Derivation of the new metric. 

The new metric is derived here by substitution of a new set of coordinates into the Schwarzchild metric. 

The Schwarzchild metric for radial motion only, using units where the speed of light c=1 and the gravitational constant G =1, is given by : 

(1)

 

The new coordinates are basically an adaptation of the Kruskal-Szekeres coordinates with the addition of a factor f defined as: 

(2)

 

These alternative versions of f work equally well: or or as can be proven by copying and pasting these alternative definitions into the definition of f above and clicking on the [!!!] icon. If a value of is used then the Kruskal-Szekeres metric is re-obtained in the final result. 

The new coordinates for T=Time and R=Radius are defined as: 

(3)

 

(4)

 

The new coordinates are uniquely defined by the two equations given above and there are no alternative forms for above and below the event horizon. 

The first step is to isolate an expression for r in terms of R and T which can later be differentiated to obtain dr for direct substitution into the Schwarzschild metric. 

(5)

 

(6)

 

(7)

 

(8)

 

Note that r is undefined for X<0 which corresponds to T^2 > R^2.  This is the region in the North and South quadrants (grey regions) of the diagram below which illustrates the new coordinate system. The significance of this is that there is no alternative universe or white hole present in the coordinates.  

Note that there are two solutions for r in the new coordinates even when T^2 < R^2, but after the final substitutions at step 30 onwards, it turns out that this makes no difference to the final result.Here we continue with the primary solution for r. 

Interestingly, the primary solution for r is identical Kruskal-Szekeres coordinates and the new coordinates.  

(9)

 

(10)

 

The following just reinserts (R^2-T^2) for X to obtain r in terms of R and T. 

The variables R and T have to be unassigned to prevent them expanding into there fulf definition in the following equations. 

(11)

 

Now r is differeretiated to obtain an expression for dr. 

(12)

 

The next step is to isolate an expression for t in terms of R and T which can later be differentiated to obtain dt for direct substitution into the Schwarzschild metric. 

R and T are reasssigned their initial expanded definitions. 

(13)

 

(14)

 

(15)

 

(16)

 

(17)

 

Now t is differeretiated to obtain an expression for dt. 

(18)

 

Now that alternative expressions for dr and dt have been obtained they can be directly substituted into the simplified radial Schwarzschild metric, which after some substitutions and further simplifications yields the new metrics: 

(19)

 

(20)

 

(21)

 

(22)

 

(23)

 

Which is the new metric, which can be expressed as #              

For comparison the Kruskal-Szekeres metric is: #                         

Copyright (c) 2008 KevPegrume