Exact solutions of Einstein's equations

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Curator: Malcolm A.H. MacCallum

Exact Solutions of Einstein's Equations

Einstein's General Relativity is the leading theory of space-time and gravity: it is highly nonlinear. Exact solutions of Einstein's equations thus model gravitating systems and enable exploration of the mathematics and physics of the theory.



Einstein's field equations of general relativity are 10 nonlinear partial differential equations in 4 independent variables. This complicated system cannot be generally integrated, although it has been reformulated as a self-coupled integral equation (Sciama, Waylen and Gilman, 1969). Analytic and numerical approximations can be used to explore physical situations. Exact solutions, though obtained by imposing simplifying assumptions, complement such approaches in several ways. They embody the full nonlinearity, allowing study of strong field regimes; they provide backgrounds on which perturbative approximations can be built; and they enable checks of numerical accuracy.

The term 'exact solution' is not well-defined: usually it means a solution where all quantities are expressed by elementary functions or the well-known special functions, but sometimes it is extended to include solutions known only up to solution of one or more differential equations. The known exact solutions are obtained from a wide variety of assumptions, the most important of these being the imposition of symmetry groups or special forms of the curvature tensor. Among the known solutions, some have been of particular importance physically or mathematically.

A number of books provide surveys of exact solutions, and should be consulted if fuller details are desired. For a general survey of solutions containing the simple energy-momenta given by vacuum, electromagnetism and perfect fluids see Stephani et al. (2003), for inhomogeneous cosmological solutions (defined as those containing as a special case one of the FLRW models discussed below) see Krasi$\acute{\rm n}$ski (1997), and for detailed surveys of some special classes see Griffiths (1991) and Belinskii and Verdaguer (2001).

For physical interpretations of many important solutions see Bi$\check{\rm c}\acute{\rm a}$k (2000) and Griffiths and Podolsk$\acute{\rm y}$ (2009). It should be noted that an exact solution does not necessarily have a unique interpretation. For instance, among the examples given later the Schwarzschild solution can be interpreted as representing either the exterior region of a spherical mass, or the interaction region following the collision of two particular plane waves. A related point is that different sources may give rise to the same exact solution.

Einstein's equations

Einstein's General Relativity theory generalizes Newton's gravity theory to one compatible with special relativity. It models space and time points as a (pseudo-)Riemannian four-dimensional manifold with a metric \(g_{ab}\) of signature \(\pm 2\) (the sign choice is conventional). Test particles are assumed to move on the geodesics of this manifold and tidal gravitational forces are described by its curvature.

The formulae relating the metric, the connection \(\Gamma ^a{}_{bc}\), and the Riemannian curvature, in coordinate components, are: \[ \begin{array}{cl} \Gamma^a{}_{bc} &= g^{ad}(g_{bd,c}+g_{dc,b}-g_{bc,d})/2,\\ R^a{}_{bcd}&=\Gamma ^a{}_{bd,c}-\Gamma ^a{}_{bc,d}+\Gamma^e{}_{bd}\Gamma ^a{}_{ec}-\Gamma ^e{}_{bc}\Gamma ^a{}_{ed},\end{array} \] where \(g^{ad}\) is the inverse of \(g_{bc}\) and the comma denotes a partial derivative (so \(f_{,a}=\partial f/\partial x^a\)). The metric and its inverse are used to raise and lower indices.

To generalize Newton's theory, the curvature must be related to the spacetime distribution of the energy-momentum tensor of the matter content, \(T_{ab}\). Taking a weak field, slow motion, limit, comparison of the geodesic equation with Newtonian free fall identifies corrections to an approximating flat (special-relativistic) metric with the Newtonian gravitational potential \(\Phi\). One thus desires equations relating the second derivative of the metric to \(T_{ab}\). Defining the Ricci tensor \(R_{ab}\) and the Ricci scalar \(R\) by \[R_{bd}:= R^a{}_{bad}, \qquad R := g^{ab}R_{ab},\] Einstein's field equations (EFE) \[\tag{1} G_{ab} := R_{ab} - \tfrac{1}{2} R g_{ab}=\kappa_0 T_{ab} + \Lambda g_{ab}\] achieve this. Here \(\Lambda\) is the cosmological constant and \(\kappa_0= 8\pi G/c^4\) where \(G\) is Newton's gravitational constant and \(c\) is the speed of light. \(G_{ab}\) is called the Einstein tensor.

The left hand side of Eq. (1) is linear in the second derivatives of \(g_{ab}\). If the denominators arising in inverting to \(g^{ab}\) are cleared, it is of order 8 in \(g_{ab}\) and its derivatives, and thus highly nonlinear.

The equations have been introduced in terms of a coordinate basis but are frequently written in the form obtained by assuming a tetrad (a choice of basis of the tangent vector space whose basis vectors have constant scalar products), or in terms of the spin-coefficient formalism.

Because by starting from a different set of characterizing assumptions one may arrive at the same solution in different coordinates, the 'equivalence problem' of deciding when two manifolds are (locally) the same, i.e. isometric, is of importance. This is formally undecideable but in practice can usually be resolved using methods based on ideas of Cartan (see chapter 9 of Stephani et al. (2003)).

The same equations (mutatis mutandis) have been used and solved in higher dimensions (see Black ring), with some of the same techniques, but so far very little of the full landscape of possible solutions in 5 or more dimensions has been explored.

Making the equations tractable

Authors sometimes assume a metric form and use Eq. (1) to calculate the energy-momentum (this is the deprecated \(g\)-method described by Synge (1971)). Since no equation is actually solved, the outcome does not merit being called a solution. However, exact solutions are generally obtained by less extreme forms of simplification which, for a given form of energy-momentum, may automatically ensure some of the equations are true while leaving others to be solved.

Symmetry groups

Suppose a transformation \(\Psi\) of a manifold \(M\) maps \(p \in M\) to \(\Psi(p) \in M\), is invertible, and preserves continuity and differentiability. In spacetime it thus induces a transformation of the metric at \(p\) to a metric at \(\Psi(p)\). If this metric agrees with the original one at \(\Psi(p)\) the transformation is called an isometry (or motion). Assuming the existence of a group, usually a Lie group, of motions typically leads to a special form of the metric expressed in coordinates adapted to the symmetry.

When the group is a Lie group, its generators define vector fields on the manifold representing infinitesimal transformations. These are known as Killing vectors and form a Lie algebra: they obey \[ v_{a;b} =0 \] where for any tensor \(A_{(ab)}:=\tfrac{1}{2}(A_{ab}+A_{ba})\) [and similarly \(A_{[ab]}:=\tfrac{1}{2}(A_{ab}-A_{ba})\)] and the semicolon denotes a covariant derivative. Assumptions on the existence of an isometry group are often described via the Lie algebra and the surfaces to which its Killing vectors are tangent.

For example, an assumption of spherical symmetry implies that there are three linearly independent Killing vector fields which are tangent to spheres. The metric can then be written as \[ \text{d}s^2 = Y^2[\text{d}\theta^2 + \sin^2\theta\, \text{d}\phi^2]+\text{e}^{2\lambda}(\text{d}x^3)^2\mp \text{e}^{2\nu} (\text{d}x^4)^2 \tag{2} \] where \(Y\), \(\lambda\) and \(\nu\) are functions of \(x^3\) and \(x^4\). The angular coordinates have their usual ranges: $0\leq \theta \leq \pi$ and $0 \leq \phi \leq 2\pi$, where $\phi=0$ and $\phi=2\pi$ are identified. If the gradient of \(Y\) is spacelike one can set \(Y=x^3\) and rename \(x^3\) as \(r\), and it is then usually assumed that $r\geq 0$.

Having a specialized form of the metric such as Eq. (2), one can then make assumptions on those components of the energy-momentum for which the corresponding terms in \(G_{ab}\) are not automatically determined, and solve the remaining equations. If in the spherically symmetric case one assumes a vacuum \((T_{ab} =0,~\Lambda=0)\), the unique solution tending to the flat metric of special relativity as \(r\rightarrow \infty\) is the Schwarzschild solution.

Solutions obtained by such assumptions are covered by Part II of Stephani et al. (2003): see also Griffiths (1991), Belinski and Verdaguer (2009) and Bolejko et al (2010).

"Algebraically special" solutions

The full curvature can be expressed, in four dimensions, as \[ R^{ab}{}_{cd}=C^{ab}{}_{cd}-\tfrac 13R\delta _{[c}^a\delta _{d]}^b+2\delta _{[c}^{[a}R_{\;\;d]}^{b]}.\] The Weyl tensor \({C^a}_{bcd}\) thus defined is conformally invariant. When the Weyl tensor is zero, the spacetime is conformally flat (i.e. the metric is a multiple, in general a position-dependent multiple, of the flat spacetime metric of special relativity).

A non-zero Weyl tensor has the property that there are four "principal null directions" (PNDs), defined by null vectors obeying \[ \tag{3} k_{[e}C_{a]bc[d}k_{f]}k^bk^c=0. \] The algebraic structure of the Weyl tensor is then characterized by whether two or more of the PNDs coincide. When at least two do, then, providing a suitable energy-momentum is assumed, the metric tensor can be simplified. Such spacetimes are known as `algebraically special', and can be classified into the Petrov types by the numbers of coincident PNDs: the details of the possible cases are given in the article on the spin-coefficient formalism. For the energy-momenta usually considered in such spacetimes, the vector field of the repeated PND is geodesic and shearfree by the Kundt-Thompson theorem, which (see Stephani et al (2003), theorem 7.5) generalises the Goldberg-Sachs theorem. When just two PNDs coincide, the spacetime is of Petrov type II. In the article on the spin-coefficient formalism the example of the Robinson-Trautman solutions (Petrov type II metrics in which the field of repeated PNDs is twist-free) is derived in detail.

As another example, metrics of Petrov type D (where the PNDs form two pairs of coincident directions) and in which there may be a non-null electromagnetic field whose similarly defined principal directions are aligned with those of the Weyl tensor, can be written as \[\tag{4} \begin{array} \text{d}s^2 &= (1-pq)^{-2}\left[(p^2+q^2)\text d p^2/X + X(\text{d} \tau+q^2\text{d} \sigma)^2/(p^2+q^2)\right.\\ &\phantom{= (1-pq)^{-2}}\left. +(p^2+q^2)\text d q^2/Y - Y(\text d \tau-p^2\text d \sigma)^2/(p^2+q^2)\right],\\ X & = X(p) = (-g^2+\gamma-\Lambda/6)+2lp-\varepsilon p^2+ 2mp^3-(e^2+\gamma +\Lambda /6)p^4, \\ Y & = Y(q) = (e^2+\gamma-\Lambda/6)-2mq+\varepsilon q^2 - 2lq^3+(g^2-\gamma -\Lambda /6)q^4, \end{array}\] (with constants \(e\), \(g\), \(l\), \(m\), \(q\) and \(\gamma\) and a range for $p$ such that $X>0$) or limits thereof (e.g. for $Y=0$).

The known algebraically special solutions are discussed in part III of Stephani et al. (2003). There is naturally an overlap with solutions obtained by assuming symmetry groups. For example, all spherically symmetric solutions are of Petrov type D or, as a special case, conformally flat.

Other simplifying assumptions

Some other specializations of interest arise from the following assumptions

  • there exist constant vector or tensor fields
  • the curvature is recurrent, complex recurrent or symmetric (these are conditions on, e.g., \(R_{abcd;e}\))
  • there is a Killing or Killing-Yano tensor
  • the spacetime admits conformal motions or collineations (vector fields generating a transformation under which the metric is mapped to a multiple of itself or the curvature to itself)
  • the spacetime contains surfaces with special properties (for example, flat three-dimensional slices)
  • the spacetime has special embedding properties

A particularly common case is where there is a conformal motion for which the multiple is a constant: such transformations are called homotheties. Their generating vector fields obey \[ v_{(a;b)} = C g_{ab}\] where \(C\) is a constant. A significant number of known solutions admit homotheties, though many of these were discovered without the presence of the homothety being assumed.

Solving the equations

Once one has simplified the metric and introduced a suitable energy-momentum tensor, the remaining non-trivial equations will form a system of differential equations (or in the case of spacetime homogeneity, algebraic equations). There is no general algorithm for all cases but some methods used in other areas have proved useful.

Lie point symmetries of the system of equations, although useful in many situations (see e.g. Stephani (1989) or Olver (1986)), usually reduce in the spacetime context to diffeomorphisms of the manifold (just saying that the results are coordinate invariant) or to isometric or homothetic motions. However, there are cases (for example, spherically symmetric shearfree perfect fluids) where Lie point symmetries have been helpful in finding exact solutions. Generalized symmetries, prolongation and linearization can also be of help.

In particular, solutions with two commuting Killing vectors (acting on spacelike or timelike two-dimensional surfaces), and containing matter with suitable energy-momentum, are amenable to methods from the theory of integrable systems, such as harmonic maps (potential space symmetries), Bäcklund transformations, inverse scattering, and Riemann-Hilbert problems. For instance all stationary axisymmetric vacuum spacetimes can be obtained using such generating techniques starting from flat space. Among the outcomes are solitonic solutions.

Some important solutions

A great many solutions are known, as perusal of the references cited in the Summary will show, and many of these have not been fully interpreted physically. Knowing the metric in closed form, elucidation of its physical properties may still be difficult (see Griffiths and Podolsk$\acute{\rm y}$ (2009)): for example, the geodesic equations, whose solutions give the possible tracks of test particles and light rays, may be intractable even for simple metrics. Among the most important solutions are those now briefly described. (Note that although the selected solutions are all algebraically special and several are spherically symmetric, this is far from being the case for all solutions.) The original papers in which the selected solutions were first derived are all readily available, having, except for the first plane waves paper, been included in the "Golden Oldies" series.

The Schwarzschild and Kerr solutions

In the coordinates of Eq. (2) with \(Y=x^3=r\) and \(x^4=t\) the Schwarzshild solution takes the form \[\tag{5}\text{d}s^2=r^2(\text{d}\theta ^2+\sin ^2\theta \,% \text{d}\phi ^2)+(1-2m/r)^{-1}\text{d}r^2-(1-2m/r)\text{d}t^2, \] where \(m\) is the mass of the central object (defined at \(r \rightarrow \infty\) by the comparison with the gravitational effect of a Newtonian mass at the centre: here mass is measured in geometric units (i.e. units such that \(c=G=1\)). A number of other coordinate systems are in frequent use.

The Schwarzschild metric is the unique external solution for a spherically symmetric body in a surrounding empty space. This suggests that General Relativity shares with Newtonian gravity the property that the external field of any spherical body depends only on its total mass and not on the radial distribution of the matter. However, interpretation of the solution as the same as that of a point mass at the centre is unsatisfactory because the form above is suitable only in \(r>2m\). In the early years after the discovery of the solution, researchers were not clear whether \(r=2m\), where the metric of Eq. (5) clearly has a singular coefficient, represented a true singularity. It is now well-understood that it is an "event horizon", the boundary of a black hole, and that the complete analytic continuation of the solution is singular at \(r=0\). For historical information see Eisenstaedt (1982) and for a general discussion of global properties of spacetimes, including those discussed here, see Hawking and Ellis (1973). The Schwarzschild solution provided a pattern for later investigations of singularities and of black holes.

This solution's uniqueness shows that General Relativity does not admit monopolar gravitational waves. It is also the lowest order approximation to the field of real astronomical bodies such as the Earth and the Sun. Calculating geodesics in this field has enabled accurate predictions of light-bending by the Sun and the advance of the perihelion of Mercury, two of the "classical tests" of general relativity theory.

The Schwarzschild solution is a special case of the Kerr solution (found in 1963) which represents the external field of a rotating black hole. This can be written as an instance of Eq. (4) with \(e=g=l=\Lambda=0\) and it is usual to write \(a^2:=\gamma\). The ratio of spin to mass (in geometrized units) is then \(a/m\). The Schwarzschild and Kerr solutions provide the background for studies of the physics in the field of black holes, which are used in modelling X-ray binary sources and active galactic nuclei in astronomy. Observations of radiation from matter near black holes enables us to infer that there are astronomical black holes with \(a/m > 0.95\): see Black holes.

The Schwarzschild and Kerr black holes can be readily generalized to include non-zero electromagnetic charges and (using Eq. (4) for example) non-zero \(l\) and \(\Lambda\). There are uniqueness theorems showing (with some technical caveats) that these families are the unique stationary black holes with spherical topology of a non-singular event horizon.

The Friedmann-Lemaître-Robertson-Walker (FLRW) solutions

These solutions give the geometry of the "standard model" in modern cosmology, and thus provide the background for an enormous number of papers studying cosmological physics, including perturbations of the solutions. Their geometry was clarified by Robertson and Walker, independently, in the 1930s, and the most frequently used specific solutions were found by Friedmann and by Lemaître in the 1920s: hence the lengthy name.

These solutions have the metric form \[\tag{6} \text{d}{s^2}=-\text{d}{t^2} + a^2(t)[\text{d}{r^2}+ \Sigma^2(r,k) (\text{d}{\vartheta^2}+\sin^2\vartheta\, \text{d}{\varphi^2})], \] where \(\Sigma(r,k)=\sin r,\,r\) or \(\sinh r\), respectively, when \(k=1,\,0\) or \(-1\). Here \(k/a^2\) is the curvature scalar of the three-dimensional surfaces \(t=\) constant. These spacetimes are spherically symmetric and conformally flat, hence exemplifying both the major types of simplification of the Einstein equations, and must contain a energy-momentum tensor of perfect fluid type. They always admit a 6-dimensional group of isometries acting on surfaces \(t=\) constant, so they are isotropic and spatially-homogeneous.

The Einstein field equations are (assuming $\dot{a}\neq 0$, since a constant $a(t)$ gives only the Einstein static solution and flat space) \[\tag{7} \begin{array}{rl} 3\dot{a}^2&=\kappa_0\mu a^2+\Lambda a^2-3k, \\ \dot{\mu}&+ (\mu+p)\dot{a}/a=0, \end{array} \] where \(\mu\) and \(p\) are the density and pressure of the perfect fluid. The first of Eq. (7) is called the "Friedmann equation". Many explicit solutions are known. (Indeed Ellis and Madsen (1991) pointed out that for massive scalar fields one can choose a function \(a(t)\) and use Eq. (7) to obtain the potential for the field: this is another version of the Synge \(g\)-method).

Lemaître-Tolman-Bondi (LTB) solutions

These spherically symmetric solutions are the solutions for Eq. (2) containing "dust" (a perfect fluid with \(p=0\)) with \(\Lambda=0\). They generalize the FLRW solutions for dust to inhomogeneous solutions. Since dust is believed to be an appropriate representation of the universe's matter content on the large scale at the present time, LTB solutions have been much used to provide exact models of structures in the universe (see Bolejko et al (2010)). They contain as special cases both the Schwarzschild and dust FLRW solutions.

The metric form is \[\tag{8} \text{d}s^2=Y^2(r,t)[{\rm d}\vartheta^2 +\Sigma^2(\vartheta,k){\rm d}\varphi^2] + Y'^2 {\rm d}r^2[1- \varepsilon f^2(r)]^{-1}- {\rm d}t^2, \] where $m$, $f$ and $t_0$ are arbitrary functions of \(r\) and \(Y\) is given by \[ \tag{9}t-t_0(r)=\pm \tfrac{2}{3}Y^{3/2}[2m(r)]^{-1/2},\] for \(\varepsilon =0\) and \[\tag{10} \begin{array}{rcl} t-t_0(r) & = & \pm h(\eta )m(r)f^{-3}(r),\quad Y=h^{\prime }(\eta )m(r)f^{-2}(r),\\ h(\eta ) & = & \left\{ \eta -\sin \eta ,\,\sinh \eta -\eta \right\} \quad\text{for}\quad\varepsilon =\left\{+1,-1\right\}, \end{array} \] for $\varepsilon \neq 0$.

Plane waves

The Einstein-Maxwell plane wave solutions were first found by Baldwin and Jeffrey in 1926. Using complex coordinates $\zeta$ and $\bar{\zeta}$, the metric takes the form \[ \tag{11}\text{d}s^2=2\text{d}\zeta \text{d}\bar \zeta -2\text{d}u\,% \text{d}v-2\left[ A(u)\zeta ^2+\overline{A}(u)\bar \zeta ^2+B(u)\zeta \bar \zeta \right] \text{d}u^2 \] ($A(u)$ complex, $B(u)$ real); any linear function of $\zeta $ and $\bar \zeta $ in $H$ can be removed by a change of coordinates. Here $B(u)$, the electromagnetic term, can alternatively represent other forms of aligned null radiation. Plane waves admit a 5-dimensional isometry group containing an Abelian 3-dimensional subgroup acting on null hypersurfaces. They are spacetime homogeneous (i.e. admit an additional Killing vector such that the 6 independent Killing vectors act on the whole spacetime) if either $A(u)=A_0\text{e}^{2\text{i}\kappa u}$, $B(u)=B_0$ or $A(u)=A_0u^{2\text{i}\kappa -2}$, $B(u)=B_0u^{-2}$ ($A_0$, $B_0$ real constants).

These spacetimes provide an important example of unexpected global structure. If one joins a plane wave to flat spaces either side of some range of \(u\), forming a "sandwich wave", then the light cone from a point on one side refocuses on the other side, as found by Penrose (1965). The sandwich wave structure resolved the issue of whether the gravitational waves first found, using approximations, by Einstein could be merely coordinate effects: Bondi, Pirani and Robinson (1959) showed that free test particles are relatively accelerated by passage through the wave region, implying that the wave must carry energy.

Plane waves are the first approximation for gravitational radiation far from a source in an otherwise empty space. They are a special case of the more general pp-waves, solutions with a covariantly constant null Killing vector reresenting plane-fronted gravitational waves with parallel rays and found in 1925 by Brinkman. This whole class is of Petrov type N (all four PNDs coincident) or conformally flat.

The Taub-NUT family

The solution known as the Taub-NUT solution is given by \[\tag{12} \text{d}s^{2}= U^{-1}\text{d}\tau^{2} -(2\ell)^{2} U (\text{d}\psi + {\rm cos} \theta\, \text{d}\phi)^{2}+ (\tau^{2} + \ell^{2}) (\text{d}\theta^{2}+ {\rm sin}^{2} \theta\, \text{d}\phi^{2}), \] where \(U\) is given by \[ U(\tau)= 1-2\: \frac {m\tau+\ell^{2}} {\tau^{2}+\ell^{2}}, \] with constant parameters \(m > 0\) and \(\ell > 0\). Here \(0 \leq \theta < \pi\), \(0 \leq \phi < 2 \pi\) and \(0 \leq \psi < 4 \pi\). With changed coordinates, the limit $\ell \rightarrow 0$ can be seen to be the Schwarzschild solution. This vacuum solution can be generalized to non-vacuum cases.

For vacuum the portion with \(U<0\), where $\tau$ is a timelike coordinate, was first given by Taub in 1951. It is spatially homogeneous and is axisymmetric at each point ("locally rotationally symmetric"), thus admitting 4 independent Killing vectors acting on surfaces \(\tau=\) constant. Subsequently (1963) Newman, Tamburino and Unti (whose initials were rearranged to NUT) found the full solution including the portions with \(U>0\). This metric is one member of a wider family in which $\varepsilon$ gives the curvature of the two-dimensional spaces $\tau =$ constant and $\psi =$ constant, $U(\tau)= \varepsilon-2(m\tau+\ell^{2})/ ({\tau^{2}+\ell^{2}})$, and other coefficients in the metric form Eq. (12) have to be generalized to accommodate $\varepsilon \neq 1$.

Taub-NUT spacetime has very unexpected global properties. The NUT region contains closed timelike lines and no sensible Cauchy surfaces, there are two inequivalent maximal analytic extensions of the Taub region (or one non-Hausdorff manifold with both extensions), the spacetime is nonsingular in the sense of a curvature singularity, and there are geodesics of finite affine parameter length. These properties gave rise to the title of Misner's 1963 paper (some of these properties are shared by the other Taub-NUT metrics). The solution had a great influence on studies of exact solutions and cosmological models which are spatially-homogeneous, and more generally on those which are hypersurface-homogeneous and self-similar, on cosmology in general, and on our understanding of global analysis and singularities in space-times.


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See also

Black hole, Black ring, Cosmological Constant, General relativity, Spin-coefficient formalism

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