By Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt
A totally revised and up to date version of this vintage textual content, masking very important new equipment and lots of lately came across suggestions. This version includes new chapters on new release equipment and their program, type of metrics through invariants, and coverings of homothetic motions and techniques from dynamical structures concept. additionally it is colliding waves, inhomogeneous cosmological recommendations, and spacetimes containing unique subspaces.
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Extra resources for Exact Solutions of Einstein's Field Equations
11 Fibre bundles 29 Jet bundles J (n) (X, U ) are another type of ﬁbre bundle of interest in exact solutions (see Chapter 10). They are used in describing systems of diﬀerential equations. Here the base manifold X is the space of independent variables and the ﬁbre is the set U of values of the dependent variables and their ﬁrst n partial derivatives with respect to the X. A (partial) diﬀerential equation, or a system of such equations, speciﬁes a submanifold in a jet bundle. 1 Introduction In Chapter 2 we treated diﬀerential geometry without a metric.
If α is a p-form (p ≥ 1) and dα = 0, then there is a (p − 1)-form β such that α = dβ. In components, α[i1 ···ip ,j] = 0 ⇔ αi1 ···ip = β[i1 ···ip−1 ,ip ] . 2 (Frobenius’s theorem). Let σ 1 , . . , σ r be r 1-forms linearly independent at a point p ∈ M. Suppose there are 1-forms τ A B (A, B = 1, . . , r) satisfying dσ A = τ A B ∧ σ B . Then in a neighbourhood of p there are functions f A B , hA such that σ A = f A B dhB . Other formulations of Frobenius’s theorem. Introducing the r-form Σ ≡ σ 1 ∧ · · · ∧ σ r , we can replace the condition dσ A = τ A B ∧ σ B by either of the two equivalent conditions: (i) dσ A ∧ Σ = 0, (ii) there exists a 1-form λ such that dΣ = λ ∧ Σ .
R be r 1-forms linearly independent at a point p ∈ M. Suppose there are 1-forms τ A B (A, B = 1, . . , r) satisfying dσ A = τ A B ∧ σ B . Then in a neighbourhood of p there are functions f A B , hA such that σ A = f A B dhB . Other formulations of Frobenius’s theorem. Introducing the r-form Σ ≡ σ 1 ∧ · · · ∧ σ r , we can replace the condition dσ A = τ A B ∧ σ B by either of the two equivalent conditions: (i) dσ A ∧ Σ = 0, (ii) there exists a 1-form λ such that dΣ = λ ∧ Σ . 44) σ[a,b σc] = 0 ⇔ σa = f h,a .
Exact Solutions of Einstein's Field Equations by Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt