^{1}

^{*}

^{2}

^{*}

^{3}

^{*}

^{3}

^{4}

^{*}

A REDUCE code for the Newman-Janis algorithm is described. This algorithm is intended to include rotation into nonrotating solutions of the Einstein field equations with spherically symmetry or perturbed spherically symmetry and has been successfully applied to many spacetimes. The applicability of the code is restricted to metrics containing potentials of the form 1/r.

In 1965, Newman and Janis [

Despite still not being fully understood and the fact that a complete satisfactory explanation of why it works has not been given yet [

To perform the NJ transformation, a REDUCE program [

The method is easily described as a series of steps to be followed once one has the seed metric to which the algorithm is meant to be applied.

1) The seed metric in spherical coordinates needs to be transformed to the advanced null coordinates, also known as Eddington-Finkelstein coordinates [

2) The next step is to find the null tetrad system that satisfies the contravariant metric.

3) Once the null tetrads are obtained, the Newman-Janis trick is used. The trick goes as follows, the radial coordinate of your metric is allowed to belong to a complex domain, this meaning merely that it can acquire complex values, but is required specifically that it must be always real, therefore terms of the form

where

4) Then, perform the NJ complex transformation on the advanced and radial coordinates:

where a is the rotation parameter.

5) Finally, it is applied the Boyer-Lindquist coordinate transformation [

In the following section, these different steps will be described as the code makes the calculation.

The code Newman-Janis.red is explained in detail as we go through the above exposed steps. The metric that the code needs to run, has the form:

where one has to define explicitly the metric components

in terms of the seed metric.

Now, we have the metric in terms of the advanced null coordinates:

where

The code enlists the components of this new metric tensor, writes it in matrix notation to calculate the inverse matrix. Then, the null tetrads are computed in terms of the components of this metric. To avoid errors the program computes the contravariant metric and verifies that both the tetrads and the metric components fulfill the following relation

The step 3 of the Newman-Janis procedure can only be done by hand, this is because it is cumbersome to do it with REDUCE [

The next step is to apply the Newman-Janis transformation (1) to the latter obtained null tetrads, which is the key step in the whole process. For the sake of simplicity in notation the code displays the following quantity in all the tetrads expressions

Then, the new contravariant metric components are obtained using Equation (5). The expression for it is of the form

The new covariant metric is determined from the contravariant one. The code computes again the new covariant metric in a more compacted way and confirms that both expressions are equivalent by performing the difference between them.

Next, the transformation to the generalized Boyer-Lindquist coordinates is performed by the program in order to display the final metric in the standard form. The code rewrites the expressions in a simpler and standard way:

and compares them to avoid mistakes. In (8) we have used

We tested the program for the Schwarzschild [

where

where

The second metric is given by

where

The resulting metric [

where

This REDUCE program is very useful to include rotation to metrics with spherical symmetry. It should not be used for metrics with cosmological constant and with no spherical symmetry. The inputs to the program that the user has to provide, are the metric and the change in term like

CarlosGutiérrez-Chávez,FranciscoFrutos-Alfaro,IvánCordero-García,JavierBonatti-González, (2015) A Computer Program for the Newman-Janis Algorithm. Journal of Modern Physics,06,2226-2230. doi: 10.4236/jmp.2015.615227

The output for main result in the case of the Schwarzschild metric is: