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gravitational collapse results in black holes which contain singularities .
the collapse process is not described by an analytical solution of the field equations , and the gravitational field is so strong that perturbation theory is not helpful either . so numerical simulation is needed .
but even here , the full process of gravitational collapse is not usually examined : some simulations examine the formation of black holes but not singularities @xcite while others examine singularities but not black holes @xcite .
note , however , that there has been some recent progress in numerical simulations capable of examining the formation of both black holes and singularities.@xcite in the case where black holes are examined , the simulation sometimes ends when the black hole forms @xcite or the simulation goes on , but the interior of the black hole is excised @xcite . in the case where the singularity is examined ,
one usually imposes periodic boundary conditions , which means that what is studied is singularity formation in a closed cosmology .
there is a longstanding conjecture due to belinskii , khalatnikov , and lifschitz ( bkl ) as to the general behavior of spacetime singularities.@xcite essentially the bkl conjecture is that as a generic singularity is approached in a comoving coordinate system , the field equations will be dominated by terms containing derivatives with respect to time , and all other terms can be neglected .
the bkl conjecture has been verified by numerical simulations@xcite for the case of singularity formation in a closed cosmology .
since bkl dynamics are local , it seems likely that the bkl conjecture is also true in the case of singularity formation inside a black hole ( at least for part of the singularity , see @xcite for a conjecture about another part of the singularity ) .
however , one would like to verify the bkl conjecture by a treatment of the singularities inside black holes .
one of the earliest studies of gravitational collapse , that of oppenheimer and snyder @xcite was capable of also examining the singularity . in @xcite gravitational collapse
is modeled as a spherically symmetric spacetime containing two parts : the schwarzschild metric on the outside and a friedmann - robertson - walker ( frw ) spacetime with non - relativistic matter ( dust ) on the inside .
the schwarzschild and frw spacetimes have two different coordinate systems which must be matched along the boundary .
this difficulty can be overcome @xcite through a coordinate transformation which allows the full spacetime to be covered by a single coordinate system .
however a more serious difficulty is that pressureless dust is not a good model of how matter behaves in the extreme conditions of singularity formation .
so the properties of dust singularities tell us very little about the properties of generic singularities .
ideally , one would like to check the bkl conjecture by treating the gravitational collapse of a more realistic matter model , like a scalar field , using a single coordinate system that follows the formation of the black hole and the singularity .
as a preliminary step in that direction , we perform numerical simulations of the behavior of a test massive scalar field on the spacetime of @xcite using the coordinates of @xcite .
the relevant equations and numerical methods are described in section 2 , while results are presented and discussed in section 3 .
a spherically symmetric spacetime can be put in the form @xcite @xmath0 here , @xmath1 is the usual area coordinate , while @xmath2 is a time coordinate chosen so that @xmath3 .
we now show how to put the schwarzschild and frw metrics in this form .
the schwarzschild metric in the usual coordinates is @xmath4 we wish to express @xmath5 as a function of @xmath2 and @xmath1 so that the metric takes the form of eqn .
( [ pgmetric ] ) .
some straightforward but tedious algebra shows that the expression @xmath6\ ] ] yields a metric of the form of eqn .
( [ pgmetric ] ) with @xmath7 the spatially flat dust frw spacetime has the metric @xmath8 ) \label{frw}\ ] ] with the scale factor @xmath9 given by @xmath10 where @xmath11 is a constant .
we wish to perform a coordinate transformation that gives this metric the form of eqn .
( [ pgmetric ] ) . comparing the angular parts of the metric
, it is clear that we must have @xmath12 .
it is then straightforward to show that with @xmath13 the metric of eqn .
( [ frw ] ) takes the form of eqn .
( [ pgmetric ] ) with @xmath14    the collapse takes place at negative times and is complete at @xmath15 .
the boundary between the interior frw part of the spacetime and the exterior schwarzschild part is given by @xmath16 .
for some function @xmath17 . since the metric , and
therefore @xmath18 must be continuous across the boundary , we have @xmath19 thus the spacetime metric is given by eqn .
( [ pgmetric ] ) with @xmath20 given by eqn ( [ schpsi ] ) for @xmath21 and @xmath20 given by eqn .
( [ frwpsi ] ) for @xmath22 .
we wish to simulate the klein - gordon equation on this spacetime assuming spherical symmetry for the scalar field .
that is , we want a scalar field @xmath23 that satisfies @xmath24 for a metric of the form of eqn .
( [ pgmetric ] ) the klein - gordon equation for a spherically symmetric scalar field becomes @xmath25 for the numerical method used , we put this equation in first order form : we introduce the quantities @xmath26 and @xmath27 given by @xmath28 from the definitions of @xmath26 and @xmath27 it immediately follows that @xmath29 while eqn .
( [ kg ] ) becomes @xmath30    the equations to be evolved are eqns .
( [ dtphi]-[kg1 ] ) .
we evolve using finite differences , where each scalar @xmath31 is represented by the quantities @xmath32 .
spatial derivatives are evaluated using standard centered differences , that is @xmath33 however , to maintain accuracy near the origin , the quantity @xmath34 requires special treatment .
we define @xmath35 and @xmath36 by @xmath37 and @xmath38 .
correspondingly we define @xmath39 and @xmath40 .
then we use the finite difference @xmath41 the quantity @xmath18 and its derivatives are evaluated analytically using the formulas in eqns .
( [ schpsi ] ) and ( [ frwpsi ] ) .
( [ dtphi]-[kg1 ] ) are evolved using a three step iterated crank - nicholson method @xcite with kreiss - oliger dissipation @xcite .    at the origin , ( gridpoint @xmath42 ) smoothness
requires that @xmath27 vanish and that @xmath43 and @xmath26 have vanishing derivative with respect to @xmath1 .
we implement this condition as @xmath44 at the outer boundary , we impose the condition that @xmath45 and @xmath26 all vanish .
@xmath43 _ vs _ @xmath1 at the initial time @xmath46 ]    @xmath43 _ vs _ @xmath1 for @xmath47 at @xmath48 ]    @xmath43 _ vs _ @xmath1 for @xmath49 at @xmath48 ]    in the simulations , let @xmath50 be the number of gridpoints and @xmath51 be the maximum value of @xmath1
. then we have @xmath52 .
@xmath53 was used for the runs resulting in figures [ fig1]-[fig3 ] .
@xmath54 was used for the runs resulting in figures [ fig4]-[fig11 ] .
we choose initial data for the scalar field at a time @xmath55 to take the form @xmath56 and @xmath57 with @xmath27 given by the derivative of this expression . here ,
@xmath58 and @xmath59 are constants .
for all the results displayed , the parameters are @xmath60 , and @xmath61 where @xmath62 is the mass of the schwarzschild black hole .
this initial configuration of the scalar field is shown in figure [ fig1 ] .
we run the simulations to a time near the singularity .
figures [ fig2 ] and [ fig3 ] show the configuration of the scalar field near this final time .
both figures are for @xmath48 with @xmath47 for figure [ fig2 ] and @xmath49 for figure [ fig3 ]
. the initial scalar field can be thought of as a linear combination of a left mover and a right mover .
the right mover escapes from the black hole , while the left mover plunges towards it . some part of the left mover scatters off of the black hole , while another part is captured and approaches the singularity .
we are particularly interested in finding the behavior of the scalar field as the singularity is approached . in our coordinate system ,
the singularity is approached as @xmath63 but only in the frw portion of the spacetime . to examine the scalar field behavior near the singularity
, we plot the scalar field at three different times near @xmath15 and deduce a trend from those plots .
the scalar field @xmath43 is plotted as a function of @xmath1 in a small region near @xmath64 for the times @xmath65 and @xmath66 for the @xmath47 case in figure [ fig4 ] and for the @xmath49 case in figure [ fig5 ] .
it is clear that the trend is that as the singularity is approached , the scalar field gets a larger amplitude and a steeper profile . to see whether this behavior can be understood quantitatively
, we turn to the bkl conjecture : the claim that as a generic singularity is approached in a comoving coordinate system , the field equations will be dominated by terms containing derivatives with respect to time , and all other terms can be neglected . to apply this conjecture , we consider the klein - gordon equation ( [ kleingordon ] ) in the untransformed frw metric and coordinates given by eqns .
( [ frw ] ) and ( [ adust ] ) . keeping only terms containing time derivatives , we find @xmath67 the general solution of this equation is @xmath68    @xmath43 _ vs _
@xmath1 for @xmath47 at @xmath69 ( solid)@xmath70 ( dashed ) and @xmath66 ( dotted ) ]    @xmath43 _ vs _
@xmath1 for @xmath49 at @xmath69 ( solid)@xmath70 ( dashed ) and @xmath66 ( dotted ) ]    @xmath71 _ vs _
@xmath72 for @xmath47 at @xmath69 ( solid)@xmath70 ( dashed ) and @xmath66 ( dotted ) ]    @xmath71 _ vs _
@xmath72 for @xmath49 at @xmath69 ( solid)@xmath70 ( dashed ) and @xmath66 ( dotted ) ]    for some functions @xmath73 and @xmath74 .
in particular , the bkl conjecture leads us to expect that as the singularity is approached , @xmath75 should approach a function of @xmath76 .
note , that expressed in terms of the @xmath77 coordinates of our simulation , we have @xmath78 and @xmath79 .
thus , we are led to expect that as the singularity is approached , if we plot @xmath71 _ vs _ @xmath72 the plots should approach some limiting profile as @xmath80 . to see whether that expectation is realized ,
such plots are given in figure [ fig6 ] for @xmath47 and [ fig7 ] for @xmath49 .
these figures are consistent with @xmath71 approaching a limit as the singularity is approached . however , in order to be more definite , we need to approach the singularity more closely .
the results of simulations which show the scalar field at times closer to the singularity are shown in figures [ fig8]-[fig11 ] . here
figure [ fig8 ] shows @xmath43 _ vs _
@xmath1 for @xmath47 and the times @xmath81 and @xmath82 .
figure [ fig9 ] shows @xmath43 _ vs _ @xmath1 at the same times for the @xmath49 case .
the corresponding rescaled quantities , that is @xmath71 _ vs _ @xmath72 are shown in figure [ fig10 ] for @xmath47 and in figure [ fig11 ] for @xmath49 .
@xmath43 _ vs _
@xmath1 for @xmath47 at @xmath83 ( solid)@xmath84 ( dashed ) and @xmath82 ( dotted ) ]    @xmath43 _ vs _
@xmath1 for @xmath49 at @xmath83 ( solid)@xmath84 ( dashed ) and @xmath82 ( dotted ) ]    @xmath71 _ vs _
@xmath72 for @xmath47 at @xmath83 ( solid)@xmath84 ( dashed ) and @xmath82 ( dotted ) ]    @xmath71 _ vs _
@xmath72 for @xmath49 at @xmath83 ( solid)@xmath84 ( dashed ) and @xmath82 ( dotted ) ]    it is clear from figures [ fig10 ] and [ fig11 ] that @xmath71 approaches a limit as the singularity is approached . since this is just what one would expect from the bkl conjecture , our simulations lend support to the notion that the bkl conjecture holds , not only in the extensively studied case of closed cosmologies , but also in the more physically interesting case of asymptotically flat spacetimes . to explore this issue
further , it would be helpful to go beyond the case of a test scalar field on a background spacetime .
in particular , one could treat the collapse of a self - gravitating scalar field using the method of@xcite and see whether the singularity produced in the collapse satisfies the bkl conjecture .
the work of dg was supported by nsf grant phy-0855532 to oakland university .
rs and ra are supported by a grant from the us department of energy .
numerical simulations are performed of a test scalar field in a spacetime undergoing gravitational collapse . the behavior of the scalar field near the singularity is examined and implications for generic singularities are discussed . in particular , 
 our example is the first confirmation of the bkl conjecture for an asymptotically flat spacetime .