BSSN from ADM to evolution - GRTLCollaboration/engrenage GitHub Wiki

Having learned the ADM decomposition, you may think you are now ready to code up the equations and evolve them. However, in their current form they are not suitable for evolving numerically and will lead to unstable simulations in which the constraints quickly diverge.

The equations are recast into a form suitable for numerical evolution by way of several tricks:

  1. The ADM quantities, in particular the intrinsic curvature $\gamma_{ij}$ and extrinsic curvature $K_{ij}$, are decomposed into new variables which result in a more stable numerical evolution.

    For the intrinsic curvature we separate out the metric $\gamma_{ij}$ into a conformal part $\bar \gamma_{ij}$ and a conformal factor $\phi$ as follows:

$$ \gamma_{ij} = e^{4\phi} \bar \gamma_{ij}$$

For the extrinsic curvature we separate out $K_{ij}$ into its trace (also the mean curvature of the slice) $K$ and traceless part $A_{ij}$, which is then further decomposed into a conformal part $\bar A_{ij}$ in the same way as the intrinsic curvature:

$$ K_{ij} = e^{4\phi} (\bar A_{ij} + \frac{1}{3} K \bar \gamma_{ij} )$$

  1. Multiples of the constraints (which are equal to zero, so can always be freely added) are added to the evolution equation for the extrinsic curvature $\partial_t K_{ij}$ in order to make it well posed. Being well-posed means that deviations in the constraints remain bounded in an evolution, which is a minimum for the system to be solvable numerically.

  2. Another change required for well-posedness is to promote a quantity related to the (3D) Christoffel symbol for the conformal part of the intrinsic metric $\bar \gamma_{ij}$ to a new evolution variable $\bar\Gamma^k = \bar \gamma^{ij} \bar \Gamma^k_{ij}$. Since we know the evolution equation for $\bar \gamma_{ij}$, we can fairly easily find (although with quite a bit of maths) the equation for the evolution of $\bar \Gamma^k$ (which is related to its spatial derivative in each direction). On the face of it this seems like extra work for no gain, but replacing mixed second derivatives of the spatial metric with first derivatives of $\bar \Gamma^k$ casts the equations into a hyperbolic form that is stable for evolution.

  3. A dynamical gauge is used in which the lapse $\alpha$ and shift $\beta^i$ evolve in time, rather than being fixed (e.g. as in Schwarzschild metric). This change (especially for the lapse) is essential for stability of the evolution. Again one might naively think that this makes things a lot more complicated and is unnecessary, but in a dynamical spacetime it makes sense that the gauge also evolves, so that our coordinate observers remain at a roughly fixed proper distance from any singularities during the evolution.

In classes, I discuss these issues in turn and try to give some physical intuition for each. My slides are here: ADMtoBSSN.pdf. These issues are discussed in all the main NR texts in further detail, but for the purposes of using the code the main thing is to remember the names of the new conformal and traceless variables and how they relate to the ADM ones. They will be further modified as part of the engrenage code, as described in Technical details specific to Engrenage, so it is important to be familiar with them.

Note that the BSSN formulation of NR is not the only option, but is arguably one of the most common. Alternatives include General Harmonic Coordinates (GHC) and null coordinate methods.