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Open Preview See a Problem? Details if other :. Thanks for telling us about the problem. Return to Book Page. A completely revised and updated edition of this classic text, covering important new methods and many recently discovered solutions. This edition contains new chapters on generation methods and their application, classification of metrics by invariants, and treatments of homothetic motions and methods from dynamical systems theory. It also includes colliding waves, inhomo A completely revised and updated edition of this classic text, covering important new methods and many recently discovered solutions.

It also includes colliding waves, inhomogeneous cosmological solutions, and spacetimes containing special subspaces.

Get A Copy. Hardcover , pages. More Details Original Title. Other Editions 6. Friend Reviews. To see what your friends thought of this book, please sign up. Lists with This Book. This book is not yet featured on Listopia. Community Reviews. Showing This edition contains new chapters on generation methods and their application, classification of metrics by invariants, and treatments of homothetic motions and methods from dynamical systems theory.

It also includes colliding waves, inhomo A completely revised and updated edition of this classic text, covering important new methods and many recently discovered solutions. It also includes colliding waves, inhomogeneous cosmological solutions, and spacetimes containing special subspaces.

Get A Copy. Hardcover , pages. More Details Original Title. Other Editions 6. Friend Reviews. To see what your friends thought of this book, please sign up. Lists with This Book. This book is not yet featured on Listopia. Community Reviews. Showing Rating details. All Languages. More filters. Sort order. Ingemar Eriksson rated it really liked it Jan 08, Yuri Popov rated it really liked it Apr 04, 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. 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 General Relativity theory generalizes Newton's gravity theory to one compatible with special relativity.

Test particles are assumed to move on the geodesics of this manifold and tidal gravitational forces are described by its curvature. The metric and its inverse are used to raise and lower indices. The left hand side of Eq. 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. 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.

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. Authors sometimes assume a metric form and use Eq.

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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. 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.

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. Having a specialized form of the metric such as Eq. Solutions obtained by such assumptions are covered by Part II of Stephani et al. When the Weyl tensor is zero, the spacetime is conformally flat i. When at least two do, then, providing a suitable energy-momentum is assumed, the metric tensor can be simplified.

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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 , theorem 7. 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. The known algebraically special solutions are discussed in part III of Stephani et al. 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. A particularly common case is where there is a conformal motion for which the multiple is a constant: such transformations are called homotheties.

## Exact solutions of einsteins field equations pdf download

A significant number of known solutions admit homotheties, though many of these were discovered without the presence of the homothety being assumed. 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.

Stephani or Olver , 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. For instance all stationary axisymmetric vacuum spacetimes can be obtained using such generating techniques starting from flat space. Among the outcomes are solitonic solutions.

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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. 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.

In the coordinates of Eq.