Asymptotic analysis of field commutators for Einstein-Rosen gravitational waves

Journal of Mathematical Physics

Volumn 45, Issue 9, 2004, Pages 3498-3532

  Barbero G , J Fernando ^a   Marugán, Guillermo A Mena ^a   Villaseñor, Eduardo J S ^b  

^a INSTITUTO DE MATEMÁTICAS Y FÍSICA FUNDAMENTAL   (Spain)

^b UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 5644298863     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1769612     Document Type: Article

References (31)

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16. Notice that in this system of units G has dimensions of length.
17. A.
18. As shown in Ref. 6 the only real singularity that appears now corresponds to p = 1.
19. The very slow convergence of this oscillating series for large values of τ or large values of λ makes it impractical for numerical computations. This is the reason why we only give the lower portion of the plot in Fig. 2.
20. z-1 f(t). It is a holomorphic function defined on the strip α<ℛ(z)<β of the complex plane where the integral is absolutely convergent.
21. Conditions for this identity to hold can be found in Ref. 14.
22. We will use this choice throughout the paper.
23. Formula (20) is obtained, precisely, by displacing the integration contour parallel to the imaginary axis. For functions with the asymptotic behaviors considered in the theorem the only singularities are poles whose residues give the asymptotic expansion.
24. More precisely, an asymptotic sequence given by inverse powers of log τ is appropriate to capture the behavior of our integral in τ.
25. The asymptotic analysis of multiple integrals is greatly simplified by the identification of the critical points, those points that give the dominant contributions. For Laplace or Fourier types of integrals these are easy to identify and, in practice, they are just a finite number of isolated points that can be singled out and studied by using neutralizers.
26. n it is sometimes useful to consider neutralizers that depend only on ∥q∥.
27. It is important to point out that the asymptotic behavior of an improper integral over [0,∞) may be very different from the limit of the asymptotic expansion of the integral over [0, A] when A → ∞.
28. The reason why we use neutralizers depending only on q is, precisely, to exploit this freedom.
29. 2 may be zero even though the partial derivatives of Φ are different from zero.
30. iθ factor comes from the measure in the integral.
31. 1>Q>log τ in the neutralizers introduced before.