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Zero-lapse loci in asymptotically flat maximally foliated spacetime manifolds
Volume 84A, number 2
PHYSICS LETTERS
13 July 1981
ZERO-LAPSE LOCI IN ASYMPTOTICALLY FLAT MAXIMALLY FOLIATED SPACETIME MANIFOLDS Mauro CARFORA Department of Physics F02.3, University of Texas at Dallas, Richardson, TX 75080, USA Received 23 February 1981
3, in an asymptotically
We settle the question of the occurrence ofzero-lapse loci for maximal foliations {S~J,S~ A flat spacetime.
Let (V4 ,g) be the Cauchy development [1,2] of a regular, asymptotically euclidean initial data set (S 0, 3 is the three-manifold carrier h0 , K0), where S0 andA (h of the initial data, 0 , K0) are tensor fields on S 0 representing, in the final spacetime, the induced niemannian three-metric on S0 and its initial rate of deformation, respectively. We shall assume thattrace (S0,Kh0) is harmonically em4,g), viz, bedded in(V = 0. Under con4 g) can be partially0 foliated in asuch suitable ditions (V neighbourhood of S 0, by a one-parameter family maximal hypersurfaces{S~}diffeomorphic to eachof other. The lapse function N characterizing such foliation (the proper-time separation between two nearby slices of{S~})is uniquely defined to be the solution, in each slice S~,of the second-order elliptic equation i k +i 1 [(8irk/c4 )(T.knn 2T)+4KIkK
and consider the maximal foliation {S~}as defining an Then incompressible and irrotational “fluid of field reference” I’. we can associate with{S~}a scalan Usa 2 logN and a space-like vector field G ~i~l U c (~~dfsa gradf + n(f)n being Cattaneo’s transverse —
gradient [4]), in which one easily recognizes the relative scalar gravitational potential and the relative dragging gravitational field with respect to F, respectively. In terms of Uand G eq. (1) can be rewritten as 2)(nmnkTik + ~ T) ~ U ~c2k.kkik _(871k1c + c2G.G1 (2) .
Eq. (2) simply tells us what instantaneous acceleration (A —G) must be provided to the observers of F in order to balance the focusing effect of matter and gra-
(1)
vitation in such a way to maintain, in time, the incom-
with tI: Laplace—Beltrami operator in the given S~, n: the unit time-like future-pointing normal field conresponding to the foliation{S~},T: the energy—momentum tensor describing matter fields present, K: the trace-free part of thethe rate-of-deformation tenson (the shear tensor), k: the newtonian constant, and i, k, = 1,2,3,4. Of course eq. (1) must be provided with suitable asymptotic conditions [N—i = O(r~), h = O(r1),K = O(r2), rbeing the euclidean distance in the region external to some compact set], and with inner boundary conditions on the two surfaces separating, in each S~,matter from vacuum, We shall adopt a well-known interpretation [3,4]
pressibility condition (trK— 0). One should note that, according to eq. (2), such an acceleration results from an instantaneous balance between the focusing inducing terms [1] nmnkTjk +1~T and KikK~’,and energy associated with the the defocusing density (4irk) acceleration fieldG.G itself. This typical relativistic feedback is responsible for the peculiar behaviour of maximal foliations in strong field regions. As a direct consequence of the strong maximum principle [5] which holds for eq. (1), and of Hawking’s strong energy condition [1] (Tlknhnk + ~T> 0 for any time-like vector u), one can show [6]that U is a nonnegative function and that the flux of G over any twosphere, in S~,is a non-positive quantity. Consequently,
tIN
...,
jkl
—
1N 0,
.
..
.
.
—
53
Volume 84A, number 2
PHYSICS LETTERS
m~= —(4irk)~~ G~’d~,
(3)
S2 2 is the two-sphere intercepted at spatial infiwhere S nity by the given S~,is non-negative and uniformly bounded for any (s~~ h, K). One applying, over a given slice S~,Gauss’s theorem to eq. (2) one obtains an allspace integral representation for this functional: =
f
-
(4irkc2)’G~G’]dS~
going to zero abruptly, outside some compact set. and that it and K~,CK”~, and their respective gradients, are uniformly bounded in each S . Then, on applying a theorem due to Yau [9] (theorem 3, pp. 216). it is not difficult to verify that corresponding to a solution
of eq. (1) we have the gradient bound: gradN~~B(N+ b)
(5)
,
where B is a constant depending on the upper bound (in absolute value) of the Ricci curvature of the given S~and b is any constant such that b + inf~N> 0.
~.
[p + Tic~+ c2(l6xk)_lKjkKik
SN —
13 July 1981
(4)
where jic2 and r, respectively, are the relative energy density and the trace of the relative stress tensor of the sources with respect to the frame F. According to eq. (4) and to the interpretation of eq. (2) we can think of m~as representing the total focusing mass associated with a given slice S~.The properties of m~have been previously exploited for quite different purposes by various authors, notably Pirani, Komar, Misner, and Cattaneo (see ref. [7] for extensive references). Notice that in general m is not conserved. The maximal character of the foliation {S~}ensures that the frame F does not develop coordinate singularities by focusing effects. As a consequence we may use it to propagate the initial data off S~,via Einstein’s equations, provided that U remains bounded or, equivalently, if the lapse function N does not go to zero during the propagation process (for other caveats see ref. [8]). As soon as this occurs the evolution, with respect to F, is actually halted in the region where N-÷0 forcing us to introduce a new frame of reference in order to follow the data into the future. Because of the importance of maximal slicings in investigating, either analytically or numerically, the dynamics of the gravitational field, attention has been paid to the question of the occurrence of their zerolapse loci. Here we settle this question when S~ H 3, without making use of particular examples or heuristic models and without making assumptions on the nature of singularities present in the spacetime considered [2,3,81. Given a slice S~we shall denote by ~ the domain bounded by the generic equipotential two-surface U =
Successive applications of Gauss’s theorem to eq. (1) over the regions S~and SN\~Nyield, taking into account (5), the two estimates 0 ~ fflf~N~ m~ X
f
—2
-.
[P + nc2 + c2(l6~k)
KikK~1dS~
Sx
(6)
(4irk/c2)in~
B(NI~~+ c) — [meas(a~~)]’
—
f N [(8xk/c4)(ninkTik
+ ~ T) + ~ ~kK1k1
dS~
> B inf N (7) N where b + inf~N is positive but otherwise arbitrary. The first estimate (6) suggests that the ratio between the effective focusing mass mA and the “bare mass”
f
[(~
+ nc2) + c2(l6lrk)lK.kKth]dSx
,
(8)
1
const. For the sake of simplicity we shall assume that
is the natural strength parameter which controls the vanishing of the lapse. Smarr and York [3], in their heuristic “flat space with curvature” model, use essentially a quantity analogous to (8) in discussing the qualitative behaviour of N. As m~is uniformly bounded, we would expect that the lapse function corresponding to a maximal foliation tends to vanish when the bare mass (8) grows larger and larger, i.e. when the source fields are, with respect to the frame F, in an extreme relativistic regime (p -+ oo) and/or when the (shear)-deformation
~
energy c4(l61rk)1KlfrK~/’grows unboundedly large.
is topologically a two-sphere. We shall further sup-
pose that nmnkTjk + ~T vanishes smoothly, instead of 54
Admittedly, such a criterion leads to a somewhat am-
Volume 84A, number 2
PHYSICS LETTERS
biguous characterization of the vanishing of the lapse. For the kind of regime in which p, i-, and K are depends largely on the particular relative motion occurring between the frame F and the frame co-moving with the source fields. It is difficult to separate this kinematical information from the conditions imposed on F, via eq. (1), by the dynamics of the gravitational field itself (e.g. by the existence of spacetime singularities). In this connection characterizations of the vanishing of the lapse for maximal foliations as
“singularity avoidance”, suggested by many examples and also by (6), are not completely correct. Smarr and York [3], by means of their heuristic model and examining some particular samples (e.g. ref. [10]), suggested that such a behaviour of the lapse indicates
13 July 1981
supp {.t + n/c2 + c2(l 62rk)1Klk K°”} If the evolution of (p, r, K) with respect to F is such that meas(a~2~) ~
for some A, then the lapse function will collapse. In this way maximal slices avoid those regions for which the negative potential energy associated with the acceleration field G would exceed the positive contribution to m~due to nmn~cTik+ ~T andKik K”. Finally we may remark that this particular behaviour is not peculiar to maximal foliations only. For it is manifested by more general classes of foliations (cf. ref. [6] for a discussion of this point), as, for instance, those defined by
that maximal slices try to avoid global regions of small
volume.
8~(trK)=0, trK
That this is in a suitable sense the correct charactenization follows in a simple way from our second estimate (7). First, let us suppose that the main contribution to the bare focusing mass (8) comes from a sequence of domains E~C S~of smaller and smaller surface area (imagine, for instance, that the matter is concentrated in smaller and smaller regions). Under this assumption the volume integral over SN\~xappeaning in (7) can be neglected and, since NIaEX ~ 1, (7) yields 1(47rk/c2)m~+ e ~ inf~N. (9) [Bmeas(a~~)] This latter estimate implies that as soon as the surface area of one of the domains ~ is such that meas (a~~) ~ B1 (4irk/c2 )m~, (10)
0f(x) where f(x) is an assigned function on the initial slice 5~ Thisis proved by observing that the lapse function governing such foliations is determined again as a solution of eq. (1)(withK in place of K). I am deeply grateful to W. Rindler for his warm encouragement and for some useful discussions. This work has been supported in part by a research grant from the Italian National Research Council (CNR), and by Organized Research Funds of the University of Texas at Dallas.
—
we shall have inf~N ~
.
That is, inf~Nwill attain arbitrarily small values and the lapse function will eventually collapse. According to (10) the effective focusing mass m~ is the quantity which sets a limit to the actual size of a compact region where a maximal slice can fail to exist. The density of bare focusing mass, appearing as integrand in (8), tells us if this limit region is actually reached. For instance, let us suppose that such a mass
density has compact support, ~
in S~
References [1] S.W. Hawking and G.F.R. Ellis, The large scale structure of space—time (Cambridge, 1973). [2] J.W. York Jr., in: Sources of gravitational radiation, ed. L. Smarr (Cambridge, 1979) p. 83. [3] L. Smarr and J.W. York Jr., Phys. Rev. 17D (1978)
2529. [4] C. Cattaneo, Formulation relative des lois physiques en relativité gén~ra1e,publications du College de France (1961). [5] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order (Springer, Berlin, 1977). [6] M. Carfora, Poisson gauged foliations, in preparation. [7] C. Cattaneo, Ann. Inst. H. Poincaré 4A (1966) 1. [8] J.E. Marsden and F). Tipler, Phys. Rep. 66 (1980) 109. [9] S.-T. Yau, Commun. Pure AppI. Math. 28 (1975) 201. [10] D. Eardley and L. Smarr, Phys. Rev. 19D (1979) 2239.
55
PHYSICS LETTERS
13 July 1981
ZERO-LAPSE LOCI IN ASYMPTOTICALLY FLAT MAXIMALLY FOLIATED SPACETIME MANIFOLDS Mauro CARFORA Department of Physics F02.3, University of Texas at Dallas, Richardson, TX 75080, USA Received 23 February 1981
3, in an asymptotically
We settle the question of the occurrence ofzero-lapse loci for maximal foliations {S~J,S~ A flat spacetime.
Let (V4 ,g) be the Cauchy development [1,2] of a regular, asymptotically euclidean initial data set (S 0, 3 is the three-manifold carrier h0 , K0), where S0 andA (h of the initial data, 0 , K0) are tensor fields on S 0 representing, in the final spacetime, the induced niemannian three-metric on S0 and its initial rate of deformation, respectively. We shall assume thattrace (S0,Kh0) is harmonically em4,g), viz, bedded in(V = 0. Under con4 g) can be partially0 foliated in asuch suitable ditions (V neighbourhood of S 0, by a one-parameter family maximal hypersurfaces{S~}diffeomorphic to eachof other. The lapse function N characterizing such foliation (the proper-time separation between two nearby slices of{S~})is uniquely defined to be the solution, in each slice S~,of the second-order elliptic equation i k +i 1 [(8irk/c4 )(T.knn 2T)+4KIkK
and consider the maximal foliation {S~}as defining an Then incompressible and irrotational “fluid of field reference” I’. we can associate with{S~}a scalan Usa 2 logN and a space-like vector field G ~i~l U c (~~dfsa gradf + n(f)n being Cattaneo’s transverse —
gradient [4]), in which one easily recognizes the relative scalar gravitational potential and the relative dragging gravitational field with respect to F, respectively. In terms of Uand G eq. (1) can be rewritten as 2)(nmnkTik + ~ T) ~ U ~c2k.kkik _(871k1c + c2G.G1 (2) .
Eq. (2) simply tells us what instantaneous acceleration (A —G) must be provided to the observers of F in order to balance the focusing effect of matter and gra-
(1)
vitation in such a way to maintain, in time, the incom-
with tI: Laplace—Beltrami operator in the given S~, n: the unit time-like future-pointing normal field conresponding to the foliation{S~},T: the energy—momentum tensor describing matter fields present, K: the trace-free part of thethe rate-of-deformation tenson (the shear tensor), k: the newtonian constant, and i, k, = 1,2,3,4. Of course eq. (1) must be provided with suitable asymptotic conditions [N—i = O(r~), h = O(r1),K = O(r2), rbeing the euclidean distance in the region external to some compact set], and with inner boundary conditions on the two surfaces separating, in each S~,matter from vacuum, We shall adopt a well-known interpretation [3,4]
pressibility condition (trK— 0). One should note that, according to eq. (2), such an acceleration results from an instantaneous balance between the focusing inducing terms [1] nmnkTjk +1~T and KikK~’,and energy associated with the the defocusing density (4irk) acceleration fieldG.G itself. This typical relativistic feedback is responsible for the peculiar behaviour of maximal foliations in strong field regions. As a direct consequence of the strong maximum principle [5] which holds for eq. (1), and of Hawking’s strong energy condition [1] (Tlknhnk + ~T> 0 for any time-like vector u), one can show [6]that U is a nonnegative function and that the flux of G over any twosphere, in S~,is a non-positive quantity. Consequently,
tIN
...,
jkl
—
1N 0,
.
..
.
.
—
53
Volume 84A, number 2
PHYSICS LETTERS
m~= —(4irk)~~ G~’d~,
(3)
S2 2 is the two-sphere intercepted at spatial infiwhere S nity by the given S~,is non-negative and uniformly bounded for any (s~~ h, K). One applying, over a given slice S~,Gauss’s theorem to eq. (2) one obtains an allspace integral representation for this functional: =
f
-
(4irkc2)’G~G’]dS~
going to zero abruptly, outside some compact set. and that it and K~,CK”~, and their respective gradients, are uniformly bounded in each S . Then, on applying a theorem due to Yau [9] (theorem 3, pp. 216). it is not difficult to verify that corresponding to a solution
of eq. (1) we have the gradient bound: gradN~~B(N+ b)
(5)
,
where B is a constant depending on the upper bound (in absolute value) of the Ricci curvature of the given S~and b is any constant such that b + inf~N> 0.
~.
[p + Tic~+ c2(l6xk)_lKjkKik
SN —
13 July 1981
(4)
where jic2 and r, respectively, are the relative energy density and the trace of the relative stress tensor of the sources with respect to the frame F. According to eq. (4) and to the interpretation of eq. (2) we can think of m~as representing the total focusing mass associated with a given slice S~.The properties of m~have been previously exploited for quite different purposes by various authors, notably Pirani, Komar, Misner, and Cattaneo (see ref. [7] for extensive references). Notice that in general m is not conserved. The maximal character of the foliation {S~}ensures that the frame F does not develop coordinate singularities by focusing effects. As a consequence we may use it to propagate the initial data off S~,via Einstein’s equations, provided that U remains bounded or, equivalently, if the lapse function N does not go to zero during the propagation process (for other caveats see ref. [8]). As soon as this occurs the evolution, with respect to F, is actually halted in the region where N-÷0 forcing us to introduce a new frame of reference in order to follow the data into the future. Because of the importance of maximal slicings in investigating, either analytically or numerically, the dynamics of the gravitational field, attention has been paid to the question of the occurrence of their zerolapse loci. Here we settle this question when S~ H 3, without making use of particular examples or heuristic models and without making assumptions on the nature of singularities present in the spacetime considered [2,3,81. Given a slice S~we shall denote by ~ the domain bounded by the generic equipotential two-surface U =
Successive applications of Gauss’s theorem to eq. (1) over the regions S~and SN\~Nyield, taking into account (5), the two estimates 0 ~ fflf~N~ m~ X
f
—2
-.
[P + nc2 + c2(l6~k)
KikK~1dS~
Sx
(6)
(4irk/c2)in~
B(NI~~+ c) — [meas(a~~)]’
—
f N [(8xk/c4)(ninkTik
+ ~ T) + ~ ~kK1k1
dS~
> B inf N (7) N where b + inf~N is positive but otherwise arbitrary. The first estimate (6) suggests that the ratio between the effective focusing mass mA and the “bare mass”
f
[(~
+ nc2) + c2(l6lrk)lK.kKth]dSx
,
(8)
1
const. For the sake of simplicity we shall assume that
is the natural strength parameter which controls the vanishing of the lapse. Smarr and York [3], in their heuristic “flat space with curvature” model, use essentially a quantity analogous to (8) in discussing the qualitative behaviour of N. As m~is uniformly bounded, we would expect that the lapse function corresponding to a maximal foliation tends to vanish when the bare mass (8) grows larger and larger, i.e. when the source fields are, with respect to the frame F, in an extreme relativistic regime (p -+ oo) and/or when the (shear)-deformation
~
energy c4(l61rk)1KlfrK~/’grows unboundedly large.
is topologically a two-sphere. We shall further sup-
pose that nmnkTjk + ~T vanishes smoothly, instead of 54
Admittedly, such a criterion leads to a somewhat am-
Volume 84A, number 2
PHYSICS LETTERS
biguous characterization of the vanishing of the lapse. For the kind of regime in which p, i-, and K are depends largely on the particular relative motion occurring between the frame F and the frame co-moving with the source fields. It is difficult to separate this kinematical information from the conditions imposed on F, via eq. (1), by the dynamics of the gravitational field itself (e.g. by the existence of spacetime singularities). In this connection characterizations of the vanishing of the lapse for maximal foliations as
“singularity avoidance”, suggested by many examples and also by (6), are not completely correct. Smarr and York [3], by means of their heuristic model and examining some particular samples (e.g. ref. [10]), suggested that such a behaviour of the lapse indicates
13 July 1981
supp {.t + n/c2 + c2(l 62rk)1Klk K°”} If the evolution of (p, r, K) with respect to F is such that meas(a~2~) ~
for some A, then the lapse function will collapse. In this way maximal slices avoid those regions for which the negative potential energy associated with the acceleration field G would exceed the positive contribution to m~due to nmn~cTik+ ~T andKik K”. Finally we may remark that this particular behaviour is not peculiar to maximal foliations only. For it is manifested by more general classes of foliations (cf. ref. [6] for a discussion of this point), as, for instance, those defined by
that maximal slices try to avoid global regions of small
volume.
8~(trK)=0, trK
That this is in a suitable sense the correct charactenization follows in a simple way from our second estimate (7). First, let us suppose that the main contribution to the bare focusing mass (8) comes from a sequence of domains E~C S~of smaller and smaller surface area (imagine, for instance, that the matter is concentrated in smaller and smaller regions). Under this assumption the volume integral over SN\~xappeaning in (7) can be neglected and, since NIaEX ~ 1, (7) yields 1(47rk/c2)m~+ e ~ inf~N. (9) [Bmeas(a~~)] This latter estimate implies that as soon as the surface area of one of the domains ~ is such that meas (a~~) ~ B1 (4irk/c2 )m~, (10)
0f(x) where f(x) is an assigned function on the initial slice 5~ Thisis proved by observing that the lapse function governing such foliations is determined again as a solution of eq. (1)(withK in place of K). I am deeply grateful to W. Rindler for his warm encouragement and for some useful discussions. This work has been supported in part by a research grant from the Italian National Research Council (CNR), and by Organized Research Funds of the University of Texas at Dallas.
—
we shall have inf~N ~
.
That is, inf~Nwill attain arbitrarily small values and the lapse function will eventually collapse. According to (10) the effective focusing mass m~ is the quantity which sets a limit to the actual size of a compact region where a maximal slice can fail to exist. The density of bare focusing mass, appearing as integrand in (8), tells us if this limit region is actually reached. For instance, let us suppose that such a mass
density has compact support, ~
in S~
References [1] S.W. Hawking and G.F.R. Ellis, The large scale structure of space—time (Cambridge, 1973). [2] J.W. York Jr., in: Sources of gravitational radiation, ed. L. Smarr (Cambridge, 1979) p. 83. [3] L. Smarr and J.W. York Jr., Phys. Rev. 17D (1978)
2529. [4] C. Cattaneo, Formulation relative des lois physiques en relativité gén~ra1e,publications du College de France (1961). [5] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order (Springer, Berlin, 1977). [6] M. Carfora, Poisson gauged foliations, in preparation. [7] C. Cattaneo, Ann. Inst. H. Poincaré 4A (1966) 1. [8] J.E. Marsden and F). Tipler, Phys. Rep. 66 (1980) 109. [9] S.-T. Yau, Commun. Pure AppI. Math. 28 (1975) 201. [10] D. Eardley and L. Smarr, Phys. Rev. 19D (1979) 2239.
55