General Relativity and Cosmology for Undergraduates by Norbury J.

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By Norbury J.

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Compute A − B) we must first put them at the same spacetime point. Thus in order to calculate Aµ (x + dx) − Aµ (x) we must first define what is meant by parallel transport in a general curved space. When we parallel transport a vector in flat space its components don’t change when we move it around, but they do change in curved space. Imagine standing on the curved surface of the Earth, say in Paris, holding a giant arrow (let’s call this vector A) vertically upward. If you walk from Paris to Moscow and keep the arrow pointed upward at all times (in other words transport the vector parallel to itself), then an astronaut viewing the arrow from a stationary position in space will notice that the arrow points in different directions in Moscow compared to Paris, even though according to you, you have parallel transported the vector and it still points vertically upward from the Earth.

70) meaning that which is constant. 72) which is an inflationary solution, valid for any V . Warning We have found that if k = Λ = 0 and if ρ ∝ R1m then R∝t2 for any value of m. All of this is correct. 7. COSMOLOGY WITH THE SCALAR FIELD 59 ˙ 1 1 dR dt and say R R ∝ t giving R dt dt ∝ t which yields ln R ∝ lnt and thus 2/m R∝t . The result R ∝t is wrong because we have left out an important constant. R˙ Actually if R = ct then ln R = c ln t = ln tc giving R ∝ tc instead of R∝t. 2 2/m t2/m Let’s keep our constants then.

If a vector is parallel transported from an ’absolute’ point of view (the astronaut’s window), then it must still be the same vector A, except now moved to a different point (Moscow). Let’s denote δAµ as the change produced in vector Aµ (xα ) located at xα by an infinitessimal parallel transport by a distance dxα . We expect δAµ to be directly proportional to dxα . 54) We also expect δAµ to be directly proportional to Aµ ; the bigger our arrow, the more noticeable its change will be. 56) where Γνµα are called Christoffel symbols or coefficients of affine connection or simply connection coefficients .

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