An exact analytical solution to the two-body problem with a separable central and tensor interaction

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

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The two-body problem: an effective-one-body approach

10.1093/oso/9780198786399.003.0056 ◽

2018 ◽

Author(s):

Nathalie Deruelle ◽

Jean-Philippe Uzan

Keyword(s):

Body Problem ◽

Equations Of Motion ◽

Reaction Force ◽

Kerr Black Hole ◽

Radiation Reaction ◽

Spherically Symmetric ◽

Geodesic Motion ◽

Two Body Problem ◽

Symmetric Spacetime ◽

Radiation Reaction Force

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.

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The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux

Scientific Reports ◽

10.1038/s41598-020-73086-0 ◽

2020 ◽

Vol 10 (1) ◽

Author(s):

Hamdy M. Youssef ◽

Najat A. Alghamdi

Keyword(s):

Heat Flux ◽

Analytical Solution ◽

Exact Analytical Solution ◽

Constant Heat Flux ◽

Bioheat Transfer ◽

Skin Tissue ◽

Phase Lag ◽

Dual Phase ◽

Constant Heat ◽

Two Temperature

Abstract This work is dealing with the temperature reaction and response of skin tissue due to constant surface heat flux. The exact analytical solution has been obtained for the two-temperature dual-phase-lag (TTDPL) of bioheat transfer. We assumed that the skin tissue is subjected to a constant heat flux on the bounding plane of the skin surface. The separation of variables for the governing equations as a finite domain is employed. The transition temperature responses have been obtained and discussed. The results represent that the dual-phase-lag time parameter, heat flux value, and two-temperature parameter have significant effects on the dynamical and conductive temperature increment of the skin tissue. The Two-temperature dual-phase-lag (TTDPL) bioheat transfer model is a successful model to describe the behavior of the thermal wave through the skin tissue.

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