- Pp-wave spacetime
general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutionsof Einstein's field equation. These solutions model radiationmoving at the speed of light. This radiation may consist of:
* "massless" radiation associated with some hypothetical distinct type relativistic classical field,
or any combination of these, so long as the radiation is all moving in the "same" direction.
A special type of pp-wave spacetime, the
plane wave spacetimes,provide the most general analog in general relativity of the plane wavesfamiliar to students of electromagnetism.In particular, in general relativity, we must take into account the gravitational effectsof the energy density of the electromagnetic fielditself. When we do this,"purely electromagnetic plane waves" provide the direct generalization of ordinaryplane wave solutions in Maxwell's theory.
Furthermore, in general relativity, disturbances in the gravitational field itself canpropagate, at the speed of light, as "wrinkles" in the curvature of spacetime. Such"gravitational radiation" is the gravitational field analog of electromagnetic radiation.In general relativity, the gravitational analogue of electromagnetic plane wavesare precisely the vacuum solutions among the plane wave spacetimes.They are called
gravitational plane waves.
There are physically important examples of pp-wave spacetimes which are "not" plane wave spacetimes.In particular, the physical experience of an observer who whizzes by a gravitating object (such as a star or a black hole) at nearly the speed of light can be modelled by an"impulsive" pp-wave spacetime called the
Aichelburg-Sexl ultraboost.The gravitational field of a beam of light is modelled, in general relativity, by a certain axi-symmetricpp-wave.
Pp-waves were introduced by
Hans Brinkmannin 1925 and have been rediscovered many times since, most notably by Albert Einsteinand Nathan Rosenin 1937.The term "pp" stands for "plane-fronted waves with parallel propagation", and was introduced in 1962 by Jürgen Ehlersand Wolfgang Kundt.
A "pp-wave spacetime" is any
Lorentzian manifoldwhose metric tensorcan be described, with respect to Brinkmann coordinates, in the form
where is any
smooth function. This was the original definition of Brinkmann, and it has the virtue of being easy to understand.
The definition which is now standard in the literature is more sophisticated.It makes no reference to any coordinate chart, so it is a
coordinate-freedefinition.It states that any Lorentzian manifoldwhich admits a "covariantly constant" null vectorfield is called a pp-wave spacetime. That is, the covariant derivativeof must vanish identically:
This definition was introduced by Ehlers and Kundt in 1962. To relate Brinkmann's definition to this one, take , the
coordinate vectororthogonal to the hypersurfaces . In the "index-gymnastics" notation for tensor equations, the condition on can be written .
Neither of these definitions make any mention of any field equation; in fact,they are "entirely independent of physics". In this sense, the notion of a pp-wave spacetime is entirely mathematical and belongs to the study of
pseudo-Riemannian geometry.In the next section, we will turn to the "physical interpretation" of pp-waves.
Ehlers and Kundt gave several more coordinate-free characterizations, including:
* A Lorentzian manifold is a pp-wave if and only if it admits a one-parameter subgroup of isometries having null orbits, and whose curvature tensor has vanishing eigenvalues.
* A Lorentzian manifold with nonvanishing curvature is a (nontrivial) pp-wave if and only if it admits a covariantly constant
bivector. (If so, this bivector is a null bivector.)
It is a purely mathematical fact that the
characteristic polynomialof the Einstein tensorof any pp-wave spacetime vanishes identically. Equivalently, we can find a Newman/Penrose tetradsuch that the Ricci spinor(describing any matter or nongravitational fields which may be present in a spacetime)and the Weyl spinor(describing any gravitational field which may be present)each have only one nonvanishing component. Specifically, with respect to the NP tetrad:::the only nonvanishing component of the Ricci spinor is:and the only nonvanishing component of the Weyl spinor is:This means that any pp-wave spacetime can be interpreted, in the context of general relativity,as a null dust solution. Also, the Weyl tensoralways has Petrov type N as may be verified by using the Bel criteria.
In other words, pp-waves model various kinds of "classical" and "massless"
radiationtraveling at the local speed of light. This radiation can be gravitational, electromagnetic, some hypothetical kind of massless radiation other than these two, or any combination of these. All this radiation is traveling in the same direction, and the null vector plays the role of a wave vector.
Relation to other classes of exact solutions
Unfortunately, the terminology concerning pp-waves, while fairly standard, is highly confusing and tends to promote misunderstanding.
In any pp-wave spacetime,the covariantly constant vector field always has identically vanishing
optical scalars. Therefore, pp-waves belong to the "Kundt class" (the class of Lorentzian manifolds admitting a null congruencewith vanishing optical scalars).
Going in the other direction, pp-waves include several important special cases.
From the form of Ricci spinor given in the preceding section, it is immediately apparent that a pp-wave spacetime (written in the Brinkmann chart) is a
vacuum solutionif and only if is a harmonic function(with respect to the spatial coordinates ). Physically, these represent purely gravitational radiation propagating along the null rays .
Ehlers and Kundt and Sippel and Gönner have classified vacuum pp-wave spacetimes by their
autometry group, or group of "self-isometries". This is always a Lie group, and as usual it is easier to classify the underlying Lie algebrasof Killing vector fields. It turns out that the most general pp-wave spacetime has only one Killing vector field, the null geodesic congruence . However, for various special forms of , there are additional Killing vector fields.
The most important class of particularly symmetric pp-waves are the
plane wave spacetimes, which were first studied by Baldwin and Jeffery.A plane wave is a pp-wave in which is quadratic, and can hence be transformed to the simple form
Here, are arbitrary smooth functions of .Physically speaking, describe the wave profiles of the two linearly independent
polarization modesof gravitational radiation which may be present,while describes the wave profile of any nongravitational radiation.If , we have the vacuum plane waves, which are often called plane gravitational waves.
Equivalently, a plane-wave is a pp-wave with at least a five dimensional Lie algebra of Killing vector fields , including and four more which have the form
Intuitively, the distinction is that the wavefronts of plane waves are truly "planar"; all points on a given two-dimensional wavefront are equivalent. This not quite true for more general pp-waves.Plane waves are important for many reasons; to mention just one, they are essentialfor the beautiful topic of
colliding plane waves.
A more general subclass consists of the axisymmetric pp-waves, which in general have a two dimensional
AbelianLie algebra of Killing vector fields. These are also called "SG2 plane waves", because they are the second type in the symmetryclassification of Sippel and Gönner.A limiting case of certain axisymmetric pp-waves yieldsthe Aichelburg/Sexl ultraboost modeling an ultrarelativistic encounter with an isolated sphericallysymmetric object.
(See also the article on
plane wave spacetimesfor a discussion of physically important special cases of plane waves.)
J. D. Steele has introduced the notion of generalised pp-wave spacetimes.These are nonflat Lorentzian spacetimes which admit a
self-dualcovariantly constant null bivector field.The name is potentially misleading, since as Steele points out, these are nominally a "special case" of nonflat pp-waves in the sense defined above. They are only a generalization in the sense that although the Brinkmann metric form is preserved, they are not necessarily the vacuum solutions studied by Ehlers and Kundt, Sippel and Gönner, etc.
Another important special class of pp-waves are the
sandwich waves. These have vanishing curvature except on some range , and represent a gravitational wave moving through a Minkowski spacetimebackground.
Relation to other theories
Since they constitute a very simple and natural class of Lorentzian manifolds, defined in terms of a null congruence, it is not very surprising that they are also important in other relativistic
classical field theoriesof gravitation. In particular, pp-waves are exact solutions in the Brans-Dicke theory,various higher curvature theoriesand Kaluza-Klein theories,and certain gravitation theories of J. W. Moffat.Indeed, B. O. J. Tupperhas shown that the "common" vacuum solutions in general relativity and in the Brans/Dicke theory are precisely the vacuum pp-waves (but the Brans/Dicke theory admits further wavelike solutions). Hans-Jürgen Schmidthas reformulated the theory of (four-dimensional) pp-waves in terms of a "two-dimensional" metric-dilaton theory of gravity.
Pp-waves also play an important role in the search for
quantum gravity, because as Gary Gibbonshas pointed out, all loop termquantum corrections vanish identically for any pp-wave spacetime. This means that studying tree-levelquantizations of pp-wave spacetimes offers a glimpse into the yet unknown world of quantum gravity.
It is natural to generalize pp-waves to higher dimensions, where they enjoy similar properties to those we have discussed.
C. M. Hullhas shown that such "higher dimensional pp-waves" are essential building blocks for eleven-dimensional supergravity.
Geometric and physical properties
PP-waves enjoy numerous striking properties. Some of their more abstract mathematical properties have already been mentioned. In this section we can discuss only a few additional properties.
Consider an inertial observer in Minkowski spacetime who encounters a sandwich plane wave. Such an observer will experience some interesting optical effects. If he looks into the "oncoming" wavefronts at distant galaxies which have already encountered the wave, he will see their images undistorted. This must be the case, since he cannot know the wave is coming until it reaches his location, for it is traveling at the speed of light. However, this can be confirmed by direct computation of the optical scalars of the null congruence . Now suppose that after the wave passes, our observer turns about face and looks through the "departing" wavefronts at distant galaxies which the wave has not yet reached. Now he sees their optical images sheared and magnified (or demagnified) in a time-dependent manner. If the wave happens to be a
polarized"gravitational plane wave", he will see circular images alternately squeezed horizontally while expanded vertically, and squeezed vertically while expanded horizontally. This directly exhibits the characteristic effect of a gravitational wave in general relativity on light.
The effect of a passing polarized gravitational plane wave on the relative positions of a cloud of (initially static) test particles will be qualitatively very similar. We might mention here that in general, the motion of test particles in pp-wave spacetimes can exhibit
The fact that Einstein's field equation is
nonlinearis well-known. This implies that if you have two exact solutions, there is almost never any way to linearly superimposethem. PP waves provide a rare exception to this rule: if you have two PP waves sharing the same covariantly constant null vector (the same geodesic null congruence, i.e. the same wave vector field), with metric functions respectively, then gives a third exact solution. Roger Penrosehas observed that near a null geodesic, "every Lorentzian spacetime looks like a plane wave". To show this, he used techniques imported from algebraic geometry to "blow up" the spacetime so that the given null geodesic becomes the convariantly constant null geodesic congruence of a plane wave. This construction is called a Penrose limit.
Penrose also pointed out that in a pp-wave spacetime, all the
polynomial scalar invariantsof the Riemann tensor "vanish identically", yet the curvature is almost never zero. If you view the Riemann tensor as a second rank tensor acting on bivectors, this phenomenon is analogous to the fact that a nonzero null vector has vanishing squared length.
Penrose was also the first to understand the strange nature of causality inpp-sandwich wave spacetimes. He showed that some or all of the null geodesics emitted at a given event will be refocused at a later event (or string of events). The details depend upon whether the wave is purely gravitational, purely electromagnetic, or neither.
Every pp-wave admits many different Brinkmann charts. These are related by
coordinate transformations, which in this context may be considered to be gauge transformations. In the case of plane waves, these gauge transformations allow us to always regard two colliding plane waves to have "parallel wavefronts", and thus the waves can be said to "collide head-on".This is an exact result in fully nonlinear general relativity which is analogous to a similar result concerning electromagnetic plane waves as treated in special relativity.
There are many noteworthy "explicit" examples of pp-waves.("Explicit" means that the metric functions can be written down in terms of
elementary functionsor perhaps well-known special functionssuch as Mathieu functions.)
Explicit examples of "axisymmetric pp-waves" include
Aichelburg-Sexl ultraboostis an impulsive plane wavewhich models the physical experience of an observer who whizzes by a spherically symmetric gravitating object at nearly the speed of light,
Bonnor beamis an axisymmetric plane wave which models the gravitational field of an infinitely long beam of incoherent electromagnetic radiation.
Explicit examples of "plane wave spacetimes" include
monochromatic gravitational plane waveand monochromatic electromagnetic plane wavesolutions, which generalize solutions which are well-known from weak-field approximation,
Schwarzschild generating plane wave, a gravitational plane wave which, should it collide head-on with a twin, will produce in the "interaction zone" of the resulting colliding plane wave solution a region which is locally isometricto part of the "interior" of a Schwarzschild black hole, thereby permitting a classical peek at the local geometry "inside" the event horizon,
uniform electromagnetic plane wave; this spacetime is foliated by spacelike hyperslices which are isometric to ,
wave of deathis a gravitational plane wave exhibiting a "strong nonscalar null curvature singularity", which propagates through an initially flat spacetime, progressively destroying the universe,
homogeneous plane waves, or "SG11 plane waves" (type 11 in the Sippel and Gönner symmetry classification), which exhibit a "weak nonscalar null curvature singularity" and which arise as the Penrose limits of an appropriate null geodesicapproaching the curvature singularity which is present in many physically important solutions, including the Schwarzchild black holes and FRW cosmological models.
* "See Section 24.5"
* "See Section 2-5"
* [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=307203 Pp-wave on arxiv.org]
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