Scale-Covariant Theory of Gravitation Through Self-Similarity

Abstract

We study the scale-covariant theory of gravitation proposed by Canuto under the self-similar hypothesis. By considering the matter collineation approach, we deduce the form of the main quantities and obtain some restrictions on the behavior of the gauge function, φ(t). We apply the obtained results to study the Bianchi I, VII0, IX and Kantowsky Sachs models, and arrive at the conclusion that instead of considering a varying G theory, the solution is only valid for a particular value of the parameter of the equation of state, γ,. This leads us to obtain φ(t) = const, and by taking into account the external condition, Gφ2 = const, we may arrive at the conclusion that G = const. Therefore, all the obtained solutions are the same as the ones obtained in the framework of general relativity.

Dimensional analysis in physics and the Buckingham theorem

“…The dimension of the physical quantity indicates its physical nature, because regardless of whether we express the measured distance in feet or metres, it is a measurement of length. The symbols that are commonly used to designate the dimensions of basic physical quantities such as length, mass and time are L, M and T. 6 Base physical quantities by mutual multiplication and division give new, derived physical quantities with corresponding dimensions. Thus the ratio of the travelled distance and the time interval gives a new physical quantity (speed), whose dimension is L/T.…”

“…Areas in which it is successfully applied are difficult to enumerate. Some of them are astrophysics, electrodynamics, aerodynamics, design and construction of ships, mass and heat transfer, mechanics of elastic and plastic structures, simulation of nuclear reactors, biology and economy [4][5][6]. Dimensional analysis is particularly useful in studying the new phenomena for which the appropriate equations and boundary conditions are not yet fully known.…”

“…For example, the time interval between two successive sunrises can be expressed as 1 day, as 24 h, as 1440 min or 86 400 s. We see that the numeric value changes depending on the unit of time, although it is always one and the same interval of time 5. Height of Mount Everest (h = 8.848 km) and the radius of the Earth (R = 6370 km) are obviously dimensional quantities, but their ratio, h/R = 0.0014, is dimensionless, and therefore, independent of the unit system 6. The dimension of volume of electric current is denoted by I, and the dimension of absolute temperature by .0143-0807/10/040893+14$30.00 c 2010 IOP Publishing Ltd Printed in the UK & the USA…”