Description

Gauss-Coddazzi-Mainardi system and frame equations

To bypass the corrugation process from primitive embedding, an alternative is to solve both Gauss-Coddazzi-Mainardi (GCM) and frame equations. Indeed, the local existence of an $mathcal{C}^3$-isometric embedding is ensured by the existence of a local solution of GCM equations (cite{han2006isometric}), as well as the frame equations (see cite{palais} and cite{spivak1975comprehensive}). In the following, we will present a numerical resolution of GCM equations which can be transformed in a quasi-linear partial derivative system. We subsequently solve the frame equations (Frobenius partial derivative type). Boundary conditions for the resolution of the GCM equations are deduced from symmetric considerations. The latter are also used to deduce the initial conditions of the frame equations.

Firstly, the program solve Gauss-Mainardi-Codazzi (CGM) systeme of semi-linear partial derivative for poloidal submanifold. To be solve the equation will be putted in the following form :

\[d_l U + A(r,l,U) d_s U = S(r,l,U)\]

where U two dimensionnal vector defined on \([0,s_m]\times[0,\pi/2]\). This equation is solved using finite second order element method ; (details in paper subsection called : Rewrite the Gauss-Mainardi-Codazzi system).

The initial condition :
  • for \(cin = 1\) used \(u(s,l=0)=0\) and \(v(s,l=0)=1\) ;
  • for \(cin = 2\) modify line 432 of the FunctionElem.f90 file to test other types of initial conditions.

The error will be estimated by numerically calculating the standard 2 of the right-hand side member of (55).

OUTPUT : Second form :
  • two dimensionnal file and the associated interpolation columns for the quantities
    associated with second fondamental form u,v,L,M,N ;
  • two dimensionnal file and the associated interpolation columns for the quantities \(e_{ij}\), \(||e||_2\)
    associated to the numerical error when solving the CGM system.

Gauss-Coddazzi-Mainardi system numerical resolution

Quasi-linear form of GCM equations

Since an analytical solution is difficult to provide, we decided to solve GCM and frame equations with numerical methods. On each step, the numerical errors is estimate to reach the convergence. The dedicated developed program called textit{Isopol}footnote{url{https://perso.imcce.fr/frederic-dauvergne/isopol/index.html}} is provided, with documentation, as an open source license.

\[\partial_\lambda\boldsymbol{U}+\boldsymbol{A}\left(s,\lambda,\boldsymbol{U}\right)\partial_s \boldsymbol{U}=\boldsymbol{S}\left(s,\lambda,\boldsymbol{U}\right)\,,\]

where \(\boldsymbol{U}\) is a two dimensionnal vector linked to the second fundamental form and defined \([0,s_m]\times[0,\pi/2]\). This equation is solved using a second order finite difference method (One can see in the paper for more details).

Using the following initial condition :
  • for \(cin = 1\) used \(u(s,l=0)=0\) and \(v(s,l=0)=1\) ;
  • for \(cin = 2\) modify line 432 of the FunctionElem.f90 file to test other types of initial conditions.

The main input parameters are, begin{itemize} item the $n_s$ number, item $r_{rm max}$ value (associated with a value of $s_{rm max}$), item the spin of the black hole $a$, item a reference value $(Delta lambda/Delta s)_{rm ref}$ for the ratio between $Delta lambda$ and $Delta s$ in the grid. end{itemize} In the solution, we choose $n_s=3000$, $s_{rm max}=20$ and $(Delta lambda/Delta s)_{rm ref} = 0.01$. This choice is a good compromise between numerical stability and computational time.

numerically solved

The error will be estimated by numerically calculating the standard 2 of the right-hand side member of (55).
The main input parameters are,
  • the $n_s$ number,
  • $r_{rm max}$ value (associated with a value of $s_{rm max}$),
  • the spin of the black hole $a$,
  • a reference value $(Delta lambda/Delta s)_{rm ref}$ for the ratio between $Delta lambda$ and $Delta s$ in the grid.

In the solution, we choose $n_s=3000$, $s_{rm max}=20$ and $(Delta lambda/Delta s)_{rm ref} = 0.01$. This choice is a good compromise between numerical stability and computational time.

Boundary conditions for GCM system

We are seeking for a $mathcal{C}^3$ isometric embedding $(s,lambda)mapsto f inR^3$ which respects a particular symmetry. Indeed, the transformation $lambdaleftrightarrow -lambda$ corresponds to an orthogonal symmetry with respect to a plane. The coordinate system of $R^3$ will be fixed so that the symmetry plane is $0_{xz}$. Then the matrix of this symmetry is,

Consequently it implies,

\[\begin{split}\label{Eq-BoundaryCondition-1}\left\{ \begin{array}{cc} M(s,\lambda=0)&=0 \\ \partial_\lambda L(s,\lambda=0)&=0 \\ \partial_\lambda N(s,\lambda=0)&=0 \end{array}\right.\,.\end{split}\]

GCM equations resolution

Firstly, the program solves the Eqs.(ref{Eq-uvGCM-SystemQuasiLinear-General2}) with the source term and the matrix explicitly given in Eq.(ref{eq-matrix-GCM-reduced}) and Eq.(ref{eq-source-GCM-reduced}). The integration along $lambda$ is numerically done using a second order finite difference method. We use a regular grid of $n_s$ points for $sin[0,s_{rm max}]$ and $n_lambda$ points for $lambdain[0,pi/2]$. The value of $n_lambda$ is evaluated from a reference parameter $(Deltalambda/Delta s)_{ref}$. We ensure that $n_lambda$ is an integer by setting $n_lambda=displaystyleleftlfloorfrac{pi}{2}left(frac{Delta lambda}{Delta s}right)_{ref}frac{s_{rm max}}{n_s}rightrfloor$.

Frame equation solver

Once the interpolations of the L, M and N solutions of the CGM equations have been obtained, we are able to integrate, using a Runge-Kutta 4 method, the equations governing the evolution of the local natural reference frames and of the surface itself. Indeed these equations are of the Frobenius type. Thus the code produces an embedding encoded by 3 tables X(nl,ns),Y(nl,ns),Z(nl,ns) that can be interpolated with L(nl) and S(ns).

The error obtained on the induced metric is estimated by calculating the induced metric calculated from a numerical derivation of the X,Y,Z tables.

OUTPUT : Frame and Bonnet :
  • 3 two dimensionnal file and the associated interpolation columns for the quantities X,Y,Z wich is the seeked embedding ;
  • 3 two dimensionnal file and the associated interpolation columns for the quantities \((f*-g)_{ij}\) wich is the error between the induced metrics of our embedding and the target metrics.

Frame equations resolution

Once solved the GCM system Eq.(ref{Eq-CM-1}), we used the pseudo boundary conditions to get the frame $boldsymbol{mathcal{R}}$ and the surface $f$.

Then the Frobenius-type equations are transformed into an ordinary differential equation by setting the value of $s$. Indeed, the frame equations could be written in this form,

\[\begin{split}\frac{\rm d}{{\rm d}\lambda}\left|\begin{array}{c} \boldsymbol{f}\\ \boldsymbol{\partial}_s\\ \boldsymbol{\partial}_\lambda \end{array}\right. =\left|\begin{array}{c} \boldsymbol{\partial}_\lambda\\ \Gamma_{s\lambda}^s \boldsymbol{\partial}_s+\Gamma_{s\lambda}^\lambda \boldsymbol{\partial}_\lambda +M \frac{\boldsymbol{\partial}_s\times\boldsymbol{\partial}_\lambda}{||\boldsymbol{\partial}_s\times\boldsymbol{\partial}_\lambda||} \\ \Gamma_{\lambda\lambda}^s \boldsymbol{\partial}_s+\Gamma_{\lambda\lambda}^\lambda \boldsymbol{\partial}_\lambda +N \frac{\boldsymbol{\partial}_s\times\boldsymbol{\partial}_\lambda}{||\boldsymbol{\partial}_s\times\boldsymbol{\partial}_\lambda||} \end{array}\right. \,.\end{split}\]

Frame initial conditions

For each abscissa $s$, on the grid, the integration will be done along the $lambda$ direction. The resolution of the differential equations is solved by a 4th order Runge-Kutta with initial conditions deduced from the pseudo-boundary conditions,

\[\label{Eq-InitialConditionFrame} \forall s\geq0, \quad \left\{ \eqalign{\boldsymbol{f}(s,0)&=\int_0^s \boldsymbol{u}(\psi(s)){\rm d}s \cr \boldsymbol{\partial}_s(s,\lambda=0)&=\boldsymbol{u}(\psi(s))\cr \boldsymbol{\partial}_\lambda(s,\lambda=0)&=r(s)\boldsymbol{e}_y \cr }\right. \, .\]

Once this integration is done, the isometric default of the induced metric could be estimated by a discrete derivative on ${bf f}$ in order to get the value of $(boldsymbol{partial}_s,boldsymbol{partial}_lambda)$, and then we get induce metrics and the corresponding isometric default.