Gauge equations, conditions, and methods

set of scripts and methods used inSpEC binary black hole evolutions in order to perform the excision and transition to a black hole evolution. This method relies on locating the apparent horizon at several timesteps. The kinematics of the apparent horizon over these several steps provides enough information to initialize the coordinate mappings necessary to lock the grid boundary onto the rapidly expanding horizon. For our evolutions, we choose 8 steps with a ∆t= 0.25 (except for the H3 model, where we must use ∆t = 0.10 to avoid the code crashing before enough horizon finds occur). Since we do not know precisely when an apparent horizon can first be found, we choose to start looking for an apparent horizon whenρb,max/ρb,max|^t=0= 20. In practice, we could search for an apparent horizon at all points during the simulation, however the apparent horizon finder can be computationally expensive (especially when it does not converge, which will always occur when there is no horizon), and thus we employ the stated density threshold.

Then, a specification for H^a in terms of coordinate and metric variables corresponds to a choice of gauge. This is the basis for the ‘generalized harmonic evolution system’ employed in SpEC[18].

Additionally, Lindblom et al. [17, 19] have developed numerical gauge ‘drivers’ which allow one to specify the gauge source function in a manner which corresponds to gauge choices commonly used in other formulations of general relativity (see Ch.4 of the text by Baumgarte & Shapiro[6] for a discussion of common gauge choices).

7.3.2 Damped harmonic gauge for numerical evolutions

We find that a ‘damped’ harmonic gauge condition is necessary for robust collapse. We use the same prescription as in Szil´agyi, Lindblom & Scheel [31] (hereafter as SLS) for the damped harmonic gauge.

The spatial part of a generalized harmonic gauge can be written as:

∇^c∇^cxⁱ=Hⁱ, (7.18)

where Hⁱ is the spatial part of H^a. The coordinate dynamics are ‘damped’ by choosing Hⁱ such that it represents a damping term for the above equation. In SLS, they choose:

Hⁱ=µ_stⁱ=−µ_Sβⁱ/α, (7.19)

where 1/µS is the time scale of the damping. Equation 7.19 governs the evolution of the spatial coordinates and we refer to it as the damped harmonicshift condition.

For the time component of the generalized harmonic gauge, SLS note the gauge constraint yields the equation (that is, expanding the LHS of Eq. 7.17 in terms of 3+1 metric variables one obtains):

t^aHa=t^a∂alog √g

α

−α⁻¹∂kβ^k. (7.20)

As with the spatial part of the generalized harmonic gauge source function, the time component of the gauge source function may be chosen as a damping term, this time for√g/α:

t^aH_a=−µ_Llog √g

α

, (7.21)

where again,µL is a damping factor. As Eq. 7.21 determines the temporal evolution of the coordi- nates; we refer to it as the damped harmonicslicing condition. SLS find that for highly dynamical spacetimes, that choosing:

µS =µL =µ0

log

√g α

²

, (7.22)

is effective for their binary black hole evolutions. Hereµ₀ is a smooth order-unity function of time

which may be specified as a function of space and/or time to adjust the strength of the damping.

Equations 7.19 and 7.21 correspond to a complete choice of coordinates by fixing the spatial and time components of the gauge source function, H^a, respectively. We will refer to a gauge source function chosen in this manner asH_D^a where the ‘D’ denotes it is adamped and therefore adynamical gauge source function.

7.3.3 Imposing damped harmonic gauge in SpEC simulations

For an arbitrary choice of initial data, the gauge source function, H^a, is fully determined by the coordinates of the initial data and thus has some initial value, which we will refer to asH_I^a. When the initial data is interpolated to the evolution grid and the evolution is started, the gauge source functionH^a must not change discontinuously from its initial value. Therefore, our specification of the gauge (which corresponds to prescribingH^aanalytically) must take into account the initial value of the gauge source functionH_I^a. This is done inSpEC simulations by specifying the gauge in the following way:

H^a(t) =H_I^ae^{−(t/τoff)4}+H_D^a

1−e^−(t/τon)4

, (7.23)

where H_D^a is the dynamical damped harmonic gauge condition defined by Eqs. 7.19-7.22 and τ_off

&τ_on are damping timescales chosen for the physical problem at hand. Note thatτ_off controls the rate at which the initial gauge is ‘rolled off’ to zero, andτoncontrols the rate at which the damped gauge condition is ‘rolled on’.

We may decompose Eq. 7.23 into its time and space components:

Hⁱ(t) =H_Iⁱe^{−(t/τs,off)4}+H_Dⁱ

1−e^{−(t/τs,on)4}

, (7.24)

t^aHa(t) =t^aHIae^{−(t/τt,off)4}+t^aHDa

1−e^{−(t/τt,on)4}

. (7.25)

This allows us to independently specify the ‘roll-on’ and ‘roll-off’ timescales for the damped harmonic shift, and damped harmonicslicing conditions, Eqs. 7.19 and 7.21 respectively.

We have investigated a series of five different gauge conditions in order to investigate the coordi- nate dynamics during gravitational collapse and attempt to determine what condition will be lead to the most robust simulation of black hole formation. The conditions are denoted: ‘froze’ for a frozen gauge whereH^a(t) =H_I^a, ‘harm’ for a pureharmonic gauge (i.eH^a= 0), ‘shift’ for a gauge where only the damped harmonic shift condition Eq. 7.19 is rolled on, ‘slice’ for a gauge where only the damped harmonic slicing condition Eq. 7.21 is rolled on, and ‘full’ for a fully harmonic gauge in which both the damped harmonic shift and slicing conditions are used. These gauges are

Gauge Roll-off timescale Roll-on timescale τs,off τt,off τs,on τt,on

froze ∞ ∞ ∞ ∞

harm 10 10 ∞ ∞

shift 10 10 25 ∞

slice 10 10 ∞ 25

full 10 10 25 25

Table 7.1: List of gauge conditions examined. Timescales are are from Eqs. 7.24 & 7.25 and are specified in code units. A value of ∞ corresponds to setting the exponential term corresponding to the timescale equal to unity. The subscript s stands for the spatial (shift) condition, and the subscript t stands for thetemporal (slicing) condition.

listed in Tab. 7.1 In all cases but the frozen gauge, the initial gauge is rolled off by choosing a value for τ_s,off = τ_t,off = 10.0 (in coordinate time). This results in H_I^a being rolled off to zero within roundoff precision aftert= 30.0.³ For the shift only, slicing only, and fully damped harmonic gauge conditions, we use aτon of 25.0 code units which is about half of the time to black hole formation, so that our damped harmonic gauge condition has fully ‘kicked in’ by the time of collapse and black hole formation.