Gauge conditions for black hole spacetimes. Miguel Alcubierre ICN-UNAM, Mexico. Desirable properties of gauges for black hole evolutions. Desirable properties of gauges are: Avoid physical and coordinate singularities. Keep coordinate lines from falling down the holes.
To specify a foliation one needs to prescribe a way to calculate the lapse. There are many ways of doing this:
Notice that some of these classes might overlap. For example, harmonic slicing can also be seen as a prescribed densitized lapse.
Foliations of spacetime can go wrong for serveral reasons:
The standard example of an elliptic slicing condition is the “K-freezing” condition:
Which in the particular case when trK=0 reduces to maximal slicing, which is strongly singularity avoiding.
The K-freezing condition results in an elliptic equation for the lapse:
Singularity avoiding with zero shift leads to “grid stretching” (exponential growth of the metric in the region close to the horizon … but a shift can help to reduce this.
The Bona-Masso (BM) family of slicing conditions has the form
Using the evolution equation for Kijwe can easily find that
The lapse function then obeys a wave equation with sources. The wave speed along a fixed spatial direction xi is given by
For f=1, this is equal to the speed of light, but for other choices of f it can be smaller or larger than the speed of light.
A gauge speed larger than that of light introduces no causality problems, since this is just the speed of propagation of the coordinate system.
BM slicing Myth: Having all characteristic speeds equal to the speed of light is a good thing. FALSE! This is a prejudice. Why should this be good? In fact, experience shows that having gauge speeds larger than the speed of light is better (1+log).
In general, with the BM slicing condition one has the following relation between the lapse and the spatial volume elements
Or in integral form
A short calculation shows that the Bona-Masso family of slicing conditions can be written in 4-covariant form as a generalized wave equation on the time function T in the following way
With n the unit normal vector to the spatial hypersurfaces:
If we take f = 1 we see that T obeys the simple wave equation, so T it is a harmonic function. This is why this case is known as harmonic slicing.
We define a focusing singularity as a place where the spatial volume elements vanish at a bounded rate. If the singularity occurs after a proper time s (measured by normal observers), the elapsed coordinate time will be
We will say that the singularity is of order m if the volume elements vanish as
Notice that m must be positive for there to be a singularity at all, and it must be larger than or equal to 1 for the singularity to be approached at a bounded rate.
As the volume elements 1/2 go to zero, the lapse can do one of 3 things:
1) It can remain finite, 2) it can vanish with 1/2, 3) it can vanish before 1/2.
CASE I: One can easily see that case 1 can not happen with the BM slicing conditions as long as f 0. The lapse always collapses as 1/2 goes to zero.
CASE II: If the lapse collapses with 1/2 we can hit the singularity after a finite or infinite coordinate time, depending on how fast the lapse collapses as we approach the singularity. If we reach the singularity in an infinite coordinate time we say that we have “marginal singularity avoidance”.
CASE III: If the lapse collapses before 1/2 then the time slices stop advancing before the singularity is reached (the slices can in fact move back in some cases). In this case we say that we have “strong singularity avoidance”.
Shocks: Discontinuous solutions to non-linear hyperbolic PDE’s that arise from smooth data and are characterized by the crossing of characteristic lines.
The Einstein equations can be written (in some gauges) as a linearly degenerate system, so physical shocks are not expected!
One can have traveling discontinuities called “contact discontinuities”, but these do not arise from smooth data and travel along null lines.
T = 0
T = 100
When using hyperbolic gauge conditions, shocks associated with the propagation of the gauge can arise from smooth initial data.
These “gauge shocks” are a particular form of coordinate singularity where the spatial slices develop a kink.
The no-shock condition (linear degeneracy) for the Bona-Masso family of slicings implies that
This clearly contains harmonic slicing as a particular case (k = 0).
For non-zero k this is not a very good slicing condition since for small it can allow the lapse to become negative. To see this notice that if we use this solution in the Bona-Masso slicing condition we obtain, for small :
However, we can still find useful slicings if we look only for approximate solutions to the gauge avoidance equation.
Assume the lapse is of the form =1+ with << 1, and expand f as
I we want to satisfy the condition for shock avoidance to zero order in we must have
One particular family of solutions that has such an expansion is
The case a0 = 1 corresponds to harmonic slicing.
Another case of special interest is a0 = 2, for which we find: f = 2/.
This specific member of the 1+log family is precisely the one that has been found empirically to be very robust in black hole and Brill wave simulations!
To satisfy the condition for shock avoidance to first order in we must take
One way to achieve this is to use
Again, for a0 = 1 we recover harmonic slicing. If we take a0 = 4/3 we obtain
For small this behaves as a member of the 1+log family. Moreover, it satisfies the shock avoidance condition to higher order that f = 2/ and its gauge speed in the asymptotically flat region is only 1.15 instead of 1.41.
Could this be a more robust slicing condition?
Pros: They are hyperbolic, so well posed. Cons: No idea what they do in real life, nor how to choose parameters. Do they produce gauge shocks? Any other type of singularity?
In BSSN one can consider families of elliptic, parabolic and hyperbolic shift conditions that relate the shift choice to the evolution of the BSSN conformal connection functions.
An elliptic shift condition is obtained by asking for the conformal connection functions to be time-independent:
This “Gamma-freezing” condition is closely related to the “minimal distortion” shift condition (the principal part of the elliptic operator acting on the shift is identical in both cases).
One can obtain parabolic and hyperbolic shift conditions by making the first or second time derivatives of the shift proportional to the elliptic operator contained in the above condition.
Such parabolic or hyperbolic conditions are called “drivers”.
The “hyperbolic Gamma-driver” shift condition has the following form:
where and are positive functions of position and possibly of time.
It is important to add a damping term to reduce the oscillations in the shift (this is not numerical dissipation!).
We have found that by choosing an adequate form for the function and fine-tuning the value of the dissipation coefficient we can almost freeze the evolution of a black hole system at late times!
Why does this work better than minimal distortion? Beats me, it just does (I guess we are lucky). But we really don’t know almost anything about the analytical properties of this shift condition.
When evolving with zero shift, a black hole can’t move in coordinate space (the horizon just grows in place). Moving a black hole requires large shifts, and introduces large artificial dynamics.
Why force a black hole to move if it naturally wants to stay in place?
A better alternative is to use co-moving coordinates …
How to choose a shift vector that gives us co-moving coordinates?
Answer: easy, zero shift does precisely this. So we just need to make sure that the shift goes to zero somewhere inside the black-hole.
When using puncture initial data, a simple way to achieve this is to use the hyperbolic Gamma-driver shift with:
With the time independent conformal factor from the initial data, which is infinite at the punctures contained inside all black holes.