7 Appendix: Conformal Rescalings And Conformal Infinity5 Acknowledgements

6 Appendix: Reduction Of The Conformal Field Equations 

In this appendix we show how to perform the reduction process for the conformal field equations to obtain the symmetric hyperbolic system of evolution equations and the constraints. We assume that we are given a time-like unit vector tex2html_wrap_inline4584 with respect to which the reduction is done. Any vector tex2html_wrap_inline5384 may be decomposed into parts perpendicular and parallel to tex2html_wrap_inline4584,

displaymath5388

and similarly, for one-forms tex2html_wrap_inline5390 . We call a vector spatial with respect to tex2html_wrap_inline4584 if it is orthogonal to tex2html_wrap_inline4584 . In particular, the metric tex2html_wrap_inline3923 gives rise to a spatial metric tex2html_wrap_inline4612 by the decomposition

equation1005

The volume four-form tex2html_wrap_inline5400 which is defined by the metric also gives rise to a decomposition as follows:

  equation1010

The covariant derivative operator tex2html_wrap_inline4099 is written analogously,

displaymath5404

thus defining two derivative operators tex2html_wrap_inline5406 with tex2html_wrap_inline5408 and tex2html_wrap_inline5410 . The covariant derivative of tex2html_wrap_inline4584 itself is an important field. It gives rise to two component fields defined by

  equation1018

Note that tex2html_wrap_inline5414 is spatial in both its indices and that there is no symmetry implied between the two indices. Similarly, tex2html_wrap_inline5416 is automatically spatial.

It is useful to define two new derivative operators tex2html_wrap_inline4618 and tex2html_wrap_inline4594 by the following relations:

  equation1023

These operators have the property that they are compatible with the spatial metric tex2html_wrap_inline4612 and that they annihilate tex2html_wrap_inline4584 and tex2html_wrap_inline4598 . If tex2html_wrap_inline4584 is the unit normal of a hypersurface, i.e., if tex2html_wrap_inline4598 is hypersurface orthogonal, then tex2html_wrap_inline4606 is symmetric, tex2html_wrap_inline4612 is the induced (negative definite) metric on the hypersurface, and tex2html_wrap_inline4594 is its Levi-Civita connection. In general this is not the case and so the operator tex2html_wrap_inline4594 possesses torsion. In particular, we obtain the following commutators (acting on scalars and spatial vectors):

  equation1032

  equation1039

  equation1054

  equation1059

These commutators are obtained from the commutators between the derivative operators tex2html_wrap_inline5406 and D by expressing them in terms of tex2html_wrap_inline4594 and tex2html_wrap_inline4618 on the one hand, and by the four-dimensional connection tex2html_wrap_inline4099 on the other hand. This procedure yields two equations for the derivatives of tex2html_wrap_inline4584,

  equation1069

  equation1081

The information contained in the commutator relations and in the Equations (45Popup Equation) and (46Popup Equation) is completely equivalent to the Cartan equations for tex2html_wrap_inline4099 which define the curvature and torsion tensors.

This completes the preliminaries and we can now go on to perform the splitting of the equations. Out intention is to end up with a system of equations for all the spatial parts of the fields. In order not to introduce too many different kinds of indices, all indices refer to the four-dimensional space-time, but they are all spatial, i.e., any transvection with tex2html_wrap_inline4598 and tex2html_wrap_inline4584 vanishes. If we introduce hypersurfaces with normal vector tex2html_wrap_inline4584 then there exists an isomorphism between the tensor algebra on the hypersurfaces and the subalgebra of spatial four-dimensional tensors.

We start with the tensorial part of the equations. To this end we decompose the fields into various spatial parts and insert these decompositions into the conformal field equations defined by (17Popup Equation, 18Popup Equation, 19Popup Equation, 20Popup Equation, 21Popup Equation). The fields are decomposed as follows:

equation1097

equation1115

equation1122

The function tex2html_wrap_inline4803 is fixed in terms of tex2html_wrap_inline4843 because tex2html_wrap_inline4340 is trace-free.

Inserting the decomposition of tex2html_wrap_inline3877 into Equation (18Popup Equation), decomposing the equations into various spatial parts and expressing derivatives in terms of the operators tex2html_wrap_inline4594 and tex2html_wrap_inline4618 yields four equations:

eqnarray1128

eqnarray1145

Here we have defined tex2html_wrap_inline5472 . Treating the other fields and equations in a similar way, we obtain Equation (17Popup Equation) in the form of four equations:

eqnarray1183

The equation (19Popup Equation) for the conformal factor are rather straightforward. We obtain

eqnarray1221

while Equation (20Popup Equation) yields four equations:

eqnarray1224

Finally, the equation (21Popup Equation) for S gives two equations

eqnarray1233

This completes the gauge independent part of the equations. In order to deal with the gauges we now have to introduce an arbitrary tetrad and arbitrary coordinates. We extend the time-like unit vector to a complete tetrad tex2html_wrap_inline5476 with tex2html_wrap_inline5478 for i =1,2,3. Let tex2html_wrap_inline5482 with tex2html_wrap_inline5484 be four arbitrary functions which we use as coordinates. Application of tex2html_wrap_inline4618 and tex2html_wrap_inline4594 to the coordinates yields

equation1236

The four functions tex2html_wrap_inline5490 and the four one-forms tex2html_wrap_inline5492 may be regarded as the 16 expansion coefficients of the tetrad vectors in terms of the coordinate basis tex2html_wrap_inline5494 because of the identity

eqnarray1238

In a similar spirit, we apply the derivative operators to the tetrad and obtain

eqnarray1240

Again, transvection with tex2html_wrap_inline4584 on any index of tex2html_wrap_inline5498 and tex2html_wrap_inline5500 vanishes. Furthermore, both tex2html_wrap_inline5502 and tex2html_wrap_inline5504 are antisymmetric in their (last two) indices. Together with the 12 components of tex2html_wrap_inline5506 and tex2html_wrap_inline5414 these fields provide additional 12 components which account for the 24 connection coefficients of the four-dimensional connection tex2html_wrap_inline4099 with respect to the chosen tetrad.

Note that these fields are not tensor fields. They do not transform as tensors under the change of tetrad. Since we will keep the tetrad fixed here, we may, however, regard them as defining tensorfields whose components happen to coincide with them in the specified tetrad.

In order to extract the contents of the first of Cartan's structure equations one needs to apply the commutators (41Popup Equation) and (43Popup Equation) to the coordinates to obtain

eqnarray1253

Similarly, the second of Cartan's structure equations is exploited by applying the commutators to the tetrad vectors. Equation (16Popup Equation) is then used to substitute for the Riemann tensor in terms of the gravitational field, the trace-free part of the Ricci tensor, and the scalar curvature. Apart from the Equations (45Popup Equation) and (46Popup Equation), which come from acting on tex2html_wrap_inline4584, this procedure yields

eqnarray1260

Now we have collected all the equations which can be extracted from the conformal field equations and Cartan's structure equations. What remains to do is to separate them into constraints and evolution equations. Before doing so, we notice that we do not have enough evolution equations for the tetrad components and the connection coefficients. The remedy to this situation is explained in Section  3.2 . It amounts to adding appropriate ``divergence equations''. We obtain these by computing the ``gauge source functions''. The missing equations for the coordinates are obtained by applying the d'Alembert operator to the coordinates. Expressing the wave operator in terms of tex2html_wrap_inline4618 and tex2html_wrap_inline4594 yields the additional equations

equation1294

In order to find the missing equations for the tetrad we need to compute the gauge source functions

  equation1296

In a similar way as explained above, we may regard these functions as components of tensorfields tex2html_wrap_inline4628 and tex2html_wrap_inline4626 whose components happen to agree with them in the specified basis. Thus,

displaymath5522

Computing these tensorfields from (76Popup Equation) gives

eqnarray1312

Now we are ready to collect the constraints:

  equation1330

  eqnarray1335

  equation1356

  equation1370

  eqnarray1384

  equation1402

  equation1413

  equation1424

  equation1428

  equation1437

  equation1442

Finally, we collect the evolution equations:

  equation1446

  equation1451

  eqnarray1455

  equation1473

  equation1486

  equation1497

  equation1507

  eqnarray1514

  eqnarray1532

  eqnarray1553

  equation1574

  equation1578

  equation1582

  equation1586

This is the complete system of evolution equation which can be extracted from the conformal field equations. As it is written, this system is symmetric hyperbolic. This is not entirely obvious but rather straightforward to verify. It is important to keep in mind that with our conventions the spatial metric tex2html_wrap_inline4612 is negative . Altogether these are 65 equations.



7 Appendix: Conformal Rescalings And Conformal Infinity5 Acknowledgements

image Conformal Infinity
Jörg Frauendiener
http://www.livingreviews.org/lrr-2000-4
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