# BSSN Time-Evolution Equations for the Gauge Fields $\alpha$ and $\beta^i$¶

## Authors: Zach Etienne & Terrence Pierre Jacques¶

### Formatting improvements courtesy Brandon Clark¶

Notebook Status: Validated

Validation Notes: All expressions generated in this module have been validated against a trusted code (the original NRPy+/SENR code, which itself was validated against Baumgarte's code).

## Introduction:¶

This tutorial notebook constructs SymPy expressions for the right-hand sides of the time-evolution equations for the gauge fields $\alpha$ (the lapse, governing how much proper time elapses at each point between one timestep in a 3+1 solution to Einstein's equations and the next) and $\beta^i$ (the shift, governing how much proper distance numerical grid points move from one timestep in a 3+1 solution to Einstein's equations and the next).

Though we are completely free to choose gauge conditions (i.e., free to choose the form of the right-hand sides of the gauge time evolution equations), very few have been found robust in the presence of (puncture) black holes. So we focus here only on a few of the most stable choices.

$$\label{toc}$$

This notebook is organized as follows

1. Step 1: Initialize needed Python/NRPy+ modules
2. Step 2: Right-hand side of $\partial_t \alpha$
1. Step 2.a: $1+\log$ lapse
2. Step 2.b: Harmonic slicing
3. Step 2.c: Frozen lapse
4. Step 2.d: Alternative 1+log condition for Static Trumpet initial data
3. Step 3: Right-hand side of $\partial_t \beta^i$: Second-order Gamma-driving shift conditions
1. Step 3.a: Original, non-covariant Gamma-driving shift condition
2. Step 3.b: Brown's suggested covariant Gamma-driving shift condition
1. Step 3.b.i: The right-hand side of the $\partial_t \beta^i$ equation
2. Step 3.b.ii: The right-hand side of the $\partial_t B^i$ equation
3. Step 3.c: Non-advecting Gamma-driving shift condition (used for evolving "Static Trumpet" initial data)
4. Step 4: Rescale right-hand sides of BSSN gauge equations
5. Step 5: Code Validation against BSSN.BSSN_gauge_RHSs NRPy+ module
6. Step 6: Output this notebook to $\LaTeX$-formatted PDF file

$$\label{initializenrpy}$$

Let's start by importing all the needed modules from Python/NRPy+:

In :
# Step 1: Import all needed modules from NRPy+:
import sympy as sp                # SymPy: The Python computer algebra package upon which NRPy+ depends
import NRPy_param_funcs as par    # NRPy+: Parameter interface
import grid as gri                # NRPy+: Functions having to do with numerical grids
import indexedexp as ixp          # NRPy+: Symbolic indexed expression (e.g., tensors, vectors, etc.) support
import reference_metric as rfm    # NRPy+: Reference metric support
import BSSN.BSSN_quantities as Bq # NRPy+: Computes useful BSSN quantities
import BSSN.BSSN_RHSs as Brhs     # NRPy+: Constructs BSSN right-hand-side expressions
import sys                        # Standard Python modules for multiplatform OS-level functions

# Step 1.c: Declare/initialize parameters for this module
thismodule = "BSSN_gauge_RHSs"
par.initialize_param(par.glb_param("char", thismodule, "LapseEvolutionOption", "OnePlusLog"))

# 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: 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.f: Define BSSN scalars & tensors (in terms of rescaled BSSN quantities)
import BSSN.BSSN_quantities as Bq
Bq.BSSN_basic_tensors()
Bq.betaU_derivs()

import BSSN.BSSN_RHSs as Brhs
Brhs.BSSN_RHSs()


# Step 2: Right-hand side of $\partial_t \alpha$ [Back to top]¶

$$\label{lapseconditions}$$

## Step 2.a: $1+\log$ lapse [Back to top]¶

$$\label{onepluslog}$$

The $1=\log$ lapse condition is a member of the Bona-Masso family of lapse choices, which has the desirable property of singularity avoidance. As is common (e.g., see Campanelli et al (2005)), we make the replacement $\partial_t \to \partial_0 = \partial_t - \beta^i \partial_i$ to ensure lapse characteristics advect with the shift. The bracketed term in the $1+\log$ lapse condition below encodes the shift advection term:

\begin{align} \partial_0 \alpha &= -2 \alpha K \\ \implies \partial_t \alpha &= \left[\beta^i \partial_i \alpha\right] - 2 \alpha K \end{align}
In :
# Step 2.a: The 1+log lapse condition:
#   \partial_t \alpha = \beta^i \alpha_{,i} - 2*\alpha*K
# First import expressions from BSSN_quantities
cf    = Bq.cf
trK   = Bq.trK
alpha = Bq.alpha
betaU = Bq.betaU

# Implement the 1+log lapse condition
if par.parval_from_str(thismodule+"::LapseEvolutionOption") == "OnePlusLog":
alpha_rhs = -2*alpha*trK
alpha_dupD = ixp.declarerank1("alpha_dupD")
for i in range(DIM):
alpha_rhs += betaU[i]*alpha_dupD[i]


$$\label{harmonicslicing}$$

As defined on Pg 2 of https://arxiv.org/pdf/gr-qc/9902024.pdf , this is given by

$$\partial_t \alpha = \partial_t e^{6 \phi} = 6 e^{6 \phi} \partial_t \phi$$

If

$$\text{cf} = W = e^{-2 \phi},$$

then

$$6 e^{6 \phi} \partial_t \phi = 6 W^{-3} \partial_t \phi.$$

However, $$\partial_t \phi = -\partial_t \text{cf} / (2 \text{cf})$$

(as described above), so if cf$=W$, then \begin{align} \partial_t \alpha &= 6 e^{6 \phi} \partial_t \phi \\ &= 6 W^{-3} \left(-\frac{\partial_t W}{2 W}\right) \\ &= -3 \text{cf}^{-4} \text{cf}\_\text{rhs} \end{align}

Exercise to students: Implement Harmonic slicing for cf$=\chi$

In :
# Step 2.b: Implement the harmonic slicing lapse condition
if par.parval_from_str(thismodule+"::LapseEvolutionOption") == "HarmonicSlicing":
if par.parval_from_str("BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "W":
alpha_rhs = -3*cf**(-4)*Brhs.cf_rhs
elif par.parval_from_str("BSSN.BSSN_quantities::EvolvedConformalFactor_cf") == "phi":
alpha_rhs = 6*sp.exp(6*cf)*Brhs.cf_rhs
else:
print("Error LapseEvolutionOption==HarmonicSlicing unsupported for EvolvedConformalFactor_cf!=(W or phi)")
sys.exit(1)


$$\label{frozen}$$

This slicing condition is given by $$\partial_t \alpha = 0,$$

which is rarely a stable lapse condition.

In :
# Step 2.c: Frozen lapse
#    \partial_t \alpha = 0
if par.parval_from_str(thismodule+"::LapseEvolutionOption") == "Frozen":
alpha_rhs = sp.sympify(0)


## Step 2.d: Alternative $1+\log$ lapse for Static Trumpet initial data [Back to top]¶

$$\label{statictrumpet_onepluslog}$$

An alternative to the standard 1+log condition to be used with Static Trumpet initial data, the lapse is evolved according to a condition consistent with staticity, given by equation 67 in Ruchlin, Etienne, & Baumgarte (2018)

\begin{align} \partial_0 \alpha &= -\alpha(1 - \alpha) K \\ \implies \partial_t \alpha &= \left[\beta^i \partial_i \alpha\right] -\alpha(1 - \alpha) K \end{align}
In :
# Step 2.d: Alternative 1+log lapse condition:
#   \partial_t \alpha = \beta^i \alpha_{,i} -\alpha*(1 - \alpha)*K

# Implement the alternative 1+log lapse condition
if par.parval_from_str(thismodule+"::LapseEvolutionOption") == "OnePlusLogAlt":
alpha_rhs = -alpha*(1 - alpha)*trK
alpha_dupD = ixp.declarerank1("alpha_dupD")
for i in range(DIM):
alpha_rhs += betaU[i]*alpha_dupD[i]


# Step 3: Right-hand side of $\partial_t \beta^i$: Second-order Gamma-driving shift conditions [Back to top]¶

$$\label{shiftconditions}$$

The motivation behind Gamma-driving shift conditions are well documented in the book Numerical Relativity by Baumgarte & Shapiro.

$$\label{origgammadriving}$$

Option 1: Non-Covariant, Second-Order Shift

We adopt the shifting (i.e., advecting) shift non-covariant, second-order shift condition: \begin{align} \partial_0 \beta^i &= B^{i} \\ \partial_0 B^i &= \frac{3}{4} \partial_{0} \bar{\Lambda}^{i} - \eta B^{i} \\ \implies \partial_t \beta^i &= \left[\beta^j \partial_j \beta^i\right] + B^{i} \\ \partial_t B^i &= \left[\beta^j \partial_j B^i\right] + \frac{3}{4} \partial_{0} \bar{\Lambda}^{i} - \eta B^{i}, \end{align} where $\eta$ is the shift damping parameter, and $\partial_{0} \bar{\Lambda}^{i}$ in the right-hand side of the $\partial_{0} B^{i}$ equation is computed by adding $\beta^j \partial_j \bar{\Lambda}^i$ to the right-hand side expression given for $\partial_t \bar{\Lambda}^i$ in the BSSN time-evolution equations as listed here, so no explicit time dependence occurs in the right-hand sides of the BSSN evolution equations and the Method of Lines can be applied directly.

In :
# Step 3.a: Set \partial_t \beta^i
# First import expressions from BSSN_quantities
BU         = Bq.BU
betU       = Bq.betU
betaU_dupD = Bq.betaU_dupD
# Define needed quantities
beta_rhsU = ixp.zerorank1()
B_rhsU = ixp.zerorank1()
# Step 3.a.i: Compute right-hand side of beta^i
# *  \partial_t \beta^i = \beta^j \beta^i_{,j} + B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]
for j in range(DIM):
beta_rhsU[i] += betaU[j]*betaU_dupD[i][j]
# Compute right-hand side of B^i:
eta = par.Cparameters("REAL", thismodule, ["eta"],2.0)

# Step 3.a.ii: Compute right-hand side of B^i
# *  \partial_t B^i     = \beta^j B^i_{,j} + 3/4 * \partial_0 \Lambda^i - eta B^i
# Step 3.a.iii: Define BU_dupD, in terms of derivative of rescaled variable \bet^i
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD","nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j]*rfm.ReU[i] + betU[i]*rfm.ReUdD[i][j]

# Step 3.a.iv: Compute \partial_0 \bar{\Lambda}^i = (\partial_t - \beta^i \partial_i) \bar{\Lambda}^j
Lambdabar_partial0 = ixp.zerorank1()
for i in range(DIM):
Lambdabar_partial0[i] = Brhs.Lambdabar_rhsU[i]
for i in range(DIM):
for j in range(DIM):
Lambdabar_partial0[j] += -betaU[i]*Brhs.LambdabarU_dupD[j][i]

# Step 3.a.v: Evaluate RHS of B^i:
for i in range(DIM):
B_rhsU[i] += sp.Rational(3,4)*Lambdabar_partial0[i] - eta*BU[i]
for j in range(DIM):
B_rhsU[i] += betaU[j]*BU_dupD[i][j]


$$\label{covgammadriving}$$

### Step 3.b.i: The right-hand side of the $\partial_t \beta^i$ equation [Back to top]¶

$$\label{partial_beta}$$

This is Brown's suggested formulation (Eq. 20b; note that Eq. 20a is the same as our lapse condition, as $\bar{D}_j \alpha = \partial_j \alpha$ for scalar $\alpha$): $$\partial_t \beta^i = \left[\beta^j \bar{D}_j \beta^i\right] + B^{i}$$ Based on the definition of covariant derivative, we have $$\bar{D}_{j} \beta^{i} = \beta^i_{,j} + \bar{\Gamma}^i_{mj} \beta^m,$$ so the above equation becomes \begin{align} \partial_t \beta^i &= \left[\beta^j \left(\beta^i_{,j} + \bar{\Gamma}^i_{mj} \beta^m\right)\right] + B^{i}\\ &= {\underbrace {\textstyle \beta^j \beta^i_{,j}}_{\text{Term 1}}} + {\underbrace {\textstyle \beta^j \bar{\Gamma}^i_{mj} \beta^m}_{\text{Term 2}}} + {\underbrace {\textstyle B^i}_{\text{Term 3}}} \end{align}

In :
# Step 3.b: The right-hand side of the \partial_t \beta^i equation
# Step 3.b Option 2: \partial_t \beta^i = \left[\beta^j \bar{D}_j \beta^i\right] + B^{i}
# First we need GammabarUDD, defined in Bq.gammabar__inverse_and_derivs()
Bq.gammabar__inverse_and_derivs()
GammabarUDD = Bq.GammabarUDD
# Then compute right-hand side:
# Term 1: \beta^j \beta^i_{,j}
for i in range(DIM):
for j in range(DIM):
beta_rhsU[i] += betaU[j]*betaU_dupD[i][j]

# Term 2: \beta^j \bar{\Gamma}^i_{mj} \beta^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
beta_rhsU[i] += betaU[j]*GammabarUDD[i][m][j]*betaU[m]
# Term 3: B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]


### Step 3.b.ii: The right-hand side of the $\partial_t B^i$ equation [Back to top]¶

$$\label{partial_upper_b}$$$$\partial_t B^i = \left[\beta^j \bar{D}_j B^i\right] + \frac{3}{4}\left( \partial_t \bar{\Lambda}^{i} - \beta^j \bar{D}_j \bar{\Lambda}^{i} \right) - \eta B^{i}$$

Based on the definition of covariant derivative, we have for vector $V^i$ $$\bar{D}_{j} V^{i} = V^i_{,j} + \bar{\Gamma}^i_{mj} V^m,$$ so the above equation becomes \begin{align} \partial_t B^i &= \left[\beta^j \left(B^i_{,j} + \bar{\Gamma}^i_{mj} B^m\right)\right] + \frac{3}{4}\left[ \partial_t \bar{\Lambda}^{i} - \beta^j \left(\bar{\Lambda}^i_{,j} + \bar{\Gamma}^i_{mj} \bar{\Lambda}^m\right) \right] - \eta B^{i} \\ &= {\underbrace {\textstyle \beta^j B^i_{,j}}_{\text{Term 1}}} + {\underbrace {\textstyle \beta^j \bar{\Gamma}^i_{mj} B^m}_{\text{Term 2}}} + {\underbrace {\textstyle \frac{3}{4}\partial_t \bar{\Lambda}^{i}}_{\text{Term 3}}} - {\underbrace {\textstyle \frac{3}{4}\beta^j \bar{\Lambda}^i_{,j}}_{\text{Term 4}}} - {\underbrace {\textstyle \frac{3}{4}\beta^j \bar{\Gamma}^i_{mj} \bar{\Lambda}^m}_{\text{Term 5}}} - {\underbrace {\textstyle \eta B^i}_{\text{Term 6}}} \end{align}

In :
if par.parval_from_str(thismodule+"::ShiftEvolutionOption") == "GammaDriving2ndOrder_Covariant":
# Step 3.c: Covariant option:
#  \partial_t B^i = \beta^j \bar{D}_j B^i
#               + \frac{3}{4} ( \partial_t \bar{\Lambda}^{i} - \beta^j \bar{D}_j \bar{\Lambda}^{i} )
#               - \eta B^{i}
#                 = \beta^j B^i_{,j} + \beta^j \bar{\Gamma}^i_{mj} B^m
#               + \frac{3}{4}[ \partial_t \bar{\Lambda}^{i}
#                            - \beta^j (\bar{\Lambda}^i_{,j} + \bar{\Gamma}^i_{mj} \bar{\Lambda}^m)]
#               - \eta B^{i}
# Term 1, part a: First compute B^i_{,j} using upwinded derivative
BU_dupD = ixp.zerorank2()
betU_dupD = ixp.declarerank2("betU_dupD","nosym")
for i in range(DIM):
for j in range(DIM):
BU_dupD[i][j] = betU_dupD[i][j]*rfm.ReU[i] + betU[i]*rfm.ReUdD[i][j]
# Term 1: \beta^j B^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += betaU[j]*BU_dupD[i][j]
# Term 2: \beta^j \bar{\Gamma}^i_{mj} B^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += betaU[j]*GammabarUDD[i][m][j]*BU[m]
# Term 3: \frac{3}{4}\partial_t \bar{\Lambda}^{i}
for i in range(DIM):
B_rhsU[i] += sp.Rational(3,4)*Brhs.Lambdabar_rhsU[i]
# Term 4: -\frac{3}{4}\beta^j \bar{\Lambda}^i_{,j}
for i in range(DIM):
for j in range(DIM):
B_rhsU[i] += -sp.Rational(3,4)*betaU[j]*Brhs.LambdabarU_dupD[i][j]
# Term 5: -\frac{3}{4}\beta^j \bar{\Gamma}^i_{mj} \bar{\Lambda}^m
for i in range(DIM):
for j in range(DIM):
for m in range(DIM):
B_rhsU[i] += -sp.Rational(3,4)*betaU[j]*GammabarUDD[i][m][j]*Bq.LambdabarU[m]
# Term 6: - \eta B^i
# eta is a free parameter; we declare it here:
eta = par.Cparameters("REAL", thismodule, ["eta"],2.0)
for i in range(DIM):
B_rhsU[i] += -eta*BU[i]


$$\label{statictrumpet_nonadvecgammadriving}$$

For the shift vector evolution equation, we desire only that the right-hand sides vanish analytically (although numerical error is expected to result in specious evolution). To this end, we adopt the nonadvecting Gamma-driver condition, given by equations 68a and 68b in Ruchlin, Etienne, & Baumgarte (2018)

\begin{align} \partial_t \beta^i &= B^{i} \\ \partial_t B^i &= \frac{3}{4} \partial_{t} \bar{\Lambda}^{i} - \eta B^{i}, \end{align}
In :
# Step 3.c: Set \partial_t \beta^i

# Step 3.c.i: Compute right-hand side of beta^i
# *  \partial_t \beta^i = B^i
for i in range(DIM):
beta_rhsU[i] += BU[i]

# Compute right-hand side of B^i:
eta = par.Cparameters("REAL", thismodule, ["eta"],2.0)

# Step 3.c.ii: Compute right-hand side of B^i
# *  \partial_t B^i     = 3/4 * \partial_t \Lambda^i - eta B^i
# Step 3.c.iii: Evaluate RHS of B^i:
for i in range(DIM):
B_rhsU[i] += sp.Rational(3,4)*Brhs.Lambdabar_rhsU[i] - eta*BU[i]


$$\label{rescale}$$

Next we rescale the right-hand sides of the BSSN equations so that the evolved variables are $\left\{h_{i j},a_{i j},\text{cf}, K, \lambda^{i}, \alpha, \mathcal{V}^i, \mathcal{B}^i\right\}$

In :
# Step 4: Rescale the BSSN gauge RHS quantities so that the evolved
#         variables may remain smooth across coord singularities
vet_rhsU    = ixp.zerorank1()
bet_rhsU    = ixp.zerorank1()
for i in range(DIM):
vet_rhsU[i]    =   beta_rhsU[i] / rfm.ReU[i]
bet_rhsU[i]    =      B_rhsU[i] / rfm.ReU[i]
#print(str(Abar_rhsDD).replace("**","^").replace("_","").replace("xx","x").replace("sin(x2)","Sin[x2]").replace("sin(2*x2)","Sin[2*x2]").replace("cos(x2)","Cos[x2]").replace("detgbaroverdetghat","detg"))
#print(str(Dbarbetacontraction).replace("**","^").replace("_","").replace("xx","x").replace("sin(x2)","Sin[x2]").replace("detgbaroverdetghat","detg"))
#print(betaU_dD)
#print(str(trK_rhs).replace("xx2","xx3").replace("xx1","xx2").replace("xx0","xx1").replace("**","^").replace("_","").replace("sin(xx2)","Sinx2").replace("xx","x").replace("sin(2*x2)","Sin2x2").replace("cos(x2)","Cosx2").replace("detgbaroverdetghat","detg"))
#print(str(bet_rhsU).replace("xx2","xx3").replace("xx1","xx2").replace("xx0","xx1").replace("**","^").replace("_","").replace("sin(xx2)","Sinx2").replace("xx","x").replace("sin(2*x2)","Sin2x2").replace("cos(x2)","Cosx2").replace("detgbaroverdetghat","detg"))


# Step 5: Code Validation against BSSN.BSSN_gauge_RHSs NRPy+ module [Back to top]¶

$$\label{code_validation}$$

Here, as a code validation check, we verify agreement in the SymPy expressions for the RHSs of the BSSN gauge equations between

1. this tutorial and
2. the NRPy+ BSSN.BSSN_gauge_RHSs module.

By default, we analyze the RHSs in Spherical coordinates and with the covariant Gamma-driving second-order shift condition, though other coordinate systems & gauge conditions may be chosen.

In :
# Step 5: We already have SymPy expressions for BSSN gauge RHS expressions
#         in terms of other SymPy variables. Even if we reset the
#         list of NRPy+ gridfunctions, these *SymPy* expressions for
#         BSSN RHS variables *will remain unaffected*.
#
#         Here, we will use the above-defined BSSN gauge RHS expressions
#         to validate against the same expressions in the
#         BSSN/BSSN_gauge_RHSs.py file, to ensure consistency between
#         this tutorial and the module itself.
#
# Reset the list of gridfunctions, as registering a gridfunction
#   twice will spawn an error.
gri.glb_gridfcs_list = []

# Step 5.a: Call the BSSN_gauge_RHSs() function from within the
#           BSSN/BSSN_gauge_RHSs.py module,
#           which should generate exactly the same expressions as above.
import BSSN.BSSN_gauge_RHSs as Bgrhs
Bgrhs.BSSN_gauge_RHSs()

print("Consistency check between BSSN.BSSN_gauge_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.")

print("alpha_rhs - bssnrhs.alpha_rhs = " + str(alpha_rhs - Bgrhs.alpha_rhs))

for i in range(DIM):
print("vet_rhsU["+str(i)+"] - bssnrhs.vet_rhsU["+str(i)+"] = " + str(vet_rhsU[i] - Bgrhs.vet_rhsU[i]))
print("bet_rhsU["+str(i)+"] - bssnrhs.bet_rhsU["+str(i)+"] = " + str(bet_rhsU[i] - Bgrhs.bet_rhsU[i]))

Consistency check between BSSN.BSSN_gauge_RHSs tutorial and NRPy+ module: ALL SHOULD BE ZERO.
alpha_rhs - bssnrhs.alpha_rhs = 0
vet_rhsU - bssnrhs.vet_rhsU = 0
bet_rhsU - bssnrhs.bet_rhsU = 0
vet_rhsU - bssnrhs.vet_rhsU = 0
bet_rhsU - bssnrhs.bet_rhsU = 0
vet_rhsU - bssnrhs.vet_rhsU = 0
bet_rhsU - bssnrhs.bet_rhsU = 0


# Step 6: Output this notebook to $\LaTeX$-formatted PDF file [Back to top]¶

$$\label{latex_pdf_output}$$

The following code cell converts this Jupyter notebook into a proper, clickable $\LaTeX$-formatted PDF file. After the cell is successfully run, the generated PDF may be found in the root NRPy+ tutorial directory, with filename Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.pdf (Note that clicking on this link may not work; you may need to open the PDF file through another means.)

In :
import cmdline_helper as cmd    # NRPy+: Multi-platform Python command-line interface
cmd.output_Jupyter_notebook_to_LaTeXed_PDF("Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs")

Created Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.tex, and compiled
LaTeX file to PDF file Tutorial-BSSN_time_evolution-BSSN_gauge_RHSs.pdf