Equations of General Relativistic Magnetohydrodynamics (GRMHD)¶

Author: Zach Etienne¶

This notebook documents and constructs a number of quantities useful for building symbolic (SymPy) expressions for the equations of general relativistic magnetohydrodynamics (GRMHD), using the same (Valencia-like) formalism as IllinoisGRMHD¶

Notebook Status: Self-Validated; induction equation not yet implemented

Validation Notes: This tutorial notebook has been confirmed to be self-consistent with its corresponding NRPy+ module, as documented below. Additional validation tests may have been performed, but are as yet, undocumented. (TODO)

Introduction¶

We write the equations of general relativistic magnetohydrodynamics in conservative form as follows (Eqs. 41-44 of Duez et al:

where

We represent $T^{\mu\nu}$ as the sum of the stress-energy tensor of a perfect fluid $T^{\mu\nu}_{\rm GRHD}$, plus the stress-energy associated with the electromagnetic fields in the force-free electrodynamics approximation $T^{\mu\nu}_{\rm GRFFE}$ (equivalently, $T^{\mu\nu}_{\rm em}$ in Duez et al):

where

• $T^{\mu\nu}_{\rm GRHD}$ is constructed from rest-mass density $\rho_0$, pressure $P$, internal energy $\epsilon$, 4-velocity $u^\mu$, and ADM metric quantities as described in the NRPy+ GRHD equations tutorial notebook; and
• $T^{\mu\nu}_{\rm GRFFE}$ is constructed from the magnetic field vector $B^i$ and ADM metric quantities as described in the NRPy+ GRFFE equations tutorial notebook.

All quantities can be written in terms of the full GRMHD stress-energy tensor $T^{\mu\nu}$ in precisely the same way they are defined in the GRHD equations. *Therefore, we will not define special functions for generating these quantities, and instead refer the user to the appropriate functions in the GRHD module* Namely,

• The GRMHD conservative variables:
  • $\rho_* = \alpha\sqrt{\gamma} \rho_0 u^0$, via GRHD.compute_rho_star(alpha, sqrtgammaDET, rho_b,u4U)
  • $\tilde{\tau} = \alpha^2\sqrt{\gamma} T^{00} - \rho_*$, via GRHD.compute_tau_tilde(alpha, sqrtgammaDET, T4UU,rho_star)
  • $\tilde{S}_i = \alpha \sqrt{\gamma} T^0{}_i$, via GRHD.compute_S_tildeD(alpha, sqrtgammaDET, T4UD)
• The GRMHD fluxes:
  • $\rho_*$ flux: $\left(\rho_* v^j\right)$, via GRHD.compute_rho_star_fluxU(vU, rho_star)
  • $\tilde{\tau}$ flux: $\left(\alpha^2 \sqrt{\gamma} T^{0j} - \rho_* v^j \right)$, via GRHD.compute_tau_tilde_fluxU(alpha, sqrtgammaDET, vU,T4UU,rho_star)
  • $\tilde{S}_i$ flux: $\left(\alpha \sqrt{\gamma} T^j{}_i \right)$, via GRHD.compute_S_tilde_fluxUD(alpha, sqrtgammaDET, T4UD)
• GRMHD source terms:
  • $\tilde{\tau}$ source term $s$: defined above, via GRHD.compute_s_source_term(KDD,betaU,alpha, sqrtgammaDET,alpha_dD, T4UU)
  • $\tilde{S}_i$ source term: $\frac{1}{2} \alpha\sqrt{\gamma} T^{\mu\nu} g_{\mu\nu,i}$, via GRHD.compute_S_tilde_source_termD(alpha, sqrtgammaDET,g4DD_zerotimederiv_dD, T4UU)

In summary, all terms in the GRMHD equations can be constructed once the full GRMHD stress energy tensor $T^{\mu\nu} = T^{\mu\nu}_{\rm GRHD} + T^{\mu\nu}_{\rm GRFFE}$ is constructed. For completeness, the full set of input variables include:

• Spacetime quantities:
  • ADM quantities $\alpha$, $\beta^i$, $\gamma_{ij}$, $K_{ij}$
• Hydrodynamical quantities:
  • Rest-mass density $\rho_0$
  • Pressure $P$
  • Internal energy $\epsilon$
  • 4-velocity $u^\mu$
• Electrodynamical quantities
  • Magnetic field $B^i= \tilde{B}^i / \gamma$

A Note on Notation¶

As is standard in NRPy+,

• Greek indices refer to four-dimensional quantities where the zeroth component indicates temporal (time) component.
• Latin indices refer to three-dimensional quantities. This is somewhat counterintuitive since Python always indexes its lists starting from 0. As a result, the zeroth component of three-dimensional quantities will necessarily indicate the first spatial direction.

For instance, in calculating the first term of $b^2 u^\mu u^\nu$, we use Greek indices:

T4EMUU = ixp.zerorank2(DIM=4)
for mu in range(4):
    for nu in range(4):
        # Term 1: b^2 u^{\mu} u^{\nu}
        T4EMUU[mu][nu] = smallb2*u4U[mu]*u4U[nu]

When we calculate $\beta_i = \gamma_{ij} \beta^j$, we use Latin indices:

betaD = ixp.zerorank1(DIM=3)
for i in range(3):
    for j in range(3):
        betaD[i] += gammaDD[i][j] * betaU[j]

As a corollary, any expressions involving mixed Greek and Latin indices will need to offset one set of indices by one: A Latin index in a four-vector will be incremented and a Greek index in a three-vector will be decremented (however, the latter case does not occur in this tutorial notebook). This can be seen when we handle $\frac{1}{2} \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu}$:

# \alpha \sqrt{\gamma} T^{\mu \nu}_{\rm EM} \partial_i g_{\mu \nu} / 2
for i in range(3):
    for mu in range(4):
        for nu in range(4):
            S_tilde_rhsD[i] += alpsqrtgam * T4EMUU[mu][nu] * g4DD_zerotimederiv_dD[mu][nu][i+1] / 2

Table of Contents¶

Each family of quantities is constructed within a given function (boldfaced below). This notebook is organized as follows

1. Step 1: Import needed NRPy+ & Python modules
2. Step 2: Define the GRMHD stress-energy tensor $T^{\mu\nu}$ and $T^\mu{}_\nu$:
   • compute_T4UU(), compute_T4UD():
3. Step 3: Construct $T^{\mu\nu}$ from GRHD & GRFFE modules with ADM and GRMHD input variables, and construct GRMHD equations from the full GRMHD stress-energy tensor.
4. Step 4: Code Validation against GRMHD.equations NRPy+ module
5. Step 5: Output this notebook to $\LaTeX$-formatted PDF file

Step 1: Import needed NRPy+ & Python modules [Back to top]¶

Step 2: Define the GRMHD stress-energy tensor $T^{\mu\nu}$ and $T^\mu{}_\nu$ [Back to top]¶

Recall from above that

where

• $T^{\mu\nu}_{\rm GRHD}$ is constructed from the GRHD.compute_T4UU(gammaDD,betaU,alpha, rho_b,P,epsilon,u4U) GRHD (tutorial) function, and
• $T^{\mu\nu}_{\rm GRFFE}$ is constructed from the GRFFE.compute_TEM4UU(gammaDD,betaU,alpha, smallb4U, smallbsquared,u4U) GRFFE (tutorial) function

Since a lowering operation on a sum of tensors is equivalent to the lowering operation applied to the individual tensors in the sum,

where

• $T^\mu{}_{\nu}{}^{\rm GRHD}$ is constructed from the GRHD.compute_T4UD(gammaDD,betaU,alpha, T4UU) GRHD (tutorial) function, and
• $T^{\mu\nu}_{\rm GRFFE}$ is constructed from the GRFFE.compute_TEM4UD(gammaDD,betaU,alpha, TEM4UU) GRFFE (tutorial) function.

Step 3: Declare ADM and hydrodynamical input variables, and construct all terms in GRMHD equations [Back to top]¶

Step 4: Code Validation against GRMHD.equations NRPy+ module [Back to top]¶

As a code validation check, we verify agreement in the SymPy expressions for the GRHD equations generated in

1. this tutorial versus
2. the NRPy+ GRMHD.equations module.

Step 5: Output this notebook to $\LaTeX$-formatted PDF file [Back to top]¶

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-GRMHD_Equations-Cartesian.pdf (Note that clicking on this link may not work; you may need to open the PDF file through another means.)