU_dD[k][m] * detgammabar_dD[k] +
betaU[k] * detgammabar_dDD[k][m]) / (2 * detgammabar) \
- betaU[k] * detgammabar_dD[k] * detgammabar_dD[m] / (
2 * detgammabar * detgammabar)
for i in range(DIM):
for m in range(DIM):
Lambdabar_rhsU[i] += sp.Rational(
1, 3) * gammabarUU[i][m] * Dbarbetacontraction_dBarD[m]
# Step 6.e: Term 5 of \partial_t \bar{\Lambda}^i:
# - 2 \bar{A}^{i j} (\partial_{j} \alpha - 6 \alpha \partial_{j} \phi)
for i in range(DIM):
for j in range(DIM):
Lambdabar_rhsU[i] += -2 * AbarUU[i][j] * (alpha_dD[j] -
6 * alpha * phi_dD[j])
# Step 6.f: Term 6 of \partial_t \bar{\Lambda}^i:
# 2 \alpha \bar{A}^{j k} \Delta^{i}_{j k}
DGammaUDD = Bq.DGammaUDD # From RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
Lambdabar_rhsU[
i] += 2 * alpha * AbarUU[j][k] * DGammaUDD[i][j][k]
# Step 6.g: Term 7 of \partial_t \bar{\Lambda}^i:
# -\frac{4}{3} \alpha \bar{\gamma}^{i j} \partial_{j} K
trK_dD = ixp.declarerank1("trK_dD")
for i in range(DIM):
for j in range(DIM):
Lambdabar_rhsU[i] += -sp.Rational(
4, 3) * alpha * gammabarUU[i][j] * trK_dD[j]
# Step 7: Rescale the RHS quantities so that the evolved
# variables are smooth across coord singularities
global h_rhsDD, a_rhsDD, lambda_rhsU
h_rhsDD = ixp.zerorank2()
a_rhsDD = ixp.zerorank2()
lambda_rhsU = ixp.zerorank1()
for i in range(DIM):
lambda_rhsU[i] = Lambdabar_rhsU[i] / rfm.ReU[i]
for j in range(DIM):
h_rhsDD[i][j] = gammabar_rhsDD[i][j] / rfm.ReDD[i][j]
a_rhsDD[i][j] = Abar_rhsDD[i][j] / rfm.ReDD[i][j]
def BSSN_constraints(add_T4UUmunu_source_terms=False):
# Step 1.a: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step 1.b: Given the chosen coordinate system, set up
# corresponding reference metric and needed
# reference metric quantities
# The following function call sets up the reference metric
# and related quantities, including rescaling matrices ReDD,
# ReU, and hatted quantities.
rfm.reference_metric()
# Step 2: Hamiltonian constraint.
# First declare all needed variables
Bq.declare_BSSN_gridfunctions_if_not_declared_already() # Sets trK
Bq.BSSN_basic_tensors() # Sets AbarDD
Bq.gammabar__inverse_and_derivs() # Sets gammabarUU
Bq.AbarUU_AbarUD_trAbar_AbarDD_dD() # Sets AbarUU and AbarDD_dD
Bq.RicciBar__gammabarDD_dHatD__DGammaUDD__DGammaU() # Sets RbarDD
Bq.phi_and_derivs() # Sets phi_dBarD & phi_dBarDD
###############################
###############################
# HAMILTONIAN CONSTRAINT
###############################
###############################
# Term 1: 2/3 K^2
global H
H = sp.Rational(2, 3) * Bq.trK**2
# Term 2: -A_{ij} A^{ij}
for i in range(DIM):
for j in range(DIM):
H += -Bq.AbarDD[i][j] * Bq.AbarUU[i][j]
# Term 3a: trace(Rbar)
Rbartrace = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
Rbartrace += Bq.gammabarUU[i][j] * Bq.RbarDD[i][j]
# Term 3b: -8 \bar{\gamma}^{ij} \bar{D}_i \phi \bar{D}_j \phi = -8*phi_dBar_times_phi_dBar
# Term 3c: -8 \bar{\gamma}^{ij} \bar{D}_i \bar{D}_j \phi = -8*phi_dBarDD_contraction
phi_dBar_times_phi_dBar = sp.sympify(0) # Term 3b
phi_dBarDD_contraction = sp.sympify(0) # Term 3c
for i in range(DIM):
for j in range(DIM):
phi_dBar_times_phi_dBar += Bq.gammabarUU[i][j] * Bq.phi_dBarD[
i] * Bq.phi_dBarD[j]
phi_dBarDD_contraction += Bq.gammabarUU[i][j] * Bq.phi_dBarDD[i][j]
# Add Term 3:
H += Bq.exp_m4phi * (Rbartrace - 8 *
(phi_dBar_times_phi_dBar + phi_dBarDD_contraction))
if add_T4UUmunu_source_terms:
M_PI = par.Cparameters("#define", thismodule, "M_PI",
"") # M_PI is pi as defined in C
BTmunu.define_BSSN_T4UUmunu_rescaled_source_terms()
rho = BTmunu.rho
H += -16 * M_PI * rho
# FIXME: ADD T4UUmunu SOURCE TERMS TO MOMENTUM CONSTRAINT!
# Step 3: M^i, the momentum constraint
###############################
###############################
# MOMENTUM CONSTRAINT
###############################
###############################
# SEE Tutorial-BSSN_constraints.ipynb for full documentation.
global MU
MU = ixp.zerorank1()
# Term 2: 6 A^{ij} \partial_j \phi:
for i in range(DIM):
for j in range(DIM):
MU[i] += 6 * Bq.AbarUU[i][j] * Bq.phi_dD[j]
# Term 3: -2/3 \bar{\gamma}^{ij} K_{,j}
trK_dD = ixp.declarerank1(
"trK_dD") # Not defined in BSSN_RHSs; only trK_dupD is defined there.
for i in range(DIM):
for j in range(DIM):
MU[i] += -sp.Rational(2, 3) * Bq.gammabarUU[i][j] * trK_dD[j]
# First define aDD_dD:
aDD_dD = ixp.declarerank3("aDD_dD", "sym01")
# Then evaluate the conformal covariant derivative \bar{D}_j \bar{A}_{lm}
AbarDD_dBarD = ixp.zerorank3()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
AbarDD_dBarD[i][j][k] = Bq.AbarDD_dD[i][j][k]
for l in range(DIM):
AbarDD_dBarD[i][j][
k] += -Bq.GammabarUDD[l][k][i] * Bq.AbarDD[l][j]
AbarDD_dBarD[i][j][
k] += -Bq.GammabarUDD[l][k][j] * Bq.AbarDD[i][l]
# Term 1: Contract twice with the metric to make \bar{D}_{j} \bar{A}^{ij}
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
MU[i] += Bq.gammabarUU[i][k] * Bq.gammabarUU[j][
l] * AbarDD_dBarD[k][l][j]
# Finally, we multiply by e^{-4 phi} and rescale the momentum constraint:
for i in range(DIM):
MU[i] *= Bq.exp_m4phi / rfm.ReU[i]
def ScalarField_RHSs():
# Step B.4: Set spatial dimension (must be 3 for BSSN, as BSSN is
# a 3+1-dimensional decomposition of the general
# relativistic field equations)
DIM = 3
# Step B.5: Import all basic (unrescaled) BSSN scalars & tensors
Bq.BSSN_basic_tensors()
trK = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU
Bq.gammabar__inverse_and_derivs()
gammabarUU = Bq.gammabarUU
global sf_rhs, sfM_rhs
# Step B.5.a: Declare grid functions for varphi and Pi
sf, sfM = sfgfs.declare_scalar_field_gridfunctions_if_not_declared_already(
)
# Step 2.a: Add Term 1 to sf_rhs: -alpha*Pi
sf_rhs = -alpha * sfM
# Step 2.b: Add Term 2 to sf_rhs: beta^{i}\partial_{i}\varphi
sf_dupD = ixp.declarerank1("sf_dupD")
for i in range(DIM):
sf_rhs += betaU[i] * sf_dupD[i]
# Step 3a: Add Term 1 to sfM_rhs: alpha * K * Pi
sfM_rhs = alpha * trK * sfM
# Step 3b: Add Term 2 to sfM_rhs: beta^{i}\partial_{i}Pi
sfM_dupD = ixp.declarerank1("sfM_dupD")
for i in range(DIM):
sfM_rhs += betaU[i] * sfM_dupD[i]
# Step 3c: Adding Term 3 to sfM_rhs
# Step 3c.i: Term 3a: gammabar^{ij}\partial_{i}\alpha\partial_{j}\varphi
alpha_dD = ixp.declarerank1("alpha_dD")
sf_dD = ixp.declarerank1("sf_dD")
sfMrhsTerm3 = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -gammabarUU[i][j] * alpha_dD[i] * sf_dD[j]
# Step 3c.ii: Term 3b: \alpha*gammabar^{ij}\partial_{i}\partial_{j}\varphi
sf_dDD = ixp.declarerank2("sf_dDD", "sym01")
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -alpha * gammabarUU[i][j] * sf_dDD[i][j]
# Step 3c.iii: Term 3c: 2*alpha*gammabar^{ij}\partial_{j}\varphi\partial_{i}\phi
Bq.phi_and_derivs(
) # sets exp^{-4phi} = exp_m4phi and \partial_{i}phi = phi_dD[i]
for i in range(DIM):
for j in range(DIM):
sfMrhsTerm3 += -2 * alpha * gammabarUU[i][j] * sf_dD[
j] * Bq.phi_dD[i]
# Step 3c.iv: Multiplying Term 3 by e^{-4phi} and adding it to sfM_rhs
sfMrhsTerm3 *= Bq.exp_m4phi
sfM_rhs += sfMrhsTerm3
# Step 3d: Adding Term 4 to sfM_rhs
# Step 3d.i: Term 4a: \alpha \bar\Lambda^{i}\partial_{i}\varphi
LambdabarU = Bq.LambdabarU
sfMrhsTerm4 = sp.sympify(0)
for i in range(DIM):
sfMrhsTerm4 += alpha * LambdabarU[i] * sf_dD[i]
# Step 3d.ii: Evaluating \bar\gamma^{ij}\hat\Gamma^{k}_{ij}
GammahatUDD = rfm.GammahatUDD
gammabarGammahatContractionU = ixp.zerorank1()
for k in range(DIM):
for i in range(DIM):
for j in range(DIM):
gammabarGammahatContractionU[
k] += gammabarUU[i][j] * GammahatUDD[k][i][j]
# Step 3d.iii: Term 4b: \alpha \bar\gamma^{ij}\hat\Gamma^{k}_{ij}\partial_{k}\varphi
for i in range(DIM):
sfMrhsTerm4 += alpha * gammabarGammahatContractionU[i] * sf_dD[i]
# Step 3d.iii: Multplying Term 4 by e^{-4phi} and adding it to sfM_rhs
sfMrhsTerm4 *= Bq.exp_m4phi
sfM_rhs += sfMrhsTerm4
return