By Ta-Pei Cheng

This complicated undergraduate textual content introduces Einstein's basic conception of relativity. the themes coated contain geometric formula of precise relativity, the primary of equivalence, Einstein's box equation and its spherical-symmetric answer, in addition to cosmology. An emphasis is put on actual examples and easy purposes with out the total tensor equipment. It starts off by means of analyzing the physics of the equivalence precept and appears at the way it encouraged Einstein's concept of curved spacetime because the gravitational box. At a extra mathematically obtainable point, it presents a metric description of a warped area, permitting the reader to review many attention-grabbing phenomena reminiscent of gravitational time dilation, GPS operation, mild deflection, precession of Mercury's perihelion, and black holes. a variety of sleek themes in cosmology are mentioned from primordial inflation and cosmic microwave history to the darkish power that propels an accelerating universe.

Building on Cheng's prior e-book, 'Relativity, Gravitation and Cosmology: A simple Introduction', this article has been adapted to the complex pupil. It concentrates at the center parts of the topic making it appropriate for a one-semester direction on the undergraduate point. it could possibly additionally function an obtainable creation of common relativity and cosmology for these readers who are looking to learn the topic on their lonesome. the correct tensor formula of Einstein's box equation is gifted in an appendix bankruptcy for these wishing to glimpse extra on the mathematical information.

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41), and the second is zero by conservation of relativistic energy γ mc2 . There are three equations for momentum and one for energy conservation. They can be written compactly in terms of 4-momenta: pμA + pμB = pμC + pμD . 45) Being a tensor equation (with every term a 4-vector), it is manifestly covariant; and these relations are valid in every inertial frame. 14 We will use the same criteria when we discuss particle energy and momentum in general relativity. Massless particles always travel at speed c When m = 0, we can no longer define 4-velocity as x˙ μ , because the concept of proper time does not apply (see the discussion below); thus the 4-momentum cannot be pμ = m˙xμ .

4, we present two worked examples illustrating the phenomena of time dilation and length contraction, as well as the issues of reciprocity of relativity and relativity of event order. 3 The twin paradox The twin paradox is a well-known reciprocity puzzle that sheds light on several basic concepts in relativity. 15 The paradox as a reciprocity puzzle Two brothers, Al and Bill, have the same age. Al goes on a long journey in a spaceship at high speed; Bill stays at home. Al’s biological clock appears to the stay-at-home Bill to run slow.

16) We say that tensor indices can be lowered or raised through contractions with the metric tensor or inverse metric tensor. 1 Product of symmetric and antisymmetric tensors Use the fact that dummy indices can be freely relabeled to prove that the contraction of a symmetric tensor (Sμν = +Sνμ ) and an antisymmetric tensor (Aμν = –Aνμ ) vanishes: Sμν Aμν = 0. , ijklmn) for 3D Euclidean tensors (or the spatial components of Minkowski tensors). Consider a 4D space whose position coordinates are the contravariant components xμ with the Greek index7 μ ∈ {0, 1, 2, 3} so that xμ = (x0 , x1 , x2 , x3 ) = (ct, x, y, z).

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