Step one: Beginning with Einstein's field equations, derive the Scharzschild radius (equation 13 below):
Next, we call on Pythagoras and a right triangle to derive a basic metric equation (equation 15 below):
Using the same right triangle we derive the Lorentz factor (equation 19 below):
Now check out equation 20:
Because of equation 20, we can make a substitution and derive equations 22 and 23:
Equations 22 and 23 allow us to make more substitutions. The result is something that resembles the Schwarzschild metric (equation 24):
Here's the actual Scharzschild metric:
We can replace equation 24's cdt' with dr (differential radius) to get the following:
It would be great if the middle term (vdt)^2 had a plus sign instead of a minus sign in front of it. With some trigonometric slight of hand we change the minus sign to a plus. The result is equation 29:
So far we've used a triangle with only two space dimensions. We are one dimension short, but we can fix that:
Each dimension in space is a hypotenuse of a right triangle with two other dimensions which can replace the hypotenuse. We make a final substitution and we get the Schwarzschild metric (equation 32). Schwarzschild used spherical coordinates. For clarity and to help you visualize this type of coordinate system, I provide the diagrams below. The first two diagrams show the front and side view of a sphere of spacetime with a mass in the center. The position in spacetime is given by the radius (r), the first angle (top diagram), and a shorter radius (rsin[first angle]) and the second angle (bottom diagram).
The variables used in the Schwarzschild metric, however, are differential--a tiny piece of the radius and each angle. The value of each space variable is indicated in red below:
If we take the limit of these variables we get a point in spacetime indicated by the red dot in the diagram below:
The objective is to figure out the spacetime curvature in that tiny (red) region of space. To solve the field equations, we need to know the metric tensor components; i.e., the g's.
We can find the value of each g component within the Scharzschild metric:
Thus the metric tensor is as follows:
With the information we have, we can derive equation 42 below.
We can solve for R44--the spacetime curvature--by plugging in the mass (m) of a star, planet or black hole; the volume (V) of the space, mass and energy within an imaginary sphere with radius (r); and radius (r).