Note - This is a compilation of my comments about the subject matter.

Isaac Newton's experiments with the torsion bar and the lead weights demonstrated a gravitational field and the warping of space-time by masses. Gravitational fields are equivalent to acceleration when it comes to exerting forces on masses, and hence gravity causes a bending of the fabric of space-time. Mass and relativity theory have their validities as far as explaining the behaviour of gravitation, at least the mathematics of it in Roger Penrose's "The Road to Reality" looks sound enough.

The mass density by volume of the Earth is non-uniform and thereby creates a non-uniform gravitational field near the surface (variations ranging at +/-60 mGal whereby 1 milliGal is 10^-5 m/sec^2) that is more defined at close proximities than further away. Actually, at a far enough distance all planets look to be or can be more appropriately approximated as point masses at their center of mass (the point about which all unhinged mass distributions rotate).

Computationally, you can model the mass density by volume distribution of a given mass-body by discretization via breaking it up into cubic grid sections and then approximating the mass for the centroid of the individual equal volume segments, and then summing the gravitational field from each volumetric segment at any point in space to get the total superposition of the gravitational field at all points in the model. If the density of the mass-body is uniform, or varies linearly then you could use an analytical integration of the density over volume method instead of a finite element approximation method.

The hydrostatic distribution of many small masses that make up a fluid is determined by a statistical energy distribution based primarily on gravity, and second more locally on temperature and pressure, and in some cases electrical fields certainly can play a role in that distribution as well. (note - photon absorption and thereby electromagnetic heating is what causes thermals and wind currents in our atmosphere. At small scales, Brownian motion, electrostatic intermolecular forces, and the mean free path also plays an essential role in weather) Buoyancy is a result of gravitational energy and mass density distributions also.

In a planetary ring system, gravitation dictates that the lower ring masses orbit a planet in a faster time period than the higher ring masses. In conventional Newtonian physics, masses are held fixed for doing the calculations of the gravitational forces between them, and computing orbital trajectories is simple for a 2 body problem, however gets quite complicated for a 3 or more body problem. It takes complex differential equations or computational iterative simulations (like Euler's Method or Runge Kutta with boundary IC's or initial state IC's) to compute the multiple trajectories of the stars using Newton's or Einstein's equations, and at that they may or may or may not be completely correct either. Here is an interesting video on the situation:

Symmetry and Simulation: How Geometry Affects Scientific Computing from the Solar System to your Microwave Oven

According to quantum gravitation theory, gravitons are responsible for the presence of a gravitational field, however this phenomenon hasn't been experimentally documented, nonetheless Richard Feynman hypothesized this to be the case in his "The Feynman Lectures on Physics" series. The main theory (until demonstrated otherwise) is that mass exerts a force-field with an intensity that falls off as an inverse square law, just like any other type of particle-wave radiation. Currently, there is perhaps evidence that gravitational fields can be warped by intense magnetic fields. Are Dark Matter and Dark Energy true entities or are they just made up entities to force the stars and galaxies in the observable universe to fit with Einstein's gravitational theories and an expanding universe? Is the gravitational constant ‘G’ really a constant or does it depend on electromagnetic fields or on other things too?

There are no known experimental demonstrations of anti-gravity that are unclassified, however according to one hypothetical theory, an anti-gravity device would be driven by anti-mass, such that a gravitational field exerts the exact opposite direction of force than it does with regular mass. In such a system, like masses would attract each other and opposite masses would repel each other.

Black Holes might be created by intense mass collisions in particle accelerators, or by intense magnetic fields that warp gravitation, or by some variation of strong-weak nuclear interactions. Perhaps the space-time anomalies generated by miniature black holes, or the evaporation of them thereof, could spawn anti-mass which would be repelled by ordinary gravitational fields. (all in hypothetical speculation at this point, but perhaps those huge jets of particles escaping black holes are anti-mass)

Minkowski space-time can be curved by using a linear transformation matrix on the basis vector (x,y,z,t), which may be used to represent gravitational lensing as well as Non-Euclidean Geometries. This can be used to reflect reality, say if the coordinates of space-time aren't flat but instead elliptical or hyperbolic and using a linear transformation in order to correct flat Euclidean measurements, and also can be utilized to explain the apparent uniform expansion and contraction of the universe via a cosmological constant as well, assuming that constant actually exists.

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