The existence of gravitational waves was predicted by Einstein [35] shortly after he had found the general theory of relativity. However, due to the mathematical and physical complexity of the theory of gravitation there was confusion for a long time about whether the field equations really rigorously do admit solutions with a wave-like character. For instance, Rosen [123] came to the conclusion that there were no such solutions because in the class of plane symmetric waves every non-flat solution of the field equations became singular on a two-dimensional submanifold. This result was, however, due to the lack of understanding of the different kinds of singularities which can occur in a covariant theory and the singularity appearing in the plane wave space-times later turned out to be a coordinate singularity.
Thus, one of the early problems in the research area of
gravitational waves was the invariant characterization of
radiation. In 1957, Pirani [119] started the investigation with the suggestion that the
algebraic properties of the Riemann (more specifically the Weyl)
tensor should be considered as indicating the presence of
radiation. In particular, under the assumption that gravitational
radiation can in fact be characterized by the curvature tensor
and that it propagates with the local speed of light, he proposed
the definition that gravitational radiation is present if the
curvature tensor has Petrov types
or
. He arrived at this characterization by the observation that a
gravitational wave-front would manifest itself as a discontinuity
of the Riemann tensor across null hypersurfaces, these being the
characteristics for the Einstein equations. This result had
previously been obtained by Lichnerowicz [90]. In his article, Pirani did not enforce the field equations,
but towards the end of the paper he proposed to look at the
equations
, which follow from the Bianchi identity for vacuum space-times.
This is the first hint at the importance of the Bianchi identity
for the study of gravitational waves. In [91] Lichnerowicz proposed a similar definition for a pure
gravitational radiation field.
The next important step in the development of the subject was Trautman's study of the question of boundary conditions for the gravitational field equations [140]. He wanted to obtain conditions which were general enough to allow for gravitational radiation of an isolated system of matter, but still strong enough to guarantee uniqueness (appropriately defined) of the solution given ``reasonable'' initial data. He gave an asymptotic fall-off condition for the metric coefficients with respect to a certain class of coordinate systems. It was obtained by analogy to the situation with the scalar wave equation and Maxwell theory [139], where the fields can be required to satisfy Sommerfeld's ``Ausstrahlungsbedingung''. In those cases it is known that there exist unique solutions for given initial data, while in the gravitational case this was not known at the time and, in fact, remained unknown until only quite recently.
Trautman then went on to discuss some consequences of this boundary condition. He defined an energy-momentum four-vector at infinity which is well defined as a consequence of the boundary condition. It is obtained as the limit of an integral of the energy-momentum pseudo-tensor over a space-like manifold with boundary as it stretches out to infinity. By application of Stokes' theorem, the three-dimensional integral can be converted to a surface integral over the boundary, the sphere at infinity, of certain components of the so-called ``superpotentials'' for the energy-momentum pseudo-tensor. Nowadays these are recognized as certain special cases of the Nester-Witten two-form [37] and, hence, Trautmans energy-momentum integral coincides with the Bondi-Sachs expression or the ADM expression depending on how the limit to infinity is taken. This, however, is not explicitly specified in the paper. He considered the difference of energy-momentum between two space-like hypersurfaces and concluded that it must be due to radiation crossing the time-like cylinder which together with the two hypersurfaces bounds a four-dimensional volume. An estimate for the amount of radiation showed that it is non-negative. If the limit would have taken out to infinity along null directions, then this result would coincide with the Bondi-Sachs mass-loss formula.
Finally, Trautman observed that the definitions of pure radiation fields given by Pirani and Lichnerowicz are obeyed not exactly but only asymptotically by gravitational fields satisfying his boundary condition. Hence, he concluded that such solutions approach radiation fields in the limit of large distances to the source.
Based on the ideas of Pirani and Trautman and guided by his
own investigations [130] of the structure of retarded linearized gravitational fields of
particles, Sachs [125] proposed an invariant condition for outgoing gravitational
waves. The intuitive idea was that at large distances from the
source, the gravitational field, i.e. the Riemann tensor, of
outgoing radiation should have approximately the same algebraic
structure as does the Riemann tensor for a plane wave. As one
approaches the source, deviations from the plane wave should
appear. Sachs analyzed these deviations in detail and obtained
rather pleasing qualitative insights into the behaviour of the
curvature in the asymptotic regime.
In contrast to the earlier work, Sachs used more advanced
geometrical methods. Based on his experience in the study of
algebraically special metrics [34,
124,
86] he first analyzed the geometry of congruences of null curves.
After the introduction of an appropriate null tetrad he used the
Bianchi identity for the curvature tensor in a form which follows
from the vacuum field equations to obtain the characteristic
fall-off behaviour of the curvature components which has been
termed the ``peeling property''. The Riemann tensor of a vacuum
space-time has this property, if for any given null geodesic with
an affine parameter
r
which extends to infinity, the curvature falls off along the
curve in such a way that to order 1/
r
it is null (Petrov type N or
) with a quadruple PND (principal null direction) along the
curve. To order
it has type 3 (
) with the triple PND pointing along the curve, and to order
it has type 2 (
) with the double PND oriented along the curve. To order
it is algebraically general (type 1 or
) but one of the PND's lies in the direction of the geodesic. To
order
the curvature is not related to the geodesic. Symbolically one
can express this behaviour in the form
where
C
on the left hand side denotes the Weyl tensor, and each
on the right hand side stands for a tensor which has an
r
-fold PND along the null geodesic and which is independent of
r
. The important point is that the part of the curvature which has
no relation to the null direction of the outgoing geodesic goes
as
.
Sachs postulated the outgoing radiation condition to mean that
a bounded source field is free of mixed (i.e. a non-linear
superposition of in- and outgoing) radiation at large distances
if and only if the field has the peeling property. Later it was
realized [26,
102] that this condition does not exclude ingoing radiation. Instead
it is possible to have an ingoing wave profile provided that it
falls off sufficiently fast as a function of an advanced time
parameter.
The study of gravitational waves and the related questions was
the main area of research of the group around H. Bondi and
F. Pirani at King's College, London, in the years between
1955 and 1967. In a series of papers [92,
93,
24,
120,
94,
125,
25,
127
] they analyzed several problems related to gravitational waves
of increasing complexity. The most important of their results
certainly is the work on axisymmetric radiating systems by Bondi,
van der Burg and Metzner [25]. They used a different approach to the problem of outgoing
gravitational waves. Instead of looking at null geodesics they
focused on null hypersurfaces, and instead of analyzing the
algebraic structure of the curvature using the Bianchi identity
they considered the full vacuum field equations. Their work was
concerned with axi-symmetric systems but shortly afterwards
Sachs [127] removed this additional assumption.
The essential new ingredient was the use of a retarded time function. This is a scalar function u whose level surfaces are null hypersurfaces opening up towards the future. Based on the assumption that such a function exists, one can introduce an adapted coordinate system, so-called Bondi coordinates, by labeling the generators of the null hypersurfaces with coordinates on the two-sphere and introducing the luminosity distance r (essentially the square root of the area of outgoing wave fronts) along the null generators. The metric, when written in this kind of coordinate system, contains only six free functions.
Asymptotic conditions were imposed to the effect that one should be able to follow the null geodesics outwards into the future for arbitrarily large values of r . Then the metric was required to approach the flat metric in the limit of infinite distance. Additionally, it was assumed that the metric functions and other quantities of interest (in particular the curvature) were analytic functions of 1/ r .
The field equations in Bondi coordinates have a rather nice
hierarchical structure which is symptomatic for the use of null
coordinates and which allows for a simpler formal analysis
compared to the related Cauchy problem. Bondi et al. and Sachs
were able to solve the field equations asymptotically for large
distances. In essence their procedure amounts to the formulation
of a certain characteristic initial value problem (see [128,
129]) and the identification of the free data. It turns out that the
freely specifiable data are two functions, essentially components
of the metric, on an initial null hypersurface
and two similar functions at ``
''. These latter functions are Bondi's
news functions
whose non-vanishing is taken to indicate the presence of
gravitational radiation.
The results of this analysis were very satisfactory and
physically reasonable. The most important consequence is the
demonstration that outgoing gravitational waves carry away energy
from the source and hence diminish its mass. This is the
consequence of the Bondi-Sachs mass loss formula which relates
the rate of the mass decrease to the integral over the absolute
value of the news. Another consequence of the analysis was the
peeling property: For space-times which satisfy the vacuum field
equations and the Bondi-Sachs boundary conditions, the curvature
necessarily has the asymptotic behaviour (1) as predicted by Sachs' direct analysis of the Riemann tensor
using the vacuum Bianchi identity. Thus, the Bondi-Sachs
conditions imply the covariant outgoing radiation condition of
Sachs and also the boundary condition proposed by Trautman.
The group of coordinate transformations which preserve the
form of the metric and the boundary conditions was determined.
This infinite dimensional group which became known as the BMS
group is isomorphic to the semi-direct product of the homogeneous
Lorentz group with the Abelian group of so-called
super-translations. The emergence of this group came as a
surprise because one would have expected the Poincaré group as
the asymptotic symmetry group but one obtained a strictly larger
group. However, the structure of the BMS group is sufficiently
similar to the Poincaré group. In particular, it contains a
unique Abelian normal subgroup of four dimensions, which can be
identified with the translation group. This result forms the
basis of further investigations into the nature of
energy-momentum in general relativity. The BMS group makes no
reference to the metric which was used to derive it. Therefore,
it can be interpreted as the invariance group of some universal
structure which comes with every space-time satisfying the
Bondi-Sachs boundary conditions. The BMS group has been the
subject of numerous further investigations since then. For some
of them we refer to [95,
96,
97,
101,
111,
118
,
126].
At about the same time Newman and Penrose [100] had formulated what has become known as the NP formalism. It
combined the spinor methods which had been developed earlier by
Penrose [107] with the (null-)tetrad calculus used hitherto. Newman and
Penrose applied their formalism to the problem of gravitational
radiation. In particular, they constructed a coordinate system
which was very similar to the ones used by Bondi et al. and
Sachs. The only difference was their use of an affine parameter
instead of luminosity distance along the generators of the null
hypersurfaces of constant retarded time. Based on these
coordinates and an adapted null frame they showed that the single
assumption
(and the technical assumption of the uniformity of the angular
derivatives) already implied the peeling property as stated by
Sachs. The use of
as the quantity whose properties are specified on a null
hypersurface was in accordance with a general study of
characteristic initial value problems for spinor equations and in
particular for general relativity undertaken by Penrose [113].
An important point in this work was the realization that the Bianchi identity could be regarded as a field equation for the Weyl tensor. It might be useful here to point out that it is a misconception to consider the Bianchi identity as simply a tautology and to ignore it as contributing no further information, as it is done even today. It is an important piece of the structure on a Riemannian or Lorentzian manifold which relates the (derivatives of the) Ricci and Weyl tensors. If the Ricci tensor is restricted by the Einstein equations to equal the energy-momentum tensor, then the Bianchi identity provides a differential equation for the Weyl tensor. Its structure is very similar to the familiar zero rest-mass equation for a particle with spin 2. In fact, in a sense one can consider this equation as the essence of the gravitational theory.
Newman and Unti [104] carried the calculations started in [100] further and managed to solve the full vacuum field equations
asymptotically for large distances. The condition of asymptotic
flatness was imposed not on the metric but directly on the Weyl
tensor in the form suggested by Newman and Penrose, namely that
the component
of the Weyl tensor should have the asymptotic behaviour
. From this assumption alone (and some technical requirements
similar to the ones mentioned above) they obtained the correct
peeling behaviour of the curvature, the form of the metric up to
the order of
, in particular its flatness at large distances, and also the
Bondi-Sachs mass loss formula. Later, the procedure developed by
Newman and Unti to integrate the vacuum field equations
asymptotically has been analyzed by Dixon [33], who showed that it can be carried out consistently to all
orders of 1/
r
.
It is remarkable how much progress could be made within such a short time (only about four years). The trigger seems to have been the use of the structure of the light cones in one form or another in order to directly describe the properties of the radiation field: the introduction of the retarded time function, the use of an adapted null-tetrad, and the idea to ``follow the field along null directions''. This put the emphasis onto the conformal structure of space-times.
The importance of the conformal structure became more and more
obvious. Schücking had emphasized the conformal invariance of the
massless free fields, a fact which had been established much
earlier by Bateman [15] and Cunningham [31] for the wave equation and the free Maxwell field, and by
McLennan [98] for general spin. This had led to the idea that conformal
invariance might play a role also in general relativity and, in
particular, in the asymptotic behaviour of the gravitational
radiation field (see [116] for a personal account of the development of these ideas).
Finally, Penrose [108] outlined a completely different point of view on the subject,
arrived at by taking the conformal structure of space-time as
fundamental. He showed that if one regarded the metric of
Minkowski space-time to be specified only up to conformal
rescalings
for some arbitrary function
, then one could treat points at infinity on the same basis as
finite points. Minkowski space-time could be completed to a
highly symmetrical conformal manifold by adding a ``null-cone at
infinity''. The well known zero rest-mass fields which transform
covariantly under conformal rescalings of the metric are well
defined on this space and the condition that they be finite on
the null-cone at infinity translates into reasonable fall-off
conditions for the fields on Minkowski space. On the infinite
null-cone one could prescribe characteristic data for the fields
which correspond to the strength of their radiation field. He
suggested that asymptotically flat space-times should share at
least some of these properties.
This point of view proved successful. In a further paper by
Penrose [109], the basic qualitative picture we have today is developed.
Roughly speaking (see the next section for a detailed account),
the general idea is to attach boundary points to the ``physical''
space-time manifold which idealize the end-points at infinity
reached by infinitely extended null geodesics. This produces a
manifold with boundary, the ``unphysical'' manifold, whose
interior is diffeomorphic to the physical manifold. Its boundary
is a regular hypersurface whose causal character depends on the
cosmological constant. The unphysical manifold is equipped with a
metric which is conformal to the physical metric with a conformal
factor
which vanishes on the boundary. In addition, the structure of
the conformal boundary is uniquely determined by the physical
space-time.
Let us illustrate this with a simple example. The metric of Minkowski space-time in polar coordinates is
where
is the metric of the unit sphere. To perform the conformal
rescaling we introduce null coordinates
u
=
t
-
r
and
v
=
t
+
r
. This puts the Minkowski line-element into the form
where
is the metric of the unit sphere. The coordinates
u
and
v
each range over the complete real line, subject only to the
condition
. This infinite range is compactified by transforming with an
appropriate function, e.g.
thus introducing new null coordinates U and V, in terms of which the metric takes the form
The coordinates
U,
V
both range over the open interval
with the restriction
. Obviously, the Minkowski metric is not defined at points with
or
. Any extension of the metric in this form will be singular.
Now we define a different metric
, conformally related to
by the conformal factor
. Thus,
This metric is perfectly regular at the points mentioned above
and, in fact,
g
is the metric of the Einstein cylinder
. This can be verified by defining an appropriate time and
radius coordinate. With
T
=
U
+
V,
R
=
V
-
U
we have
Thus, we may consider Minkowski space to be conformally
embedded into the Einstein cylinder. This is shown in
Figure
1
. The Minkowski metric determines the structure of the boundary,
namely the two 3-dimensional null-hypersurfaces
and
which represent (future and past) ``null-infinity''. This is
where null-geodesics ``arrive''. They are given by the conditions
(
) and
(
). The points
are given by
. They represent ``future and past time-like infinity'', the
start respectively the end point of time-like geodesics, while
is a point with
. It is the start and end point of all space-like geodesics,
hence it represents ``space-like infinity''.
The conformal boundary of Minkowski space-time consists of the
pieces
,
, and
. These are fixed by the Minkowski metric. In contrast to this,
the conformal manifold into which Minkowski space-time is
embedded (here the Einstein cylinder) is not fixed by the metric.
Obviously, had we chosen a different conformal factor
with some arbitrary positive function
we would not have obtained the metric of the Einstein cylinder
but a different one. We see from this that although the conformal
boundary is unique, the
conformal extension
beyond the boundary is not.
The conformal compactification process is useful for several reasons. First of all it simplifies the discussion of problems at infinity which would involve complicated limit procedures when viewed with respect to the physical metric. Transforming to the unphysical metric attaches a boundary to the manifold so that issues which arise at infinity with respect to the physical metric can be analyzed by local differential geometric arguments in the neighbourhood of the boundary.
This is particularly useful for discussing solutions of conformally invariant field equations on space-time. The basic idea is the following: Consider a space-time which allows us to attach a conformal boundary, thus defining an unphysical manifold conformally related to the given space-time. Suppose we are also given a solution of a conformally invariant equation on this unphysical manifold. Because of the conformal invariance of the equation, there exists a rescaling of that unphysical field with a power of the conformal factor, which produces a solution of the equation on the physical manifold. Now suppose that the unphysical field is smooth on the boundary. Then the physical solution will have a characteristic asymptotic behaviour which is entirely governed by the conformal weight of the field, i.e. by the power of the conformal factor used for the rescaling. Thus, the regularity requirement of the unphysical field translates into a characteristic asymptotic fall-off or growth behaviour of the physical field, depending on its conformal weight.
Penrose used this idea to show that solutions of the zero
rest-mass equations for arbitrary spin on a space-time, which can
be compactified by a conformal rescaling, exhibit the peeling
property in close analogy to the gravitational case as discovered
by Sachs. Take as an example the spin-2 zero rest-mass equation
for a tensor
with the algebraic properties of the Weyl tensor
This is the equation for linear perturbations of the
gravitational fields propagating on a fixed background. It is
conformally covariant, in the sense that it remains unchanged
provided the field is rescaled as
, i.e., it is a conformal density with weight -1.
Using the geometric technique of conformal compactification, Penrose was able to establish the peeling property also for general (non-linear) gravitational fields. We will discuss this result explicitly in the following section. Furthermore, he showed that the group of transformations of the conformal boundary leaving the essential structure invariant was exactly the BMS group. This geometric point of view suggested that the asymptotic behaviour of the gravitational field of an isolated radiating gravitational system can be described entirely in terms of its conformal structure. The support for this suggestion was overwhelming from an aesthetical point of view, but a rigorous support for this claim was provided essentially only from the examination of the formal expansion type solutions of Bondi-Sachs and Newman-Unti and the analysis of explicit stationary solutions of the field equations.
The geometric point of view outlined above is the foundation on which many modern developments within general relativity are based. Let us now discuss the notion of asymptotically flat space-times and some of their properties in more detail.
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Conformal Infinity
Jörg Frauendiener http://www.livingreviews.org/lrr-2000-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |