Thursday, May 30, 2013

xTras v1.1.3

I've just posted a new version of my Mathematica package xTras. It fixes a couple of bugs and introduces some new functions. One of these functions is MakeTraceless, which takes any tensorial expression and returns its traceless version. For example:
  In:   MakeTraceless[RiemannCD[-a,-b,-c,-d]]
Out:   \( R_{abcd} + \frac{2 R \underset{1234}{Sym}(g_{ac} g_{bd})}{2 -3 d + d^2} + - \frac{4 \underset{1234}{Sym}(g_{bd} R_{ac})}{-2 + d} \)
The output uses the implicit symmetrizations of the SymManipulator package. But we can also symmetrize explicitly by expanding the symmetries:
  In:   ToCanonical@ExpandSym@MakeTraceless[RiemannCD[-a, -b, -c, -d]]
Out:   \(- \frac{g_{bd} R_{ac}}{-2 + d} + \frac{g_{bc} R_{ad}}{-2 + d} + \frac{g_{ad} R_{bc}}{-2 + d}  - \frac{g_{ac} R_{bd}}{-2 + d}  - \frac{g_{ad} g_{bc} R}{2  -3 d + d^2} + \frac{g_{ac} g_{bd} R}{2  -3 d + d^2} + R_{abcd} \)
Of course, this is just the Weyl tensor in d dimensions:
  In:   Simplification@ RiemannToWeyl@ExpandSym@MakeTraceless[RiemannCD[-a, -b, -c, -d]]
Out:  \(W_{abcd} \)
Another new function is ConstructDDIs, which construct dimensional dependent identities (DDIs). Say we have a two-dimensional manifold. We can then ask for a list of all DDIs which have one curvature tensor and two free indices:
  In:   ConstructDDIs[RiemannCD[a,b,c,d],IndexList[a,b]
Out:   \({R^{ab}  - \tfrac{1}{2} g^{ab} R} \)
It returns just one DDI, namely the vanishing of the Einstein tensor. This is well know fact, namely that gravity in two dimensions is purely topological, because the Einstein-Hilbert action is equal to the two-dimensional Euler density.

As always, you can grab the newest version of xTras from its, or have a look at the changelog for all the new features.