#### Bergische Universität Wuppertal Fakultät für Mathematik und NaturwissenschaftenAngewandte Informatik - Algorithmik

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# Project "Newton's constant of gravitation and verified numerical quadrature"

Oliver Holzmann
Bruno Lang

### Duration and funding

about 1995 to 2000

### Description

Newton's constant of gravitation, G, is one of the fundamental quantities in physics. While other constants, such as the speed of light, are known to at least eight significant digits, the precision of G is much lower. In fact, even the claim of knowing this constant to four decimals is dubious, since different experiments lead to values of G differing in the third digit, implying that some of the published values must contain unknown errors. It is therefore essential to address all sources of errors in the process of obtaining a value for G from a physical experiment.

In an experiment carried out at the University of Wuppertal, two pendulums are positioned between two heavy "field masses." Since the gravitational force on a pendulum depends on its distance to the field masses, moving the latter with spindles also increases or decreases the distance of the pendulums from each other.

The constant G can be determined from this movement of the pendulums relative to each other. This process involves computing sixtuple integrals of the form

where the integration is over the volumina of one field mass and (a part of) one pendulum with densities ρ, and d denotes the directed distance between the respective points. Due to the complicated geometry of the pendulums they are subdivided into segments S( i, k ).

In this project we have developed adaptive techniques that yield the value of a multi-dimensional integral with a guaranteed prescribed maximum error. Using these methods we were able to show that

• the numerical evaluation of the integrals is no obstacle to the desired goal, which is to determine G to at least four, or even five, decimals,
• most of the geometric tolerances in the experiment are also harmless, but
• some of the current tolerances must be reduced significantly in order to achieve the desired accuracy. In particular, the temperature must be monitored very carefully because it has a major influence on the position of the field masses.

### Project-related publications

 [1] Bruno Lang. Derivative-based subdivision in multi-dimensional verified Gaussian quadrature. In Götz Alefeld, Jiři Rohn, Siegfried Rump, and Tetsuro Yamamoto, editors, Symbolic Algebraic Methods and Verification Methods, pages 145--152, Wien, 2001. Springer-Verlag. [ Abstract ] [2] Bruno Lang. A comparison of subdivision strategies for verified multi-dimensional Gaussian quadrature. In Tibor Csendes, editor, Developments in Reliable Computing --- SCAN-98 Proceedings, pages 67--75, Dordrecht, The Netherlands, 1999. Kluwer Academic Publishers. [ Abstract ] [3] Bruno Lang. Verified quadrature in determining Newton's constant of gravitation. J. Univers. Comput. Sci., 4(1):16--24, 1998. [ Abstract ] [4] Oliver Holzmann, Bruno Lang, and Holger Schütt. Newton's constant of gravitation and verified numerical quadrature. Reliab. Comput., 2(3):229--239, November 1996. [ Abstract ]

### Project-related theses

 [1] Oliver Holzmann. Untersuchungen zur Integration bei der Messung der Newtonschen Gravitationskonstanten. Diploma thesis, Bergische Universität Gesamthochschule Wuppertal, Germany, May 1996.

### See also

the other projects involving result-verifying techniques on the Research page

 University of Wuppertal Faculty of Mathematics and Natural Sciences Department of Mathematics and Computer Science Applied Computer Science Group

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