12 research outputs found
Frame dragging and bending of Light in Kerr and Kerr-(anti) de Sitter spacetimes
The equations of general relativity in the form of timelike and null
geodesics that describe motion of test particles and photons in Kerr spacetime
are solved exactly including the contribution from the cosmological constant.
We then perform a systematic application of the exact solutions obtained to the
following cases. The exact solutions derived for null, spherical, polar and
non-polar orbits are applied for the calculation of frame dragging
(Lense-Thirring effect) for the orbit of a photon around the galactic centre,
assuming that the latter is a Kerr black hole for various values of the Kerr
parameter including those supported by recent observations. Unbound null polar
orbits are investigated, and an analytical expression for the deviation angle
of a polar photon orbit from the gravitational Kerr field is derived. In
addition, we present the exact solution for timelike and null equatorial
orbits. In the former case, we derive an analytical expression for the
precession of the point of closest approach (perihelion, periastron) for the
orbit of a test particle around a rotating mass whose surrounding curved
spacetime geometry is described by the Kerr field. In the latter case, we
calculate an exact expression for the deflection angle for a light ray in the
gravitational field of a rotating mass (the Kerr field). We apply this
calculation for the bending of light from the gravitational field of the
galactic centre for various values of the Kerr parameter and the impact factor.Comment: LaTeX file, 45 pages 1 figure, typos fixed, v3 published in Classical
and Quantum Gravity 22 (2005) 4391-442
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Die Verdoppelung des Quadrats in Platos ”Menon“ — die erste Mathematikstunde in der Menschheitsgeschichte
Iterated elliptic and hypergeometric integrals for Feynman diagrams
We calculate 3-loop master integrals for heavy quark correlators and the 3-loop quantum chromodynamics corrections to the -parameter. They obey non-factorizing differential equations of second order with more than three singularities, which cannot be factorized in Mellin-N space either. The solution of the homogeneous equations is possible in terms of F Gauß hypergeometric functions at rational argument. In some cases, integrals of this type can be mapped to complete elliptic integrals at rational argument. This class of functions appears to be the next one arising in the calculation of more complicated Feynman integrals following the harmonic polylogarithms, generalized polylogarithms, cyclotomic harmonic polylogarithms, square-root valued iterated integrals, and combinations thereof, which appear in simpler cases. The inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals, where the new innermost letter itself is not an iterative integral. A new class of iterative integrals is introduced containing letters in which (multiple) definite integrals appear as factors. For the elliptic case, we also derive the solution in terms of integrals over modular functions and also modular forms, using q-product and series representations implied by Jacobi’s functions and Dedekind’s -function. The corresponding representations can be traced back to polynomials out of Lambert–Eisenstein series, having representations also as elliptic polylogarithms, a -factorial , logarithms, and polylogarithms of and their -integrals. Due to the specific form of the physical variable for different processes, different representations do usually appear. Numerical results are also presented