By Jerzy Plebanski
By Andrzej Krasinski
Publisher: Cambridge University Press
Print Publication Year: 2006
Online Publication Date:March 2010
Online ISBN:9780511617676
Hardback ISBN:9780521856232
Paperback ISBN:9781107407367
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511617676.013
Subjects: Cosmology, Relativity and Gravitation, Astrophysics
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Why Riemannian geometry?
As argued in Section 1.4, gravitational forces can be simulated by inertial forces in accelerated motion. Special relativity describes relations between objects in uniform motion with respect to inertial frames, while gravitational interactions are neglected. The metric of the Minkowski spacetime in an inertial reference frame has constant coefficients. If we transform that metric to an accelerated frame, its components will become functions. Hence, a gravitational field should have the same effect: in a gravitational field the metric should also have non-constant components. Unlike in the Minkowski spacetime, in a gravitational field it should not be possible to make the metric components constant by a coordinate transformation. This was, in great abbreviation, the basic observation that led Einstein (1916) to general relativity.
This idea had to be supplemented with equations that would generalise the Newtonian laws of gravitation, and would relate the metric form to the gravitational field. The derivation of these equations, together with several related matters, will be presented in this chapter.
Local inertial frames
Let us recall the conclusion of Chapter 1: the Universe is permeated by gravitational fields that cannot be screened. Their intensity can be decreased by going away from the sources, but one can never decrease that intensity below the minimum determined by the local mean density of matter in the Universe. For this reason, no body in the Universe moves freely in the sense of Newton's mechanics, and consequently inertial frames can be realised only approximately, with a limited precision. Moreover, there exists no natural standard of a straight line, so the departures of real motions from rectilinearity cannot be measured.
pp. i-iv
pp. v-xii
pp. xiii-xvi
The scope of this text: Read PDF
pp. xvii-xviii
pp. xix-xx
1 - How the theory of relativity came into being (a brief historical sketch): Read PDF
pp. 1-6
Part I - Elements of differential geometry: Read PDF
pp. 7-8
2 - A short sketch of 2-dimensional differential geometry: Read PDF
pp. 9-12
3 - Tensors, tensor densities: Read PDF
pp. 13-25
4 - Covariant derivatives: Read PDF
pp. 26-32
5 - Parallel transport and geodesic lines: Read PDF
pp. 33-35
6 - The curvature of a manifold; flat manifolds: Read PDF
pp. 36-47
7 - Riemannian geometry: Read PDF
pp. 48-73
8 - Symmetries of Riemann spaces, invariance of tensors: Read PDF
pp. 74-93
9 - Methods to calculate the curvature quickly – Cartan forms and algebraic computer programs: Read PDF
pp. 94-98
10 - The spatially homogeneous Bianchi type spacetimes: Read PDF
pp. 99-112
11 - * The Petrov classification by the spinor method: Read PDF
pp. 113-122
Part II - The theory of gravitation: Read PDF
pp. 123-124
12 - The Einstein equations and the sources of a gravitational field: Read PDF
pp. 125-160
13 - The Maxwell and Einstein–Maxwell equations and the Kaluza–Klein theory: Read PDF
pp. 161-167
14 - Spherically symmetric gravitational fields of isolated objects: Read PDF
pp. 168-221
15 - Relativistic hydrodynamics and thermodynamics: Read PDF
pp. 222-234
16 - Relativistic cosmology I: general geometry: Read PDF
pp. 235-260
17 - Relativistic cosmology II: the Robertson–Walker geometry: Read PDF
pp. 261-293
18 - Relativistic cosmology III: the Lemaître–Tolman geometry: Read PDF
pp. 294-366
19 - Relativistic cosmology IV: generalisations of L–T and related geometries: Read PDF
pp. 367-437
20 - The Kerr solution: Read PDF
pp. 438-497
21 - Subjects omitted from this book: Read PDF
pp. 498-500
pp. 501-517
pp. 518-534