Recoverable Identifier

arXiv:2605.19829 · detector doi_compliance · incontrovertible · 2026-05-20 04:20:28.998953+00:00

Paper page Integrity report arXiv Try DOI

Evidence text

ifm <0,(7) where cY m ℓ stands for the standard complex spherical harmonics, and the star symbol denotes complex conjugation. Similarly, for the Fourier expansion, we employ the real modes F0 = 1/ p L0, F n = p 2/L0 cos (ωnζ) ifn >0, F n = p 2/L0 sin (ωnζ) ifn <0.(8) Alternative, one can instead use complex spherical harmonics and Fourier plane waves, but then one must take into account that the mode coefficients are also complex, though related in pairs by suitable complex conjugation relations in order that the subsequent metric perturbations are real. Certainly, the final result is independent of the procedure chosen. Having explained how perturbations are decomposed, we now denote byh=h ijdxidxj the perturbation of the three-metric induced on theχ-constant sections, and byp=pijdxidxj its conjugate momentum, where indicesi, j, . . . are lowered and raised with the unperturbed three-metric. We further denote byCandB=Bidxi the perturbations of the lapse function and the shift vector, respectively, in the 3+1 decomposition of the metric associated with the foliation along theχ-direction. We then introduce our mode expansion that, with a convenient background-dependent scaling of the different perturbations (that takes into account the orthonormal relations satisfied by our mode basis [39, 40]), can be expressed as follows: hij = X n hn ij, p ij = p det(γ) X n pn ij, C=κ s p2 b|pc| L2 0 X n C n, B i =κ X n Bn i ,(9) whereκ= 16πin our choice of units, and we have made use of th

Evidence payload

{
  "printed_excerpt": "ifm <0,(7) where cY m \u2113 stands for the standard complex spherical harmonics, and the star symbol denotes complex conjugation. Similarly, for the Fourier expansion, we employ the real modes F0 = 1/ p L0, F n = p 2/L0 cos (\u03c9n\u03b6) ifn >0, F n = ",
  "reconstructed_doi": "10.13019/501100011033(Spanish",
  "ref_index": 1,
  "resolved_title": null,
  "verdict_class": "incontrovertible"
}