Liu, "3D Path planning based on nonlinear geodesic equation
," in Proceedings of the 11th IEEE International Conference on Control and Automation, pp.
In two recent papers [4, 5], I solved the scalar geodesic equation for mass-bearing particles and massless particles (photons), in the most studied particular spaces: in the space of Schwarzschild's mass-point metric, in the space of an electrically charged mass-point (the Reissner-Nordstrom metric), in the rotating space of Godel's metric (a homogeneous distribution of ideal liquid and physical vacuum), in the space of a sphere of incompressible liquid (Schwarzschild's metric), in the space of a sphere filled with physical vacuum (de Sitter's metric), and in the deforming space of Friedmann's metric (empty or filled with ideal liquid and physical vacuum).
This is as well true about the generally covariant geodesic equation.
Combining the above expression with the geodesic equation
of motion given by [du.
The second problem was solved using the geodesic equations for light-like particles (photons) which are mediators for electromagnetic radiation.
In terms of the physical observables, the isotropic geodesic equations are represented by their projections on the time line and spatial section of an observer [25-28]
We should also add that, coming from the geodesic equation
along the z-axis, which is the third equation of (61), to the simplified form (62) thereof, we omitted the harmonic term from consideration.
7) to (9) imply that not only the geodesic equation
, the Lorentz gauge, but also Maxwell's equation are satisfied.
Given that a geodesic equation
must be produced, for a monochromatic wave, the form of a photonic energy tensor should be similar to that of massive matter.
In terms of physically observable quantities--chronometric invariants (Zelmanov, 1944), which are the respective projections of four-dimensional quantities onto the time line and spatial section of a given observer--the isotropic geodesic equations
are presented with two projections onto the time line and spatial section, respectively [1-3]
We solve the second problem using the geodesic equations
for light-like particles (photons, which are mediators for microwave radiation, and for any electromagnetic radiation in general).
Thus relative accelerations of free test-particles are caused by the presence the space curvature [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and linear velocities of the particles are determined by the geodesic equations