By Shapley H.
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12) µ or, since γ AB , B = −γ AC γ B D γC D , B and X µ , A = t A , 0 = γ 1/2 [gλµ(γ AB t Aλ , B −γ AC γ B D γC D , B t Aλ + 12 γ C D γC D , B t Aλ ) + γ AB (gλµ ,κ − 12 gκλ ,µ )t Aκ t Bλ ]. 14) γ AB D B t A = 0. 14) has four independent components. 14) tangent to the world sheet are identically zero, and so only two of the components are non-trivial. 15) qµν = gµν − pµν . 17) 40 The elements of string dynamics where the right-hand side is a known function of X µ and its ﬁrst partial derivatives alone.
In view of the cylindrical symmetry of the problem, much of the analysis that follows is simpliﬁed by converting from Minkowski coordinates x µ = [t, x, y, z] to cylindrical coordinates x µ = [t, r, θ, z]. 58) where a prime denotes d/dρ and b = 12 e2 /β 2 ≡ m 2V /m 2H is the so-called Bogomol’nyi parameter. 59) where η ≡ − det(ηµν ) is the norm of the determinant of the metric tensor. 60) where X denotes any of the ﬁeld variables in the lagrangian. 61) and Q + ρ −1 Q − ρ −2 P 2 Q = 12 (Q 2 − 1)Q.
This state, known nowadays as the Big Bang, effectively marks the earliest time in the history of the Universe that could conceivably be described by the equations of classical cosmology. However, it should be stressed that it is not possible Strings as relics of the Big Bang 25 to directly observe conditions in the Universe before about 300 000 years after the Big Bang, as it was only then, at a temperature of about 3000–4000 K, that the opaque electron–proton plasma ﬁlling the Universe recombined to form an effectively transparent hydrogen gas.