Schwarzschild Metric

Part 1

Spacetime Model

 
 

Schwarzschild Metric

Here we show that the Schwarzschild Metric can be easily obtained starting from closed volumes instead of the mass concept. This demonstration does not require complex tensor manipulations as in the traditional method.

This webpage is part of the website http://www.higgs-boson.org
To understand the demonstration, it is necessary to read the whole theory.


Minkowski Metric

The expression of the Minkowski Metric, in spherical coordinates, is:

equ1 - Schwarzschild Metric

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) :

equ2 - Schwarzschild Metric

Remembering that the Minkowski Equation follows the Lorentz Invariance, we know that 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:

equ3 - Schwarzschild Metric

Notes: Some authors prefer writing A(r)B(r) = K with K=c2. In that case, the term c2 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 do not follow this rule in this webpage because a simple number, "1" in this case, does not have a dimensional quantity like c2 or G, [L2/T2] or [L3/MT2] respectively.


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):

fig1 - Schwarzschild Metric

where :

  • drout is an elementary differential radial variation outside of any mass,
  • drin is an elementary differential radial variation inside a Schwarzschild space,
  • r is the point of measurement.

As in the Newton Law webpage, we have:

equ4 - Schwarzschild Metric

where:

  • e is a coefficient of the increase of spacetime curvature at distance r,
  • DR is the initial curvature of spacetime produced by the closed volume,

The order of magnitude of e being 10E-39, we can use the first order approximation:

equ5 - Schwarzschild Metric

Since e is a simple coefficient, we can calculate the relation between two differential elementary radius drout and drin, out and in a gravitational field:

equ6 - Schwarzschild Metric

Since e << 1, the equivalent formula is:

equ7 - Schwarzschild Metric

or, elevating in square:

equ8 - Schwarzschild Metric

Developing the denominator (1 - e)2 = 1 - 2e + e2 and ignoring the last term e2, we obtain:

equ9 - Schwarzschild Metric

This result is nothing but the radial component of the Schwarzschild Metric, that is to say the function A(r) of dr2 in equation 2. The calculation of B(r) is immediate using equation 3:

equ10 - Schwarzschild Metric
equ11 - Schwarzschild Metric

So, equation 2 becomes:

equ12 - Schwarzschild Metric

In the Newton Law webpage, we have obtained the following result: DR = KM (equ. 13), where K = G/c2 (equ. 20). So, equation 5 can be rewritten as:

equ13 - Schwarzschild Metric

Finally, porting this expression in equation 12 gives:

equ14 - Schwarzschild Metric

As we have here demonstrated,


 
The proposed theory, based on closed
volumes, gives an easy and consistent
explanation of the Schwarzschild Metric.