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					 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.
• If possible, minimize changes in metric (co-moving, co-rotating, etc.)
• If possible, follow Killing fields when the exist (at late times).

A way to address these points is to relate the gauge choice to the change in
time of geometric quantities. One can do this in different ways:

• Force the change of some geometric quantities to be zero. This typically
leads to elliptic gauge conditions.
• Make the change of the gauge functions proportional to the change of
some geometric quantities. This leads to parabolic or hyperbolic conditions.
      Specifying a foliation of spacetime
To specify a foliation one needs to prescribe a way to calculate the lapse. There are
many ways of doing this:

• Prescribed lapse (or prescribed densitized lapse): Lapse given as a known
function of xi and t.
                                = 1 (geodesic slicing).
                         = lapse from known exact solution.
• Algebraic lapse: Lapse given as function of geometric variables.
                           = 1/2 (harmonic slicing).
• Elliptic lapse condition: Lapse obtained by solving elliptic equation.
                        2  =  Kij Kij (maximal slicing).
• Time derivative lapse condition: Time derivative of lapse given as function of
geometric variables.
             t  =  2 trK (differential form of harmonic slicing).

Notice that some of these classes might overlap. For example, harmonic slicing can
also be seen as a prescribed densitized lapse.
               How can a foliation of
               spacetime go wrong?
Foliations of spacetime can go wrong for serveral reasons:


• The slices can hit a physical singularity (black holes).


• The slices can hit a coordinate singularity where the spatial
  volume elements vanish (focusing of normal observers).

• The slices can become non-smooth at a point (gauge shocks).

• The slices can remain smooth but stop being spacelike (e.g. they
  can become null at a point).

• Etcetera.
             Elliptic slicing conditions
                                            The standard example of an elliptic slicing
                                            condition is the “K-freezing” condition:

                                                            t trK  0
                                            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:

                                               2   i i trK   K ij Kij

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.
      Maximal slicing pros and cons
Pros:
• Maximal slicing produces nice and smooth lapses, and avoids singularities
   very well. When it can be used, experience shows that it is much more
   accurate and less prone to instabilities than other common choices (1+log).
• It eliminates one degree of freedom (trK), which in BSSN means one
   variable less to evolve, and hence less chance of instabilities!

Cons:
• Maximal slicing is slow to solve. In 3D, and with a good elliptic solver
  (BAM), one typically still spends about 90% of the CPU time solving this
  single equation.
• With excision we have currently no good idea for a boundary condition at
  the excision region.

Excision myth:
• We don’t need singularity avoidance with excision. FALSE, we do!
   We need to avoid coordinate singularities too, not just the physical
   singularity. And coordinate singularities can appear anywhere …
               The Bona-Masso family of
                   slicing conditions
The Bona-Masso (BM) family of slicing conditions has the form
                       d
                            ( t  L )    2 f ( ) K
                       dt
With f() > 0 but otherwise arbitrary.
Things to notice:
• This family was introduced in the context of the BM hyperbolic re-formulation of
the evolution equations, but it can be used with any form of the equations.
• If one prefers to use a densitized lapse of the form Q =   /2, then the BM
slicing condition takes the form
                    d
                       Q  ( t  L ) Q  Q 2  / 2 ( f   ) K
                    dt
• The shift terms in the BM slicing condition guarantee that we will have the same
foliation for any shift. This seems natural but other generalizations are possible.
           Wave equation for the lapse
Using the evolution equation for Kij we can easily find that
        d2
        dt 2
                                        
                2 f  2   3 f K ij K ij  2 f   f  K 2   
The lapse function then obeys a wave equation with sources. The
wave speed along a fixed spatial direction xi is given by

                         vg   f  ii
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).
  Some particular cases of BM slicings
From the ADM equations one can easily show that the evolution equation for
the spatial volume elements is
                            d 1/ 2
                                   1/ 2 K
                            dt
• Consider the case when f = N , with N a constant. Comparing the last
equation with the BM slicing condition, we can find  as a function of 1/2:

                              h( x i )  N / 2
with h an arbitrary time independent function. The case N = 1 (that is f = 1) is
known as “harmonic slicing”.
• Take now f = N/. In that case we find the “1+log” family

                         h( x i )  ln   N / 2 
             General relation between
           lapse and volume elements
In general, with the BM slicing condition one has the following relation
between the lapse and the spatial volume elements

                                              d
                           d ln    1/2
                                          
                                             f ( )
Or in integral form


                                          d 
                   1/ 2
                            F ( x ) exp 
                                    i
                                                    
                                           f ( ) 
    Generalized wave equation for the
    time function: the foliation equation
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

                        1   
                g  1 
                                 n n     T  0
                        f ( ) 
                                      
With n the unit normal vector to the spatial hypersurfaces:

                                      T
                    n  g
                                       t 
                                        
                                               1/ 2



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.
                    Focusing singularities
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
                                            s
                                                 d
                                    t  
                                            0
                                                 
We will say that the singularity is of order m if the volume elements vanish as


                                           s  
                                   1/ 2               m




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.
                      Strong and marginal
                      singularity avoidance
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”.
   Singularity avoidance: Conclusions

The final result can be summarized as follows:


If f() behaves as f = An for small , and we have a singularity of
order m, then
• For n<0 we have strong singularity avoidance.
• For n=0 and mA1 we have marginal singularity avoidance.
• For n>0 , or n=0 with mA<1, we do not have singularity avoidance,
even though the lapse collapses to zero at the singularity.


If we have a singularity of order m=1 , then harmonic slicing (n=0, A=1)
marks the boundary between avoiding and hitting the singularity.
                           Gauge shocks
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.
                Avoiding gauge shocks
The no-shock condition (linear degeneracy) for the Bona-Masso family of
slicings implies that
                            f  1 k / 2
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 :

                             d
                                  k K
                             dt

However, we can still find useful slicings if we look only for approximate
solutions to the gauge avoidance equation.
           Zero order shock avoidance
Assume the lapse is of the form =1+ with  << 1, and expand f as

                     f  a0  a1  a2 2  
I we want to satisfy the condition for shock avoidance to zero order in 
we must have
                          a1  2 1  a0 
One particular family of solutions that has such an expansion is
                        2                    2
                       a0                   a0
            f                  
                a0  2 a0  1 2  a0   2 a0  1
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!
              First order shock avoidance
To satisfy the condition for shock avoidance to first order in  we must take
                           a1  2 1  a0 
                           a2  3a1 / 2  3 1  a0 
One way to achieve this is to use
                                              3
                                            a0
                      f  2
                         a0  2a0 1  a0   1  a0 4  a0  2
                                              3
                                             a0
                        
                          4  3a0    1  a0  4  a0   8
Again, for a0 = 1 we recover harmonic slicing. If we take a0 = 4/3 we obtain

                                f  8 / 3 3   
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?
                        Shift conditions
•   Far less is known about good shift conditions that about slicing conditions.

•   Experience shows that traditional elliptic conditions (minimal distortion) do
    not work very well with BH’s. Why? Your guess is as good as mine.

•   Several hyperbolic shift conditions with many free parameters have been
    put forward recently (Lindblom-Scheel, Bona-Palenzuela).
    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?

•   Good to remember: Well posedness only guarantees good behavior for a
    FINITE time, which in practice can be rather small. We need to know
    more …
                   BSSN shift conditions
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:

         ~           ~                ~ ~         2~                ~        ~        
      t  i   2 A ij  j  2   ijk A jk   ij  j K  6 A ij  j   ij S j 
                                                  3                                   
                       ~ ~                2~             ~                 1~
               j  j  i   j  j  i   i  j  j   jk  j  k  i   ij  j  k  k  0
                                          3                                3
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”.
   Hyperbolic Gamma-Driver Shift
The “hyperbolic Gamma-driver” shift condition has the following form:

                                   ~i
                           t   t  i
                         2
                         t
                             i        n


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.
                 Co-moving shift vector
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:

                                   k/         n


With  the time independent conformal factor from the initial data, which is
infinite at the punctures contained inside all black holes.
              Co-rotating shift vector
Good idea when evolving BH’s in orbital configurations.

• How do we do this?
With a hyperbolic shift condition choose as initial a rigid rotation (with some
guess for the angular velocity), that goes to zero at the punctures:
                      y                        x
            x          ,           y 
                       2
                                                2
 … then just let the evolution equation for the shift take care of the rest.

• What is a good initial guess for the angular velocity?
Use some analytic information, and/or simple trial and error …

• What about the light-cylinder? (rotation speed = c)
Irrelevant for stability, the only relevant condition for stability is the CFL
condition: if the shift is too large far away, just use a smaller time step.
• But … boundary conditions on a cube are messy and probably inconsistent.
It is probably a good idea to use a cylinder as the boundary.

				
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