# Penyelesaian tepat dalam kerelatifan am

Dalam kerelatifan am, suatu penyelesaian tepat adalah manifold Lorentz yang dilengkapi dengan medan tensor tertentu yang dibawa kepada keadaan model jirim biasa, seperti suatu cecair, atau medan bukankegravitian klasik seperti medan elektromagnet. Medan tensor ini harus mematuhi mana-mana hukum fizikal berkenaan (contohnya, mana-mana medan elektromagnet harus memuaskan persamaan Maxwell). Mengikuti suatu resipi piawai yang mana digunakan secara luas dalam fizik matematik, medan tensor ini juga harus memberikan sumbangan khusus pada tensor tenaga-tekanan $T^{ab}$ . (Mengikut akal, bila-bila sahaja suatu medan dijelaskan oleh suatu Lagrangian, mengubah dengan hal medan harus memberikan persamaan medan dan mengubah dengan hal pada metrik harus memberikan sumbangan tenaga-tekanan oleh kerana medan.)

Kerelatifan am
$G_{\mu \nu }+\Lambda g_{\mu \nu }={8\pi G \over c^{4}}T_{\mu \nu }$ Sumber

Akhirnya, apabila semua sumbangan pada tensor tenaga-tekanan ditambah, keputusannya harus memuaskan persamaan medan Einstein (ditulis dalam unit geometri)

$G^{ab}=8\pi \,T^{ab}.$ Dalam persamaan yang di atas, medan tensor di bahagian tangan kiri, tensor Einstein, dikira secara unik dari tensor metrik yang adalah sebahagian dari takrifan sebuah pancarongga Lorentz. Oleh kerana memberikan tensor Einstein tidak memberikan penentuan penuh tensor Riemann, tetapi meninggalkan tensor Weyl tidak ditentukan (lihat penguraian Ricci), persamaan Einstein dapat dianggap suatu jenis keadaan keserasian: geometri ruangmasa harus tekal dengan jumlah dan pergerakan mana-mana jirim atau medan bukan kegravitian, dari segi yang kehadiran segera "sini dan sekarang" tenaga-momentum bukan kegravitian menyebabkan suatu jumlah seimbang dengan kelengkungan Ricci "sini dan sekarang". Tambahan, mengambil terbitan kovarian persamaan medan dan menggunakan identiti Bianchi, didapati bahawa suatu jumlah/pergerakan berbeza tenaga-momentum bukan kegravitian yang sesuai dapat menyebabkan riak dalam kelengkungan untuk merambat sebagai sinaran graviti, walaupun merentasi rantau vakum, yang tidak mengandungi jirim atau medan bukan kegravitian.

## Kesukaran dengan takrifan

Ambil mana-mana manifold Lorentz, kira tensor Einsteinnya $G^{ab}$ , which is a purely mathematical operation, divide by $8\pi$ , and declare the resulting symmetric second rank tensor field to be the stress-energy tensor $T^{ab}$ . Thus any Lorentzian manifold is a solution of the Einstein field equation with some right hand side. Which of course doesn't make general relativity useless, but only shows that there are two complementary ways to use it. One can fix the form of the stress-energy tensor (from some physical reasons, say) and study the solutions of the Einstein equations with such right hand side (for example, if the stress-energy tensor is chosen to be that of the perfect fluid, a spherically symmetric solution can serve as a stellar model). Alternatively, one can fix some geometrical properties of a spacetime and look for a matter source that could provide these properties. This is what cosmologists have done for the last 5-10 years: they assume that the Universe is homogenous, isotropic, and accelerating and try to realize what matter (called dark energy) can support such a structure.

Within the first approach the alleged stress-energy tensor must arise in the standard way from a "reasonable" matter distribution or nongravitational field. In practice, this notion is pretty clear, especially if you restrict the admissible nongravitational fields to the only one known in 1916, the electromagnetic field. But ideally we would like to have some mathematical characterization that states some purely mathematical test which we can apply to any putative "stress-energy tensor", which passes everything which might arise from a "reasonable" physical scenario, and rejects everything else. Unfortunately, no such characterization is known. Instead, we have crude tests known as the energy conditions, which are similar to placing restrictions on the eigenvalues and eigenvectors of a linear operator. But these conditions, it seems, can satisfy no-one. On the one hand, they are far too permissive: they would admit "solutions" which almost no-one believes are physically reasonable. On the other, they may be far too restrictive: the most popular energy conditions are apparently violated by the Casimir effect.

Einstein also recognized another element of the definition of an exact solution: it should be a Lorentzian manifold (meeting additional criteria), i.e. a smooth manifold. But in working with general relativity, it turns out to be very useful to admit solutions which are not everywhere smooth; examples include many solutions created by matching a perfect fluid interior solution to a vacuum exterior solution, and impulsive plane waves. Once again, the creative tension between elegance and convenience, respectively, has proven difficult to resolve satisfactorily.

In addition to such local objections, we have the far more challenging problem that there are very many exact solutions which are locally unobjectionable, but globally exhibit causally suspect features such as closed timelike curves. Some of the best known exact solutions, in fact, have this character.

## Jenis jawapan tepat

Many well-known exact solutions belong to one of several types, depending upon the intended physical interpretation of the stress-energy tensor:

• vacuum solutions: $T^{ab}=0$ ; these describe regions in which no matter or nongravitational fields are present,
• electrovacuum solutions: $T^{ab}$  must arise entirely from an electromagnetic field which solves the source-free Maxwell equations on the given curved Lorentzian manifold; this means that the only source for the gravitational field is the field energy (and momentum) of the electromagnetic field,
• null dust solutions: $T^{ab}$  must correspond to a stress-energy tensor which can be interpreted as arising from incoherent electromagnetic radiation, without necessarily solving the Maxwell field equations on the given Lorentzian manifold,
• fluid solutions: $T^{ab}$  must arise entirely from the stress-energy tensor of a fluid (often taken to be a perfect fluid); the only source for the gravitational field is the energy, momentum, and stress (pressure and shear stress) of the matter comprising the fluid.

In addition to such well established phenomena as fluids or electromagnetic waves, one can contemplate models in which the gravitational field is produced entirely by the field energy of various exotic hypothetical fields:

• scalar field solutions: $T^{ab}$  must arise entirely from a scalar field (often a massless scalar field); these can arise in classical field theory treatments of meson beams, or as quintessence,
• Lambdavacuum solutions (not a standard term, but a standard concept for which no name yet exists): $T^{ab}$  arises entirely from a nonzero cosmological constant.

One possibility which has received little attention (perhaps because the mathematics is so challenging) is the problem of modeling an elastic solid. Presently, it seems that no exact solutions for this specific type are known.

Below we have sketched a classification by physical interpretation. This is probably more useful for most readers than the Segre classification of the possible algebraic symmetries of the Ricci tensor, but for completeness we note the following facts:

• nonnull electrovacuums have Segre type $\{\,(1,1)(11)\}$  and isotropy group SO(1,1) x SO(2),
• null electrovacuums and null dusts have Segre type $\{\,(2,11)\}$  and isotropy group E(2),
• perfect fluids have Segre type $\{\,1,(111)\}$  and isotropy group SO(3),
• Lambdavacuums have Segre type $\{\,(1,111)\}$  and isotropy group SO(1,3).

The remaining Segre types have no particular physical interpretation and most of them cannot correspond to any known type of contribution to the stress-energy tensor.

## Membina jawapan

The Einstein field equation, when fully written out as a system of partial differential equations, takes the form of a rather complicated system of coupled, nonlinear partial differential equations. As such, in general, it is very hard to solve.

Nonetheless, several effective techniques for obtaining exact solutions are available.

The simplest involves imposing symmetry conditions on the metric tensor, such as stationarity (symmetry under time translation) or axisymmetry (symmetry under rotation about some symmetry axis). With sufficiently clever assumptions of this sort, it is often possible to reduce the Einstein field equation to a much simpler system of equations, even a single partial differential equation (as happens in the case of stationary axisymmetric vacuum solutions, which are characterized by the Ernst equation) or a system of ordinary differential equations (as happens in the case of the Schwarzschild vacuum).

This naive approach usually works best if one uses a frame field rather than a coordinate basis.

A related idea involves imposing algebraic symmetry conditions on the Weyl tensor, Ricci tensor, or Riemann tensor. These are often stated in terms of the Petrov classification of the possible symmetries of the Weyl tensor, or the Segre classification of the possible symmetries of the Ricci tensor. As will be apparent from the discussion above, such Ansätze often do have some physical content, although this might not be apparent from their mathematical form.

This second kind of symmetry approach has often been used with the Newman-Penrose formalism, which uses spinorial quantities for more efficient bookkeeping.

Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS).

But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations. Recall too that solutions of the heat equation can be found by assuming a scaling Ansatz. These notions are merely special cases of Sophus Lie's notion of the point symmetry of a differential equation (or system of equations), and as Lie showed, this can provide an avenue of attack upon any differential equation which has a nontrivial symmetry group. Indeed, both the Ernst equation and the NLS have nontrivial symmetry groups, and some solutions can be found by taking advantage of their symmetries. These symmetry groups are often infinite dimensional, but this is not always a useful feature.

Emmy Noether showed that a slight but profound generalization of Lie's notion of symmetry can result in an even more powerful method of attack. This turns out to be closely related to the discovery that some equations, which are said to be completely integrable, enjoy an infinite sequence of conservation laws. Quite remarkably, both the Ernst equation (which arises several ways in the studies of exact solutions) and the NLS turn out to be completely integrable. They are therefore susceptible to solution by techniques resembling the inverse scattering transform which was originally developed to solve the Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which arises in the theory of solitons, and which is also completely integrable. Unfortunately, the solutions obtained by these methods are often not as nice as one would like. For example, in a manner analogous to the way that one obtains a multiple soliton solution of the KdV from the single soliton solution (which can be found from Lie's notion of point symmetry), one can obtain a multiple Kerr object solution, but unfortunately, this has some features which make it physically implausible.

There are also various transformations which can transform (for example) a vacuum solution found by other means into a new vacuum solution, or into an electrovacuum solution, or a fluid solution. These are analogous to the Bäcklund transformations known from the theory of certain partial differential equations, including some famous examples of soliton equations. This is no coincidence, since this phenomenon is also related to the notions of Noether and Lie regarding symmetry. Unfortunately, even when applied to a "well understood", globally admissible solution, these transformations often yield a solution which is poorly understood, or even globally objectionable.

Given the difficulty of constructing explicit small families of solutions, much less presenting something like a "general" solution to the Einstein field equation, or even a "general" solution to the vacuum field equation, a very reasonable approach is to try to find qualitative properties which hold for all solutions, or at least for all vacuum solutions. One of the most basic questions one can ask is: do solutions exist, and if so, how many?

To get started, we should adopt a suitable initial value formulation of the field equation, which gives two new systems of equations, one giving a constraint on the initial data, and the other giving a procedure for evolving this initial data into a solution. Then, one can prove that solutions exist at least locally, using ideas not terribly dissimilar from those encountered in studying other differential equations.

To get some idea of "how many" solutions we might optimistically expect, we can appeal to Einstein's constraint counting method. A typical conclusion from this style of argument is that a generic vacuum solution to the Einstein field equation can be specified by giving four arbitrary functions of three variables and six arbitrary functions of two variables. These functions specify initial data, from which a unique vacuum solution can be evolved. (In contrast, the Ernst vacuums, the family of all stationary axisymmetric vacuum solutions, are specified by giving just two functions of two variables, which are not even arbitrary, but must satisfy a system of two coupled nonlinear partial differential equations. This may give some idea of how just tiny a typical "large" family of exact solutions really is, in the grand scheme of things.)

However, this crude analysis falls far short of the much more difficult question of global existence of solutions. The global existence results which are known so far turn out to involve another idea.

## Catatan

1. ^ Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; & Herlt, E. (2003). Exact Solutions of Einstein's Field Equations (Edisi ke-2). Cambridge: Cambridge University Press. ISBN 0-521-46136-7.Selenggaraan CS1: pelbagai nama: senarai pengarang (link) Sumber bertakrifan untuk penyelesaian tepat pada umumnya.
2. ^ Belinski, V.; & Verdaguer, E. (2001). Gravitational solitons. Cambridge: Cambridge University Press. ISBN 0-521-80586-4.Selenggaraan CS1: pelbagai nama: senarai pengarang (link) A monograph on the use of soliton methods to produce stationary axisymmetric vacuum solutions, colliding gravitational plane waves, and so forth.

## Rujukan

• Krasiński, A. (1997). Inhomogeneous Cosmological Models. Cambridge: Cambridge University Press. ISBN 0-521-48180-5.
• MacCallum, M. A. H. "Finding and using exact solutions of the Einstein Equations". arXiv eprint server. Dicapai pada February 5 2006. Parameter |dateformat= yang tidak diketahui diendahkan (bantuan); Periksa date values in: |accessdate= (bantuan) An up-to-date review article, but too brief, compared to the review articles by Bičák or Bonnor et al. (see below).
• Rendall, Alan M. "Local and Global Existence Theorems for the Einstein Equations". Living Reviews in Relativity. Dicapai pada August 11 2005. Parameter |dateformat= yang tidak diketahui diendahkan (bantuan); Periksa date values in: |accessdate= (bantuan) A thorough and up-to-date review article.
• Friedrich, Helmut. "Is general relativity essentially understood' ?". arXiv eprint server. Dicapai pada August 11 2005. Parameter |dateformat= yang tidak diketahui diendahkan (bantuan); Periksa date values in: |accessdate= (bantuan) An excellent and more concise review.
• Bičák, Jiří (2000). "Selected exact solutions of Einstein's field equations: their role in general relativity and astrophysics". Lect. Notes Phys. 540: 1–126. doi:10.1007/3-540-46580-4_1. See also the "eprint version". arXiv. Dicapai pada June 23 2005. Parameter |dateformat= yang tidak diketahui diendahkan (bantuan); Periksa date values in: |accessdate=` (bantuan) An excellent modern survey.
• Bonnor, W. B.; Griffiths, J. B.; & MacCallum, M. A. H. (1994). "Physical interpretation of vacuum solutions of Einstein's equations. Part II. Time-dependent solutions". Gen. Rel. Grav. 26: 637–729. doi:10.1007/BF02116958.Selenggaraan CS1: pelbagai nama: senarai pengarang (link)
• Bonnor, W. B. (1992). "Physical interpretation of vacuum solutions of Einstein's equations. Part I. Time-independent solutions". Gen. Rel. Grav. 24: 551–573. doi:10.1007/BF00760137. A wise review, first of two parts.
• Griffiths, J. B. (1991). Colliding Plane Waves in General Relativity. Oxford: Clarendon Press. ISBN 0-19-853209-1. The definitive resource on colliding plane waves, but also useful to anyone interested in other exact solutions.available online by the author
• Hoenselaers, C.; & Dietz, W. (1985). Solutions of Einstein's Equations: Techniques and Results. New York: Springer. ISBN 3-540-13366-6.Selenggaraan CS1: pelbagai nama: senarai pengarang (link)
• Ehlers, Jürgen; & Kundt, Wolfgang (1962). "Exact solutions of the gravitational field equations". In Witten, L.. Gravitation: An Introduction to Current Research. New York: Wiley. m/s. 49–101. A classic survey, including important original work such as the symmetry classification of vacuum pp-wave spacetimes.