Calculate Killing vectors

[Daniel Volinski]

Hi All,

Given a space-time metric, is there a way to calculate its Killing vectors?

Thanks,

Daniel Volinski

[Eberhard Schruefer]

Well, you could define a coframe for your space-time in Excalc and calculate the Lie-derivative of the metric (you would need to do this by calculating the Lie-derivative of the basis of the coframe). This would give you differential equations for the components of the Killing vector which you need to solve some how.

Eberhard

[Oscar Castillo-Felisola]

Hi Eberhard, Could you please provide an example on how to obtain the set of PDEs? Thank you.

Oscar

[Daniel Volinski]

Thanks Eberhard, I understand.

I was hoping there is already a command in REDUCE that calculate the Killing vectors, given the coframe.

Daniel

[Eberhard Schruefer]

Hi Oscar,
There is an example in the Excalc test file for calculating the PDE's for the Killing vectors of the 2-sphere:

2: coframe e th = d th,e ph = sin th*d ph;

3: killing_vector u;

      ph        th
{@  *u   + @  *u  ,
  ph        th

       th
 {@  (u  ),
   th

       ph         2        th
  @  (u  )*sin(th)  + @  (u  ),
   th                  ph

           th        ph
  cos(th)*u   + @  (u  )*sin(th)}}
                 ph

The first entry of the returned list gives the explicit form of the Killing vector with undetermined coefficients and the rest are the PDE's for its coefficients.

Eberhard

[Remigiusz Durka]

Dear Eberhard, could you provide whole file/script for this?

[Eberhard Schruefer]

There is nothing special to do. Just start Reduce and then type the following into it

load excalc;
coframe e th = d th, e ph = sin th*d ph;
killing_vector u;

That's it.
Eberhard

[Daniel Volinski]

En domingo, 25 de abril de 2021 21:53:19 GMT+3, Eberhard Schruefer <eschruefer@users.sourceforge.net> escribió:

There is nothing special to do. Just start Reduce and then type the following into it

load excalc;
coframe e th = d th, e ph = sin th*d ph;
killing_vector u;

That's it.
Eberhard

Calculate Killing vectors

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[Remigiusz Durka]

I had old version of Reduce (from 2019). With update it worked like a charm! Thanks!