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Conformal time slicing condition in three dimensional numerical relativity
0083-6656/93 $24.00 © 1993 Pergamon PressLtd
Viatoa in Astronomy, Vol. 37, pp. 445--448, 1993 Printed in Great Britain. All rights reset'veal.
CONFORMAL TIME SLICING CONDITION IN THREE DIMENSIONAL NUMERICAL RELATIVITY M a s a r u Shibata* and Takashi N a k a m u r a t *Department of Physics, Kyoto University, Kyoto 606-01, Japan tYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, J a p a n
ABSTRACT. A new time slicing condition which may be useful to estimate gravitational waves in three dimensional(3D) numerical relativity is proposed. In this time slice, the lapse function is a function of the conformal factor of metric. Spherical symmetric dust collapse is investigated using this time slice. It is found that the exterior vacuum region is nearly static, which suggests that this slice is convenient to estimate gravitational waves in two and three dimensional numerical relativity.
INTRODUCTION The projects of the gravitational wave detection, such as LIGO (Laser Interferometer Gravity Wave Observatory) project, are now active in the world and detectors with sensitivity high enough to observe gravitational waves emitted by coalescing binary neutron stars at Virgo cluster, will be in operation by the end of this century. Hence it is urgent to perform the reliable calculations of coalescing binary neutron stars or the stellar core collapse of a massive star. In such phenomena general relativity plays an essential role. This means we must calculate the amplitude and the wave pattern of gravitational waves in fully general relativistic simulations. However there are only a few examples of fully general relativistic simulations. The most crucial reason why the general relativistic simulation has been hardly performed until now is that we do not have sufficient knowledges of the coordinate conditions. In this article, we propose a new time coordinate condition which may be useful to calculate gravitational waves as a first step to find appropriate gauge conditions. We call this time slice conformal slice.
446
M. Shibata and T. Nakamura C O N F O R M A L T I M E SLICING C O N D I T I O N
Usually we require the following property to a time slice. 1) In the central region, singularities should be avoided until gravitational waves pass through the outer boundary of numerical calculation. 2) In the exterior vacuum region, the metric should become stationary except for the wave parts. The maximal slicing is a well-known slice to have the singularity avoidance property [1] • However it does not satisfy the property 2) and for that reason it is not so appropriate to estimate gravitational waves. Taking the case of the spherically symmetric space-time in the above gauge, let us explain the reason. In the maximal slicing, the vacuum exterior satisfies the following relation;
-- 1 + M. + O(r_4)' 1 - MI2r = 1 + M / 2 r + Clr + O ( r - 2 ) '
(01)
where M is the gravitational mass of the system and (71 is a time dependent constant. T h e source terms of the evolution equations of the extrinsic curvature in the vacuum region become
--sit d + aRij ~ O(r-3).
(02)
This means K o and 7ij become ,-, t / r 3 and ~ t2/r 3, respectively, with time evolution• In the two or three dimensional simulations, the amplitude of gravitational waves is proportional to r -1. According to the above estimation, the non-wave parts of the metric has t2/r 3 dependence in the maximal slicing. When gravitational waves pass through the outer boundary of numerical grid, t is larger than r. Hence the non-wave parts of the metric become O(r-1). Therefore it seems to b e difficult to estimate gravitational waves in the maximal slice because of the same r dependence in the non-wave and the wave parts of the metric. To overcome above difficulties, we propose the following slice. We define e -- ~ - 1, where ~ is appropriately determined conformal factor. If the background metric is the Schwarzschild metric, ~ = M/2r. In order to fix the exterior vacuum geometry as a Schwarzschild metric, we choose the lapse function as follows. 2
$
$
a = e-2'-~ ~ - ~ .
(03)
T h e main source term of the evolution equation of the extrinsic curvature is aRij - alij. In this lapse function, it is written as
ot~'j +
2a[~(1
+ 2e + 2e 2 + 2es + 2~4 +
~5)
T
~J(3
+ 6~ + 6 ~ 2 + 6~3 + 6~4 + 12~ 5 + 10~ 6 + 8 ~ r + 6~8 + 4~ 9 + 2~ t°)
~,,k4,'k,, 7ijT(i + 2t + 2E2 + 2t3 + 2t 4 + 2~5) -- 7ij
(04)
.
],
where 7/i is defined by ~/j = ~-47/j, V denotes the derivative operator with respect to $/i
Conformal Time Slicing Condition and Rij is the Ricci tensor of 7ij- In the gauge conditions such as the minimal distortion gauge [21 or the isothermal gauge [31 , the leading parts of $/j are the wave parts and the first term of the right hand side of Eq.(04)denotes the source of gravitational waves. Therefore other terms of Eq.(04)are regarded as the source for the non-wave parts, the lowest order part is the spherically symmetric part. At first let us consider the spherically symmetric case. In this case, ~ = l + M / 2 r + O ( r - n ) , n > 4, for large r, so that the source terms of the evolution equations of Kij becomes r -n+2. This means that Kij ~ t / r n+2 and ~fij "~ t2/r n+2 < 0(r-4) for large r. In the two and three dimensional case, ~ = 1 + M / 2 r + IijzizJ/r 5 Jr O(r-4), so that we must consider the next order of the spherically symmetric component. However we can easily show that the higher order terms contribute to the source terms at most 0 ( r - 6 ) . Therefore as for the non-wave parts, Kij "~ t / r 6 and 7ij "~ t2/r 6. Since the time dependent non-wave parts have only ,,~ O(r -4) amplitude when gravitational waves reach the outer boundary, we can calculate the amplitude of gravitational waves up to O(r~n~ ) accuracy, where rmaz is the maximum distance in unit of mass of the system from the origin in numerical calculations. As for the highly relativistic region, e ~ 1. This means ct ~ 5 x 10 -2. For e = 1.5, c~ = 2 x 10 -4. Hence the lapse function will sufficiently freeze in the dense region.
N U M E R I C A L T E S T OF C O N F O R M A L SLICING To investigate the conformal slice, we perform the numerical calculation of the spherically symmetric dust collapse. The detailed equations are shown in another paper [4] . As an initial condition, we use the time symmetric initial condition, Ji = 0 and K~ = 0, and give a properly determined dust configuration. In figure 1 we show the proper time at the origin as a function of coordinate time T. It indicates that the proper time at the origin freezes for the sufficient large coordinate time. This is favorable to estimate gravitational waves because gravitational waves will be able to propagate in the wave zone while at the origin the singularity can be avoided. In figure 2 we show how the lapse function(a) behaves at the origin when the conformal h c t o r ( $ ) becomes large in both the conformal slice and the maximal slice. From this figure it is found that the lapse function in our slice freezes faster than that in the maximal slice, so it seems to be useful to avoid the singularity. In figure 3 we show the components of the extrinsic curvature I(rr -- K1. It falls r - " with n ,~ 10, and remains small. This means that in this slice the outer region is guaranteed to be nearly static. To compare our slice with maximal slice, we show Krr calculated in the maximal slice with the same initial condition as in our slice in figure 4. Near the outer most grid Krr oc r -3. Thus in this slice - 1 - M / 2 r behaves ,~ t2/r3ma~: ~> r ~ in the outer most grid. Therefore in the maximal slicing condition it seems to be difficult to analyze gravitational waves accurately.
SUMMARY In this paper we suggest a new slicing condition in which it may be possible to analyze gravitational waves in two and three dimensional numerical relativity. In our slice we can
447
448
M. S ~
and T. Naknmura
keep the time variation of the non-wave parts of the metric to O ( r ~ ) , to the maxima] slice in which we can keep them only to O ( r ~ ) .
n _> 4, in contrast
This work was partly supported by a Grant-in-Aid Scientific Rest'arch on Priority Area of Ministry of Education, Science and Culture(04234104).
REFERENCES 1. D. M. Eardley and L. Smarr, Phys. Rev. D 1 9 (1979), 2239 2. L. Smarr and J. W. York Jr. Phys. Rev. D 1 7 (1978), 1945 3. J. M, Baxdeen and T. Piran, Phy. Rep 96(1983), 205 4. M. Shibata and T. Nakamura, Prog. Theor. Phys. to be published 14.0
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, . , . I
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nformal slice
. . . .
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2.0
.
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2.5
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,t
Fig.1
Fig.2
K1 I
10-3
- - -T=5.1 T=19.0 . . . . . T=30.0 - - -T=46.1
-K1
10"2
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! Io.0
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~ 1
K1 ~_"._.
10"49.
'
10.0 r (log scale) Fig.4
,
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,
,
,
3.5
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4.0
T=5.4 - - - -T=19.2 . . . . . T=28.2 , . . . . . T=30.6
12.5
Viatoa in Astronomy, Vol. 37, pp. 445--448, 1993 Printed in Great Britain. All rights reset'veal.
CONFORMAL TIME SLICING CONDITION IN THREE DIMENSIONAL NUMERICAL RELATIVITY M a s a r u Shibata* and Takashi N a k a m u r a t *Department of Physics, Kyoto University, Kyoto 606-01, Japan tYukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-01, J a p a n
ABSTRACT. A new time slicing condition which may be useful to estimate gravitational waves in three dimensional(3D) numerical relativity is proposed. In this time slice, the lapse function is a function of the conformal factor of metric. Spherical symmetric dust collapse is investigated using this time slice. It is found that the exterior vacuum region is nearly static, which suggests that this slice is convenient to estimate gravitational waves in two and three dimensional numerical relativity.
INTRODUCTION The projects of the gravitational wave detection, such as LIGO (Laser Interferometer Gravity Wave Observatory) project, are now active in the world and detectors with sensitivity high enough to observe gravitational waves emitted by coalescing binary neutron stars at Virgo cluster, will be in operation by the end of this century. Hence it is urgent to perform the reliable calculations of coalescing binary neutron stars or the stellar core collapse of a massive star. In such phenomena general relativity plays an essential role. This means we must calculate the amplitude and the wave pattern of gravitational waves in fully general relativistic simulations. However there are only a few examples of fully general relativistic simulations. The most crucial reason why the general relativistic simulation has been hardly performed until now is that we do not have sufficient knowledges of the coordinate conditions. In this article, we propose a new time coordinate condition which may be useful to calculate gravitational waves as a first step to find appropriate gauge conditions. We call this time slice conformal slice.
446
M. Shibata and T. Nakamura C O N F O R M A L T I M E SLICING C O N D I T I O N
Usually we require the following property to a time slice. 1) In the central region, singularities should be avoided until gravitational waves pass through the outer boundary of numerical calculation. 2) In the exterior vacuum region, the metric should become stationary except for the wave parts. The maximal slicing is a well-known slice to have the singularity avoidance property [1] • However it does not satisfy the property 2) and for that reason it is not so appropriate to estimate gravitational waves. Taking the case of the spherically symmetric space-time in the above gauge, let us explain the reason. In the maximal slicing, the vacuum exterior satisfies the following relation;
-- 1 + M. + O(r_4)' 1 - MI2r = 1 + M / 2 r + Clr + O ( r - 2 ) '
(01)
where M is the gravitational mass of the system and (71 is a time dependent constant. T h e source terms of the evolution equations of the extrinsic curvature in the vacuum region become
--sit d + aRij ~ O(r-3).
(02)
This means K o and 7ij become ,-, t / r 3 and ~ t2/r 3, respectively, with time evolution• In the two or three dimensional simulations, the amplitude of gravitational waves is proportional to r -1. According to the above estimation, the non-wave parts of the metric has t2/r 3 dependence in the maximal slicing. When gravitational waves pass through the outer boundary of numerical grid, t is larger than r. Hence the non-wave parts of the metric become O(r-1). Therefore it seems to b e difficult to estimate gravitational waves in the maximal slice because of the same r dependence in the non-wave and the wave parts of the metric. To overcome above difficulties, we propose the following slice. We define e -- ~ - 1, where ~ is appropriately determined conformal factor. If the background metric is the Schwarzschild metric, ~ = M/2r. In order to fix the exterior vacuum geometry as a Schwarzschild metric, we choose the lapse function as follows. 2
$
$
a = e-2'-~ ~ - ~ .
(03)
T h e main source term of the evolution equation of the extrinsic curvature is aRij - alij. In this lapse function, it is written as
ot~'j +
2a[~(1
+ 2e + 2e 2 + 2es + 2~4 +
~5)
T
~J(3
+ 6~ + 6 ~ 2 + 6~3 + 6~4 + 12~ 5 + 10~ 6 + 8 ~ r + 6~8 + 4~ 9 + 2~ t°)
~,,k4,'k,, 7ijT(i + 2t + 2E2 + 2t3 + 2t 4 + 2~5) -- 7ij
(04)
.
],
where 7/i is defined by ~/j = ~-47/j, V denotes the derivative operator with respect to $/i
Conformal Time Slicing Condition and Rij is the Ricci tensor of 7ij- In the gauge conditions such as the minimal distortion gauge [21 or the isothermal gauge [31 , the leading parts of $/j are the wave parts and the first term of the right hand side of Eq.(04)denotes the source of gravitational waves. Therefore other terms of Eq.(04)are regarded as the source for the non-wave parts, the lowest order part is the spherically symmetric part. At first let us consider the spherically symmetric case. In this case, ~ = l + M / 2 r + O ( r - n ) , n > 4, for large r, so that the source terms of the evolution equations of Kij becomes r -n+2. This means that Kij ~ t / r n+2 and ~fij "~ t2/r n+2 < 0(r-4) for large r. In the two and three dimensional case, ~ = 1 + M / 2 r + IijzizJ/r 5 Jr O(r-4), so that we must consider the next order of the spherically symmetric component. However we can easily show that the higher order terms contribute to the source terms at most 0 ( r - 6 ) . Therefore as for the non-wave parts, Kij "~ t / r 6 and 7ij "~ t2/r 6. Since the time dependent non-wave parts have only ,,~ O(r -4) amplitude when gravitational waves reach the outer boundary, we can calculate the amplitude of gravitational waves up to O(r~n~ ) accuracy, where rmaz is the maximum distance in unit of mass of the system from the origin in numerical calculations. As for the highly relativistic region, e ~ 1. This means ct ~ 5 x 10 -2. For e = 1.5, c~ = 2 x 10 -4. Hence the lapse function will sufficiently freeze in the dense region.
N U M E R I C A L T E S T OF C O N F O R M A L SLICING To investigate the conformal slice, we perform the numerical calculation of the spherically symmetric dust collapse. The detailed equations are shown in another paper [4] . As an initial condition, we use the time symmetric initial condition, Ji = 0 and K~ = 0, and give a properly determined dust configuration. In figure 1 we show the proper time at the origin as a function of coordinate time T. It indicates that the proper time at the origin freezes for the sufficient large coordinate time. This is favorable to estimate gravitational waves because gravitational waves will be able to propagate in the wave zone while at the origin the singularity can be avoided. In figure 2 we show how the lapse function(a) behaves at the origin when the conformal h c t o r ( $ ) becomes large in both the conformal slice and the maximal slice. From this figure it is found that the lapse function in our slice freezes faster than that in the maximal slice, so it seems to be useful to avoid the singularity. In figure 3 we show the components of the extrinsic curvature I(rr -- K1. It falls r - " with n ,~ 10, and remains small. This means that in this slice the outer region is guaranteed to be nearly static. To compare our slice with maximal slice, we show Krr calculated in the maximal slice with the same initial condition as in our slice in figure 4. Near the outer most grid Krr oc r -3. Thus in this slice - 1 - M / 2 r behaves ,~ t2/r3ma~: ~> r ~ in the outer most grid. Therefore in the maximal slicing condition it seems to be difficult to analyze gravitational waves accurately.
SUMMARY In this paper we suggest a new slicing condition in which it may be possible to analyze gravitational waves in two and three dimensional numerical relativity. In our slice we can
447
448
M. S ~
and T. Naknmura
keep the time variation of the non-wave parts of the metric to O ( r ~ ) , to the maxima] slice in which we can keep them only to O ( r ~ ) .
n _> 4, in contrast
This work was partly supported by a Grant-in-Aid Scientific Rest'arch on Priority Area of Ministry of Education, Science and Culture(04234104).
REFERENCES 1. D. M. Eardley and L. Smarr, Phys. Rev. D 1 9 (1979), 2239 2. L. Smarr and J. W. York Jr. Phys. Rev. D 1 7 (1978), 1945 3. J. M, Baxdeen and T. Piran, Phy. Rep 96(1983), 205 4. M. Shibata and T. Nakamura, Prog. Theor. Phys. to be published 14.0
o~ 0.7
Proper time at the origin ,
.
l
l
,
.
.
l
,
.
,
|
,
,
.
i
.
.
l
12.0
0.6
10.0
0.5
8.0
0.4
6.0
0.3
4.0
0.2
2.0
0.1
O.Ooo
.
.
l
l
l
t
9.2
l
-K1 10-2
t
t
i
18.4
l
i
i
27.6
l
l
l
36.8
I
'46.0
o.~
~x \~
maximal slice.
\
, . , . I
1.5
nformal slice
. . . .
t,"~',~
2.0
.
I'~,,
2.5
T
,t
Fig.1
Fig.2
K1 I
10-3
- - -T=5.1 T=19.0 . . . . . T=30.0 - - -T=46.1
-K1
10"2
10"4
10"4
lO-S
1O'S
10-~ 0
~bVS(X
! Io.0
2.5
r (log scale) Fig.3
~ 1
K1 ~_"._.
10"49.
'
10.0 r (log scale) Fig.4
,
3.0
,
,
,
3.5
R
4.0
T=5.4 - - - -T=19.2 . . . . . T=28.2 , . . . . . T=30.6
12.5