In
general relativity
, the
pp-wave spacetimes
, or
pp-waves
for short, are an important family of
exact solutions
of
Einstein's field equation
. The term
pp
stands for
plane-fronted waves with parallel propagation
, and was introduced in 1962 by
Jurgen Ehlers
and
Wolfgang Kundt
.
Overview
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]
The pp-waves solutions model
radiation
moving at the
speed of light
. This radiation may consist of:
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 analogue in general relativity of the
plane waves
familiar to students of
electromagnetism
.
In particular, in general relativity, we must take into account the gravitational effects of the energy density of the
electromagnetic field
itself. When we do this,
purely electromagnetic plane waves
provide the direct generalization of ordinary plane wave solutions in
Maxwell's theory
.
Furthermore, in general relativity, disturbances in the gravitational field itself can propagate, at the speed of light, as "wrinkles" in the curvature of spacetime. Such
gravitational radiation
is the gravitational field analogue of electromagnetic radiation.
In general relativity, the gravitational analogue of electromagnetic plane waves are 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-symmetric
pp-wave.
An example of pp-wave given when gravity is in presence of matter is the gravitational field surrounding a neutral Weyl fermion: the system consists in a gravitational field that is a pp-wave, no electrodynamic radiation, and a massless spinor exhibiting axial symmetry. In the
Weyl-Lewis-Papapetrou
spacetime, there exists a complete set of exact solutions for both gravity and matter.
[1]
Pp-waves were introduced by
Hans Brinkmann
in 1925 and have been rediscovered many times since, most notably by
Albert Einstein
and
Nathan Rosen
in 1937.
Mathematical definition
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A
pp-wave spacetime
is any
Lorentzian manifold
whose
metric tensor
can 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-free
definition.
It states that any
Lorentzian manifold
which admits a
covariantly constant
null vector
field
is called a pp-wave spacetime. That is, the
covariant derivative
of
must vanish identically:
This definition was introduced by Ehlers and Kundt in 1962. To relate Brinkmann's definition to this one, take
, the
coordinate vector
orthogonal 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
. The vacuum Einstein equations are very simple for pp waves, and in fact linear: the metric
obeys these equations if and only if
. But the definition of a pp-wave spacetime does not impose this equation, so it is entirely mathematical and belongs to the study of
pseudo-Riemannian geometry
. In the next section we turn to
physical interpretations
of pp-wave spacetimes.
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.)
Physical interpretation
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It is a purely mathematical fact that the
characteristic polynomial
of the
Einstein tensor
of any pp-wave spacetime vanishes identically. Equivalently, we can find a
Newman?Penrose complex null tetrad
such that the
Ricci-NP scalars
(describing any matter or nongravitational fields which may be present in a spacetime) and the
Weyl-NP scalars
(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 tensor
always 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
radiation
traveling at the local
speed of light
. This radiation can be gravitational, electromagnetic, Weyl fermions, or some hypothetical kind of massless radiation other than these three, 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
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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 congruence
with 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 solution
if 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 Gonner 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 algebras
of
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 modes
of 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
where
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 essential for the beautiful topic of
colliding plane waves
.
A more general subclass consists of the
axisymmetric pp-waves
, which in general have a two-dimensional
Abelian
Lie algebra of Killing vector fields.
These are also called
SG2 plane waves
, because they are the second type in the symmetry classification of Sippel and Gonner.
A limiting case of certain axisymmetric pp-waves yields the Aichelburg/Sexl ultraboost modeling an ultrarelativistic encounter with an isolated spherically symmetric object.
(See also the article on
plane wave spacetimes
for 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-dual
covariantly 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 Gonner, 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 spacetime
background.
Relation to other theories
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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 theories
of
gravitation
. In particular, pp-waves are exact solutions in the
Brans?Dicke theory
,
various
higher curvature theories
and
Kaluza?Klein theories
, and certain gravitation theories of
J. W. Moffat
.
Indeed,
B. O. J. Tupper
has 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-Jurgen Schmidt
has 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 Gibbons
has pointed out, all
loop term
quantum corrections vanish identically for any pp-wave spacetime. This means that studying
tree-level
quantizations 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. Hull
has shown that such
higher-dimensional pp-waves
are essential building blocks for eleven-dimensional
supergravity
.
Geometric and physical properties
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PP-waves enjoy numerous striking properties. Some of their more abstract mathematical properties have already been mentioned. In this section a few additional properties are presented.
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
chaos
.
The fact that Einstein's field equation is
nonlinear
[
disambiguation needed
]
is well known. This implies that if you have two exact solutions, there is almost never any way to
linearly superimpose
them. 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 Penrose
has 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 covariantly 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 invariants
of the
Riemann tensor
vanish identically
, yet the curvature is almost never zero. This is because in four-dimension all pp-waves belong to the class of
VSI spacetimes
. Such statement does not hold in higher-dimensions since there are higher-dimensional pp-waves of algebraic type II with non-vanishing
polynomial scalar invariants
. If you view the Riemann tensor as a second rank tensor acting on bivectors, the vanishing of invariants 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 in pp-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
.
Examples
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There are many noteworthy
explicit
examples of pp-waves.
("Explicit" means that the metric functions can be written down in terms of
elementary functions
or perhaps well-known
special functions
such as
Mathieu functions
.)
Explicit examples of
axisymmetric pp-waves
include
Explicit examples of
plane wave spacetimes
include
- exact
monochromatic gravitational plane wave
and
monochromatic electromagnetic plane wave
solutions, which generalize solutions which are well-known from
weak-field approximation
,
- exact solutions of the
gravitational
field of a
Weyl fermion
,
- the
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 isometric
to part of the
interior
of a
Schwarzschild black hole
, thereby permitting a classical peek at the local geometry
inside
the
event horizon
,
- the
uniform electromagnetic plane wave
; this spacetime is foliated by spacelike hyperslices which are isometric to
,
- the
wave of death
is 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 Gonner symmetry classification), which exhibit a
weak nonscalar null curvature singularity
and which arise as the
Penrose limits
of an appropriate
null geodesic
approaching the curvature singularity which is present in many physically important solutions, including the
Schwarzschild black holes
and
FRW cosmological models
.
See also
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Notes
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- ^
Cianci, R.; Fabbri, L.; Vignolo S., Exact solutions for Weyl fermions with gravity
References
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]
- "On Generalised P.P. Waves"
(PDF)
.
J. D. Steele
. Retrieved
June 12,
2005
.
- Hall, Graham (2004).
Symmetries and Curvature Structure in General Relativity (World Scientific Lecture Notes in Physics)
. Singapore: World Scientific Pub. Co.
ISBN
981-02-1051-5
.
- Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard (2003).
Exact Solutions of Einstein's Field Equations
. Cambridge:
Cambridge University Press
.
ISBN
0-521-46136-7
.
See Section 24.5
- Sippel, R. & Gonner, H. (1986). "Symmetry classes of pp waves".
Gen. Rel. Grav
.
12
: 1129?1243.
- Penrose, Roger (1976). "Any spacetime has a plane wave as a limit".
Differential Geometry and Relativity
. pp. 271?275.
- Tupper, B. O. J. (1974). "Common solutions of the Einstein and Brans-Dicke theories".
Int. J. Theor. Phys
.
11
(5): 353?356.
Bibcode
:
1974IJTP...11..353T
.
doi
:
10.1007/BF01808090
.
S2CID
122456995
.
- Penrose, Roger (1965). "A remarkable property of plane waves in general relativity".
Rev. Mod. Phys
.
37
(1): 215?220.
Bibcode
:
1965RvMP...37..215P
.
doi
:
10.1103/RevModPhys.37.215
.
- Ehlers, Jurgen & Kundt, Wolfgang (1962). "Exact solutions of the gravitational field equations".
Gravitation: an Introduction to Current Research
. pp. 49?101.
See Section 2-5
- Baldwin, O. R. & Jeffery, G. B. (1926).
"The relativity theory of plane waves"
.
Proc. R. Soc. Lond. A
.
111
(757): 95.
Bibcode
:
1926RSPSA.111...95B
.
doi
:
10.1098/rspa.1926.0051
.
- H. W. Brinkmann (1925). "Einstein spaces which are mapped conformally on each other".
Math. Ann
.
18
: 119?145.
doi
:
10.1007/BF01208647
.
S2CID
121619009
.
- Yi-Fei Chen and J.X. Lu (2004), "
Generating a dynamical M2 brane from super-gravitons in a pp-wave background
"
- Bum-Hoon Lee (2005), "
D-branes in the pp-wave background
"
- H.-J. Schmidt (1998). "A two-dimensional representation of four-dimensional gravitational waves,"
Int. J. Mod. Phys.
D7
(1998) 215?224 (arXiv:gr-qc/9712034).
- Albert Einstein
, "On Gravitational Waves,"
J. Franklin Inst.
223
(1937).
43?54.
External links
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