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# General Relativity

The research on gravity theory focuses on problems, related to the implementation of new mathematical techniques and especially in relativistic reference systems, the theory of which is widely implemented and is of great importance for future astrophysical experiments such as GAIA and LISA. The problems treated are:

1. Manoff’s generalized deviation equation and its possible applications in relativistic astrometry, celestial mechanics and Relativistic Reference Systems – geocentric and barycentric.
2. Algebraic geometry approach in gravitational theory. New solutions of the Einstein’s equations in terms of elliptic functions within the framework of extended gravitational theories.
3. Application of the algebraic geometry approach and of the anisotropic length scale in anisotropic cosmological models, also in isotropic cosmological models of the Friedman-Robertson-Walker type with the purpose of finding new solutions of the corresponding nonlinear equations.

### Theory of GPS

Our research is focused on the “time transfer” between atomic clocks on satellites, moving on two adjacent orbits. For example, in the satellite experiment GRACE (Gravity Recover and Climate Experiment) the two satellites are 220 km apart, but the range rate between the two satellites must be known to better than $10^{-6} m s^{-1}$. A new mathematical model has been created in which the time transfer between the satellites takes into account the ellipticity of the orbits. A new physical conception has been implemented, according to which for the synchronization of clocks it is of key importance to have an inverse time transformation, which in the case is ensured by elliptic integrals and Jacobi elliptic functions, possessing this property For values of a typical GPS-orbit, the range of the proper and coordinate time is calculated to be $28136.7619\times 10^{-6} s$ and $-18.7924\times 10^{-6}s$.

GPS (Global Positioning System), based on satellite technology, has a wide range of applications – sea and land navigation, low Earth orbit (LEO) satellite determination, static and kinematic positioning etc. The breakthrough in geophysical applications of atomic clocks have stimulated the advanced research in relativistic geodesy. The obtained overall fractional frequency uncertainty of for atomic clocks enables to perform height difference on the Earth surface with unprecedented absolute level of 1 cm, providing the opportunity thus to compare different see-level changes – probably caused by melting continental ice sheets (satellite GOCE- “the Gravity field and steady-state ocean circulation explorer”).

### References

1. Bogdan G. Dimitrov, “Cubic algebraic equations in gravity theory, parametrization with the Weierstrass function and nonarithmetic theory of algebraic equations”, Journ. of Math. Phys. 44 (2003) 2542-2578.
2. Bogdan G. Dimitrov, “Elliptic Curves, Algebraic Geometry Appoach in Gravity Theory And Uniformization Of Multivariable Cubic Algebraiс Equations“, Int. J. Geom. Meth. Mod. Phys. 5 (2008) 677-698.
3. Bogdan G. Dimitrov, “Algebraic geometry approach in gravity theory and new relations between the parameters in type I low-energy string theory action in theories with extra dimensions”, Ann. Der Physik (Berlin) 19 № 3‐5 (2010) 254‐257.
4. S.S. Manoff, and B. G. Dimitrov, “Flows and particles with shear-free and expansion-free velocities in (Ln, g)- and Weyl spaces”, Class. Quant. Grav. 19 (2002) 4377 – 4397.
5. S.S. Manoff, and B. G. Dimitrov,” On the Existence of a Gyroscope in Spaces with Affine Connections and Metrics”, Gen. Rel. Gravit. 35 No.1 (2003) 25‐33.
6. Bogdan G. Dimitrov, Manoff’s Generalized Deviation Equation and its Possible Applications in Celestial Mechanics and Relativistic Astrometry, Bulgarian Chemical Communications, Volume 42 (pp. 1–7) 2015.