problem with Gdelta

[Christopher Herzog]

Hi.

I'm having a problem with using Gdelta to compute the Gauss-Bonnet term.

While the command

Gdelta[-a1, -a2, -a3, -a4, b1, b2, b3, b4] RiemannCD[a1, 

   a2, -b1, -b2] RiemannCD[a3, a4, -b3, -b4] // CollectTensors

produces the Gauss-Bonnet term with Riemann^2:Ricci^2:RicciScalar^2 in the correct ratio

4:(-16):4, the seemingly identical command

Gdelta[-a1, -b1, -a2, -b2, c1, d1, c2, d2] RiemannCD[a1, 

   b1, -c1, -d1] RiemannCD[a2, b2, -c2, -d2] // CollectTensors

produces the the curvature squared terms in the incorrect ratio (-4):8:(-4).

(The preamble, before issuing the above two commands, in my Mathematica notebook is 

Quit

<< xAct`xTensor`

<< xAct`xTras`;

DefManifold[M, 

  dim, {a, b, c, d, a1, b1, c1, d1, a2, b2, c2, d2, a3, b3, a4, b4}];

DefMetric[-1, g[-a, -b], CD, {";", "\[EmptyDownTriangle]"}];

)

Best,

Chris

[Leo Stein]

Hi Chris,

I agree that you have found a bug. To make it clearer what is going on, we can compare what happens when the contraction with Gdelta is automatically expanded (using an internal function contractGdeltaTimes) versus what happens if first expanding with ExpandGdelta:

In[] := 

  Gdelta[-a1, -a2, -a3, -a4, b1, b2, b3, b4]
  RiemannCD[a1, a2, -b1, -b2] RiemannCD[a3, a4, -b3, -b4]

  == (Gdelta[-a1, -a2, -a3, -a4, b1, b2, b3, b4] // 
      ExpandGdelta ) RiemannCD[a1, a2, -b1, -b2]
  RiemannCD[a3, a4, -b3, -b4] // ContractMetric // ToCanonical
Out[] = True

In[] :=

  Gdelta[-a1, -b1, -a2, -b2, c1, d1, c2, d2]
  RiemannCD[a1, b1, -c1, -d1] RiemannCD[a2, b2, -c2, -d2]

  == (Gdelta[-a1, -b1, -a2, -b2, c1, d1, c2, d2] // 
      ExpandGdelta) RiemannCD[a1, b1, -c1, -d1]
  RiemannCD[a2, b2, -c2, -d2] // ContractMetric // ToCanonical
 Out[] = (not true)

The function ExpandGdelta is so simple that it is clearly not at fault. Therefore there is a bug within contractGdeltaTimes, but tracking it down requires understanding that code.

I you would like to disable that code, you can write:

Gdelta /: (f_ /; f === Times)[expr1___, gdelta_Gdelta, expr2___] =.

after you load xAct`xTensor`. This will be slightly slower and you will have to explicitly call ExpandGdelta whenever you want to expand it, but at least it will be correct until somebody finds the ultimate cause of this bug.

L

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