From: Chris Hillman Subject: Re: gravitational field of a moving mass Date: 6 Nov 2001 02:52:59 GMT Newsgroups: sci.physics.research Summary: Aichelburg-Sexl field equations for moving masses On 29 Oct 2001, Corrado Massa wrote: > The question: "what is the field of a rapidly moving electric charge > in special relativity?" is treated and answered in any standard > textbook of special relativity. OK. But...I'm in trouble with the > analogous gravitational question: "what is the field of a rapidly > moving (neutral) mass in general relativity? " My GR textbooks do not > answer. Can someone help me? Best regards and many thanks Corrado > > [Moderator's note: use Google to search under > "Aichelburg-Sexl" - jb] The original paper is: author = {P. C. Aichelburg and P. U. Sexl}, title = {On the gravitational field of a massless particle}, journal = {Gen. Rel. Grav.}, volume = 2, year = 1971, pages ={303}} and it was motivated by the observation of P. G. Bergmann about the ultrarelativistic boost of a charged particle in Minkowksi spacetime (which becomes an EM impulsive plane wave), i.e. the very result mentioned by Massa. Several more recent works summarize their method, including the invaluable monograph: author = {Valeri P. Frolov and Igor D. Novikov}, title = {Black Hole Physics: Basic Concepts and New Developments}, publisher = {Kluwer}, series = {Fundamental Theories of Physics}, volume = 96, year = 1998} The basic idea is to adopt the "spatially isotropic" coordinate chart for the exterior Schwarzschild vacuum ds^2 = -((1-m/2/R)/(1+m/2/R))^2 dt^2 + (1+m/2/R)^4 (dx^2 + dy^2 + dz^2) R = sqrt(x^2 + y^2 + z^2) then "boost along the z axis" by the naive transformation: T = (t+vz)/sqrt(1-v^2) Z = (z+vt)/sqrt(1-v^2) X = x Y = y then rewrite the resulting chart in the form ds^2 = (1-m/2/R)^2 [-dT^2 + dZ^2 + dX^2 + dY^2] + [((1-m/2/R)/(1+m/2/R))^2 - (1-m/2/R)^4] (dT - v dX)^2/(1-v^2) Fix mu = m/sqrt(1-v^2) and take the limit v -> 1. Further transformations put the resulting chart in the form ds^2 = 4 mu delta(U) log(sqrt(X^2+Y^2) -2 dU dV + dX^2 + dY^2 or ds^2 = 2 mu delta(U) log(R) - 2 dU dV + dR^2 + R^2 dTheta^2 -infty < U, V < infty, 0 < R < infty, -Pi < Theta < Pi Even from this brief description it should be clear that this is not the only possible limiting method, and that different methods can give different results! What the results have in common is that we obtain an -impulsive- PP wave, i.e. the spacetime is flat except on a single planar wavefront where the curvature is concentrated, as expressed by the "Dirac delta function" which appears above. Of course, we can then consider this to be the limit of a "Gaussian pulse", an ordinary PP wave in the cartesian harmonic chart ds^2 = -H(U,X,Y) dU^2 - 2 dU dV + dX^2 + dY^2, -infty < U,V,X,Y < infty where H(U,X,Y) is a Gaussian pulse and where for a vacuum (gravitational) PP wave H is harmonic in X,Y, i.e. H_(XX) + H_(YY) = 0 Specifically, again adopting a cylindrical harmonic chart, our Gaussian pulse can be written -2 m exp(-a^2 U^2/2) ds^2 = ------------------- log R dU^2 sqrt(2 \pi) a - 2 dU dV + dR^2 + R^2 dTheta^2, -infty < U,V < \infty, 0 < R < \infty, -Pi < \Theta < Pi For a bit of background on ordinary PP waves see author = {Jiri Bic\'ak}, title = {Selected solutions of {E}instein's field equations: their role in general relativity and astrophysics}, booktitle = {{E}instein Field Equations and Their Physical Implications (Selected essays in honour of {J}uergen {E}hlers)}, editor = {Bernd G. Schmidt}, publisher = {Springer-Verlag}, year = 2000, series = {Lecture Notes in Physics}, volume = 540, note = {gr-qc/0004016}} or author = {D. Kramer and H. Stephani and E. Herlt and M. MacCallum}, title = {Exact Solutions of {E}instein's Field Equations}, publisher = {Cambridge University Press}, year = 1980} Also, for a nice survey of radiative spacetimes in general see author = {Jiri Bic\'ak}, title = {Exact radiative spacetimes: some recent developments}, journal = {Annalen Phys.}, volume = 9, year = 2000, pages = {206--217}, note = {gr-qc/0004031}} As Pirani already knew in 1959, author = {F. A. E. Pirani}, title = {Gravitational waves in general relativity. IV. The gravitational field of a fast-moving particle.}, journal = {Proc. Roy. Soc. London. Ser. A}, volume = 252, year = 1959, pages = {96--101}} under an "ultraboost", the Petrov type of the Weyl tensor changes from D (pure Coulomb tidal field) to N (transverse gravitational wave). Later Aichelburg and Belasin observed that it is appropriate to extend the usual notion of symmetries (isometries generated by Killing vector fields) to include distributional transformations, and this motivated them to extend the classification by Ehlers and Kundt of the symmetries of PP waves author = {J\"urgen Ehlers and Wolfgang Kundt}, title = {Exact Solutions of the Gravitational Field Equations}, booktitle = {Gravitation: an Introduction to Current Research}, editor = {Louis Witten}, publisher = {Wiley}, year = 1962, pages = {49--101}} (see also the book by Herlt et al. cited above) to impulsive PP waves author = {Aichelburg, Peter C. and Belasin, Herbert}, title = {Symmetries of pp-waves with distributional profile}, journal = {Class. Quant. Grav.}, volume = 13, year = 1996, number = 4, pages = {723--729}, note = {gr-qc/9509025}} This work clarifies the relationship between the symmetries of the original isolated compact object and the resulting impulsive PP wave. But be careful!--- in their 1962 paper, Ehlers and Kundt only said that they -believed- their classification was complete; -every- subsequent paper I have examined omits this qualification. This is rather unfortunate, because it turns out the EK classification did in fact omit some physically very significant possibilities! I uncovered this when I studied the paper author = {R. Sippel and H. Goenner}, title = {Symmetry Classes of pp-waves}, journal = {Gen. Rel. Grav.}, volume = 18, year = 1986, number = 12, pages = {1229--1243}} which was intended to extend the EK classification to arbitrary PP waves (in the context of gtr, this means considering not only vacuum PP waves but also PP waves with a gravitational component, i.e. dropping the requirement that H above be harmonic in X,Y). When I considered the vacuum case of each SG type, it was obvious that (contrary to the assertion of Sippel and Goenner) several of their types -do- have vacuum specializations, and these are not included in the EK classification. Classification in mathematics is notoriously tricky--- as a rule, very few papers which attempt a classification get it perfectly right the first time around. Prof. Branko Grunbaum of the UW Math. Department has published many papers giving very careful classifications of things like geometric configurations of lines, and often found errors in previous works, including papers by very famous mathematicians. I am not courageous enough to attempt to check whether the SG classification is indeed complete (I seem to have lost track of the reference, but a later author claimed, without giving details, to have verified their classification by another method). Curiously enough, in perusing the literature I have noticed a marked -decline- in standards of care from 1962 to the present! For example, EK are careful to give the intended ranges of the variables in their coordinate charts, as I always am--- this is very good practice and should -always- be followed by -everyone- who writes down a coordinate chart, since much confusion in the literature results from omitting to specify the ranges, i.e. incompletely specifying the coordinate chart. Furthermore, unlike the other papers I have seen, EK 1962 explain very clearly exactly what they are trying to classify--- they are -not- trying to classify the vacuum PP waves up to local isometry; that would require running the Karlhede algorithm (an streamlined version of Cartan's general algorithm for detecting local isometries between two coordinate charts in possibly different semi-Riemannian manifolds). Rather, EK attempted to classify the vacuum PP waves up to coordinate transformations which preserve the form of the cartesian harmonic coordinate chart given above, i.e. a type of "gauge transformation" for PP waves. They prove that their types are distinct in this sense by providing invariants for these gauge transformations. (These "invariants" are -not- invariant under general coordinate transformations, of course!--- see remark in the paragraph after the next paragraph.) I would also mention that in another paper in the 1962 book edited by Louis Witten, which is still valuable, the author points out something which surprised me very much--- Friedmann and Lemaitre had pointed out in the original papers on FRW models the possibility of quotient manifolds, in particular the possibility of "compactifying" the models with a family of H^3 hyperslices orthogonal to the world lines of the dust particles by taking some discrete quotient of H^3! Again, this important observation seems to have been entirely forgotten by later authors; rather recently Cornish and Weeks (in particular) have resurrected the idea and are even planning a satellite mission to look for observational evidence of such a quotient (the evidence would of course consist of recognizing that we are seeing the -same- distant galaxies in several locations on our celestial sphere). IIRC, Cornish and Weeks were not aware that Friedmann and Lemaitre had already drawn attention to this possibility, although I would welcome correction if I have misremembered. By the way, I need a volunteer with a working implementation of the Karlhede algorithm (the REDUCE implementation seems to be pretty well debugged) willing to test their implementation on the EK and SG types (I can provide real coordinate charts of each type). The scalar -curvature invariants of all orders vanish identically- for PP waves: author = {Hans-J\"urgen Schmidt}, title = {Why do all curvature invariants of a gravitational wave vanish?}, booktitle = {New Frontiers in Gravitation}, editor = {G. A. Sardanashvili}, publisher = {Hadronic Press}, address = {Palm Harbor}, year = 1994, pages = {337--344}} author = {Hans-J\"urgen Schmidt}, title = {Consequences of the noncompactness of the {L}orentz group}, journal = {Int.J.Theor.Phys.}, volume = 37, year = 1998, pages = {691--696}, note = {gr-qc/9512007}} author = {V. Pravda and J. B\'icak}, title = {Curvature invariants in algebraically special spacetimes}, note = {gr-qc/0101085}} Thus, comparing the invariant classification of the various EK and SG to check for possible local isometries provides an excellent test case of any implementation of the Karlhede algorithm. Note for example that the Edgar-Ludwig family of all conformally flat null dusts author = {S. Brian Edgar and Garry Ludwig}, title = {All conformally flat pure radiation metrics}, journal = {Class. Quant. Grav.}, volume = 14, year = 1997, pages = {L65--L68}, note = {gr-qc/9612059}} (this includes not only conformally flat plane waves, i.e. special PP waves with no gravitational radiation component but only nongravitational radiation, but further null dusts which are not PP waves) has been touted with good reason as a particularly good test case; the EK and SG classification provide excellent test cases for similar reasons. I also need a volunteer to help me find the two extra Killing vector fields for the circularly polarized EM wave I mentioned yesterday in another thread (I found three by inspection, but expect that there should be two more). If anyone is interested, please say so and I'll add an appropriate "cut-out" to my email filter (or, if your email is in the *.edu or *.gov domain, just email me). Coming back to impulsive PP waves, the deflection of geodesics of test particles or laser pulses by the wavefront where the curvature is concentrated can be treated rigorously using an extension of gtr where one considers not only C^infty metric tensors (or C^3 metric tensors) but metric tensors including distributional components. There is a growing literature on this subject, which employs the Colombeau algebra of "generalized functions"; see for example author = {R. Steinbauer}, title = {Nonlinear distributional geometry and general relativity}, note = {math-ph/0104041}} for the general theory and author = {R. Steinbauer}, title = {Geodesics and geodesic deviation for impulsive gravitational waves}, journal = {J. Math. Phys.}, volume = 39, year = 1998, number = 4, pages = {2201--2212}} for application to impulsive PP waves. As I said above, there have been numerous extensions of the original method of Aichelburg and Sexl to obtain "ultrarelativistic boosts" of various other kinds of compact objects in gtr, including 1. the Schwarzschild-de Sitter solution author = {M. Hotta and Masahiro Tanaka}, title = {Gravitational shock waves and their quantum fields in de {S}itter space}, journal = {Phys. Rev. D}, volume = 47, year = 1993, pages = {3323--3329}} author = {J. Podolsk\'y and J. B. Griffiths}, title = {Impulsive waves in de {S}itter and anti-de {S}itter space-times generated by null particles with an arbitrary multipole structure}, journal = {Class. Quant. Grav.}, volume = 15, year = 1998, pages = {453--463}, note = {gr-qc/9710049}} (note that the wavefronts considered in the latter paper are geometrically spherical, and contract to a minimum radius and then re-expand; if we consider dS = H^(1,3) as an S^2 fiber bundle over H^(1,1), which is a ruled surface with the rulings corresponding to null geodesics, then one of these rulings corresponds to the contracting and re-expanding spherical wavefront; another reason why "ultraboots" of Schwarzschild-de Sitter holes are interesting is that the "extreme" case when the cosmological and event horizons are in thermodynamic equilibrium provides a test-bed for studying creation of quantum pairs of holes), 2. the Kerr solution author = {Koichi Hayashi and Toshiharu Samura}, title = {Gravitational shock waves for {S}chwarzschild and {K}err Black Holes}, journal = {Phy. Rev. D}, volume = 50, number = 4, year = 1994, pages = {3666-3675}} author = {Herbert Belasin and Herbert Nachbagauer}, title = {The ultrarelativistic {K}err geometry and its energy momentum tensor}, journal = {Class. Quant. Grav.}, volume = 12, year = 1995, pages = {707--713}} author = {Herbert Belasin and Herbert Nachbagauer}, title = {Boosting the {K}err geometry in an arbitrary direction}, journal = {Class. Quant. Grav.}, volume = 13, year = 1996, pages = {731--737}} author = {Alexander Burinskii and Giulio Magli}, title = {{K}err-{S}child Approach to the Boosted {K}err Solution}, journal = {Phys. Rev. D}, volume = 61, year = 2000, pages = {044017}, note = {gr-qc/9904012}} 3. the Kerr-Newman solution: author = {C. O. Lousto and N. Sanchez}, title = {The ultrarelativistic limit of the {K}err-{N}ewman geometry and particle scattering at the {P}lanck scale}, journal = {Phys. Lett. B}, volume = 232, year = 1989, pages = {462--6}} 4. and "isolated objects" obtained by considering the asymptotically flat subfamily of the Weyl family of all static axisymmetric vacuum solutions to the EFE: author = {J. Podolsk\'y and J. B. Griffiths}, title = {Boosted static multipole particles as sources of impulsive gravitational waves}, journal = {Phys. Rev. D}, volume = 58, year = 1998, pages = {124024}, note = {gr-qc/9809003}} author = {C. Barrab\`es and P. A. Hogan}, title = {Aichelburg-{S}exl boost of an isolated source in general relativity}, journal = {Phys. Rev. D}, volume = 64, year = 2001, pages = {044022}, note = {gr-qc/0110032}} For multipole moments in general in the context of gtr, see the review paper by Bicak cited above for the Weyl family, and then see for example author = {H. Quevedo}, title = {Multipole moments in general relativity. Static and stationary vacuum solutions}, journal = {Forthschr. Phys.}, volume = 38, year = 1990, pages = {733--840}} author = {L. Herrera and J. L. Hernandez Pastora}, title = {Measuring multipole moments of {W}eyl metrics by means of gyroscopes}, journal = {J. Math. Phys.}, volume = 41, year = 2000, pages = {7544--7555}, note = {gr-qc/0010003}} author = {Robert Beig}, title = {Multipole Moments of Static Spacetimes}, note = {gr-qc/0005048}} and also author = {C. Barrab\`es and P. A. Hogan}, title = {Bursts of Radiation and Recoil Effects in Electromagnetism and Gravitation}, journal = {Class. Quant. Grav.}, volume = 17, year = 2000, pages = 4667, note = {gr-qc/0012024}} author = {C. Barrab\`es and G.F. Bressange and P.A. Hogan}, title = {Some Physical Consequences of Abrupt Changes in the Multipole Moments of a Gravitating Body}, journal = {Phy. Rev. D}, volume = 55, year = 1997, pages = {3477--3484}, note = {gr-qc/9701025}} The above papers on "ultraboosts" of compact isolated objects employ a variety of methods in performing the limits, and it turns out that different impulsive PP waves can be obtained from the same object by using various different notions of "ultraboosting". However, as a rule, the Petrov type of the Weyl tensor of the exact solutions mentioned above are changed by the "ultrarelativistic boost" from D to N (for the Schwarzschild, Kerr, Schwarzschild-de Sitter and Kerr-Newman). The Kerr-Newman case of course yields an impulsive PP wave with a nongravitational radiation component. The isolated static axisymmetric objects include objects with nonzero quadrupole moment (the simplest of these is the Erez-Rosen vacuum), but "ultraboosts" -along the symmetry axis- erases these distinctions! (See the above papers for details.) Part of the motivation for all this work comes from the suggestion by 't Hooft author = {'t Hooft, G.}, title = {Graviton dominance in ultra-high-energy scattering}, journal = {Phys. Lett. B}, volume = 198, year = 1987, pages = {61--63}} that the quantum scattering of two point-like "particles" at the Planck energy can be modeled by the collision of two impulsive PP waves (with two parallel wavefronts approaching each other at the speed of light). This was already of independent interest in the context of studying the gravitational radiation produced by the collision and merger of two black holes, since axisymmetric collisions yield significant simplifications and assuming the collision is also ultrarelativistic renders the problem somewhat tractable; see the book author = {P. D. D'Eath}, title = {Black Holes: Gravitational Interactions}, publisher = {Oxford University Press}, year = 1996} and references therein. More recent papers include author = {C. Barrab\`es and G.F. Bressange and P.A. Hogan}, title = {Colliding Plane Waves in {E}instein-{M}axwell Theory}, journal = {Lett. Math. Phys.}, volume = 43, year = 1998, pages = {263--265}, note = {gr-qc/9710045}} author = {B. V. Ivanov}, title = {Colliding axisymmetric pp-waves}, journal = {Phys. Rev. D}, volume = 57, year = 1998, pages = {3378--3381}, note = {gr-qc/9705061}} author = {G. Ali and J. K. Hunter}, title = {Large amplitude gravitational waves}, note = {math.AP/9812030}} author = {C. Barrab\`es and G.F. Bressange and P.A. Hogan}, title = {Colliding Plane Impulsive Gravitational Waves}, journal = {Prog. Theor. Phys.}, volume = 102, year = 1999, pages = {1085--1101}, note = {gr-qc/0002048}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {Horizon Instability in the Cross Polarized {B}ell-{S}zekeres Spacetime}, note = {gr-qc/0006038}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {The Effect of Sources on the Inner Horizon of Black Holes}, note = {gr-qc/0010112}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {Null Singularities in Colliding Waves}, note = {gr-qc/0103092}} Going back to a moment to the observations of 't Hooft, more recently Gibbons has observed that because of the fact mentioned above about scalar invariants, PP waves suffer no quantum corrections at any loop order, and thus may provide a "classical glimpse" of the long-sought quantum theory of gravitation. See: author = {Gary Gibbons}, title = {Two-loop and all-loop finite 4-metrics}, journal = {Class. Quantum Grav.}, volume = 16, year = 1999, pages = {L71--L73}} Collisions of PP waves (vacuum or not, impulsive or not) are also of independent interest because of the discovery by Penrose and Khan author = {R. Penrose and K. A. Khan}, title = {Scattering of two impulsive gravitational plane waves}, journal = {Nature}, year = 1971, volume = 229, pages = {185}} that their model of colliding impulsive PP waves develops a curvature singularity which is (almost) unavoidable: title = {Metaphysics of colliding self-gravitating plane waves}, journal = {Phys. Rev. D}, volume = 29, year = 1984, pages = {1577--1583}} This immediately raised the question of whether such singularities are generic, or can be removed by small perturbations. This question has generated a large literature; see in particular the book author = {J. B. Griffiths}, title = {Colliding plane waves in general relativity}, publisher= {Oxford University Press}, year = 1991} and references therein. Griffiths remains very active in this area; see in particular author = {J. B. Griffiths and G. A. Alekseev}, title = {Some unpolarized {G}owdy cosmologies and nonlinear colliding plane wave spacetimes}, journal = {Int. J. Mod. Phys. D}, volume = 7, year = 1998, pages = {237--247}} Globally speaking, colliding impulsive plane waves usually look pretty much like this: singularity 8888 8....8 8......8 / \....../\ / \..../ \ / \../ \ / \/ \ / /\ \ / loc. / \ loc. \ / flat /loc.\ flat \ / / flat \ \ / / \ \ fold wavefront wavefront fold where points in this conformal diagram are fibered by ordinary planes E^2, but most lightlike geodesics -projected to this base- space bend to hit the singularity; for this phenomenon and for "folds" see the paper by Matzner and Tipler. It turns out that adding things like massless scalar fields or charged massless scalar fields (!) or a null dust component to the gravitational radiation in one or both wavefronts can -sometimes- change the singularity to something milder or even remove it entirely (the Bell-Szekeres solution is the best known example; see the book by Griffiths). In the above diagram, the post-collision region of the spacetime is shaded; this is called the "interaction zone". (Pixels outside the "folds" is not part of the diagram!) By the way, the methods used to find such solutions are of independent interest. Many authors follow Penrose and Khan by first finding the solution in the interaction zone (solving therein the EFE in a specialized form first given by Peter Szekeres), and then attaching the locally flat regions in the appropriate manner. Others use the "hyperbolic characteristic formulation" to specify initial data and then solve the evolution equation. I should remark somewhere in this discussion that a fundamental feature of PP waves is that two PP waves traveling in the -same- direction can be added linearly, despite the nonlinearity of the EFE. This reflects that fact that PP waves are characterized (among many other characterizations!) by the fact that they are simultaneously solutions of the linearized EFE and of the full EFE. OTH, PP waves which collide head-on always -scatter nonlinearly- as the above diagram suggests (consider for example "sandwich waves" instead of impulsive waves). Another amusing phenomenon resulting from this fundamental asymmetry is that in the case of a single PP sandwich wave, if an observer looks through the train of oncoming wavefronts at an object behind the wave train, its image is not optically distorted (this must be the case, or else he'd be warned in advance of the speed of light limit that the wave is coming!), but if after the wave has passed he turns and looks through the departing wavefronts at objects lying ahead of the wavetrain, their images -will- be optically distorted in a manner consistent with the tidal effect of PP waves. This can be seen by computing the optical scalars of two appropriate geodesic null congruences. This principle can be summarized, following Einstein, by the slogan: "you can't watch a wave train entering the station, but you can watch it leave!" A further twist to this story is added by the discovery of a fundamental duality author = {S. Chandrasekhar and V. Ferrari}, title = {On the {N}utku-{H}alil solution for colliding impulsive gravitational waves}, journal = {Proc. Roy. Soc. London Ser. A}, volume = 396, year = 1984, number = 1810, pages = {55--74}, note = {erratum in vol. 398 (1985), no. 1815, p. 429}} which relates solutions to the Ernst equation (the family of all stationary axisymmetric vacuum solutions to the EFE) to collisions of impulsive gravitational waves. Under this duality (part of) interior region for | "interaction zone" for stationary BH, etc. | colliding impulsive plane wave ------------------------------|--------------------------------- Schwarzschild vacuum | Penrose-Khan 1971 Erez-Rosen vacuum | Ferrari-Ibanez 1987 Kerr vacuum | (case of) Chandrasekhar-Xanthopoulos BR electrovacuum | Bell-Szekeres 1974 RN electrovacuum | (case of) Chandrasekhar-Xanthopoulos Kerr-Newman electrovacuum | Chandrasekhar-Xanthopoulos 1986 On the LHS, we have spacetimes with one timelike Killing vector field and one spacelike vorticity-free Killing vector field, giving an abelian subgroup of the group of isometric automorphisms. On the RHS, we have spacetimes with two spacelike Killing vectors (generating the planar wavefronts) given a two dimensional abelian subgroup of the group of isometric automorphisms. This duality has been further generalized to at least some models with massless scalar fields. This stuff is far more interesting and intricate than the above rough summary suggests: in particular, it turns out that there is actually a local isometry between -part- of the interior of a black hole and the corresponding interaction zone, but this isometry is quite different from what you might guess! See the book by Griffiths for details. This duality also provides a deep connection between models of gravitational collapse which probe the problem of what gtr says, theoretically, about "realistic black hole interiors", because it shows that the collapse of various kinds of matter/radiation can be modeled (in the -interior- of the newly formed hole) by colliding impulsive PP wave models. In particular, the important question of whether "mass inflation" and a "caustic but non-destructive" Cauchy horizon shrinking onto the destructive spacelike singularity inside the event horizon which has been found in simplified models of black hole formation is -generic- can be attacked using models of colliding impulsive PP waves! For "realistic models of black hole interiors", including "mass inflation" and the collapsing "caustic but non-destructive" Cauchy horizon, see author = {Werner Israel}, title = {The Internal Structure of Black Holes}, booktitle = {Black Holes and Relativistic Stars}, publisher = {Chicago University Press}, year = 1989} and the book booktitle = {Internal Structure of Black Holes and Spacetime Singularities}, editor = {Lior M. Burko and Amos Ori}, publisher = {IOP Publishing}, address = {Bristol}, year = 1997} By the way, the conformal diagram for "realistic/generic BH interiors" which is suggested by the work of Poisson and Israel looks like this: singularity 88888 |* \ Cauchy horizon |* \ |** \ i^+ ("future timelike infinity") |** /\ |*** / \ scri^+ ("future null infinity") |*** / \ |***/ \ i^0 ("spatial infinity", r = infty) |*** / r = 0 |*** / |*** / |*** / |** / |** / scri^- ("past null infinity") |** / |* / |* / |*/ |/ i^- ("past timelike infinity") This diagram is fibered by spheres of varying radii (large near the right-hand border, small near the singularity and near r = 0 at left). Here, due to infalling nongravitational radiation, the CH should represent a caustic (as was pointed out long ago by Penrose), and in the diagram you can see that the CH (contrary to what you might think, it is -disjoint- from the EH!) contracts in finite time onto the strong spacelike curvature singularity, but the work of Poisson and Israel suggests that even though the curvatures diverge as a "lucky" observer approaches the CH (because it is a caustic!), this may happen so rapidly that an extended small body may not have time to react by being torn apart or even significantly distorted before it passes through the CH. Thereafter, of course, there are infinitely many possibilities for the further evolution, by definition of "Cauchy Horizon"! This should explain the intuitive meaning of the slogan "caustic but nondestructive Cauchy horizon" which was used above. For some particular colliding PP wave models, see: author = {C. Hoenselaers and F. J. Ernst}, title = {Matching pp waves to the {K}err metric}, journal = {J. Math. Phys.}, volume = 31, year = 1990, pages = {141--146}} author = {J. B. Griffiths and C. A. Hoensalaers and P. C. Ashby}, title = {Colliding plane gravitational waves with colinear polarization}, journal = {Gen. Rel. Grav.}, volume = 25, year = 1993, pages = {189--196}} author = {\"Ozay G\"urtug}, title = {A new extension of the {F}errari-{I}banez colliding wave solution}, journal = {Gen. Rel. Grav.}, volume = 27, year = 1995, pages = {651--655}} title = {Mass-Inflation in Dynamical Gravitational Collapse of a Charged Scalar-Field}, journal = {Phys. Rev. Lett.}, volume = 81, year = 1998, pages = {1554--1557}, note = {gr-qc/9803004}} author = {Shahar Hod and Tsvi Piran}, title = {The Inner Structure of Black Holes}, journal = {Gen. Rel. Grav.}, volume = 30, year = 1998, pages = {1555}, note = {gr-qc/9902008}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {Null Singularities in Colliding Waves}, note = {gr-qc/0103092}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {The Effect of Sources on the Inner Horizon of Black Holes}, note = {gr-qc/0010112}} author = {\"Ozay G\"urtug and Mustafa Halisoy}, title = {Horizon Instability in the Cross Polarized {B}ell-{S}zekeres Spacetime}, note = {gr-qc/0006038}} There is still another reason why the Chandrasekhar-Ferrari duality is extremely interesting: it turns out to provide a connection between what are by common consent the two most important of all known "solution-generating methods" in gtr, namely: a) solving the Ernst equation, for which see for example the recent monograph author = {I. Hauser and F. J. Ernst}, title = {Group structure of the solution manifold of the hyperbolic Ernst equation--- general study of the subject and detailed elaboration of mathematical proofs}, note = {gr-qc/9903104}} and references therein, b) the inverse scattering transform method ("gravitational solitons"): author = {V. A. Belinskii and V. E. Zahkarov}, title = {Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions}, journal = {Soviet Phys. JETP}, volume = 48, year = 1978, number = 6, pages = {985--994}} For the connection, see author = {J. B. Griffiths and S. Miccich\'e}, title = {The extensions of gravitational soliton solutions with real poles}, journal = {Gen. Rel. Grav.}, volume = 31, year = 1999, pages = {869--887}} There is a of course large and growing literature on solution-generating techniques for the EFE, which I won't even attempt to sketch here! But I must at least point out that one of the most interesting features of this literature is that one can reformulate the Ernst equation using matrices and then solve it using theta functions; see in particular these papers: author = {D.Korotkin and V.Matveev}, title = {Schlesinger system, {E}instein equations and hyperelliptic curves}, journal = {Lett. Math. Phys.}, volume = 49, year = 1999, pages = {145--159}, note = {gr-qc/9911003}} author = {D.Korotkin and V.Matveev}, title = {Solutions of {S}chlesinger system and {E}rnst equation in terms of theta-functions}, note = {gr-qc/9810041}} This phenomenon is related to the amazing Kraniotis-Whitehouse conjecture, which states (I think!) that the -general- solution to the EFE [recall that solving the Ernst equation gives all stationary axisymmetric vacuum solutions], at least for nonzero cosmological constant, can be written down succinctly using theta functions arising from elliptic modular curves! See author = {G. V. Kraniotis and S. B. Whitehouse}, title = {General Relativity, the Cosmological Constant and Modular forms}, note = {gr-qc/0105022}} If this were true in some form (clearly, as I stated what I think it says, it is at present too vague since one must clarify what one means by a "solution to the EFE", e.g. C^infinity, C^3, Colombeau?, allowed sources?, etc.) it would provide a deep connection between elliptic modular curves and one of the most important nonlinear PDEs in all of applied mathematics. Note that connections between "soliton solutions" and theta functions are already well established for nonlinear PDEs related to the famous Korteveg-de Vries equation. Note too that Kraniotis and Whitehouse have provided hard evidence that something like the conjecture might be true by expressing one of the most general known families of solutions to the EFE (the members in general lack any Killing vector fields) in terms of such theta functions. Still another twist: there have been some important recent developments regarding Gowdy cosmological models, and "unpolarized" Gowdy models are -also- connected via the CF duality with colliding PP waves; see for example author = {J. B. Griffiths and G. A. Alekseev}, title = {Some unpolarized {G}owdy cosmologies and nonlinear colliding plane wave spacetimes}, journal = {Int. J. Mod. Phys. D}, volume = 7, year = 1998, pages = {237--247}} Even now I have by no means exhausted the reasons by PP waves are interesting! Higher dimensional PP waves turn out to play a key role in constructing exact solutions in supergravity and superstring theories, author = {K. S. Stelle}, title = {{BPS} Branes in Supergravity}, booktitle = {Quantum Field Theory: Perspective and Prospective (Les Houches, 1998)}, publisher = {Kluwer}, year = 1999, note = {hep-th/9803116}} author = {K. S. Stelle}, title = {A lecture on super p-branes}, journal = {Fortschr. Phys.}, volume = 47, year = 1999, pages = {65--92}} they turn up in the -Newtonian- N-body problem(!) title = {Celestial mechanics, conformal structures, and gravitational waves}, author = {C. Duval and G. Gibbons and P. Horv\'athy}, journal = {Phys. Rev. D}, volume = 43, pages = {3907-3922}, year = 1991} and of course in many relativistic gravitation theories which have been proposed as competitors to gtr, and in addition there are many characterizations of PP waves among solutions to the EFE which support the notion, suggested by the comments above regarding the close interrelationship between solution generating methods, black hole interiors, Cosmic Censorship, and colliding PP waves, that PP waves (and the special case of plane waves) play a key role in the "deep structure" of the EFE. To mention just one: author = {Torre, C. G.}, title = {Gravitational Waves: Just Plane Symmetry}, note = {gr-qc/9907089}} Summing up, while PP waves at first sight to be a very specialized kind of exact solution to the EFE, in fact they appear to play a central role in gravitation theories, a role which is slowly emerging as several lines of thought begin to converge. Chris Hillman Home page: http://www.math.washington.edu/~hillman/personal.html