 Название: The Geometry of Spacetime: A Mathematical Introduction to Relativity Theory
Автор: Rainer Oloff
Издательство: Springer
Год: 2023
Страниц: 283
Язык: английский
Формат: pdf (true), epub
Размер: 21.3 MB
This book systematically develops the mathematical foundations of the theory of relativity and links them to physical relations. For this purpose, differential geometry on manifolds is introduced first, including differentiation and integration, and special relativity is presented as tensor calculus on tangential spaces. Using Einstein's field equations relating curvature to matter, the relativistic effects in the solar system including black holes are discussed in detail.
In Maxwell’s electrodynamics theory completed in 1860, it was concluded that light is an electromagnetic wave. Since usually wave phenomena require a medium, it was proposed that there is a fictitious medium called ether, which carries the electric and magnetic field strengths. In a frame which is at rest relative to this ether, the speed of light is expected the same in all directions. In a second frame that is moving with respect to the first one, the direction of motion is expected to influence the speed of light. However, scientists have failed to confirm this experimentally and in all experiments done since 1881, the speed of light was observed to be independent of the motion of the reference frame.
The validity of Newton’s laws of inertial systems gives rise to the Galilean relativity principle that velocity of intertial reference frames cannot be determined by mechanical experiments. In his 1905 published special theory of relativity, Einstein formulated his special relativity principal stating that all inertial reference frames are equivalent, that is, in all inertial reference frames all physical laws are the same.
In tensor algebra, there are always sums with often times several summation indices. To simplify the notations, the following so-called Einstein notation is commonly used: When an index occurs twice in one term, once as a subserscript and once as a subscript, then a summation is implied over the range of this index.
Contents:
1. Differentiable Manifolds
2. Tangent Vectors
3. Tensors
4. Semi-Riemannian Manifolds
5. Theory of Special Relativity
6. Differential Forms
7. The Covariant Differentiation of Vector Fields
8. Curvature
9. Matter
10. Geodesics
11. Covariant Differentiation of Tensor Fields
12. The Lie Derivative
13. Integration on Manifolds
14. Nonrotating Black Holes
15. Cosmology
16. Rotating Black Holes
17. A Glimpse of String Theory
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The Geometry of Spacetime: A Mathematical Introduction to Relativity Theory 
