Warped Space and Orbital Paths In his General Theory of Relativity (1916) Einstein suggested that Newton's “gravitational fields” should be replaced with the warping of spacetime around massive celestial objects. His rationale was based on the fact that he did not approve of the concept of “action at a distance”  a mechanism that is fundamental to fields. This concept would require that interactions occurring in one location in a field would require that information be sent "instantaneously" throughout the remainder of the field requiring speeds that exceeded the speed of light  Einstein's universal speed limit. Often used to model spacetime around a massive object a stretched or hyperbolically curved surface. Envision a circular rubber sheet stretched tightly about a circular hoop. When a massive ball is placed on its surface, a “gravitational well” is immediately created.
If a second, lighter mass, is released with a velocity tangent to the center of the well (along a gravitational equipotential) it will attempt to travel in a straight line path. However, the distortion of spacetime through which it is moving will result in the object orbiting the central celestial body and eventually spiraling down as it loses energy. Under ideal conditions smaller objects should maintain a circular orbit with a constant period.
We say that orbiting satellites are trapped in an "energy well" created by the more passive parent body distorting spacetime in its immediate vicinity. To obtain the total energy of an orbiting satellite we must add its potential and kinetic energies.
Gravitational Potential Energy
Let's first examine a satellite's potential energy. The potential energy function used for distances that exceed 1.2 radii from the center of a celestial body is
In the formula, the negative preceding the ratio indicates that on astronomical scales the potential energy of a satellite is negative until it reaches infinity where it approaches zero. In the graph shown below, the three curves represents a satellite's kinetic energy (blue), its potential energy (orange), and its total energy (green). Note the linearity of the potential energy function inside the highlighted box.
Recall that the gravitational field strength of the Earth is given by
The potential energy at the surface of the earth is
As mentioned earlier, due to the almost linear nature of our graph near the planet's surface, we can approximate the potential energy for small heights (h<< R_{E}) above the earth's surface as
The change in potential energy for a falling mass would be
