By I.B. Khriplovich
ISBN-10: 0387256431
ISBN-13: 9780387256436
ISBN-10: 0387274065
ISBN-13: 9780387274065
This e-book bargains an alternative choice to different textbooks at the topic, supplying a extra particular dialogue of various normal relativistic results for readers who've wisdom of classical mechanics and electrodynamics, together with distinct relativity. insurance comprises particularly gravitational lensing, sign retardation within the gravitational box of the solar, the Reissner-Nordstr?m answer, chosen spin results, the resonance transformation of an electromagnetic wave right into a gravitational one, and the entropy and temperature of black holes. The e-book contains quite a few difficulties at quite a few degrees of hassle, making it excellent additionally for autonomous examine by way of a wide readership of complicated scholars and researchers. I.B. Khriplovich is leader Researcher, Budker Institute of Nuclear Physics, Novosibirsk, and Chair of Theoretical Physics at Novosibirsk college. Dr. Khriplovich is a Corresponding Member of the Russian Academy of Sciences. He has been awarded the Dirac Medal ``For the development of theoretical physics'' through college of recent South Wales, Sydney, Australia, and the Pomeranchuk Prize ``For remarkable contribution to the certainty the houses of the traditional version, specifically for illuminating paintings on susceptible and powerful interactions of quarks'' by way of the Institute of Theoretical and Experimental Physics, Moscow, Russia.
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Example text
1 General Form of Equations It is natural to assume that the generally covariant equations of the gravitational field should be second-order differential equations, and that the energymomentum tensor T µν should serve as a source in them. An additional assumption is that these equations should be linear in the Riemann tensor. Then their general structure is aRµν + bg µν R + cg µν = T µν . 53) dictate that b = − a/2. In this way we arrive at the Einstein equations Rµν − 1 µν g R = 8πkT µν + Λg µν .
In a flat space the Riemann tensor vanishes. Indeed, in such a space one can choose the coordinates in such a way that Γ µνρ = 0 everywhere, and hence Rτ ρµν = 0. And a tensor vanishing in one coordinate frame, vanishes in any other one. The inverse statement is also true: if the Riemann tensor vanishes, the space is flat. Indeed, locally, at a given point one can choose a Euclidean frame in any space. And with Rτ ρµν = 0 the parallel transport of the Euclidean coordinate frame from a given point to any other one is path-independent.
4) in a gravitational field in the covariant Lorentz gauge where Aµ; µ = 0 ? 14. In the flat space-time the electromagnetic field strength Fµν satisfies the equation Fµν = 0. What is the form of the corresponding equation in a gravitational field? 41)). To investigate its other symmetry properties, it is convenient to go over from the mixed components to covariant ones: 22 3 Fundamentals of Riemann Geometry Rτ ρµν = gτ σ Rσρµν . 46) Rτ ρµν = Rµντ ρ . 46) is sufficiently obvious: it guarantees the conservation of the length of a vector under its transport along a closed contour.