By Ta-Pei Cheng
This complicated undergraduate textual content introduces Einstein's normal concept of relativity. the themes lined contain geometric formula of exact relativity, the main of equivalence, Einstein's box equation and its spherical-symmetric answer, in addition to cosmology. An emphasis is put on actual examples and easy functions with out the whole tensor gear. It starts via analyzing the physics of the equivalence precept and appears at the way it encouraged Einstein's suggestion of curved spacetime because the gravitational box. At a extra mathematically obtainable point, it offers a metric description of a warped house, permitting the reader to check many attention-grabbing phenomena akin to gravitational time dilation, GPS operation, gentle deflection, precession of Mercury's perihelion, and black holes. a number of sleek themes in cosmology are mentioned from primordial inflation and cosmic microwave heritage to the darkish power that propels an accelerating universe.
Building on Cheng's earlier ebook, 'Relativity, Gravitation and Cosmology: A easy Introduction', this article has been adapted to the complicated scholar. It concentrates at the middle components of the topic making it compatible for a one-semester direction on the undergraduate point. it may additionally function an obtainable advent of basic relativity and cosmology for these readers who are looking to learn the topic all alone. the right kind 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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Extra resources for A College Course on Relativity and Cosmology
40) These expressions will be useful when we discuss the SR limit of particle energy in general relativity when spacetime is curved. 1 Relativistic momentum conservation We have introduced the relativistic momentum through its tensor transformation property. Another approach is to show that momentum conservation in a collision process requires such an expression. 12) no longer holds in SR because it is not Lorentz-covariant. , provided that relativistic energy (γ mc2 ) is also conserved. 41) holds in one frame, it also holds in another in relative motion (v): (γA mA uA + γB mB uB ) – (γC mC uC + γD mD uD ) = 0.
The metric of a space defines its inner product. • Tensors in Minkowski space are quantities having definite transformation properties under Lorentz transformation. If a physics equation can be written in tensor form (as discussed in Chapter 1), it automatically respects the relativity principle. • Tensors with no indices (scalars) are invariant under Lorentz transformation. They include the rest mass m of a particle, the proper time τ , and the speed of light c. • Tensors with one index (4-vectors) include the position 4-vector xμ = (ct, x), with components of time and 3D position, and the momentum 4-vector pμ = (E/c, p) = γ m(c, v), with components of relativistic energy E = γ mc2 and 3D momentum p = γ mv.
Einstein’s two postulates and Lorentz transformation 17 which implies that a1 = 1 ≡ γ. 11) This is the same Lorentz factor γ we encountered in Chapter 1. This concludes Einstein’s derivation of the Lorentz transformation: x = γ (x – vt), y = y, v t = γ t – 2x . 12) We note that the above steps 1–3 set up two coordinate frames in relative motion, step 4 imposed constancy of light speed5 while step 5 is a consistency condition implied by the principle of relativity. We can rewrite the Lorentz transformation in a more symmetric form by multiplying the time coordinate by a factor of c, so it will have the same dimension as the other coordinates: x ct =γ 1 –β –β 1 x .