By Tai L. Chow
ISBN-10: 0387736298
ISBN-13: 9780387736297
Right here it's, in a nutshell: the heritage of 1 genius’s most important paintings – discoveries that have been to alter the face of contemporary physics. within the early 1900s, Albert Einstein formulated theories that will eternally switch the panorama of physics: the distinctive thought of Relativity and the final idea of Relativity. revered American educational Professor Tai Chow tells us the tale of those discoveries. He info the fundamental rules of Einstein, together with his legislations of gravitation. Deftly utilising his inimitable writing type, he is going directly to clarify the physics at the back of black holes, weaving into his account an evidence of the constitution of the universe and the technology of cosmology.
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Additional resources for Gravity, Black Holes, and the Very Early Universe: An Introduction to General Relativity and Cosmology
Sample text
32) we obtain ν αβ =− ∂ 2 f ν ∂ϕ δ ∂ϕ γ . ∂ x δ∂ x γ ∂ xα ∂ xβ The right-hand side is clearly symmetric in the indexes α and β, so that symmetric in α and β. 4 Christoffel Symbols and the Metric Tensor It is very useful to express the ’s in terms of the metric tensors. Let Aµ be any contravariant vector, Aµ = gµν Aν a covariant vector. From the definition of parallel displacement δ(Aµ Aµ ) = 0, we have δ(Aµ Aµ ) = gµν (x µ + d x µ )[Aν + δ Aν ][Aµ + δ Aµ ] − gµν (x µ )Aν Aµ = 0. Carrying out these operations gives us ∂gµν ∂xα Aµ Aν d x α + gµν Aµ δ Aν + gµν Aν δ Aµ = 0.
We now recall the conservation law of four-momentum in special relativity pµ d Pµ = 0, Pµ = µ where Pµ is the four-momentum of the system and pµ is the four-momentum of individual particle. (If the reader needs review of special relativity, Appendix II at the end of the book is there for this purpose. ) If matter-energy is continuously distributed in space, the formulas will take different forms. The continuous distributions may be considered as a limiting case of discrete particles, where in each volume element the number of point particles tends to infinity while the mass of each tends to zero.
So far, covariant and contravariant vectors have no direct connection with each other except that their inner product is an invariant. A space in which covariant and contravariant vectors exist separately is called affine. Physical quantities are independent of the particular choice of the mode of description, that is, independent of the possible choices of contravariance or covariance. In metric space, contravariant and covariant vectors can be converted into each other with the help of the fundamental tensor gµν .