The Schwarzschild Metric Introduction Here we show that the Schwarzschild Metric can be easily obtained starting from closed volumes instead of the mass concept. This demonstration doesn't require complex tensor manipulations as in the traditional method. The Minkowski Metric The expression of the Minkowski Metric, in spherical coordinates, is:
The Schwarzschild Metric refers to a static object with a spherical symmetry. It is built from a Minkowski Metric, in spherical coordinates, with two unknown functions: A(r) and B(r) :
Remembering that the Minkowski Equation follows the Lorentz Invariance, the only way to get this invariance is to set A(r) = 1/B(r). From a mathematical point of view, we get the same result developing and simplifying EFE with correct parameters. Details of calculations are described in many articles and books concerning General Relativity. This conducts to the following equality:
Notes: Some authors prefer writing A(r)B(r) = K with K=c^{2}. In that case, the term c^{2} must be excluded from the Minkowski Metric, equation 1. However, in both cases, result is identical. On the other hand, in order to simplify equations, some Authors also replace c and G by 1. To avoid inconsistent expressions, we don't follow this rule in this webpage because a simple number, "1" in this case, doesn't have a dimensional quantity like c^{2} or G, [L^{2}/T^{2}] or [L^{3}/MT^{2}] respectively. The Schwarzschild Metric To calculate the Schwarzschild Metric, we can start with the figure of the main text concerning a convex curvature of spacetime (fig. 1):
where :
As in the Newton Law webpage, we have:
where:
The order of magnitude of e being 10E39, we can use the first order approximation:
Since e is a simple coefficient, we can calculate the relation between two differential elementary radius dr_{out} and dr_{in}, out and in a gravitational field:
Since e << 1, the equivalent formula is:
or, elevating in square:
Developing the denominator (1 – e)^{2} = 1  2e + e^{2} and ignoring the last term e^{2}, we obtain:
This result is nothing but the radial component of the Schwarzschild Metric, that is to say the function A(r) of dr^{2} in equation 2. The calculation of B(r) is immediate, taking into account that A(r) B(r) = 1 from equation 3:
So, equation 2 becomes:
In the Newton Law webpage, we have obtained the following result: DR = KM (equ. 13), where K = G/c^{2} (equ. 20). So, equation 5 can be rewritten as:
Finally, porting this expression in equation 12 gives:
As we see,
