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:

equ1

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

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:

equ3

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 don't follow this rule in this webpage because a simple number, "1" in this case, doesn't have a dimensional quantity like c2 or G, [L2/T2] or [L3/MT2] 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):

fig1

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

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

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

Since e << 1, the equivalent formula is:

equ7

or, elevating in square:

equ8

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

equ9

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, taking into account that A(r) B(r) = 1 from equation 3:

equ10

equ11

So, equation 2 becomes:

equ12

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

Finally, porting this expression in equation 12 gives:

equ14

As we see,

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

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