The existence of gravitational waves was predicted by Einstein [42] 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 [140] 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 that 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 [136]
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 [106]. 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 [107] 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 [157]. He wanted to obtain conditions that were general enough to allow for gravitational radiation from 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 [156], 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 [44] and, hence, Trautman’s 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 had been taken out to infinity along null directions, then this result would have coincided 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 [147] of
the structure of retarded linearized gravitational fields of
particles, Sachs [142] 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 [41, 141, 102] 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 that 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 that extends to infinity, the curvature falls off
along the curve in such a way that to order
it is null (Petrov type N or
) with a quadruple PND (principal null direction)
along the curve. To order
it has type III (
) with the triple PND pointing along the curve, and
to order
it has type II (
) with the double PND oriented along the curve. To
order
it is algebraically general (type I 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
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 [28, 119] 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 Bondi
and Pirani at King’s College, London, in the years between 1955 and
1967. In a series of papers [108, 109, 26, 137, 110, 142, 27, 144
] they analyzed
several problems related to gravitational waves of increasing
complexity. The most important of their results certainly is the
work on axi-symmetric radiating systems by Bondi, van der Burg, and
Metzner [27]. 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 [144] removed this
additional assumption.
The essential new ingredient was the use of a
retarded time function. This is a scalar function
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
(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 .
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
.
The field equations in Bondi coordinates have a
rather nice hierarchical structure that is symptomatic for the use
of null coordinates and that 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 [145, 146]) 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 that
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 quite 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 that was
used to derive it. Therefore, it can be interpreted as the
invariance group of some universal structure that 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 [112, 113, 114, 118, 128, 135
, 143].
At about the same time, Newman and
Penrose [117] had formulated what
has become known as the NP formalism. It combined the spinor
methods, which had been developed earlier by Penrose [124], 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 that 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 [130].
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 is often done even today. It is an important piece of the structure on a Riemannian or Lorentzian manifold, relating 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 [121]
carried the calculations started in [117] 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 was analyzed by Dixon [40], who showed that
it could be carried out consistently to all orders in
.
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 that had been
established much earlier by Bateman [16] and
Cunningham [38] for the
wave equation and the free Maxwell field, and by
McLennan [115] 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 [133] for a
personal account of the development of these ideas). Finally,
Penrose [125] 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
that 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 [126], the basic
qualitative picture we have today is developed. Roughly speaking
(see the next Section 3 for a detailed account), the
general idea is to attach boundary points to the “physical”
space-time manifold that 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 that 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
whereNow we define a different metric , conformally related to
by the conformal
factor
. Thus,
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
three-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 and the
end-point, respectively, 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 that 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 that 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 that 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,
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 2.3. 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.