Gravitational collapse of K-essence Matter in Painlevé-Gullstrand coordinates
C. Danielle Leonard, Jonathan Ziprick, Gabor Kunstatter, Robert B. Mann
GGravitational collapse of K-essence Matter in Painlev´e-Gullstrand coordinates
C. Danielle Leonard,
1, 2
Jonathan Ziprick,
3, 1
Gabor Kunstatter, and Robert B. Mann
1, 3 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Department of Physics and Physical Oceanography,Memorial University of Newfoundland, St. John’s, Newfoundland, A1B 3X7, Canada Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5, Canada Department of Physics and Winnipeg Institute of Theoretical Physics,University of Winnipeg, Winnipeg, Manitoba, R3B 2E9, Canada (Dated: October 31, 2018)We conduct numerical simulations in Painlev´e-Gullstrand coordinates of a variety of K-essence-type scalar fields under spherically symmetric gravitational collapse. We write down generic con-ditions on the K-essence lagrangian that can be used to determine whether superluminality andCauchy breakdown are possible. Consistent with these conditions, for specific choices of K-essence-type fields we verify the presence of superluminality during collapse, while for other type we do not.We also demonstrate that certain choices of K-essence scalar fields present issues under gravitationalcollapse in Painlev´e-Gullstrand coordinates, such as a breakdown of the Cauchy problem.
I. INTRODUCTION
In recent years, there have been a number of models put forward to explain the accelerating expansion of theuniverse. Falling under the general rubric of dark energy, such models have included quintessence [1], Elko spinors[2], H-essence [3], quintom matter [4], phantom matter [5] and more. However, these models do not explain why theexpansion of the universe is accelerating while humans are here to observe it (the cosmic coincidence problem). Modelsthat avoid anthropic rationales for explaining cosmic coincidence generally require the fine-tuning of parameters. Itwas with the intention of avoiding both fine-tuning and anthropic arguments that K-essence [6, 7] was first proposed.K-essence models invoke dark energy in the form of a scalar field with non-linear kinetic energy. These modelsare constructed so that the scalar field develops a negative pressure once the matter dominated era begins, and sothe associated dynamical behaviour is then deemed responsible for the accelerating cosmic expansion of our universe.K-essence initially came under criticism [8] because it was shown that any such model that solved the coincidenceproblem also necessarily involved perturbations propagating faster than light. However, the proposers of K-essencedefended their model [9] by demonstrating that despite superluminality, causality was preserved for physical situations.It is due to this intrinsic superluminality that the idea of gravitational collapse of K-essence matter has becomerecently popularized. A number of K-essence-type models, characterized by non-linear kinetic energy, have emerged.These models serve the primary purpose not of solving the coincidence problem, but of exploring the idea of super-luminal information escaping from a black hole. It was shown initially in [10] that for the stationary black hole case,signals could escape from within the light horizon as superluminal perturbations. More recently, a dynamic collapsescenario was considered and numerically simulated in maximal slicing coordinates [11]. This study found evidence ofsuperluminality in certain Lagrangian choices for K-essence. Other choices of Lagrangian, however, showed no suchevidence. This study also demonstrated that in some choices of Lagrangian we are limited by the eventual breakdownof the Cauchy problem.Because the well-posedness of the Cauchy problem is dependent on the time-coordinate remaining globally valid, theauthors of [11] were unable to determine whether the observed breakdown was a result of the choice of Lagrangian orof coordinate systems. To this end, we now examine a selection of choices of Lagrangians in Painlev´e-Gullstrand (P-G)coordinates, which are regular across future luminal horizons. We provide general analytic expressions for determiningwhether superliminal propagation and/or Cauchy breakdown occur for specific classes of K-essence fields. BecauseP-G coordinates offer an indisputably distinct time coordinate from maximal slicing coordinates, simulations in P-Gshould illuminate the coordinate dependence of Cauchy breakdown or lack thereof. Using P-G coordinates is also astrong check to ensure the coordinate independence of the results of the maximal slicing simulations in general.Through a series of numerical simulations in P-G coordinates of three models of K-essence, we demonstrate thatthe gravitational collapse of K-essence can produce either superluminal or subluminal sonic horizons. Hence we verifythe qualitative results of [11] and therefore provide evidence for the coordinate independence of these results. Wefurther examine the breakdown (or lack thereof) of the Cauchy problem for these three models. We find that onemodel experiences a breakdown of this problem in many parameter choices, and that the other two models do notexperience Cauchy breakdown. This corroborates the results found in [11]. Therefore, our results are indicative thatthe issue of Cauchy breakdown for these models of K-essence is coordinate independent. a r X i v : . [ g r- q c ] J un II. METHODS
We start with a general action for K-essence coupled to gravity, given by I = (cid:90) √ g d x (cid:18) (4) R πG + L ( X ) (cid:19) (1)where (4) R is the spacetime scalar curvature, G is Newton’s gravitational constant in four dimensions, and X = − ∇ a ψ ∇ a ψ where ψ is the K-essence scalar field.If we vary the action with respect to the metric, we obtain the Einstein field equation G ab = κT ab (2)where we have denoted denoted the Einstein tensor as G ab , and we note that for K-essence we have T ab = L X ∇ a ψ ∇ b ψ + L g ab (3)Above and throughout this paper we use L X to mean d L /dX and L XX to mean d L /dX . We vary the action withrespect to ψ , the K-essence scalar field, and obtain equations of motion for K-essence:˜ g ab ∇ a ∇ b ψ = 0 (4)Note that in the above we have used ˜ g ab , which is the effective (inverse) metric governing propagations of perturbationsof the K-essence field, given by:˜ g ab = L X g ab − L XX ∇ a ψ ∇ b ψ = ⇒ ˜ g ab = 1 L X g ab + c s L XX L X ∇ a ψ ∇ b ψ (5)where c s = L X L X + 2 X L XX is the speed of sound.The luminal apparent horizon forms when surfaces of constant r become null, implying g ab ∇ a r ∇ b r = 0 = ⇒ g rr = 0 (6)The right hand side assumes only a coordinate basis with r as the spatial coordinate.We are now able to examine general conditions for superluminal propagation and Cauchy breakdown. The conditionfor the formation of the sonic horizon requires that the same general condition as (6) be satisfied for the effectivemetric of K-essence, that is ˜ g ab ∇ a r ∇ b r = 0 (7)which, in analogy to the luminal case leads to: ˜ g rr = 0 (8)A calculation reveals that in general˜ g rr = L X g rr − L XX ( ∇ r ψ ) = g rr ( L X + 2 X L XX ) − L XX r ( − gl ) ( ˙ ψ ) = L X c s g rr − L XX r ( − gl ) ( ˙ ψ ) = 0 (9)where in the last line g is the determinant of the 2-metric as given in equation (24), and l is an arbitrary length scale.Note that g is negative, so − g is positive. Eq.(9) shows that the relative sign difference between the sonic horizoncondition and the luminal horizon condition is determined by the signs of L X and L XX . Superluminal escape requiresthe sign of ˜ g rr to be positive just inside a black hole horizon, where g rr is negative. We assume that L X is positive,which is necessary for the kinetic term in the stress energy tensor to be positive (cf. Eq.(3)). Thus, for the right handside of equation (9) to be positive, it is necessary (but not sufficient) that L XX <
0. When this condition is satisfied,we can generically say that superluminal escape is possible. This will be verified in the numerical studies below.Another key aspect of our method is the examination of the potential breakdown of the Cauchy problem. Abreakdown of the Cauchy problem occurs when the surfaces of constant time (5) become null with respect to theeffective metric. We can write this as ˜ g ab ∇ a t ∇ b t = 0 = ⇒ ˜ g tt = 0 (10)which using (39), may be suggestively rewritten as˜ g tt = L X g tt − L XX (cid:0) ∇ t ψ (cid:1) = L X c s g tt − L XX r ( − gl ) ( ψ (cid:48) ) (11)with g and l defined as in equation (9). In simulating equation (2), we require that surfaces of constant time bespacelike; in other words we require that g tt <
0. However, for K-essence we must then require that in addition tothis, the surfaces of constant time of the effective metric are also spacelike. That is, ˜ g tt <
0. Clearly, for certainLagrangian choices, ˜ g tt can go to zero independently of g tt . When surfaces of constant time are no longer spacelikewith respect to either the background or effective metric, we say that the Cauchy problem ceases to be well posedon this surface, and our simulations must halt. Eq.(11) and Eq.(9) show that the condition for Cauchy breakdownin a theory is the same as those for superluminal propagation, though the onset of when each occurs depends on therelative magnitude of | g rr /g tt | .Of course the above assumes that both L X and L XX are everywhere regular. It is also possible for pathologies tooccur if either becomes singular. This too will be illustrated in one of the examples to follow.In order to be able to distinguish physical properties of the K-essence field from a breakdown of the coordinates itis important to use spatial slicings that are regular at luminal horizons. To this end we work in Painlev´e-Gullstrandcoordinates. Our simulations are a generalization to the K-essence case of the method developed in [12]. In Painlev´e-Gullstrand coordinates the metric take the form: ds = − σ dt + (cid:32) dr + (cid:114) G M r σdt (cid:33) + r d Ω (12)where in the dynamical case both the lapse function σ and the Misner-Sharpe mass function M are functions of both r and t . Note that spatial slices ( dt = 0) are flat so that P-G coordinates are singular only at the origin, r = 0 aslong as σ and M are positive. This means that we are able to continue to evolve the simulation inside any horizonsthat form (as opposed to the situation in Schwarzschild coordinates, for example). Another reason for our choice ofcoordinates is simply to avoid issues associated with the usual choice of polar-radial coordinates. This choice would beinappropriate in this setting, because such coordinates are only valid until the first trapped surface forms. In this case,since we have superluminality potentially present, K-essence is not trapped by normal trapped surfaces, but ratherby the trapped surfaces of the effective metric. By working in P-G coordinates we instead can follow the evolutionof the system much further, providing a different qualitative perspective from that of maximal slicing coordinates, asnoted above.Since we are considering spherically symmetric collapse, the number of dynamical degrees of freedom is reduced tothat of ψ and its conjugate momentum Π ψ . These quantities obey the equations (39,40) given in appendix A, whichcontains more detailed description of our dynamical variables and specific numerical methods. For now, it will sufficeto note our initial conditions on ψ and Π ψ , which we take to be ψ ( t = 0) = Ar exp (cid:32) − ( r − r ) B (cid:33) (13)as illustrated in fig 1 and Π ψ ( t = 0) = 0 (14)for Π ψ . For all of our simulations, we chose B = 0 . r = 1 . III. RESULTS
Before proceeding to the results for each specific choice of Lagrangian, we review the behaviour of a sphericallysymmetric gravitational collapse scenario for ‘normal matter’ (where L ( X ) = X ) and note the conditions for the FIG. 1: ψ vs r at t = 0 formation of light apparent horizons and sonic horizons (horizons trapping the perturbations of the effective metric)in P-G coordinates.Let us recall that for initial data such as that we have chosen, there are a number of different possibilities for howthe collapse of normal matter proceeds. If the amplitude of the scalar field, A, as given by equation (13) is sufficientlysmall, subcritical collapse occurs. This means that no horizons are formed, and the ingoing pulse becomes, at theorigin, an outgoing pulse and the field simply ‘passes through’ itself. However, if A exceeds some critical value, thecollapse becomes supercritical. Luminal apparent horizons form, from inside which nothing can escape.K-essence matter is governed by the effective metric ˜ g , and as shown above perturbations in this matter arepermitted to exceed the speed of light. In P-G coordinates, the two distinct conditions for the formation of theluminal and sonic horizons, are respectively r − G M = 0 (15)and (cid:18) − G M r (cid:19) ( L X + 2 X L XX ) − L XX (cid:32) Π ψ πr L X + (cid:114) G M r ψ (cid:48) (cid:33) = 0 (16)Similarly we can find a general condition for the breakdown of the Cauchy problem in P-G coordinates. By notingthat this occurs when surfaces of constant t are no longer timelike (as stated above), we obtain the general condition L X + (cid:18) Π ψ πr L X (cid:19) L XX = 0 (17)Eqs.(15,16,17) are used in the numerical code to locate luminal horizons, sonic horizons and Cauchy breakdown,respectively. A. Born-Infeld Lagrangians
We examine two choices of Born-Infeld Lagrangians. These are based on the Lagrangian given in [10], whichwas proposed to demonstrate information escaping from inside a stationary luminal apparent horizon. As such theyrepresent interesting models for studying gravitational collapse since it is not immediately clear what effect thesuperluminal character of collapsing K-essence matter will have on this process.The first of this type of Lagrangian is given by L ( X ) = α [ (cid:114) Xα − − Λ (18)where α is the K-essence parameter. As α goes to ∞ , we recover L ( X ) = X , the standard Lagrangian for a masslessscalar field (‘normal’ matter). We choose Λ = 0 for our simulations. Note that for gravitational collapse it is possible FIG. 2: Sonic (dashed) and light (solid) horizon formation conditions vs r at P-G time t = 0 . L ( X ) with A=0.13and α = 3 . for X to become negative, unlike cosmological scenarios where homogeneity implies all physical quantities dependsolely on time ensuring X > t = 0 . L for a range of parameters: our values of A ran from0.01 to 0.15, and our values of α from 0.05 to 400. Within this range, there were a number of different categoriesof behaviour observed. In general terms, all parameter pairing for this choice of L may be grouped into one of thefollowing categories:1. Subcritical Collapse - no Cauchy breakdown2. Cauchy breakdown prior to any horizons forming3. Cauchy breakdown after apparent horizon formation but before sonic horizon formation4. Cauchy breakdown after both horizons form, where sonic horizon appears inside light horizon5. Cauchy breakdown after both horizons form, where within our numerical precision, sonic and apparent horizonsare indistinguishableBefore discussing specifically for which parameters each behaviour is present, we note the factors which could createdifferent behaviours in different parts of the parameter space. First, we know that as α increases we approach thebehaviour of ‘normal’ matter, and as α decreases we depart from this behaviour. Therefore, we expect breakdown ofthe Cauchy problem to occur sooner for cases of smaller α . Secondly, we know that as A increases, the collapse willgo from from subcritical to supercritical. Within the supercritical range, as A becomes bigger, collapse will be morerapid and horizon formation will occur sooner. With this in mind, we summarize our observations for L as follows.1. Subcritical collapse without Cauchy breakdown occured for A = 0 . − .
04 for relatively large α - the minimalvalue being between 0.01 and 2.0, depending on the value of A , and extending to α = 400 . FIG. 3: Map of null geodesics for L ( X ) of the background metric (dark blue solid) and the effective k-essence metric (darkgreen dashed) as well as the luminal apparent horizon (light blue solid) and sonic apparent horizon (light green dashed), forparameters A=0.12, α =2.0. The red line is the sonic event horizon, i.e. the last outgoing sound wave that does not quite escapethe sonic trapping region. The luminal event horizon is just slightly to the right of the sonic event horizon, but is too close todistinguish on this scale. as we expect - large α implies normal matter behaviour, and small A in normal matter corresponds to subcriticalcollapse.2. The breakdown of the Cauchy problem prior to any horizon formation occured at small α for all values of A .Figure 4 shows an example of this type of behaviour. Depending on the value of A , the maximum value of α forwhich this occurred was between 0.075 and 80.0. This is explained by the fact that, as stated before, at small α we have behaviour less typical of normal matter and we expect Cauchy breakdown to occur sooner.3. We observed 2 cases of Cauchy breakdown occurring between the formation of the sonic horizon and the forma-tion of the luminal apparent horizon. This was for A = 0 .
11 and α = 0 . , .
75. This behaviour is understoodeasily once we understand that in this choice of L sonic perturbations can be superluminal - the luminal apparenthorizon would then of course form before the sonic apparent horizon. In these cases, Cauchy breakdown occursbetween these formations.4. For intermediate values of α and large values of A , we observe breakdown of the Cauchy problem after luminaland sonic horizons have formed, with the sonic horizon initially occurring inside the luminal horizon. This typeof collapse was observed for A = 0 . − .
13 and for values of α ranging from 0.5 to 25.0 depending on the value of A . As we increase α within this range, the window for superluminal escape becomes smaller in time, or in otherwords the two horizons converge more rapidly. This is as expected, since as α increases we progress towardsnormal matter behaviour, so Cauchy breakdown occurs later and the two horizons become indistinguishable.The two horizons also converge more rapidly as A is increased, because increasing A increases the rapidity ofthe collapse.5. For large values of α and large values of A , we find that Cauchy breakdown occurs after both horizons haveformed. However we are unable to numerically distinguish the locations of the horizons to within our levels oftolerance. Again, this is as expected because for normal scalar field collapse there is no distinction between theluminal and sonic horizons and large α is the normal matter limit.Summarizing, for this choice of L we observe that superluminal escape is possible for a portion of the parameterrange. To observe this, we must have α sufficiently large to allow us to observe horizons prior to Cauchy breakdown,and we must also have α sufficiently small so that that the two horizons are numerically distinguishable. We also FIG. 4: Cauchy breakdown condition vs r for L at P-G time t=0.497767983 (the last timestep before Cauchy breakdown)for A=0.1 and α = 0 .
5. The point of closest approach to zero is approximately r=0.3434.FIG. 5: Sonic (dashed) and light (solid) horizon formation conditions vs r at P-G time t = 0 . L ( X ) with A=0.1and α = 2 . require A large enough to avoid subcritical collapse, but small enough to avoid the two horizons converging too rapidlyto be distinguished numerically.We also see for this choice of L that we have Cauchy breakdown occuring at small values of α for all A , in accordwith results for the MS case [11]. That we observe this breakdown for similar regions of the parameter space using anentirely different time coordinate indicates that the breakdown of the Cauchy problem in this choice of Lagrangianis likely a generic coordinate-independent feature. Conversely, for A = 0 . − .
13, the breakdown of the Cauchyproblem occurred eventually for all values of α which we examined. Therefore we see that in P-G coordinates thebreakdown of the Cauchy problem is not restricted to small α as it is in maximal slicing coordinates.We can easily modify the Lagrangian (18) to avoid the problem of Cauchy breakdown by choosing a second Born-Infeld type Lagrangian [11] L ( X ) = α [1 − (cid:114) − Xα ] − Λ (19)where again taking the limit as α goes to ∞ yields normal matter.Setting Λ to 0, in this case we find no evidence in our simulations of superluminal behaviour. Our range ofparameters observed in this case was the same as for L . For a typical choice of parameters we see in figure 5 thatthe sonic horizon forms outside of the light horizon, as we would expect for normal matter. Hence, there is nosuperluminal propagation in this case. This is further illustrated in figure 6 by mapping null geodesics and horizonsfor both background and effective metrics. As for L , we observed the two outer trapping horizons merging quickly,for the same reason, and the inner ones remaining separated. For the parameter values used in figure 5, the twohorizons were indistinguishable by P-G time t=0.779575347. FIG. 6: Map of null geodesics L ( X ) of the background metric (dark blue solid) and the effective k-essence metric (darkgreen dashed) as well as the luminal apparent horizon (light blue solid) and sonic apparent horizon (light green dashed), forparameters A=0.1, α =2.0 While we did not see any Cauchy breakdown in this case, we did encounter another issue with this choice. As can beclearly seen by the form of equation (19), it is possible for L X = (cid:113) − Xα to become infinite when X = α . However,[9] clearly states that L X should not be allowed to become infinite. Our simulations indeed become numericallyunstable in cases where X evolves to this value.In the MS case [11], simulations were reported to eventually fail not for this reason, but due to the formation ofshock waves of perturbations. This is indicated by the values of the speed of sound c s going to 0, where c s = L X L X + 2 X L XX = 1 − Xα (20)and so c s → α .The full description of behaviour for each portion of the parameter space we examined is analogous to that for L in many ways, so we present a slightly abbreviated version of our results compared to L .1. Subcritical collapse: as before, this occurred for small A and relatively large α , consistent with the limit ofnormal matter.2. We found that L X could become infinite after the formation of only a sonic horizon. This is somewhat analogousto case 3 for L , in which Cauchy breakdown occurred after the formation of only a luminal horizon. In thiscase, L X becomes infinite at small values of α for all values of A . We note that in reality this case encompassestwo distinct cases. In one case, we have this occuring for values of A which at larger α produce both horizonsprior to L X becoming infinite (see below). This case is not unexpected: we simply see L X diverging prior to theexpected luminal horizon forming. The second case is more thought provoking: this behaviour also occurs forvalues of A for which at larger α we see subcritical collapse with no horizons of any type forming (see above).This indicates there must be a threshold value of α below which subcritical behaviour yields to the formationof a sonic horizon. This behaviour is unexpected and may merit further investigation.3. For mid-range values of α and large values of A we found that L X became infinite after both horizons formed,with the sonic horizon appearing outside the light horizon. As before the two horizons were visibly separatedinitially, then converged together.4. Finally, for large α and large A , we again found both horizons forming prior to L X becoming infinite. We wereunable to distinguish between the two horizons within our numerical limits of tolerance. FIG. 7: Sonic (dashed) and light (solid) horizon formation conditions vs r at P-G time t = 0 . L ( X ) with A=0.12and C=0.8 As before, we observed larger A to imply more rapid gravitational collapse and hence horizon formation taking placesooner. While we found, as expected, no Cauchy breakdown in the parameter space for this model, we encountereda new problem: that of diverging L X . Specifically, we found that L X became infinite more rapidly for smaller valuesof α , consistent with increasing departure from the normal matter limit. Our results are commensurate with the MScase [11] and indicate that the Cauchy problem (or lack thereof) and the problem of diverging L X (or vanishing c s )are likely coordinate independent. B. Lagrangians Allowing a Global Time Coordinate
We next examine the behaviour of a choice of Lagrangian that is completely free from the problem of Cauchybreakdown by construction. Such a Lagrangian would have the left hand side of (17) be either positive everywhereor negative eveywhere. One such example is [11] L ( X ) = CX C + e X C − ∞ we recover the behaviour of normal matter ( L = X ). We canverify quite easily using a more general form of (17) that this makes the left hand side of (17) positive everywhere,therefore entirely avoiding Cauchy breakdown.Upon simulation of this model, we do not find evidence of superluminality. Commensurate with the MS case [11],we observe in certain parameter choices the sonic horizon forming outside of the apparent horizon, as in the caseof normal matter and for L ( X ). We illustrate such results for a typical choice of parameters in figure 7. Thisoffers an indication that the subluminality of this particular global- t model is independent of the coordinate choice.We also see the same behaviour as in the previous two cases with respect to the eventual merging of the horizons.For the parameter values chosen in figure 7, for example, we see the two horizons as indistinguishable by P-G timet=0.449391824. This is perhaps best illustrated in the map of null geodesics and horizons given in figure 8. We donot find Cauchy breakdown in this case, as expected by construction.For this model, we examined parameter pairings in the range of A = 0 . − .
13 and C = 0 . − .
0. We note that,as discussed in [11], the hyperbolicity requirement of equation (4) prevents us from choosing
C < .
45. Although weencountered some numerical difficulties (explained below) that forced the program to halt before completion, thesewere unrelated to the physical or mathematical characteristics of the model, in contrast to the situation for L and L . With this in mind, we give the following overview of different types of behaviour found for this case.1. Subcritical collapse was observed for small A for all choices of C .2. For mid-range values of A and small values of C , we observed a sonic horizon forming prior to the code halting,but not a luminal horizon. This is as expected from the construction of the model.3. For mid-range values of A and larger values of C , we observed neither sonic nor luminal horizons forming beforethe code halted. This is due not to a different run time before halting (which was comparable at different values0 FIG. 8: Map of null geodesics L ( X ) of the background metric (dark blue solid) and the effective k-essence metric (darkgreen dashed) as well as the luminal apparent horizon (light blue solid) and sonic apparent horizon (light green dashed), forparameters A=0.12, C=0.8 of C ) but rather due to the fact that at higher C (nearer the normal matter limit) the two horizon formationsoccur more closely together in time, so neither form before the code halts.4. For large values of A and small values of C , we observed subluminal behaviour such as that shown in figure7, with the sonic horizon forming outside the luminal horizon. This is a natural extension of the behaviour atmidsize A and small C . Here, since A is larger, the horizons form earlier, and so both have time to form beforethe code halts.5. For large values of A and large values of C , we see both horizons forming before the code halts, but they areindistinguishable within our degree of precision.We pause to comment on the numerical difficulties we encountered for this case. Because of the recursive natureof the definition of X in P-G coordinates (see appendix A), we were forced to use a numerical solver to ascertainthe value X at a given time-step. We chose a Newton-Raphson solution method, which had a very good rate ofconvergence. However the tendency of X to become unpredictably large near the origin after horizon formation madeit extremely difficult to find an initial guess that would yield convergence for the entire duration and spatial extentof the simulation. Hence we were forced to halt the code on those occasions that the Newton-Raphson method failedto converge. IV. CONCLUSIONS
Through numerical simulations of the gravitational collapse of K-essence scalar fields in Painlev´e-Gullstrand coordi-nates, we have been able to observe a number of interesting phenomena. For the Lagrangian L we have numericallyverified that sonic horizons form inside the luminal apparent horizon, allowing superluminal perturbations to escapethe black hole. For other choices of Lagrangian ( L and L ), we see that collapse proceeds with the sonic horizonoutside of the luminal horizon, indicating no possibility of superluminal escape. For all three Lagrangians we studiedthe two different horizons converge relatively quickly, as the K-essence scalar field vacates the region of the horizons.These results are fully commensurate with the MS case [11], suggesting that they are quite general and coordinateindependent.We can understand these features by examining the general form of the sonic horizon condition (16). Superluminalbehaviour generically happens when the luminal horizon forms before the sonic horizon, yielding a region in which1(at least initially) light is trapped but not sound. As explained above, when L X is positive this is definitely possibleonly when L XX is negative in the trapping region. For L , we have: L ,X = 1 (cid:112) X/α L ,XX = − α (1 + 2 X/α ) / (22)whereas for L : L ,X = 1 (cid:112) − X/α L ,XX = 1 α (1 − X/α ) / (23)Moreover, for L all three quantities are manifestly positive. Thus, superluminal escape is in principle only possiblefor L , and this is born out by our numerical calculations.It is important to note that the above considerations are slicing independent. The presence, or not, of superluminalpropagation inside a trapped surface of the luminal metric is coordinate invariant, although the precise location ofthe trapping horizons may depend on the slicing.The possibility of superluminal escape from a black hole can be considered in terms of the sonic and luminal eventhorizons of the black hole. These are defined as the last sound and light rays, respectively, to just barely escape fromthe black hole. The sonic and luminal event horizons coincide along a line of constant r , say r h , in the future afterall the matter has fallen through. For L the sonic trapping horizon is to the future of the luminal trapping horizon,as shown in Figure 3. In this case, as one moves backward in time the sonic event horizon (red dashed line) “peelsoff” towards small r slightly faster than the luminal event horizon. Thus, there will be some K-essence perturbationsjust outside the sonic event horizon that will be able to escape to infinity from within the luminal event horizon. Arelated question is whether or not sonic perturbations are able to escape from inside the luminal trapping horizon(light blue solid line). Although it is impossible to distinguish on the scale of Figure 3, we have verified numericallythat as the sonic event horizon separates from the trapping horizon, it does enter the luminal trapping horizon. Forthe parameters that we have considered the corresponding region of superluminal escape is very small. It is not clearwhether or not there is a range of initial data that allows a region of superluminal escape that is large enough toaffect the information loss problem. This is a question that is worthy of further investigation.Note that none of these three choices of Lagrangian are suitable for addressing the cosmic coincidence problem [6].For L we find that X becomes negative in certain regions of space, in contradiction with the assumed homogeneityof cosmology [11]. In the cases of L and L , superluminality is not observed in either the MS or P-G simulations andso these models cannot be used to create the dynamical behaviour used to solve the problem of cosmic coincidence[8]. Any choice of L that solves this problem must incorporate superluminality. These models are ‘K-essence’ only inthe sense of having non-linear kinetic energy.As long as L X and L XX are regular, only L permits a breakdown of the Cauchy problem and this too was verifiedby the numerical calculations. However, in the case of L , when X is too large, L X and L XX blow up. Our simulations(and those of [11]) verified that this behaviour does indeed occur in spherically symmetric collapse with L .For L , we have observed breakdown of the Cauchy problem. The smaller the value of α , the earlier this breakdownoccurs during the simulation. This indicates that the constant time surfaces in P-G coordinates fail to continue to beCauchy surfaces for either the background metric or the effective metric of K-essence, a phenomenon also observed forthe maximal slicing time coordinate [11]. This provides a compelling (albeit inconclusive) indication of the coordinateindependence of this generic breakdown of the Cauchy problem for these choices of L . Whether or not avoidanceof Cauchy breakdown is necessarily connected to subluminality remains an open question; it remains to be seenwhether another time coordinate may offer a well-posed Cauchy problem throughout for such choices of L . If no timecoordinate exists that ensures the Cauchy problem is well-posed throughout collapse for both the background andeffective metric, this would invalidate the predictive power of such models [10], and eliminate the possibility of thephysically sensible escape of information from a black hole. Acknowledgements
We thank David Garfinkle and Ryo Saotome for helpful discussions and correspondence. We are grateful toSHARCNET for providing computer resources. This work was supported by the Natural Sciences and EngineeringResearch Council of Canada.2
Appendix A
In the following, where there is potential for confusion we denote 2 dimensional quantities with a hat and leave 4dimensional quantities as they are. Since we are considering spherically symmetric collapse, we dimensionally reducethe metric from 4 dimensions to 2 via ds = 1 j ( φ ) ˆ g µν dx µ dx ν + r (cid:0) dθ + sin ( θ ) dφ (cid:1) (24)where φ = r / (4 l ) and j ( φ ) = √ φ = r/ (2 l ), with l an arbitrary length scale. Note that in the following ˆ g µν andassociated quantities refer to a 2-dimensional metric. The action (1) becomes12 ˆ G (cid:90) d x (cid:112) − ˆ g (cid:18) φ ˆ R + V ( φ ) l (cid:19) + (cid:90) d x (cid:112) − ˆ g πr j ( φ ) L ( j ˆ X ) (25)where 2 ˆ G = G/l , V ( φ ) = 1 / (2 √ φ ), and ˆ X = Xj = −
12 ˆ g µν ∂ µ ψ∂ ν ψ (26)Employing an ADM decomposition of the 2-dimensional metric in (24) d ˆ s = e ρ (cid:0) − σ dt + ( dx + N dt ) (cid:1) (27)yields after some algebraic manipulation the Hamiltonian H = (cid:90) dx (cid:32) σe ρ lφ (cid:48) (cid:0) −M (cid:48) + 4 πr ρ M (cid:1) + (cid:32) N − σ ˆ G Π ρ φ (cid:48) (cid:33) F (cid:33) (28)where M is the Hamiltonian version of the Misner-Sharpe mass function2 ˆ G M l ≡ e − ρ (cid:16) ( ˆ G Π ρ ) − ( φ (cid:48) ) (cid:17) + j ( φ ) l (29)and we define the mass energy density ρ M via4 πr ρ M = lφ (cid:48) e ρ (cid:32) Π ψ πr L X − πr e ρ j ( φ ) L ( X ) (cid:33) + l ˆ G Π ρ e ρ Π ψ ψ (cid:48) . (30)The canonical momenta are given by Π φ = 1 σ ˆ G ( ρ (cid:48) N + N (cid:48) − ˙ ρ ) (31)Π ρ = 1 σ ˆ G ( φ (cid:48) N − ˙ φ ) (32)Π ψ = 4 πr σ L X ( ˙ ψ − N ψ (cid:48) ) (33)and we have defined F = ρ (cid:48) Π ρ − Π (cid:48) ρ + φ (cid:48) Π φ + Π ψ ψ (cid:48) (34)It is important to note that the lapse, σ , and shift, N , in the above parametrization are also the lapse and shiftfunctions of the physical 4-metric as defined in (24).Choosing the r as the spatial coordinate implies both the consistency condition ˙ φ = 0 and lφ (cid:48) = j ( φ ). Using thiswe obtain the shift in terms of the lapse: N = σ ˆ G Π ρ φ (cid:48) (35)3Setting the diffeomorphism constraints strongly to zero, we eliminate Π φ , yielding H = (cid:90) dx σe ρ j ( φ ) (cid:0) −M (cid:48) + 4 πr ρ M (cid:1) + (cid:90) dx ( σ e ρ j M ) (cid:48) (36)for the partially reduced Hamiltonian upon including an appropriate boundary term. This partial coordinate choiceallows many different fully gauge fixed theories, including Schwarzschild coordinates, Painlev´e-Gullstrand (P-G) co-ordinates and maximal slicings, among others.Specializing to P-G coordinates entails fixing g rr = 1, or e ρ = j ( φ ). The consistency condition ˙ ρ = 0 fixes thelapse via σ (cid:48) σ = − l ˆ G Π ψ ψ (cid:48) (cid:113) l ˆ G M j ( φ ) (37)determining ρ and σ . Making use of the above conditions with eq (29) yields2 ˆ G M l = ( ˆ G Π ρ ) j ( φ ) (38)with Π ρ determined by the remaining Hamiltonian constraint.Finally, the field equations for the matter field independent variables ψ and Π ψ from the Hamiltonian (36) yieldsafter some manipulation ˙ ψ = σ Π ψ πr L X + (cid:115) Gl M j ( φ ) ψ (cid:48) (39)˙Π ψ = σ πr L X ψ (cid:48) + (cid:115) Gl M j ( φ ) Π ψ (cid:48) (40)The quantities M and σ as determined by the Hamiltonian constraint (38) − M (cid:48) + 4 πr ρ M ∼ σ (cid:48) σ = − l ˆ G Π ψ ψ (cid:48) (cid:113) l ˆ G M j ( φ ) (42)with ρ M given by 4 πr ρ M = l (cid:32) Π ψ πr L X − πr L ( X ) (cid:33) + (cid:115) G M lj ( φ ) Π ψ ψ (cid:48) (43)Recalling that 2 ˆ G = G/l , we note that the dynamical equations for the K-essence field and its conjugate momentummay be expressed in terms of four dimensional quantities only.As an example, for the choice (19) we have L X = 1 (cid:113) − Xα L XX = 1 α (cid:18) − Xα (cid:19) − / (44)In this case we can solve explicitly for X in terms of the phase space variables X = 12 (cid:32) Π ψ (4 πr ) L X − ( ψ (cid:48) ) (cid:33) = − α (cid:34) (4 πr ψ (cid:48) ) − Π ψ (4 πr α ) + Π ψ (cid:35) (45)4in P-G coordinates.For the purposes of our numerical simulations, we set the arbitrary length scale l to l = 1. Choosing the initialconfiguration for ψ to be that given in equation (13), we first construct a fixed, spherically symmetric spatial gridin which to evolve the system, by specifying the number of spatial steps (we use 600), the minimum space betweensteps, and the maximum space between steps. Near the origin we require a finer grid. We therefore initialize thespatial grid with the minimum space between steps for a specified number of first steps nearest to the origin (we use100 steps). Once we are 100 steps from the origin, the spatial grid is gradually expanded so that the space betweenthe grid points is at a maximum once we reach grid point 600. We select a minimum spacing of 1 . × − and amaximum spacing of 1 . × − . After specifiying the initial configuration of the spatial grid, we then initialize theconfiguration of ψ as specified above and we initialize the conjugate momentum Π ψ to be identically 0.After these initial conditions have been specified, we enter a loop, which proceeds at each timestep until a prede-termined and specfied maximal time is reached after which we anticipate no interesting evolution of the system. Notethat while the spatial grid is constant at each timestep and only specified once, the timestep is not constant. It isdetermined during each execution of the loop.Inside the loop, we first calculate the necessary derivatives of ψ and Π ψ using finite difference methods. We thencalculate L and its derivatives, as well as X and its first dervative in r. Next, we determine M and σ on the spatial gridat the current timestep using a fourth order Runge-Kutta method. We then employ the condition that determines thelocation of the luminal apparent horizon and the sonic apparent horizon, as well as checking for Cauchy breakdown.At this point in the loop, data is written to file on selected executions of the loop. We then calculate the nexttimestep, according to the speed of a local ingoing geodesic. Finally, we evolve ψ and Π ψ in time using another fourthorder Runge-Kutta method, and increase our total time by the calculated timestep. The loop then restarts.The loop continues to run until the total time has reached some predetermined and declared value as mentionedabove. Alternatively, the program may cease to run due to a check performed on the values of L and its derivativeseach time they are computed. If these values become divergent or imaginary, the program halts. This corresponds inthe case of L to Cauchy breakdown, and in the case of L to the divergence of L X . 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