def cross(vec1, vec2):
vec1_cross_vec2 = ixp.zerorank1()
LeviCivitaSymbol = ixp.LeviCivitaSymbol_dim3_rank3()
for i in range(3):
for j in range(3):
for k in range(3):
vec1_cross_vec2[
i] += LeviCivitaSymbol[i][j][k] * vec1[j] * vec2[k]
return vec1_cross_vec2
def ValenciavU_func_AR(**params):
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
unit_zU = ixp.zerorank1()
unit_zU[2] = sp.sympify(1)
r = rfm.xxSph[0]
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[i] += noif.coord_leq_bound(
r, R_NS_aligned_rotator) * LeviCivitaSymbolDDD[i][j][
k] * Omega_aligned_rotator * unit_zU[j] * rfm.xx[k]
return ValenciavU
def compute_ValenciavU_from_EU_and_BU(EU, BU):
# <a id='step3'></a>
# ### Calculate $v^i$
#
# Now, we calculate $$\mathbf{v} = \frac{\mathbf{E} \times \mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \gamma_{ij} B^i B^j$ is a trivial dot product in flat space.
# $$\label{step3}$$
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
B2 = sp.sympify(0)
for i in range(3):
# In flat spacetime, gamma_{ij} is just a Kronecker delta
B2 += BU[i]**2 # This is trivial to extend to curved spacetime
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2
return ValenciavU
def GiRaFFEfood_NRPy_Aligned_Rotator():
r = rfm.xxSph[0]
varpi = sp.sqrt(rfm.xxCart[0]**2 + rfm.xxCart[1]**2)
mu = B_p_aligned_rotator * R_NS_aligned_rotator**3 / 2
ASphD = ixp.zerorank1()
ASphD[2] = mu * varpi**2 / (
r**3) # The other components were already declared to be 0.
# <a id='step3'></a>
#
# ### Step 3: Use the Jacobian matrix to transform the vectors to Cartesian coordinates.
# $$\label{step3}$$
#
# \[Back to [top](#top)\]
#
# Now, we will use the coordinate transformation definitions provided by reference_metric.py to build the Jacobian
# $$
# \frac{\partial x_{\rm Sph}^j}{\partial x_{\rm Cart}^i},
# $$
# where $x_{\rm Sph}^j \in \{r,\theta,\phi\}$ and $x_{\rm Cart}^i \in \{x,y,z\}$. We would normally compute its inverse, but since none of the quantities we need to transform have upper indices, it is not necessary. Then, since $A_i$ and has one lower index, it will need to be multiplied by the Jacobian:
#
# $$
# A_i^{\rm Cart} = A_j^{\rm Sph} \frac{\partial x_{\rm Sph}^j}{\partial x_{\rm Cart}^i},
# $$
# Step 3: Use the Jacobian matrix to transform the vectors to Cartesian coordinates.
drrefmetric__dx_0UDmatrix = sp.Matrix([[
sp.diff(rfm.xxSph[0], rfm.xx[0]),
sp.diff(rfm.xxSph[0], rfm.xx[1]),
sp.diff(rfm.xxSph[0], rfm.xx[2])
],
[
sp.diff(rfm.xxSph[1],
rfm.xx[0]),
sp.diff(rfm.xxSph[1],
rfm.xx[1]),
sp.diff(rfm.xxSph[1], rfm.xx[2])
],
[
sp.diff(rfm.xxSph[2],
rfm.xx[0]),
sp.diff(rfm.xxSph[2],
rfm.xx[1]),
sp.diff(rfm.xxSph[2], rfm.xx[2])
]])
#dx__drrefmetric_0UDmatrix = drrefmetric__dx_0UDmatrix.inv() # We don't actually need this in this case.
global AD
AD = ixp.zerorank1(DIM=3)
for i in range(3):
for j in range(3):
AD[i] = drrefmetric__dx_0UDmatrix[(j, i)] * ASphD[j]
# <a id='step4'></a>
#
# ### Step 4: Calculate $v^i$
# $$\label{step4}$$
#
# \[Back to [top](#top)\]
#
# Here, we will calculate the drift velocity $v^i = \Omega \textbf{e}_z \times \textbf{r} = [ijk] \Omega \textbf{e}^j_z x^k$, where $[ijk]$ is the Levi-Civita permutation symbol and $\textbf{e}^i_z = (0,0,1)$. Conveniently, in flat space, the drift velocity reduces to the Valencia velocity because $\alpha = 1$ and $\beta^i = 0$.
# Step 4: Calculate v^i
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
import Min_Max_and_Piecewise_Expressions as noif
unit_zU = ixp.zerorank1()
unit_zU[2] = sp.sympify(1)
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[i] += noif.coord_leq_bound(
r, R_NS_aligned_rotator) * LeviCivitaSymbolDDD[i][j][
k] * Omega_aligned_rotator * unit_zU[j] * rfm.xx[k]
def GiRaFFEfood_NRPy_1D_tests(stagger=False):
gammamu = sp.sympify(1) / sp.sqrt(sp.sympify(1) - mu_AW**2)
# We'll use reference_metric.py to define x and y
x = rfm.xxCart[0]
y = rfm.xxCart[1]
if stagger:
x_p_half = x + sp.Rational(1, 2) * gri.dxx[0]
y_p_half = y + sp.Rational(1, 2) * gri.dxx[1]
if stagger:
g_AW = sp.cos(sp.sympify(5) * M_PI * gammamu * x_p_half) / M_PI
else:
g_AW = sp.cos(sp.sympify(5) * M_PI * gammamu * x) / M_PI
# Now, we can define the vector potential. We will create three copies of this variable, because the potential is uniquely defined in three zones. Data for $x \leq -0.1/\gamma_\mu$ shall be referred to as "left", data for $-0.1/\gamma_\mu \leq x \leq 0.1/\gamma_\mu$ as "center", and data for $x \geq 0.1/\gamma_\mu$ as "right".
global AD
AD = ixp.zerorank1()
import Min_Max_and_Piecewise_Expressions as noif
bound = sp.Rational(1, 10) / gammamu
if stagger:
Ayleft = gammamu * x_p_half - sp.Rational(15, 1000)
Aycenter = sp.Rational(115, 100) * gammamu * x_p_half - sp.Rational(
3, 100) * g_AW
Ayright = sp.Rational(13, 10) * gammamu * x_p_half - sp.Rational(
15, 1000)
else:
Ayleft = gammamu * x - sp.Rational(15, 1000)
Aycenter = sp.Rational(115, 100) * gammamu * x - sp.Rational(
3, 100) * g_AW
Ayright = sp.Rational(13, 10) * gammamu * x - sp.Rational(15, 1000)
AD[0] = sp.sympify(0)
AD[1] = noif.coord_leq_bound(x,-bound)*Ayleft\
+noif.coord_greater_bound(x,-bound)*noif.coord_leq_bound(x,bound)*Aycenter\
+noif.coord_greater_bound(x,bound)*Ayright
if stagger:
AD[2] = y_p_half - gammamu * (sp.sympify(1) -
mu_AW) * x_p_half # <a id='step2'></a>
else:
AD[2] = y - gammamu * (sp.sympify(1) - mu_AW) * x # <a id='step2'></a>
# ### Set the vectors $B^i$ and $E^i$ for the velocity
#
# Now, we will set the magnetic and electric fields that we will need to define the initial velocities. First, we need to define $$f(x)=1+\sin (5\pi x);$$ note that in the definition of $B^i$, we need $f(x')$ where $x'=\gamma_\mu x$.
# $$\label{step2}$$
xprime = gammamu * x
f_AW = 1 + sp.sin(5 * M_PI * xprime)
# We will now set the magnetic field in the wave frame:
# \begin{align}
# B'^{x'}(x') = &\ 1.0,\ B'^y(x') = 1.0, \\
# B'^z(x') = &\ \left \{ \begin{array}{lll} 1.0 & \mbox{if} & x' \leq -0.1 \\
# 1.0+0.15 f(x') & \mbox{if} & -0.1 \leq x' \leq 0.1 \\
# 1.3 & \mbox{if} & x' \geq 0.1 \end{array} \right. .
# \end{align}
#
Bzleft = sp.sympify(1)
Bzcenter = sp.sympify(1) + sp.Rational(15, 100) * f_AW
Bzright = sp.Rational(13, 10)
BpU = ixp.zerorank1()
BpU[0] = sp.sympify(1)
BpU[1] = sp.sympify(1)
BpU[2] = noif.coord_leq_bound(x,-bound)*Bzleft\
+noif.coord_greater_bound(x,-bound)*noif.coord_leq_bound(x,bound)*Bzcenter\
+noif.coord_greater_bound(x,bound)*Bzright
# Now, we will set the electric field in the wave frame:
# \begin{align}
# E'^{x'}(x') &= -B'^z(0,x'), \\
# E'^y(x') &= 0.0, \\
# E'^z(x') &= 1.0 .
# \end{align}
EpU = ixp.zerorank1()
EpU[0] = -BpU[2]
EpU[1] = sp.sympify(0)
EpU[2] = sp.sympify(1)
# Next, we must transform the the fields into the grid frame. We'll do the magnetic fields first.
# \begin{align}
# B^x(0,x) = &\ B'^{x'}(\gamma_\mu x) , \\
# B^y(0,x) = &\ \gamma_\mu [ B'^y(\gamma_\mu x) - \mu E'^z(\gamma_\mu x) ] , \\
# B^z(0,x) = &\ \gamma_\mu [ B'^z(\gamma_\mu x) + \mu E'^y(\gamma_\mu x) ] ,
# \end{align}
#
global BU
BU = ixp.zerorank1()
BU[0] = BpU[0]
BU[1] = gammamu * (BpU[1] - mu_AW * EpU[2])
BU[2] = gammamu * (BpU[2] + mu_AW * EpU[1])
# And now the electric fields:
# \begin{align}
# E^x(0,x) = &\ E'^{x'}(\gamma_\mu x) , \\
# E^y(0,x) = &\ \gamma_\mu [ E'^y(\gamma_\mu x) + \mu B'^z(\gamma_\mu x) ] ,\\
# E^z(0,x) = &\ \gamma_\mu [ E'^z(\gamma_\mu x) - \mu B'^y(\gamma_\mu x) ],
# \end{align}
#
EU = ixp.zerorank1()
EU[0] = EpU[0]
EU[1] = gammamu * (EpU[1] + mu_AW * BpU[2])
EU[2] = gammamu * (EpU[2] - mu_AW * BpU[1])
# <a id='step3'></a>
# ### Calculate $v^i$
#
# Now, we calculate $$\mathbf{v} = \frac{\mathbf{E} \times \mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \gamma_{ij} B^i B^j$ is a trivial dot product in flat space.
# $$\label{step3}$$
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
B2 = sp.sympify(0)
for i in range(3):
# In flat spacetime, gamma_{ij} is just a Kronecker delta
B2 += BU[i]**2 # This is trivial to extend to curved spacetime
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2
def GiRaFFEfood_NRPy_1D_tests_FFE_breakdown(stagger=False):
# We'll use reference_metric.py to define x and y
x = rfm.xxCart[0]
y = rfm.xxCart[1]
if stagger:
x_p_half = x + sp.Rational(1, 2) * gri.dxx[0]
y_p_half = y + sp.Rational(1, 2) * gri.dxx[1]
# Now, we can define the vector potential. We will create three copies of this variable, because the potential is uniquely defined in three zones. Data for $x \leq -0.1/\gamma_\mu$ shall be referred to as "left", data for $-0.1/\gamma_\mu \leq x \leq 0.1/\gamma_\mu$ as "center", and data for $x \geq 0.1/\gamma_\mu$ as "right".
global AD
AD = ixp.zerorank1(DIM=3)
import Min_Max_and_Piecewise_Expressions as noif
boundL = sp.sympify(0)
boundR = sp.Rational(1, 5)
if stagger:
Ayleft = x_p_half - sp.Rational(1, 5)
Aycenter = -sp.sympify(5) * x_p_half**2 + x_p_half - sp.Rational(1, 5)
Ayright = -x_p_half
else:
Ayleft = x - sp.Rational(1, 5)
Aycenter = -sp.sympify(5) * x**2 + x - sp.Rational(1, 5)
Ayright = -x
AD[0] = sp.sympify(0)
AD[1] = noif.coord_leq_bound(x,boundL)*Ayleft\
+noif.coord_greater_bound(x,boundL)*noif.coord_leq_bound(x,boundR)*Aycenter\
+noif.coord_greater_bound(x,boundR)*Ayright
if stagger:
AD[2] = y_p_half - AD[1]
else:
AD[2] = y - AD[1]
# ### Set the vectors $B^i$ and $E^i$ for the velocity
#
# Now, we will set the magnetic and electric fields that we will need to define the initial velocities. First, we need to define $$f(x)=1+\sin (5\pi x);$$ note that in the definition of $B^i$, we need $f(x')$ where $x'=\gamma_\mu x$.
# $$\label{step2}$$
zfunc = -sp.sympify(10) * x + sp.sympify(
1
) # Naming it like this to avoid any possible confusion with the coordinate.
# We will now set the magnetic field in the wave frame:
# \begin{align}
# B'^{x'}(x') = &\ 1.0,\ B'^y(x') = 1.0, \\
# B'^z(x') = &\ \left \{ \begin{array}{lll} 1.0 & \mbox{if} & x' \leq -0.1 \\
# 1.0+0.15 f(x') & \mbox{if} & -0.1 \leq x' \leq 0.1 \\
# 1.3 & \mbox{if} & x' \geq 0.1 \end{array} \right. .
# \end{align}
#
global BU
BU = ixp.zerorank1(DIM=3)
BU[0] = sp.sympify(1)
BU[1] = BU[2] = noif.coord_leq_bound(x,boundL) * sp.sympify(1)\
+noif.coord_greater_bound(x,boundL)*noif.coord_leq_bound(x,boundR)* zfunc\
+noif.coord_greater_bound(x,boundR) * -sp.sympify(1)
# Now, we will set the electric field in the wave frame:
# \begin{align}
# E'^{x'}(x') &= -B'^z(0,x'), \\
# E'^y(x') &= 0.0, \\
# E'^z(x') &= 1.0 .
# \end{align}
EU = ixp.zerorank1()
EU[0] = sp.sympify(0)
EU[1] = sp.Rational(1, 2)
EU[2] = -sp.Rational(1, 2)
# <a id='step3'></a>
# ### Calculate $v^i$
#
# Now, we calculate $$\mathbf{v} = \frac{\mathbf{E} \times \mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \gamma_{ij} B^i B^j$ is a trivial dot product in flat space.
# $$\label{step3}$$
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
B2 = sp.sympify(0)
for i in range(3):
# In flat spacetime, gamma_{ij} is just a Kronecker delta
B2 += BU[i]**2 # This is trivial to extend to curved spacetime
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2
def WeylScalars_Cartesian():
# Step 3.a: Set spatial dimension (must be 3 for BSSN)
DIM = 3
par.set_parval_from_str("grid::DIM", DIM)
# Step 3.b: declare the additional gridfunctions (i.e., functions whose values are declared
# at every grid point, either inside or outside of our SymPy expressions) needed
# for this thorn:
# * the physical metric $\gamma_{ij}$,
# * the extrinsic curvature $K_{ij}$,
# * the real and imaginary components of $\psi_4$, and
# * the Weyl curvature invariants:
gammaDD = ixp.register_gridfunctions_for_single_rank2(
"AUX", "gammaDD", "sym01") # The AUX or EVOL designation is *not*
# used in diagnostic modules.
kDD = ixp.register_gridfunctions_for_single_rank2("AUX", "kDD", "sym01")
x, y, z = gri.register_gridfunctions("AUX", ["x", "y", "z"])
global psi4r, psi4i, psi3r, psi3i, psi2r, psi2i, psi1r, psi1i, psi0r, psi0i
psi4r, psi4i, psi3r, psi3i, psi2r, psi2i, psi1r, psi1i, psi0r, psi0i = gri.register_gridfunctions(
"AUX", [
"psi4r", "psi4i", "psi3r", "psi3i", "psi2r", "psi2i", "psi1r",
"psi1i", "psi0r", "psi0i"
])
# Step 4: Set which tetrad is used; at the moment, only one supported option
# The tetrad depends in general on the inverse 3-metric gammaUU[i][j]=\gamma^{ij}
# and the determinant of the 3-metric (detgamma), which are defined in
# the following line of code from gammaDD[i][j]=\gamma_{ij}.
tmpgammaUU, detgamma = ixp.symm_matrix_inverter3x3(gammaDD)
detgamma = sp.simplify(detgamma)
gammaUU = ixp.zerorank2()
for i in range(3):
for j in range(3):
gammaUU[i][j] = sp.simplify(tmpgammaUU[i][j])
if par.parval_from_str("WeylScal4NRPy.WeylScalars_Cartesian::TetradChoice"
) == "Approx_QuasiKinnersley":
# Eqs 5.6 in https://arxiv.org/pdf/gr-qc/0104063.pdf
xmoved = x # - xorig
ymoved = y # - yorig
zmoved = z # - zorig
# Step 5.a: Choose 3 orthogonal vectors. Here, we choose one in the azimuthal
# direction, one in the radial direction, and the cross product of the two.
# Eqs 5.7
v1U = ixp.zerorank1()
v2U = ixp.zerorank1()
v3U = ixp.zerorank1()
v1U[0] = -ymoved
v1U[1] = xmoved # + offset
v1U[2] = sp.sympify(0)
v2U[0] = xmoved # + offset
v2U[1] = ymoved
v2U[2] = zmoved
LeviCivitaSymbol_rank3 = ixp.LeviCivitaSymbol_dim3_rank3()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
v3U[a] += sp.sqrt(
detgamma) * gammaUU[a][d] * LeviCivitaSymbol_rank3[
d][b][c] * v1U[b] * v2U[c]
for a in range(DIM):
v3U[a] = sp.simplify(v3U[a])
# Step 5.b: Gram-Schmidt orthonormalization of the vectors.
# The w_i^a vectors here are used to temporarily hold values on the way to the final vectors e_i^a
# e_1^a &= \frac{v_1^a}{\omega_{11}}
# e_2^a &= \frac{v_2^a - \omega_{12} e_1^a}{\omega_{22}}
# e_3^a &= \frac{v_3^a - \omega_{13} e_1^a - \omega_{23} e_2^a}{\omega_{33}},
# Normalize the first vector
w1U = ixp.zerorank1()
for a in range(DIM):
w1U[a] = v1U[a]
omega11 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega11 += w1U[a] * w1U[b] * gammaDD[a][b]
e1U = ixp.zerorank1()
for a in range(DIM):
e1U[a] = w1U[a] / sp.sqrt(omega11)
# Subtract off the portion of the first vector along the second, then normalize
omega12 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega12 += e1U[a] * v2U[b] * gammaDD[a][b]
w2U = ixp.zerorank1()
for a in range(DIM):
w2U[a] = v2U[a] - omega12 * e1U[a]
omega22 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega22 += w2U[a] * w2U[b] * gammaDD[a][b]
e2U = ixp.zerorank1()
for a in range(DIM):
e2U[a] = w2U[a] / sp.sqrt(omega22)
# Subtract off the portion of the first and second vectors along the third, then normalize
omega13 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega13 += e1U[a] * v3U[b] * gammaDD[a][b]
omega23 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega23 += e2U[a] * v3U[b] * gammaDD[a][b]
w3U = ixp.zerorank1()
for a in range(DIM):
w3U[a] = v3U[a] - omega13 * e1U[a] - omega23 * e2U[a]
omega33 = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omega33 += w3U[a] * w3U[b] * gammaDD[a][b]
e3U = ixp.zerorank1()
for a in range(DIM):
e3U[a] = w3U[a] / sp.sqrt(omega33)
# Step 5.c: Construct the tetrad itself.
# Eqs. 5.6:
# l^a &= \frac{1}{\sqrt{2}} e_2^a \\
# n^a &= -\frac{1}{\sqrt{2}} e_2^a \\
# m^a &= \frac{1}{\sqrt{2}} (e_3^a + i e_1^a) \\
# \overset{*}{m}{}^a &= \frac{1}{\sqrt{2}} (e_3^a - i e_1^a)
isqrt2 = 1 / sp.sqrt(2)
ltetU = ixp.zerorank1()
ntetU = ixp.zerorank1()
# mtetU = ixp.zerorank1()
# mtetccU = ixp.zerorank1()
remtetU = ixp.zerorank1(
) # SymPy did not like trying to take the real/imaginary parts of such a
immtetU = ixp.zerorank1(
) # complicated expression, so we do it ourselves.
for i in range(DIM):
ltetU[i] = isqrt2 * e2U[i]
ntetU[i] = -isqrt2 * e2U[i]
remtetU[i] = isqrt2 * e3U[i]
immtetU[i] = isqrt2 * e1U[i]
nn = isqrt2
else:
print("Error: TetradChoice == " + par.parval_from_str("TetradChoice") +
" unsupported!")
sys.exit(1)
# Step 5: Declare and construct the second derivative of the metric.
gammaDD_dD = ixp.declarerank3("gammaDD_dD", "sym01")
# Define the Christoffel symbols
GammaUDD = ixp.zerorank3(DIM)
for i in range(DIM):
for k in range(DIM):
for l in range(DIM):
for m in range(DIM):
GammaUDD[i][k][l] += (sp.Rational(1, 2)) * gammaUU[i][m] * \
(gammaDD_dD[m][k][l] + gammaDD_dD[m][l][k] - gammaDD_dD[k][l][m])
# Step 6.a: Declare and construct the Riemann curvature tensor:
# R_{abcd} = \frac{1}{2} (\gamma_{ad,cb}+\gamma_{bc,da}-\gamma_{ac,bd}-\gamma_{bd,ac})
# + \gamma_{je} \Gamma^{j}_{bc}\Gamma^{e}_{ad} - \gamma_{je} \Gamma^{j}_{bd} \Gamma^{e}_{ac}
gammaDD_dDD = ixp.declarerank4("gammaDD_dDD", "sym01_sym23")
RiemannDDDD = ixp.zerorank4()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
RiemannDDDD[a][b][c][d] = (gammaDD_dDD[a][d][c][b] + \
gammaDD_dDD[b][c][d][a] - \
gammaDD_dDD[a][c][b][d] - \
gammaDD_dDD[b][d][a][c]) / 2
for e in range(DIM):
for j in range(DIM):
RiemannDDDD[a][b][c][d] += gammaDD[j][e] * GammaUDD[j][b][c] * GammaUDD[e][a][d] - \
gammaDD[j][e] * GammaUDD[j][b][d] * GammaUDD[e][a][c]
# Step 6.b: We also need the extrinsic curvature tensor $K_{ij}$.
# In Cartesian coordinates, we already made the components gridfunctions.
# We will, however, need to calculate the trace of K seperately:
trK = sp.sympify(0)
for i in range(DIM):
for j in range(DIM):
trK += gammaUU[i][j] * kDD[i][j]
# Step 7: Build the formula for \psi_4.
# Gauss equation: involving the Riemann tensor and extrinsic curvature.
# GaussDDDD[i][j][k][l] =& R_{ijkl} + 2K_{i[k}K_{l]j}
GaussDDDD = ixp.zerorank4()
for i in range(DIM):
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
GaussDDDD[i][j][k][l] = RiemannDDDD[i][j][k][
l] + kDD[i][k] * kDD[l][j] - kDD[i][l] * kDD[k][j]
# Codazzi equation: involving partial derivatives of the extrinsic curvature.
# We will first need to declare derivatives of kDD
# CodazziDDD[j][k][l] =& -2 (K_{j[k,l]} + \Gamma^p_{j[k} K_{l]p})
kDD_dD = ixp.declarerank3("kDD_dD", "sym01")
CodazziDDD = ixp.zerorank3()
for j in range(DIM):
for k in range(DIM):
for l in range(DIM):
CodazziDDD[j][k][l] = kDD_dD[j][l][k] - kDD_dD[j][k][l]
for p in range(DIM):
CodazziDDD[j][k][l] += GammaUDD[p][j][l] * kDD[k][
p] - GammaUDD[p][j][k] * kDD[l][p]
# Another piece. While not associated with any particular equation,
# this is still useful for organizational purposes.
# RojoDD[j][l]} = & R_{jl} - K_{jp} K^p_l + KK_{jl} \\
# = & \gamma^{pd} R_{jpld} - K_{jp} K^p_l + KK_{jl}
RojoDD = ixp.zerorank2()
for j in range(DIM):
for l in range(DIM):
RojoDD[j][l] = trK * kDD[j][l]
for p in range(DIM):
for d in range(DIM):
RojoDD[j][l] += gammaUU[p][d] * RiemannDDDD[j][p][l][
d] - kDD[j][p] * gammaUU[p][d] * kDD[d][l]
# Now we can calculate $\psi_4$ itself! We assume l^0 = n^0 = \frac{1}{\sqrt{2}}
# and m^0 = \overset{*}{m}{}^0 = 0 to simplify these equations.
# We calculate the Weyl scalars as defined in https://arxiv.org/abs/gr-qc/0104063
# In terms of the above-defined quantites, the psis are defined as:
# \psi_4 =&\ (\text{GaussDDDD[i][j][k][l]}) n^i \overset{*}{m}{}^j n^k \overset{*}{m}{}^l \\
# &+2 (\text{CodazziDDD[j][k][l]}) n^{0} \overset{*}{m}{}^{j} n^k \overset{*}{m}{}^l \\
# &+ (\text{RojoDD[j][l]}) n^{0} \overset{*}{m}{}^{j} n^{0} \overset{*}{m}{}^{l}.
# \psi_3 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i n^j \overset{*}{m}{}^k n^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (l^{0} n^{j} \overset{*}{m}{}^k n^l - l^{j} n^{0} \overset{*}{m}{}^k n^l - l^k n^j\overset{*}{m}{}^l n^0) \\
# &- (\text{RojoDD[j][l]}) l^{0} n^{j} \overset{*}{m}{}^l n^0 - l^{j} n^{0} \overset{*}{m}{}^l n^0 \\
# \psi_2 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i m^j \overset{*}{m}{}^k n^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (l^{0} m^{j} \overset{*}{m}{}^k n^l - l^{j} m^{0} \overset{*}{m}{}^k n^l - l^k m^l \overset{*}{m}{}^l n^0) \\
# &- (\text{RojoDD[j][l]}) l^0 m^j \overset{*}{m}{}^l n^0 \\
# \psi_1 =&\ (\text{GaussDDDD[i][j][k][l]}) n^i l^j m^k l^l \\
# &+ (\text{CodazziDDD[j][k][l]}) (n^{0} l^{j} m^k l^l - n^{j} l^{0} m^k l^l - n^k l^l m^j l^0) \\
# &- (\text{RojoDD[j][l]}) (n^{0} l^{j} m^l l^0 - n^{j} l^{0} m^l l^0) \\
# \psi_0 =&\ (\text{GaussDDDD[i][j][k][l]}) l^i m^j l^k m^l \\
# &+2 (\text{CodazziDDD[j][k][l]}) (l^0 m^j l^k m^l + l^k m^l l^0 m^j) \\
# &+ (\text{RojoDD[j][l]}) l^0 m^j l^0 m^j. \\
psi4r = sp.sympify(0)
psi4i = sp.sympify(0)
psi3r = sp.sympify(0)
psi3i = sp.sympify(0)
psi2r = sp.sympify(0)
psi2i = sp.sympify(0)
psi1r = sp.sympify(0)
psi1i = sp.sympify(0)
psi0r = sp.sympify(0)
psi0i = sp.sympify(0)
for l in range(DIM):
for j in range(DIM):
psi4r += RojoDD[j][l] * nn * nn * (remtetU[j] * remtetU[l] -
immtetU[j] * immtetU[l])
psi4i += RojoDD[j][l] * nn * nn * (-remtetU[j] * immtetU[l] -
immtetU[j] * remtetU[l])
psi3r += -RojoDD[j][l] * nn * nn * (ntetU[j] -
ltetU[j]) * remtetU[l]
psi3i += RojoDD[j][l] * nn * nn * (ntetU[j] -
ltetU[j]) * immtetU[l]
psi2r += -RojoDD[j][l] * nn * nn * (remtetU[l] * remtetU[j] +
immtetU[j] * immtetU[l])
psi2i += -RojoDD[j][l] * nn * nn * (immtetU[l] * remtetU[j] -
remtetU[j] * immtetU[l])
psi1r += RojoDD[j][l] * nn * nn * (ntetU[j] * remtetU[l] -
ltetU[j] * remtetU[l])
psi1i += RojoDD[j][l] * nn * nn * (ntetU[j] * immtetU[l] -
ltetU[j] * immtetU[l])
psi0r += RojoDD[j][l] * nn * nn * (remtetU[j] * remtetU[l] -
immtetU[j] * immtetU[l])
psi0i += RojoDD[j][l] * nn * nn * (remtetU[j] * immtetU[l] +
immtetU[j] * remtetU[l])
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
psi4r += 2 * CodazziDDD[j][k][l] * ntetU[k] * nn * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi4i += 2 * CodazziDDD[j][k][l] * ntetU[k] * nn * (
-remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l])
psi3r += 1 * CodazziDDD[j][k][l] * nn * (
(ntetU[j] - ltetU[j]) * remtetU[k] * ntetU[l] -
remtetU[j] * ltetU[k] * ntetU[l])
psi3i += -1 * CodazziDDD[j][k][l] * nn * (
(ntetU[j] - ltetU[j]) * immtetU[k] * ntetU[l] -
immtetU[j] * ltetU[k] * ntetU[l])
psi2r += 1 * CodazziDDD[j][k][l] * nn * (
ntetU[l] *
(remtetU[j] * remtetU[k] + immtetU[j] * immtetU[k]) -
ltetU[k] *
(remtetU[j] * remtetU[l] + immtetU[j] * immtetU[l]))
psi2i += 1 * CodazziDDD[j][k][l] * nn * (
ntetU[l] *
(immtetU[j] * remtetU[k] - remtetU[j] * immtetU[k]) -
ltetU[k] *
(remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l]))
psi1r += 1 * CodazziDDD[j][k][l] * nn * (
ltetU[j] * remtetU[k] * ltetU[l] - remtetU[j] * ntetU[k] *
ltetU[l] - ntetU[j] * remtetU[k] * ltetU[l])
psi1i += 1 * CodazziDDD[j][k][l] * nn * (
ltetU[j] * immtetU[k] * ltetU[l] - immtetU[j] * ntetU[k] *
ltetU[l] - ntetU[j] * immtetU[k] * ltetU[l])
psi0r += 2 * CodazziDDD[j][k][l] * nn * ltetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi0i += 2 * CodazziDDD[j][k][l] * nn * ltetU[k] * (
remtetU[j] * immtetU[l] + immtetU[j] * remtetU[l])
for l in range(DIM):
for j in range(DIM):
for k in range(DIM):
for i in range(DIM):
psi4r += GaussDDDD[i][j][k][l] * ntetU[i] * ntetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi4i += GaussDDDD[i][j][k][l] * ntetU[i] * ntetU[k] * (
-remtetU[j] * immtetU[l] - immtetU[j] * remtetU[l])
psi3r += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[
j] * remtetU[k] * ntetU[l]
psi3i += -GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[
j] * immtetU[k] * ntetU[l]
psi2r += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[l] * (
remtetU[j] * remtetU[k] + immtetU[j] * immtetU[k])
psi2i += GaussDDDD[i][j][k][l] * ltetU[i] * ntetU[l] * (
immtetU[j] * remtetU[k] - remtetU[j] * immtetU[k])
psi1r += GaussDDDD[i][j][k][l] * ntetU[i] * ltetU[
j] * remtetU[k] * ltetU[l]
psi1i += GaussDDDD[i][j][k][l] * ntetU[i] * ltetU[
j] * immtetU[k] * ltetU[l]
psi0r += GaussDDDD[i][j][k][l] * ltetU[i] * ltetU[k] * (
remtetU[j] * remtetU[l] - immtetU[j] * immtetU[l])
psi0i += GaussDDDD[i][j][k][l] * ltetU[i] * ltetU[k] * (
remtetU[j] * immtetU[l] + immtetU[j] * remtetU[l])
def Psi4_tetrads():
global l4U, n4U, mre4U, mim4U
# Step 1.c: Check if tetrad choice is implemented:
if par.parval_from_str(thismodule + "::TetradChoice") != "QuasiKinnersley":
print("ERROR: " + thismodule + "::TetradChoice = " +
par.parval_from_str("TetradChoice") + " currently unsupported!")
sys.exit(1)
# Step 1.d: 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 1.e: 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.f: Import all ADM quantities as written in terms of BSSN quantities
import BSSN.ADM_in_terms_of_BSSN as AB
AB.ADM_in_terms_of_BSSN()
# Step 2.a: Declare the Cartesian x,y,z in terms of
# xx0,xx1,xx2.
x = rfm.xxCart[0]
y = rfm.xxCart[1]
z = rfm.xxCart[2]
# Step 2.b: Declare detgamma and gammaUU from
# BSSN.ADM_in_terms_of_BSSN;
# simplify detgamma & gammaUU expressions,
# which expedites Psi4 codegen.
detgamma = sp.simplify(AB.detgamma)
gammaUU = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
gammaUU[i][j] = sp.simplify(AB.gammaUU[i][j])
# Step 2.c: Define v1U and v2U
v1UCart = [-y, x, sp.sympify(0)]
v2UCart = [x, y, z]
# Step 2.d: Construct the Jacobian d x_Cart^i / d xx^j
Jac_dUCart_dDrfmUD = ixp.zerorank2()
for i in range(DIM):
for j in range(DIM):
Jac_dUCart_dDrfmUD[i][j] = sp.simplify(
sp.diff(rfm.xxCart[i], rfm.xx[j]))
# Step 2.e: Invert above Jacobian to get needed d xx^j / d x_Cart^i
Jac_dUrfm_dDCartUD, dummyDET = ixp.generic_matrix_inverter3x3(
Jac_dUCart_dDrfmUD)
# Step 2.e.i: Simplify expressions for d xx^j / d x_Cart^i:
for i in range(DIM):
for j in range(DIM):
Jac_dUrfm_dDCartUD[i][j] = sp.simplify(Jac_dUrfm_dDCartUD[i][j])
# Step 2.f: Transform v1U and v2U from the Cartesian to the xx^i basis
v1U = ixp.zerorank1()
v2U = ixp.zerorank1()
for i in range(DIM):
for j in range(DIM):
v1U[i] += Jac_dUrfm_dDCartUD[i][j] * v1UCart[j]
v2U[i] += Jac_dUrfm_dDCartUD[i][j] * v2UCart[j]
# Step 2.g: Define v3U
v3U = ixp.zerorank1()
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
for a in range(DIM):
for b in range(DIM):
for c in range(DIM):
for d in range(DIM):
v3U[a] += sp.sqrt(detgamma) * gammaUU[a][
d] * LeviCivitaSymbolDDD[d][b][c] * v1U[b] * v2U[c]
# Step 2.g.i: Simplify expressions for v1U,v2U,v3U. This greatly expedites the C code generation (~10x faster)
# Drat. Simplification with certain versions of SymPy & coord systems results in a hang. Let's just
# evaluate the expressions so the most trivial optimizations can be performed.
for a in range(DIM):
v1U[a] = v1U[a].doit() # sp.simplify(v1U[a])
v2U[a] = v2U[a].doit() # sp.simplify(v2U[a])
v3U[a] = v3U[a].doit() # sp.simplify(v3U[a])
# Step 2.h: Define omega_{ij}
omegaDD = ixp.zerorank2()
gammaDD = AB.gammaDD
def v_vectorDU(v1U, v2U, v3U, i, a):
if i == 0:
return v1U[a]
if i == 1:
return v2U[a]
if i == 2:
return v3U[a]
print("ERROR: unknown vector!")
sys.exit(1)
def update_omega(omegaDD, i, j, v1U, v2U, v3U, gammaDD):
omegaDD[i][j] = sp.sympify(0)
for a in range(DIM):
for b in range(DIM):
omegaDD[i][j] += v_vectorDU(v1U, v2U, v3U, i, a) * v_vectorDU(
v1U, v2U, v3U, j, b) * gammaDD[a][b]
# Step 2.i: Define e^a_i. Note that:
# omegaDD[0][0] = \omega_{11} above;
# omegaDD[1][1] = \omega_{22} above, etc.
# First e_1^a: Orthogonalize & normalize:
e1U = ixp.zerorank1()
update_omega(omegaDD, 0, 0, v1U, v2U, v3U, gammaDD)
for a in range(DIM):
e1U[a] = v1U[a] / sp.sqrt(omegaDD[0][0])
# Next e_2^a: First orthogonalize:
e2U = ixp.zerorank1()
update_omega(omegaDD, 0, 1, e1U, v2U, v3U, gammaDD)
for a in range(DIM):
e2U[a] = (v2U[a] - omegaDD[0][1] * e1U[a])
# Then normalize:
update_omega(omegaDD, 1, 1, e1U, e2U, v3U, gammaDD)
for a in range(DIM):
e2U[a] /= sp.sqrt(omegaDD[1][1])
# Next e_3^a: First orthogonalize:
e3U = ixp.zerorank1()
update_omega(omegaDD, 0, 2, e1U, e2U, v3U, gammaDD)
update_omega(omegaDD, 1, 2, e1U, e2U, v3U, gammaDD)
for a in range(DIM):
e3U[a] = (v3U[a] - omegaDD[0][2] * e1U[a] - omegaDD[1][2] * e2U[a])
# Then normalize:
update_omega(omegaDD, 2, 2, e1U, e2U, e3U, gammaDD)
for a in range(DIM):
e3U[a] /= sp.sqrt(omegaDD[2][2])
# Step 2.j: Construct l^mu, n^mu, and m^mu, based on r^mu, theta^mu, phi^mu, and u^mu:
r4U = ixp.zerorank1(DIM=4)
u4U = ixp.zerorank1(DIM=4)
theta4U = ixp.zerorank1(DIM=4)
phi4U = ixp.zerorank1(DIM=4)
for a in range(DIM):
r4U[a + 1] = e2U[a]
theta4U[a + 1] = e3U[a]
phi4U[a + 1] = e1U[a]
# FIXME? assumes alpha=1, beta^i = 0
if par.parval_from_str(thismodule + "::UseCorrectUnitNormal") == "False":
u4U[0] = 1
else:
# Eq. 2.116 in Baumgarte & Shapiro:
# n^mu = {1/alpha, -beta^i/alpha}. Note that n_mu = {alpha,0}, so n^mu n_mu = -1.
import BSSN.BSSN_quantities as Bq
Bq.declare_BSSN_gridfunctions_if_not_declared_already()
Bq.BSSN_basic_tensors()
u4U[0] = 1 / Bq.alpha
for i in range(DIM):
u4U[i + 1] = -Bq.betaU[i] / Bq.alpha
l4U = ixp.zerorank1(DIM=4)
n4U = ixp.zerorank1(DIM=4)
mre4U = ixp.zerorank1(DIM=4)
mim4U = ixp.zerorank1(DIM=4)
# M_SQRT1_2 = 1 / sqrt(2) (defined in math.h on Linux)
M_SQRT1_2 = par.Cparameters("#define", thismodule, "M_SQRT1_2", "")
isqrt2 = M_SQRT1_2 #1/sp.sqrt(2) <- SymPy drops precision to 15 sig. digits in unit tests
for mu in range(4):
l4U[mu] = isqrt2 * (u4U[mu] + r4U[mu])
n4U[mu] = isqrt2 * (u4U[mu] - r4U[mu])
mre4U[mu] = isqrt2 * theta4U[mu]
mim4U[mu] = isqrt2 * phi4U[mu]
def GiRaFFEfood_NRPy_1D_tests_fast_wave():
# We'll use reference_metric.py to define x and y
x = rfm.xxCart[0]
y = rfm.xxCart[1]
global AD
AD = ixp.zerorank1(DIM=3)
import Min_Max_and_Piecewise_Expressions as noif
bound = sp.Rational(1, 10)
# A_x = 0, A_y = 0
# A_z = y+ (-x-0.0075) if x <= -0.1
# (0.75x^2 - 0.85x) if -0.1 < x <= 0.1
# (-0.7x-0.0075) if x > 0.1
Azleft = y - x - sp.Rational(75, 10000)
Azcenter = y + sp.Rational(75, 100) * x * x - sp.Rational(85, 100) * x
Azright = y - sp.Rational(7, 10) * x - sp.Rational(75, 10000)
AD[0] = sp.sympify(0)
AD[1] = sp.sympify(0)
AD[2] = noif.coord_leq_bound(x,-bound)*Azleft\
+noif.coord_greater_bound(x,-bound)*noif.coord_leq_bound(x,bound)*Azcenter\
+noif.coord_greater_bound(x,bound)*Azright
# B^x(0,x) = 1.0
# B^y(0,x) = 1.0 if x <= -0.1
# 1.0-1.5(x+0.1) if -0.1 < x <= 0.1
# 0.7 if x > 0.1
# B^z(0,x) = 0
Byleft = sp.sympify(1)
Bycenter = sp.sympify(1) - sp.Rational(15, 10) * (x + sp.Rational(1, 10))
Byright = sp.Rational(7, 10)
BU = ixp.zerorank1()
BU[0] = sp.sympify(1)
BU[1] = noif.coord_leq_bound(x,-bound)*Byleft\
+noif.coord_greater_bound(x,-bound)*noif.coord_leq_bound(x,bound)*Bycenter\
+noif.coord_greater_bound(x,bound)*Byright
BU[2] = 0
# E^x(0,x) = 0.0 , E^y(x) = 0.0 , E^z(x) = -B^y(0,x)
EU = ixp.zerorank1()
EU[0] = sp.sympify(0)
EU[1] = sp.sympify(0)
EU[2] = -BU[1]
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
B2 = sp.sympify(0)
for i in range(3):
# In flat spacetime, gamma_{ij} is just a Kronecker delta
B2 += BU[i]**2 # This is trivial to extend to curved spacetime
# v^i = [ijk] (E^j B^k) / (B^2)
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2
def GiRaFFEfood_NRPy_1D_tests_three_waves(stagger=False):
# We'll use reference_metric.py to define x and y
x = rfm.xxCart[0]
y = rfm.xxCart[1]
if stagger:
x_p_half = x + sp.Rational(1, 2) * gri.dxx[0]
y_p_half = y + sp.Rational(1, 2) * gri.dxx[1]
# Now, we can define the vector potential. We will create three copies of this variable, because the potential is uniquely defined in three zones. Data for $x \leq -0.1/\gamma_\mu$ shall be referred to as "left", data for $-0.1/\gamma_\mu \leq x \leq 0.1/\gamma_\mu$ as "center", and data for $x \geq 0.1/\gamma_\mu$ as "right".
global AD
AD = ixp.zerorank1(DIM=3)
import Min_Max_and_Piecewise_Expressions as noif
AD[0] = sp.sympify(0)
if stagger:
AD[1] = sp.Rational(7, 2) * x_p_half * noif.coord_greater_bound(
-x_p_half, 0) + sp.sympify(
3) * x_p_half * noif.coord_greater_bound(x_p_half, 0)
AD[2] = y_p_half - sp.Rational(
3, 2) * x_p_half * noif.coord_greater_bound(
-x_p_half, 0) - sp.sympify(
3) * x_p_half * noif.coord_greater_bound(x_p_half, 0)
else:
AD[1] = sp.Rational(7, 2) * x * noif.coord_greater_bound(
-x, 0) + sp.sympify(3) * x * noif.coord_greater_bound(x, 0)
AD[2] = y - sp.Rational(3, 2) * x * noif.coord_greater_bound(
-x, 0) - sp.sympify(3) * x * noif.coord_greater_bound(x, 0)
# ### Set the vectors $B^i$ and $E^i$ for the velocity
#
# Now, we will set the magnetic and electric fields that we will need to define the initial velocities. First, we need to define $$f(x)=1+\sin (5\pi x);$$ note that in the definition of $B^i$, we need $f(x')$ where $x'=\gamma_\mu x$.
# $$\label{step2}$$
# We will now set the magnetic field in the wave frame:
# \begin{align}
# B'^{x'}(x') = &\ 1.0,\ B'^y(x') = 1.0, \\
# B'^z(x') = &\ \left \{ \begin{array}{lll} 1.0 & \mbox{if} & x' \leq -0.1 \\
# 1.0+0.15 f(x') & \mbox{if} & -0.1 \leq x' \leq 0.1 \\
# 1.3 & \mbox{if} & x' \geq 0.1 \end{array} \right. .
# \end{align}
#
B_aU = ixp.zerorank1(DIM=3)
E_aU = ixp.zerorank1(DIM=3)
B_pU = ixp.zerorank1(DIM=3)
E_pU = ixp.zerorank1(DIM=3)
B_mU = ixp.zerorank1(DIM=3)
E_mU = ixp.zerorank1(DIM=3)
# if stagger:
# B_aU[0] = sp.sympify(1)
# B_aU[1] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(1) + noif.coord_greater_bound(x_p_half,0) * sp.Rational(3,2)
# B_aU[2] = sp.sympify(2)
# E_aU[0] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(-1) + noif.coord_greater_bound(x_p_half,0) * sp.Rational(-3,2)
# E_aU[1] = sp.sympify(1)
# E_aU[2] = sp.sympify(0)
# B_pU[0] = sp.sympify(0)
# B_pU[1] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(0) + noif.coord_greater_bound(x_p_half,0) * sp.Rational(3,2)
# B_pU[2] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(0) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(1)
# E_pU[0] = sp.sympify(0)
# E_pU[1] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(0) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(1)
# E_pU[2] = noif.coord_leq_bound(x_p_half,0) * sp.sympify(0) + noif.coord_greater_bound(x_p_half,0) * sp.Rational(-3,2)
# B_mU[0] = sp.sympify(0)
# B_mU[1] = noif.coord_leq_bound(x_p_half,0) * sp.Rational(1,2) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(0)
# B_mU[2] = noif.coord_leq_bound(x_p_half,0) * sp.Rational(3,2) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(0)
# E_mU[0] = sp.sympify(0)
# E_mU[1] = noif.coord_leq_bound(x_p_half,0) * sp.Rational(-3,2) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(0)
# E_mU[2] = noif.coord_leq_bound(x_p_half,0) * sp.Rational(1,2) + noif.coord_greater_bound(x_p_half,0) * sp.sympify(0)
# else:
B_aU[0] = sp.sympify(1)
B_aU[1] = noif.coord_leq_bound(x, 0) * sp.sympify(
1) + noif.coord_greater_bound(x, 0) * sp.Rational(3, 2)
B_aU[2] = sp.sympify(2)
E_aU[0] = noif.coord_leq_bound(x, 0) * sp.sympify(
-1) + noif.coord_greater_bound(x, 0) * sp.Rational(-3, 2)
E_aU[1] = sp.sympify(1)
E_aU[2] = sp.sympify(0)
B_pU[0] = sp.sympify(0)
B_pU[1] = noif.coord_leq_bound(x, 0) * sp.sympify(
0) + noif.coord_greater_bound(x, 0) * sp.Rational(3, 2)
B_pU[2] = noif.coord_leq_bound(
x, 0) * sp.sympify(0) + noif.coord_greater_bound(x, 0) * sp.sympify(1)
E_pU[0] = sp.sympify(0)
E_pU[1] = noif.coord_leq_bound(
x, 0) * sp.sympify(0) + noif.coord_greater_bound(x, 0) * sp.sympify(1)
E_pU[2] = noif.coord_leq_bound(x, 0) * sp.sympify(
0) + noif.coord_greater_bound(x, 0) * sp.Rational(-3, 2)
B_mU[0] = sp.sympify(0)
B_mU[1] = noif.coord_leq_bound(x, 0) * sp.Rational(
1, 2) + noif.coord_greater_bound(x, 0) * sp.sympify(0)
B_mU[2] = noif.coord_leq_bound(x, 0) * sp.Rational(
3, 2) + noif.coord_greater_bound(x, 0) * sp.sympify(0)
E_mU[0] = sp.sympify(0)
E_mU[1] = noif.coord_leq_bound(x, 0) * sp.Rational(
-3, 2) + noif.coord_greater_bound(x, 0) * sp.sympify(0)
E_mU[2] = noif.coord_leq_bound(x, 0) * sp.Rational(
1, 2) + noif.coord_greater_bound(x, 0) * sp.sympify(0)
global BU
BU = ixp.zerorank1(DIM=3)
EU = ixp.zerorank1(DIM=3)
for i in range(3):
BU[i] = B_aU[i] + B_pU[i] + B_mU[i]
EU[i] = E_aU[i] + E_pU[i] + E_mU[i]
# <a id='step3'></a>
# ### Calculate $v^i$
#
# Now, we calculate $$\mathbf{v} = \frac{\mathbf{E} \times \mathbf{B}}{B^2},$$ which is equivalent to $$v^i = [ijk] \frac{E^j B^k}{B^2},$$ where $[ijk]$ is the Levi-Civita symbol and $B^2 = \gamma_{ij} B^i B^j$ is a trivial dot product in flat space.
# $$\label{step3}$$
LeviCivitaSymbolDDD = ixp.LeviCivitaSymbol_dim3_rank3()
B2 = sp.sympify(0)
for i in range(3):
# In flat spacetime, gamma_{ij} is just a Kronecker delta
B2 += BU[i]**2 # This is trivial to extend to curved spacetime
global ValenciavU
ValenciavU = ixp.zerorank1()
for i in range(3):
for j in range(3):
for k in range(3):
ValenciavU[
i] += LeviCivitaSymbolDDD[i][j][k] * EU[j] * BU[k] / B2