In this article, we made the mathematical explanation of the antigravitational waves, by the inspiration that we got from the observed positron in cosmic rays. Then, we analyzed the mathematical difference between positive and negative flows of gravitational waves; and we calculated the spin of the negative flow of gravitational waves, which is used to stabilize the movement of the waves. In the mathematical formulas we found that positive and negative flows move in opposite directions from each other; therefore, if we see the spin (rotation) of the waves from the planet that emits the waves, the positive flow rotates anticlockwise , while the negative flow rotates clockwise. We also investigated the possible origin of gravitational waves, and concluded that the negative flow can occur when the positive flow appears, leaving holes behind, in the gravitational field, which is triggered by the movements of a large mass of the planet.

In this research we formulated the curvature tensors with the system of spherical polar coordinates, which describe the gravitational field and gravitational waves of a black hole; and then we calculated eigenvalues of the curvature tensors to estimate the relative strengths of their components to the stress-energy tensor in Einstein's field equation. For this simulation, we assumed that the time and the distance interact with each other if we travel from Earth to the inside of the black hole, and then the result of the simulation showed that the gravitational waves carry the same components of the gravitational field of the black hole. On the other hand, when we assumed that the time and the distance are independent, which resembles the situation outside of the boundary of the black hole toward Earth, the curvature tensors are different between those of the gravitational field and the gravitational waves. Upon the results of the simulation we conclude that the gravitational waves that come from the inside of the black hole carry the information of the gravitational field inside of the black hole, if we assume that time and space are dependent each other.

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which reported the result of the numeric simulation on the artificial antigravity. This paper further describes the derivation of the idea of the artificial antigravity, and adds the simulation of angular momentum that is needed to describe the antigravity. Also, because the angular momentum is the perpendicular movement to a threedimensional curved surface in a four-dimensional space-time, this paper challenges the limit of applying the curvature tensor in quantum mechanics; while, current quantum mechanics has been established on the flat surface. The artificial rotation of a hypothetical object is simulated, in which the gravity is so strong that the time-space can be distorted. The spherical polar coordinate system is selected to describe the curvature of the space, and the curvature tensor is formulated. Then the tensor is multiplied by the Euler's rotation matrix to make the inner product for the gravitational energy and the outer cross-product for the angular momentum of the rotation. To simulate the distorted time-space, two cases are selected: the linear distortion and the non-linear distortion upon the distance from the center of the strong gravity; also, the speed of the rotation is set in two options: the slower and the faster. Then the equation of motion is set by the curvature tensor to calculate the coefficient of the gravitational energy on the surface of the sphere in the spherical polar coordinates, and to calculate the coefficient of the angular momentum in the perpendicular direction to the sphere. The result shows that the antigravity can be produced by rotating the object, and the angular momentum can show the opposite directions by the selection of the rotation speed.

In this research, we simulated the angular momentum of gravitational field of a rotating black hole and the spin momentum of gravitational waves emitted from the black hole. At first, we calculated energy densities of the rotating gravitational field and spinning gravitational waves as the vectors, which were projected on the spherical curved surface of the gravitational field and of the gravitational waves. Then we calculated the angular momentum and the spin momentum as the vectors perpendicular to the curved surface. The earlier research by Paul Dirac, published in 1964, did not select the curved surface to calculate the motion of quantum particles; but, instead, he chose the flat surface to develop the theory of quantum mechanics. However, we pursued the simulation of the gravitational waves in spherical polar coordinates that form the spherical curved surface of the gravitational waves. As a result, we found that a set of anti-symmetric vectors described the vectors that were perpendicular to the spherical curved surface, and with these vectors we simulated the angular momentum of the rotating black hole’s gravitational field and the spin momentum of gravitational waves. The obtained results describe the characteristics of the rotation of a black hole and of spinning gravitational waves.