G. F. Torres del Castillo

Bio: G. F. Torres del Castillo is an academic researcher from Benemérita Universidad Autónoma de Puebla. The author has contributed to research in topics: Einstein field equations & Field (physics). The author has an hindex of 14, co-authored 105 publications receiving 677 citations. Previous affiliations of G. F. Torres del Castillo include CINVESTAV & University of Oxford.

3-D Spinors, Spin-Weighted Functions and their Applications

Abstract: The spinor calculus employed in general relativity is a very useful tool; many expressions and computations are considerably simplified if one makes use of spinors instead of tensors. Some advantages of the spinor formalism applied in the four-dimensional space-time of general relativity come from the fact that each spinor index takes two values only, which simplifies the algebraic manipulations. Spinors for spaces of any dimension can be defined in connection with representations of orthogonal groups and in the case of spaces of dimension three, the spinor indices also take two values only, which allows us to apply some of the results found in the two-component spinor formalism of four-dimensional space-time. The spinor formalism for three-dimensional spaces has been partially developed, mainly for spaces with a definite metric, also in connection with general relativity (e.g., in space-plus-time decompositions of space-time), defining the spinors of three-dimensional space from those corresponding to four-dimensional space-time, but the spinor formalism for three-dimensional spaces considered on their own is not widely known or employed. One of the aims of this book is to give an account of the spinor formalism for three-dimensional spaces, with definite or indefinite metric, and its applications in physics and differential geometry. Another is to give an elementary treatment of the spin-weighted functions and their various applications in mathematical physics. The best-known example of the spin-weighted functions are the spin-weighted spherical harmonics, which are a generalization of the ordinary spherical harmonics and, as the latter, are very useful in the solution by separation of variables of partial differential equations. By means of the spin-weighted spherical harmonics one can give a unified treatment of fields of any spin, without requiring definitions of the vector, tensor and spinor spherical harmonics employed in electrodynamics, quantum mechanics and general relativity.

The Teukolsky–Starobinsky identities in type D vacuum backgrounds with cosmological constant

Abstract: It is shown that in all the type D solutions to the Einstein vacuum field equations with cosmological constant, the one‐variable functions appearing in the separable maximal spin‐weight components of the neutrino, electromagnetic, and gravitational perturbations are related by certain differential operators and the corresponding proportionality constants are obtained. It is also shown that analogous relations hold in the case of perturbations by a Rarita–Schwinger field if the cosmological constant vanishes.

Rarita–Schwinger fields in algebraically special vacuum space‐times

Abstract: It is shown that previous results concerning test massless fields on algebraically special vacuum backgrounds can be extended to the case of massless spin‐ (3)/(2) Rarita–Schwinger fields. A decoupled equation is derived from the Rarita–Schwinger equation on an algebraically special vacuum space‐time and it is shown that all the components of the field can be obtained from a scalar potential that obeys a wavelike equation. In the case of type D metrics, identities of the Teukolsky–Starobinsky‐type are obtained. Some relations induced by Killing spinors are also included.

The separability of Maxwell’s equations in type‐D backgrounds

Abstract: It is shown that in a type‐D space‐time that admits a two‐index Killing spinor a differential operator can be constructed that maps a solution of the Maxwell equations into another solution. By considering as a background the Plebanski–Demianski metric, which includes all the vacuum type‐D metrics, this operator is used to obtain all the components of the electromagnetic field and the vector potential. The separated functions appearing in the solutions are shown to obey identities of the Teukolsky–Starobinsky type and the separable solutions are shown to be eigensolutions of a certain differential operator with the ‘‘Starobinsky constant’’ as the eigenvalue.

Null strings and Bianchi identities

Abstract: The results of the theory of complexified V4’s which admit a null string are reexamined by using systematically the Bianchi identities. The formal reasons which permit the integration of the Gμν=0 equations to one differential constraint in the case of the HH structures are exhibited. Some new results concerning complexified Einstein–Weyl equations in the presence of a null string are given as well.

Quasinormal modes of black holes and black branes

Abstract: Quasinormal modes are eigenmodes of dissipative systems. Perturbations of classical gravitational backgrounds involving black holes or branes naturally lead to quasinormal modes. The analysis and classification of the quasinormal spectra require solving non-Hermitian eigenvalue problems for the associated linear differential equations. Within the recently developed gauge-gravity duality, these modes serve as an important tool for determining the near-equilibrium properties of strongly coupled quantum field theories, in particular their transport coefficients, such as viscosity, conductivity and diffusion constants. In astrophysics, the detection of quasinormal modes in gravitational wave experiments would allow precise measurements of the mass and spin of black holes as well as new tests of general relativity. This review is meant as an introduction to the subject, with a focus on the recent developments in the field.

Variational Principles in Mechanics

Abstract: The recognition that minimizing an integral function through variational methods (as in the last chapters) leads to the second-order differential equations of Euler-Lagrange for the minimizing function made it natural for mathematicians of the eighteenth century to ask for an integral quantity whose minimization would result in Newton’s equations of motion. With such a quantity, a new principle through which the universe acts would be obtained. The belief that “something” should be minimized was in fact a long-standing conviction of natural philosophers who felt that God had constructed the universe to operate in the most efficient manner—but how that efficiency was to be assessed was subject to interpretation. However, Fermat (1657) had already invoked such a principle successfully in declaring that light travels through a medium along the path of least time of transit. Indeed, it was by recognizing that the brachistochrone should give the least time of transit for light in an appropriate medium that Johann Bernoulli “proved” that it should be a cycloid in 1697. (See Problem 1.1.) And it was Johann Bernoulli who in 1717 suggested that static equilibrium might be characterized through requiring that the work done by the external forces during a small displacement from equilibrium should vanish. This “principle of virtual work” marked a departure from other minimizing principles in that it incorporated stationarity—even local stationarity—(tacitly) in its formulation. Efforts were made by Leibniz, by Euler, and most notably, by Lagrange to define a principle of least action (kinetic energy), but it was not until the last century that a truly satisfactory principle emerged, namely, Hamilton’s principle of stationary action (c. 1835) which was foreshadowed by Poisson (1809) and polished by Jacobi (1848) and his successors into an enduring landmark of human intellect, one, moreover, which has survived transition to both relativity and quantum mechanics. (See [L], [Fu] and Problems 8.11 8.12.)

Detection methods for stochastic gravitational-wave backgrounds: a unified treatment

Abstract: We review detection methods that are currently in use or have been proposed to search for a stochastic background of gravitational radiation. We consider both Bayesian and frequentist searches using ground-based and space-based laser interferometers, spacecraft Doppler tracking, and pulsar timing arrays; and we allow for anisotropy, non-Gaussianity, and non-standard polarization states. Our focus is on relevant data analysis issues, and not on the particular astrophysical or early Universe sources that might give rise to such backgrounds. We provide a unified treatment of these searches at the level of detector response functions, detection sensitivity curves, and, more generally, at the level of the likelihood function, since the choice of signal and noise models and prior probability distributions are actually what define the search. Pedagogical examples are given whenever possible to compare and contrast different approaches. We have tried to make the article as self-contained and comprehensive as possible, targeting graduate students and new researchers looking to enter this field.