Introduction to the 2+2 formalism

1   The 3+1 Formalism

The most frequently used formalism in Numerical Relativity is the 3+1 formalism of Arnowitt, Deser and Misner or ADM for short (see, for example, the article of York in Smarr 1979). From a geometrical viewpoint, it consists of a decomposition of space-time in which time is singled out as a privileged direction and spacetime is foliated by 3-dimensional spacelike hypersurfaces corresponding to constant time slices. Other formalisms used in Numerical Relativity include the (2+1)+1 formulation (Maeda 1980) which uses Geroch decomposition to factor out a Killing direction. The ADM approach is then used to decompose the remaining 3-dimensional spacetime into a 2+1 form. The 2+2 formalism (d'Inverno 1984) decomposes spacetime into two families of 2-dimensional spacelike hypersurfaces. It is closely related to the characteristic approach (Gomez 1986, Bishop 1992) which is specially suited to studying gravitational waves. Finally, there is the Regge calculus approach which is a discrete formulation of General Relativity where blocks of flat spacetime are `glued' together and curvature exists only on the two-dimensional edges of the blocks (Regge 1961).

The 3+1 formalism has a considerable track record of achievement in Numerical Relativity, but it possesses two major limitations. The first is that the initial data is not freely specifiable, but must satisfy the constraints. The conformal approach (O'Murchadha and York 1974) is a powerful technique for achieving this, but it does not reveal what the freely specifiable initial data-the true gravitational degrees of General Relativity-are in clear geometrical terms. The other problem is that the 3+1 approach fails if the foliation goes null (although this may be difficult to detect precisely in a numerical regime where one cannot easily characterize vanishing quantities because of rounding errors). Yet null foliations are important in their own right.

2  The 2+2 approach

Figure 1: 2-dimensional submanifold and two transvecting submanifolds

The basis of this approach is to decompose spacetime into two families of spacelike 2-surfaces. We can view this as a constructive procedure in which an initial 2-dimensional submanifold S0 is chosen in a bare manifold, together with two vector fields v1 and v2 which transvect the submanifold everywhere (Figure 1). The two vector fields can then be used to drag the initial 2-surface out into two foliations of 3-surfaces. The character of these 3-surfaces will depend in turn on the character of the two vector fields. The most important cases are when at least one of the vector fields is taken to be null. For example, if the two vector fields are null we obtain a double-null foliation (indicated schematically in Figure 2), or if one is null and the other is timelike we obtain a null-timelike foliation (Figure 3). An example of a non-null foliation is when one vector field is timelike and the other is spacelike in which case we obtain a timelike-spacelike foliation (Figure 4).

Figure 2: Double null foliation

Figure 3: Null-timelike foliation

Figure 4: Timelike-spacelike foliation

The most elegant way of proceeding is to introduce a formalism which is manifestly covariant and which uses projection operators and Lie derivatives associated with the two vector fields. The resulting formalism is called the 2+2 formalism (Smallwood 1983, d'Inverno and Smallwood 1980). When the vector fields are of a particular geometric character, then this can be refined further into a 2+(1+1) formalism. Finally, in analogy to the conformal approach of the last chapter, one extracts a conformal factor from the spacelike 2-geometries to isolate the gravitational degrees of freedom.

The resulting formalism leads to a number of advantages. First of all, it identifies the two gravitational degrees of freedom in an explicit geometrical way as residing in the conformal 2-geometry (d'Inverno and Stachel 1978). Secondly, the data is unconstrained and satisfies two dynamical equations which are simply odes along the vector fields. Most importantly, the formalism applies to situations where the foliation either is or becomes null. Such ivps are called null or characteristic ivps ( civps for short). They are the natural vehicle for studying gravitational radiation problems (since gravitational radiation propagates along null geodesics), asymptotics of isolated systems (since future and past null infinity are null hypersurfaces) and problems in cosmology (since we gain information about the universe along our past null cone). From a calculational viewpoint, this formalism allows null infinity to be incorporated into the calculational domain and so allows one to define gravitational radiation in an unambiguous manner.

The characteristic approach, however, suffers from one main drawback resulting from the fact that, in general, null hypersurfaces develop caustics. There are two quite distinct ways of proceeding. One approach is to develop techniques for generating solutions through caustics (Corkhill and Stewart 1983). The other is to restrict attention to caustic-free regimes such as the far zone in asymptotically flat regions or systems which are sufficiently close to spherical symmetry. These latter cases still include a number of astronomically interesting scenarios.

3  The 2+2 metric decomposition

As an introduction to the 2+2 formalism we show how it can be used to decompose the metric. In this section Greek indices run from 0 to 3, early Latin indices (a,b,¼) run from 0 to 1, middle Latin indices (i,j,¼) run from 2 to 3, uppercase Latin indices (A,B,¼) run from 1 to 3 (see the notation used in d'Inverno and Vickers 1995). Let M be a four-dimensional orientable manifold with metric g of signature (+1,-1,-1,-1). A foliation of codimension two can be described by two closed 1-forms n0 and n1. Thus locally
dna = 0 Û na = dfa
(1)
The two 1-forms generate hypersurfaces defined by
{S0}
:
f0(xa) = constant
{S1}
:
f1(xa) = constant
respectively. These hypersurfaces define a family of 2-surfaces {S} by
{S} = {S0}Ç{S1}
We restrict attention to the case when {S} is spacelike and denote the family of two dimensional timelike spaces orthogonal to {S} at each point by {T} (Figure 5). Let na be the dyad basis of vectors dual to na in {T}, so that
naanba = dab
(2)
Since na is a 1-form basis for {T}, the vectors na form a basis of vectors for the span of {T}. Note that, in general, [n0,n1] ¹ 0 so that {T} does not form an integrable distribution. If, however, the Lie bracket vanishes then {T} forms a 2-dimensional subspace of M and is said to be holonomic. We use the na to define a 2×2 matrix of scalars Nab by
Nab = gabnaanbb
(3)
with inverse Nab. We may use Nab to relate na and na since
naa = gabNabnbb
(4)
and
naa = gabNabnbb
(5)


We define projection operators into {S} and {T} by

Bab
=
dab-naanab
(6)
Tab
=
naanab
(7)
The 2-metric induced on {S} is given by the projection
2gab = BgaBdbggd = BadBdb = Bab
Similarly, the 2-metric induced on {T} is given by the projection
hab = TgaTdbggd = TagTgb = Tab
Note that the tetrad components of hab are just Nab since
hab = habnaanbb = gabnaanbb = Nab
In particular, the elements N00 and N11 define the lapses of {S} in {S0} and {S1}, respectively.

Figure 5: The timelike 2-space {T} orthogonal to {S} at P

We now choose a pair of vectors Ea which connect neighbouring 2-surfaces in {S}. We choose them such that

naaEba = dab
(8)
which defines Ea up to an arbitrary shift vector ba,
Ea = na+ba
(9)
with
naabca = 0
(10)
Although, in general, the na do not commute, it is always possible to choose ba so that [E0,E1] = 0. Thus, each Ea is tangent to a congruence of curves in {Sa} parametrized by fa(xa). We may, therefore, choose coordinates such that f0(xa) = x0, f1(xa) = x1 with x2 and x3 being constant along the congruence of curves.

In these coordinates

n0 = dx0,    n1 = dx1
and
E0 =
x0
 ,   E1 =
x1
so that
n0
=
E0-b0 = (1,0,b0i)
n1
=
E1-b1 = (0,1,b1i)
This results in the 2+2 decomposition of the contravariant metric
gab = æ
ç
è
Nab
-Nabbbi
-Nabbbi
2gij+Nabbaibbj
 ö
÷
ø
(11)
with inverse
gab = æ
ç
è
Nab+2gijbaibbj
2gijbaj
2gijbaj
2gij
 ö
÷
ø
(12)
where
NabNbc = dac
(13)
Note that in the 2+2 case there is a 2×2 lapse matrix and two shift vectors.

In the 2+2 formalism, the next procedure is to extract the conformal factor g given by

2gij = g 2 _
g
 
ij 
(14)
where
g = |2gij|
(15)
An analysis of the field equations goes on to show that the two gravitational degrees of freedom may be chosen to lie in the conformal 2-structure 2[`g]ij.

4  Applications

One of the key points which arises from the 2+2 formalism is that the dynamical equations consist of two equations which propagate the the conformal 2-structure. In general, choosing a particular form for the lapse matrix Nab is really equivalent to carrying out a 2+(1+1) decomposition, which is a refinement of a 2+2 decomposition. However, references (Smallwood 1983, d'Inverno and Smallwood 1980) involve a more general formalism in which {T} is kept much more on the same footing as {S}. Moreover, the Lie derivatives with respect to the two directions within {T} are encoded in a special 2-dimensional derivative called Da. The general formalism can accommodate the 4 separate cases when {S} and {T} are each either holonomic or anholonomic. Since there are 6 different types of IVP, this means that the formalism can handle all the 24 different cases of holonomic and anholonomic IVPs. In every case the 3-dimensional initial data can be chosen to be the conformal 2-structure. In (d'Inverno and Stachel 1978, Smallwood 1983) a Lagrangian formulation is presented which, in particular, gives insight into the special character of the break up of the field equations. For example, it is shown that the conformal 2-structure generates the dynamical equations in the sense that
d LG
d2 _
g
 
ij 
= Dynamical equations
(16)
where LG is the Einstein Lagrangian [Ö(-g)]R. We have only essentially discussed the question of uniqueness in the somewhat restrictive case of analytic solutions. However, there are some theorems known on the existence and stability for the double null CIVP (Muller Zum Hagen and Seiffert 1977).

The 2+2 approach served as a starting point for work on formulating General Relativity as a first order hyperbolic system (Friedrich and Stewart 1983) and related work in numerically simulating colliding plane gravitational waves (Corkhill and Stewart 1983). As far as applications of the 2+2 approach to Numerical Relativity are concerned, there is the work of the Southampton group on CCM (Cauchy-characteristic matching) which uses mixed IVPs based on the null-timelike foliation of space-time, that of Gnedin and Gnedin on the Cauchy horizon in the Reissner-Nordstrom black hole (Gnedin and Gnedin 1993) and the work of Brady et al (1996) on a version of a covariant 2+2 formalism which is aimed at investigating the Cauchy horizon of a Kerr black hole. There is also some recent work of Huebner on a 2+2 formalism for investigating conformal infinity (Huebner). We have mentioned a 2+2 Lagrangian formulation of General Relativity. There are also two Hamiltonian formulations in existence by Torre (1986) and Hayward (1993). In addition, there is the work on a 2+2 decomposition of Ashtekar's new variables (d'Inverno and Vickers 1995) which is preliminary work also aimed at a Hamiltonian formulation.

Finally, we mention the related but separate work based on the null cone formalism of Winicour and collaborators (Isaacson et al. 1983, Gómez and Winicour 1992). This is a degenerate case of a null-timelike IVP in which the innermost timelike cylinder collapses to a worldline. This approach has led to extensive applications in Numerical Relativity.

References


[1]
Bishop N T, 1992, in Approaches to Numerical Relativity, (ed.) R A d'Inverno, Cambridge University Press, 20.


[2]
Brady P.R., Droz S., Israel W., Morsink S.M., 1996, Class. Quantum Grav., 13, 2211.


[3]
Corkhill R N and Stewart J M, 1983, Proc Roy Soc, A386, 373.


[4]
d'Inverno R A and Smallwood J, 1980, Phys. Rev. D, 22, 1233.


[5]
d'Inverno R A and Stachel J, 1978, J Math Phs, 19, 2447.


[6]
d'Inverno R A and Vickers J A, 1995, Class Quant Grav, 12, 753.


[7]
d'Inverno R A, 1984, in Problems of collapse and numerical relativity, (ed.) D. Bancel and M. Signore, Reidel, Dordrecht, 221.


[8]
Friedrich H., Stewart J.M., 1983, Proc. Roy. Soc., A385, 345.


[9]
Gómez R, Isaacson R A, Welling J S and Winicour J, 1986, in Dynamical Spacetimes and Numerical Relativity, (ed.) J M Centrella, Cambridge University Press, 236.


[10]
Gómez R., and Winicour J., 1992, in Approaches to Numerical Relativity, (ed.) R.A. d'Inverno, Cambridge University Press, Cambridge, 143.


[11]
Gnedin M. L., Gnedin N. Y., 1993, Class. Quantum Grav., 10, 1083.


[12]
Hayward S. A., 1993, Class. Quantum Grav., 10, 779.


[13]
Huebner P. ``A Method for Calculating the Structure of (Singular) Spacetimes in the Large'', University of Jena, Germany.


[14]
Isaacson R. A., Welling J. S., Winicour J., 1983, J. Math. Phys., 24, 1824.


[15]
Maeda K, Sasaki M, Nakamura T and Miyama S, 1980, Prog Theo Phys, 63, 919.


[16]
Muller Zum Hagen H., Seiffert J., 1977, Gen. Rel. Grav., 8, 259.


[17]
O'Murchadha N and York J W, 1974, Phys Rev D, 10, 428.


[18]
Regge T, 1961, Nuov Cim, 19, 558.


[19]
Smallwood J, 1983, J. Math. Phys., 24, 599.


[20]
Smarr L (ed.), 1979, Sources of Gravitational Radiation, Cambridge University Press.


[21]
Torre C. G., 1986, Class. Quantum Grav., 3, 773.

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