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Table Of Integrals Series And Products

Modified Gravity and Cosmology

The status of experimental tests of general relativity and of theoretical frameworks for analyzing them is reviewed and updated. Einstein’s equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of local Lorentz invariance and clock experiments. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, the Nordtvedt effect in lunar motion, and frame-dragging. Gravitational wave damping has been detected in an amount that agrees with general relativity to better than half a percent using the Hulse-Taylor binary pulsar, and a growing family of other binary pulsar systems is yielding new tests, especially of strong-field effects. Current and future tests of relativity will center on strong gravity and gravitational waves.

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The Confrontation Between General Relativity and Experiment

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The status of experimental tests of general relativity and of theoretical frameworks for analysing them are reviewed. Einstein's equivalence principle (EEP) is well supported by experiments such as the Eotvos experiment, tests of special relativity, and the gravitational redshift experiment. Future tests of EEP and of the inverse square law will search for new interactions arising from unification or quantum gravity. Tests of general relativity at the post-Newtonian level have reached high precision, including the light deflection, the Shapiro time delay, the perihelion advance of Mercury, and the Nordtvedt effect in lunar motion. Gravitational wave damping has been detected to half a percent using the binary pulsar, and new binary pulsar systems may yield further improvements. When direct observation of gravitational radiation from astrophysical sources begins, new tests of general relativity will be possible.

/pdf/the-confrontation-between-general-relativity-and-experiment-1ruacajchm.pdf

The Confrontation between General Relativity and Experiment

The ESA mission BepiColombo will explore the planet Mercury with equipment allowing an extremely accurate tracking. While determining its orbit around Mercury, it will be possible to indirectly observe the motion of its center of mass, with an accuracy several orders of magnitude better than what is possible by radar ranging to the planet’s surface. This is an opportunity to conduct a relativity experiment which will be a modern version of the traditional tests of general relativity, based upon Mercury’s perihelion advance and the relativistic light propagation near the Sun. We define the mathematical methods to be used to extract from the data of the BepiColombo mission, as presently designed, the best constraints on the main post-Newtonian parameters, especially b,g and the Nordtvedt parameter h, but also the dynamic oblateness of the Sun J 2( and the preferred frame parameters a 1 ,a 2. We have performed a full cycle simulation of the BepiColombo radio science experiments, including this relativity experiment, with the purpose of assessing in a realistic ~as opposed to formal! way the accuracy achievable on each parameter of interest. For g the best constraint can be obtained by means of a dedicated superior conjunction experiment, with a realistic accuracy .2310 26 . For b the main problem is the very strong correlation with J 2( ; if the Nordtvedt relationship h54b2g23 is used, as it is legitimate in the metric theories of gravitation, a realistic accuracy of .2310 26 for b and .2310 29 for J 2( can be achieved, while h itself is constrained within .10 25 . If the preferred frame parameters a 1 ,a 2 are included in the analysis, they can be constrained within .8310 26 and .10 26 , respectively, at the price of some degradation in b, J 2( and h. It is also possible to test the change with time of the gravitational constant G, but the results are severely limited because of the problems of absolute calibration of the ranging transponder, to the point that the improvement as compared with other techniques ~such as lunar laser ranging! is not so important.

/pdf/testing-general-relativity-with-the-bepicolombo-radio-1ban2y8scv.pdf

Testing general relativity with the BepiColombo radio science experiment

http://astro.mff.cuni.cz/davok/papers/bc_grav.pdf

Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments

On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds

Soliton dynamics for the generalized Choquard equation

We present some new results that relate information to chaotic
dynamics. In our approach the quantity of information is measured by the
Algorithmic Information Content (Kolmogorov complexity) or by a sort of
computable version of it (Computable Information Content) in which the information
is measured by using a suitable universal data compression algorithm.
We apply these notions to the study of dynamical systems by considering the
asymptotic behavior of the quantity of information necessary to describe their
orbits. When a system is ergodic, this method provides an indicator that equals
the Kolmogorov-Sinai entropy almost everywhere. Moreover, if the entropy is
null, our method gives new indicators that measure the unpredictability of the
system and allows various kind of weak chaos to be classified. Actually, this
is the main motivation of this work. The behavior of a 0-entropy dynamical
system is far to be completely predictable except that in particular cases.
In fact there are 0-entropy systems that exhibit a sort of weak chaos, where
the information necessary to describe the orbit behavior increases with time
more than logarithmically (periodic case) even if less than linearly (positive
entropy case). Also, we believe that the above method is useful to classify
0-entropy time series. To support this point of view, we show some theoretical
and experimental results in specific cases.

Claudio Bonanno

Papers

Testing general relativity with the BepiColombo radio science experiment

Gravity field and rotation state of Mercury from the BepiColombo Radio Science Experiments

On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds

Soliton dynamics for the generalized Choquard equation

Dynamical systems and computable information