Dynamical perturbations of black-hole punctures: effects of slicing conditions

Among the most commonly used slicing conditions in numerical relativity is the Bona-Massó condition

where $\alpha$ is the lapse function, $\beta^{i}$ the shift vector, $K$ the mean curvature (i.e. the trace of the extrinsic curvature), and the Bona-Massó function $f(\alpha)$ is a function of the lapse that has yet to be determined (see [1]). The properties of the resulting slices depend, of course, on the choice for $f(\alpha)$; for $f(\alpha)=1$, for example, the slicing condition (1) is equivalent to the lapse condition in harmonic coordinates.

A particularly successful choice for the Bona-Massó function is

especially for simulations of black-hole spacetimes. In the absence of a shift vector, Eq. (1) can then be integrated to yield $\alpha=1+\log(\gamma)$, where $\gamma$ is the determinant of the spatial metric, which lends this slicing condition its name 1+log slicing (see [2, 3, 4, 5, 6, 7] for textbook discussions). Dynamical simulations with 1+log slicing render black holes in a trumpet geometry, which, in the static limit, have been analyzed by a number of different authors [8, 9, 10]. These studies, together with those of similar trumpet geometries (e.g., [11, 12, 13, 14]) have helped explain the remarkable numerical properties of these slicing conditions, especially in the context of black-hole simulations.

Even in the context of vacuum evolution calculations, however, 1+log slicing is known to lead to coordinate shocks in some circumstances (see [15, 16, 17]). Alcubierre [15, 18] therefore suggested an alternative shock-avoiding Bona-Massó slicing condition with

where $\kappa>0$ is a constant. While this condition has indeed been found to avoid some coordinate pathologies that arise in 1+log slicing, it also has some very unusual properties—in particular, it allows the lapse to become negative (see the discussion in [18], as well as Fig. 1 below for an example), which may explain why it has been adopted only rarely (see, e.g., [19]).

Despite the appearance of negative values for the lapse, shock-avoiding slicing has recently been shown to perform very similarly to 1+log slicing in terms of stability and accuracy for a number of test calculations involving black holes, neutron stars, and gravitational collapse (see [20]). One of these tests considered a Schwarzschild black hole initially represented on a slice of constant Schwarzschild time, i.e. in a wormhole geometry. These data are then evolved with the Bona-Massó slicing condition (1), which results in a coordinate transition to a trumpet geometry. In Fig. 1 we reproduce results from this test and show the values of the lapse at the black-hole puncture, i.e. at the center of the isotropic coordinate system.

Evidently, the behavior of the lapse at the black-hole puncture for 1+log versus shock-avoiding slices shows not only quantitative but also qualitative differences. For 1+log slices the lapse appears to decay approximately exponentially after a brief dynamical period, while, for shock-avoiding slices, the lapse appears to perform harmonic oscillations. At early times these oscillations appear to be damped, but at later times the amplitude remains approximately constant. We also observe that the period of the oscillations is larger for a smaller value of the constant $\kappa$ in (3).

We caution that neither the exponential decay nor the harmonic oscillation is exact. We also note that, because of the lack of differentiability at the center of the black hole, numerical error arising from finite-differencing across the black-hole puncture is large and prevents pointwise convergence. Using a completely independent code based on a multi-domain spectral method (see [21]) we found some quantitative differences resulting from the different treatment of the puncture, but the same qualitative behavior as with the finite-difference code: exponential decay, typically associated with a first-order ordinary differential equation, for 1+log slicing, versus harmonic oscillation, pointing to a second-order equation, for shock-avoiding slicing. Since both slicing conditions are imposed by the same equation, the Bona-Massó condition (1), the origin of this qualitatively different behavior is, a priori, not clear at all.

Our goal in this paper is to gain analytical insight into what causes these qualitative differences. We employ a dynamical height-function approach to describe time-dependent coordinate transformations of Schwarzschild black holes, and to explore the behavior of the lapse at the black-hole puncture. We introduce this formalism in Sec. II, and review results for static slices in Sec. III. In Sec. IV we then consider dynamical slices in the limit that they can be considered small perturbations of static slices. At large distances from the black hole, the Bona-Massó condition (1) results in well-known wave equations for the lapse, as expected. At the black-hole puncture, however, the resulting equation depends on whether or not the (unperturbed) mean curvature $K$ vanishes at the puncture. Typically, including for 1+log slicing, $K$ is positive at the puncture, in which case one obtains exponential damping. Shock-avoiding slices, however, form an exception in that $K$ vanishes at the puncture, in which case one obtains harmonic oscillation. We briefly summarize in Sec. V, concluding that the vanishing of $K$ at the black-hole puncture results in the qualitative differences observed.

We start with the Schwarzschild line element in Schwarzschild coordinates,¹¹1We adopt geometrized units with $G=1=c$ unless noted otherwise.

where $R$ is the areal radius, $F=F(R)=1-2M/R$, and $M$ is the black-hole mass.²²2We focus on uncharged Schwarzschild black holes here, but note that our calculation generalizes to charged Reissner-Nordström black holes simply by letting $F=1-2M/R+Q^{2}/R^{2}$, where $Q$ is the black-hole charge. We then transform to a new time coordinate³³3Unlike in [22], where we denoted the Schwarzschild time as $t$ and the new time coordinate as $\bar{t}$, we here adopt the opposite convention in order to reduce notational clutter for dynamical slices. $t$ by introducing a height-function $h(t,R)$ that measures how far the new time slices lift off the old time slices,

(see, e.g., [23, 24, 25, 26], as well as [4] for a textbook treatment). Unlike in many previous applications, we allow the height-function to depend on time in order to study dynamical coordinate transitions. Inserting (5) into the line element (4) results in

where the dot denotes a partial derivative with respect to time and a prime with respect to areal radius $R$. From (II) we can identify the $RR$-component of the spatial metric $\gamma_{ij}$ as

Finally we compute the mean curvature from

where $\nabla_{a}$ is the covariant derivative associated with the spacetime metric, $n^{a}$ the future-oriented normal to the spatial hypersurface, $n^{a}=\alpha^{-1}(1,-\beta^{i})$, and $g$ the determinant of the spacetime metric, $g=-\alpha^{2}\gamma_{RR}R^{4}\sin^{2}\theta$.

While the height-function approach has been adopted to study the Schwarzschild spacetime in many different coordinate systems, we focus here on transformations to trumpet geometries that satisfy the Bona-Massó slicing condition (1).

The construction of static trumpet geometries using a time-independent height function $h=h_{0}(R)$ has been discussed by a number of authors (see, e.g., [11, 8, 12, 9, 10, 13, 14, 22]), and we therefore review only some important results that are relevant for our discussion in the following sections.

For static slices, the Bona-Massó condition (1) is

the expression (7a) for the $RR$-component of the spatial metric remains unchanged, Eq. (7b) for the lapse reduces to⁴⁴4We note a typo in the corresponding Eq. (7) of [14], where the factor $F$, denoted $f_{0}$ there, should be squared in the denominator.

Inserting the above expressions together with (8) into (9) then yields an ordinary differential equation that, for many choices of the Bona-Massó function $f(\alpha)$, can be integrated in closed form. A constant of integration can be determined by imposing regularity across a singular point, making the solution unique. For some choices of $f(\alpha)$, this solution can be expressed as an explicit function $\alpha=\alpha(R)$, but for others the solution can be written in implicit form for $\alpha$ only.

In either case we may find the location $R_{0}$ of the root of the lapse, $\alpha(R_{0})=0$, which must be inside the horizon, i.e. $R_{0}<2M$, for horizon-penetrating slices. Defining

we see from (10a) that $\gamma_{RR}\simeq a_{1}^{-2}(R-R_{0})^{-2}$ close to the root of the lapse. Assuming that $a_{1}$ is positive and finite, we may integrate $ds=\gamma_{RR}^{1/2}dR$ to find that the root is located at an infinite proper distance from all points $R>R_{0}$. We therefore refer to this location as the puncture and note that, in its vicinity, the height function diverges according to

where we have adopted a negative sign in taking a square root (note also that $F(R_{0})<0$ since $R_{0}<2M$). Even for the time-dependent slices in the following sections, we will identify the puncture with a divergence of the metric component $\gamma_{RR}$, which, according to (7a), coincides with a divergence of the height function $h$ at $R<2M$. In terms of an isotropic radius $r$, which is typically employed in numerical simulations, the puncture corresponds to the origin $r=0$. For static slices the divergence of $h_{0}$ automatically coincides with a root of the lapse, but this need not be the case for time-dependent slices (see Sec. IV below). Also note that, for horizon-penetrating slices, $\alpha$ is nonzero and finite on the horizon, where $F=0$, so that (10a) indicates that the height-function necessarily has to diverge there, too.⁵⁵5This divergence could have been avoided by starting with a horizon-penetrating coordinate system in (4), rather than with Schwarzschild coordinates. Finally we note that we can compute the static mean curvature at the puncture from

where we have used (10b) and (11) in the second equality. Evidently, whether or not $K_{0}(R_{0})$ is finite depends on the behavior of $\alpha f(\alpha)$ as $\alpha\rightarrow 0$, which we have not yet evaluated in (13).

After this general discussion, we consider some examples for specific choices of the Bona-Massó function $f(\alpha)$, and summarize the key results for these static slices in Table 1.

By far the most common choice for $f(\alpha)$ is (2), which leads to 1+log slices [1]. For 1+log slicing the integral for the lapse $\alpha$ cannot be solved for $\alpha$ directly, so that the resulting equations are usually solved numerically. In particular, this yields $R_{0}\simeq 1.312M$ for the root of the lapse and $a_{1}\simeq 0.832M^{-1}$ for the derivative of the lapse at the root (see [11, 9, 10]). Finally, we use $\alpha f(\alpha)=2$ in (13) to find $K_{0}(R_{0})=0.301M^{-1}$ for the mean curvature at the puncture.

As a second example we consider the choice

which results in a completely analytical trumpet slicing of the Schwarzschild spacetime (see [13]). In this case the integral for the lapse can be solved explicitly, yielding

We evidently have $R_{0}=M$, from which we compute $a_{1}=M^{-1}$ and, using $\alpha f(\alpha)=1-\alpha$, the mean curvature at the puncture, $K_{0}(R_{0})=M^{-1}$, in agreement with Eq. (17) in [13]. Because of the simplicity of the above expressions, it is also straightforward to insert (15) into (10a), solve for $h_{0}^{\prime}$ and integrate to obtain

where we have arbitrarily chosen a constant of integration so that $h_{0}=0$ at $R=3M$. Note that $h_{0}$ diverges logarithmically both at the puncture $R_{0}=M$ as well as the horizon $R=2M$, as expected from our discussion above. In the vicinity of the puncture at $R=M$ we have $h_{0}^{\prime}\simeq-M/(R-M)$, in agreement with (12). We graph (16) in Fig. 2 together with a hypothetical time-dependent perturbation as considered in Sec. IV.

We next consider the shock-avoiding slicing condition with $f(\alpha)$ given by (3). In this case the lapse function $\alpha$ can again be expressed explicitly in terms of $R$,

where the constant of integration $C$ is given by $C=3^{3}M^{4}/2^{4}$ (see [14]). The puncture is now located at $R_{0}=3M/2$, independently of $\kappa$, and we can determine $a_{1}=2\,(6\kappa/(3\kappa-1))^{1/2}/(3M)$. In a key difference from the other slicing conditions, however, we now observe that $\alpha f(\alpha)=\alpha+\kappa/\alpha$ diverges at the puncture, so that the mean curvature (13) vanishes there, $K_{0}(R_{0})=0$.

We now consider dynamical slices in the limit that they may be considered linear perturbations of the static slices of Sec. III. Inspecting Fig. 2 we note that these perturbations need to be described in different ways in different regimes. Far from the black hole, where the slope of $h$ becomes increasingly small, the perturbed slice can be described in terms of a perturbation $\eta$ of the height function $h_{0}$ itself, so that $h(t,R)=h_{0}(R)+\eta(t,R)$. We will briefly discuss this approach in Sec. IV.1, recovering well-known wave equations for the lapse function $\alpha$. Close to the puncture, however, $h_{0}$ and its derivative diverge, so that changes in $h$ may also diverge. In this region it is more natural to describe the perturbation in terms of a small shift $\rho$ in the radius $R$. As we will show in Sec. IV.2, this approach will yield our main result concerning the dynamical behavior of different slicing conditions at the black-hole puncture.

Motivated by recent numerical experiments with shock-avoiding slicing conditions as alternatives to 1+log slicing, we explore the origins of a qualitative difference observed in these simulations: while, for the latter, the lapse function at the black-hole puncture appears to decay in an approximately exponential fashion, the former leads to approximately harmonic oscillations in the lapse. We apply a dynamical height-function approach to Schwarzschild black holes to describe time-dependent coordinate transitions, impose the Bona-Massó condition (1) with different choices for the function $f(\alpha)$, evaluate the resulting equation at the black-hole puncture, and finally consider small perturbations of a background trumpet solution.

Describing these perturbations in terms of the displacement $\rho$ of the location of the puncture, the Bona-Massó equation becomes a second-order equation for $\rho$ (see 31). The key difference between 1+log slices and shock-avoiding slices then arises from the behavior of the (unperturbed) mean curvature $K(R_{0})$ at the location of the puncture. For 1+log slicing, $K(R_{0})$ takes a nonzero, positive value, in which case $\rho$ itself drops out of the equation to linear order, resulting in a first-order equation for $\partial_{n}\rho$ that governs exponential decay. For shock-avoiding slices, on the other hand, $K(R_{0})$ vanishes, and the right-hand side of (31) ends up being dominated by $\rho$ at linear order. The equation therefore remains a second-order equation for $\rho$, resulting in harmonic oscillation. We further observe that the period of the oscillations depends on the constant $\kappa$ in (3), with larger $\kappa$ resulting in a shorter period.

While a quantitative comparison of our analytical results with the numerical findings of [20] shows some discrepancies as discussed in Sect. IV.2, we believe that these can be explained in terms of nonlinear effects and numerical error resulting from the lack of differentiability at the black-hole puncture. Despite these discrepancies, our findings provide analytical insight into the dynamical behavior of the lapse at the black-hole puncture, point to the origin of qualitative differences between different slicing conditions, and predict the dependence of decay constants and oscillation periods on the given parameters.