Derivation of the new metric. 

The new metric is derived here by substitution of a new set of coordinates into the Schwarzchild metric. This alternative derivation does not require the use of the Lambert function. 

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

(1)

 

Note: To obtain the Kruskal-Szekeres metric, define f:=1 here by deleting the # from the line below and then click the [!!!] symbol in the toolbar to execute the worksheet. 

These are the new coordinates for Time and Radius: 

(2)

 

(3)

 

There is only one set of coordinates and they are equally valid above and below the event horizon.. 

The factor f is (r-2m)/abst(r-2m), but in this derivation f is handled implicitly. 

A proof that handling f implicitly is justified here, is given later. 

For a derivation where f is handled explicitly see the alternative derivation using the LambertW function. 

To obtain the new metric the first step is to obtain expressions for dt and dr in terms of R and T. 

(4)

 

(5)

 

(6)

 

dr and dt are defined in terms of each other so far. Alternative forms for dr and dt are now obtained so that after cancellations, dr and dt are defined independently. 

(7)

 

(8)

 

Equations  




and


both equate to dr so subtracting them from each other elliminates dr and leaving dt independently defined. 

(9)

 

Equations



and


both equate to dt so subtracting them from each other elliminates dt and leaving dr independently defined. 

(10)

 

The new expressions dr and dt are now independently defined and are substituted into the Schwarzschild metric here: 

(11)

 

(12)

 

(13)

 

(14)