By Albert Eagle
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Extra resources for A practical treatise on Fourier's theorem and harmonic analysis for physicists and engineers
90) We may note, in general, that the relative importance of the matter sources with respect to the curvature is measured by the ratio (see Eq. 91) (normalized in such a way that one recovers the usual critical energy–density ratio, = = c , when is constant and the metric is isotropic). 92) which is obviously ∞ = 1 for the isotropic, radiation-dominated background with i = 1=d and constant dilaton. 39) presented in the previous subsection can be directly recovered from Eqs. 90), respectively. As before, we have invariance 1 under time re ections and under the duality transformation ai → a− i → − i .
1) i If the above equations represented a general quasi-homogeneous solution, the Z2d generalization of scale-factor duality would be explicitly respected in the form i (x) → − i (x) (for any subset of indices i), while keeping all eia (x) and ÿxed. However, the momentum constraints still have to be imposed on Eq. 1): they are trivial in the homogeneous case, but quite non-trivial in the presence of spatial gradients. These constraints preserve the simplest scale-factor duality [ i (x) → − i (x) for all i], but explicitly break its Z2d generalization .
5. Frame-independence: which is the “right” metric? 3, an S-frame in ationary solution of the string cosmology equations, when transformed to the E-frame, tends to describe a phase of accelerated contraction. This is strictly true only for isotropic backgrounds because, if the metric is anisotropic, and there is a large enough number of (S-frame) contracting dimensions, the other dimensions may keep expanding (and accelerated) in both frames. 22), corresponding to d expanding and n contracting dimensions.