Geometrical Representation of Gravity

Abstract

To use atomic clocks for the in situ determination of differences in the gravitational potential of the Earth’s gravity field was proposed for the first time by Bjerhammar (1985), Vermeer (1983), taking up Einstein’s postulation that two atomic clocks will tick at different rates due to different gravity potential values at different locations. However, this concept has not been demonstrated so far due to limitations in comparing clock frequency at ≤10−18 relative accuracy between two distant locations. Recently, a frequency transfer was demonstrated below 10-18 relative accuracy over a distance of ca. 920 km using an optical fibre (Predehl et al. 2012), with only one optical clock placed at one end of the optical fibre and a H-maser at the other end. In Svehla and Rothacher (2005b) it was proposed to use atomic clocks in space to measure the gravitational potential along an orbit, to measure together with GNSS, both position and gravity in a purely geometrical way. Here we provide the physical background to relativistic geodesy that is not given in Bjerhammar (1985) and, based on this, provide a geometrical representation of gravity and its relation to orbital motion and reference frames for time. We also show that in special cases, it is possible to measure absolute gravity potential values using quantum mechanics, which opens up new possibilities for the use of state-of-the-art optical clocks. Beyond the Standard Model in theoretical physics based on four fundamental forces, gravitation is still separated from the electromagnetic, strong nuclear, and weak nuclear interactions that are successfully related by the quantum field theory at the level of atomic, particle and high energy physics. On the other hand, general relativity brilliantly describes all observed phenomena related to gravitation in our Solar System and at galactic and cosmological scales. However, general relativity is fundamentally incomplete, because it does not include quantum effects. A unified theory relating all four known interactions will represent a step towards the unification of all fundamental forces of nature. Here we show that circular perturbations could provide an interesting representation between quantum mechanics and orbit mechanics. We try to establish an equivalence between the orbit mechanics based on circular perturbations and basic principles of quantum mechanics. We show that gravity at quantum level and at celestial level can be represented with the same property as light, i.e., gravity and light can be represented as oscillating at the equivalent rate and thus propagate at the same rate. In the essence of every orbit one could consider a wave represented by matter and time that could be modelled or represented by two geometrical rotations. We try to represent gravitational potential by two geometrical counter-rotations, with the rotation of spherical harmonic coefficients as generating functions. This dualistic concept is similar to the electromagnetic force where electricity and magnetism are elements of the same phenomenon orthogonal to each other. Following the general relativity, any form of energy that couples with spacetime creates differential geometrical forms that can describe gravity. Thus, gravitation can be considered purely as a geometrical property. However, our geometrical representation using two counter-oscillations (bi-circular orbits) can be considered as describing gravitation from the scalar point of view at the quantum as well as at the celestial level. Thus it gives geometrical and scalar properties of gravitation at the same time. This is similar to the concept of a magnetic field generated on top of an existing electric field, or similar to the concept of matter and antimatter in particle physics, where antimatter is described as material composed of antiparticles with the same mass as particles, but with opposite charge (leptons, baryons). Following recent results from the Planck mission (Planck Collaboration et al. 2013), there is strong evidence that 26.8% of the mass-energy of the Universe is made of non-baryonic dark matter particles, which should be described by the Standard Model.