New Periodic and Boundary Test Results for the Shifted Gauge Wave with Amp=0.5 with the ABIGEL Code

The ABIGEL numerical code implements the reduced harmonic Einstein equations in the flux conservative form. The principal part is chosen to be:

$$\partial_\alpha (g^{\alpha \beta} \partial_\beta \gamma_{\mu \nu})$$ (1)

The important effect is the suppression of vacuum excitation solution:

$$ds^2=e^{\lambda t}(-dt^2+dx^2)+dy^2+dz^2 , ~\hat \Gamma^\mu =0$$ (2)

This metric is a pure gauge instability of Minkowski spacetime which appears in the gauge wave test and obeys the harmonic constraints.

We are now concerned with another type of instability of Minkowski spacetime, that is constraint violating, arising in the Kerr-Schild metric:

$$g^{\mu \nu}=\eta^{\mu \nu}-(e^{\lambda t-1})k^\mu k^\nu, ~ k^\alpha \partial_\alpha=\partial_t+\partial_x$$ (3)

This instability is excited in the shifted version of the gauge wave test. The shifted gauge wave it is affected by constraint violating exponentially growing modes.

Our strategy to suppress it is to modify the non-principal terms by constraint adjustment. The harmonic constraints are:

$$C^\mu:=\Gamma^\mu - \hat \Gamma^\mu=0, ~ \Gamma^\mu=g^{\alph... ...a}= -{\frac{1}{\sqrt{-g}}} \partial_\alpha \gamma^{\alpha \mu}$$ (4)

Remarkably, for the special mode that affects the shifted gauge wave, the adjustment

$$A^{\mu\nu}= -\frac{bg^{tt}}{2C^t} C^{\mu} C^{\nu}, \, b>0 .$$ (5)

is effective by itself, for both the periodic and boundary runs, without adding anything.