Substitute metric contraction to delta

Question

Hello,

I was wondering how to make this contraction to be displayed as a delta automatically (without having to call substitute function everytime, as I don't know if there will be deltas or not until I've already done the calculation)

{a,b,c}::Indices.
g_{a b}::Metric.

d:=g_{a b} g^{b c};
eliminate_metric(_);

That is, the output is g_{a}^{c} and I'd like to get \delta_{a}^{c}.

I've tried adding

\delta_{a}^{b}::KroneckerDelta.

or

g_{a}^{b}::KroneckerDelta.

before the calculations, but none of them work.

Thanks.

Answer 1

In general, if you want to automatically run certain commands after each computation step, you need to redefine your post_process function. In your case, put

def post_process(ex):
   eliminate_metric(ex)
   collect_terms(ex)

at the top of your file. The default only contains the collect_terms(ex) line.

This way you have full control over which types of simplifications you want to run automatically.

Comments

The problem is that what I get is

g_{a}^{c}

and what I expect is

\delta_{a}^{c}

I'm updating the question accordingly.

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Add

substitute(ex, $g_{a}^{b} = \delta_{a}^{b}$)

to your post_process function.

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By the way, the property

g_{a}^{b}::KroneckerDelta.

only tells Cadabra that $g_{a}{}^{b}$ is a Kronecker delta. It does not tell it to display it as a $\delta_{a}{}^{b}$. If you want that to happen automatically, use

g_{a}^{b}::LaTeXForm("\delta").

This keeps it g_{a}^{b} internally, but displays as $\delta_{a}{}^{b}$.

---

Add

substitute(ex, $g_{a}^{b} = \delta_{a}^{b}$)

to your post_process function.

Yes, I know that will definitely work, but I was thinking about why does it give that answer, as there does not exist a metric with one index of each kind, it's a Kronecker delta (and it's treated like that by cadabra).

My point is: don't you think that the substitution should be automatic?

Anyway, thanks a lot for your quick answer. =)

If you want that to happen automatically, use

g_{a}^{b}::LaTeXForm("\delta").

This keeps it g_{a}^{b} internally, but displays as δab.

I tried

{a,b,c}::Indices.
g_{a b}::Metric.
g_{a}^{b}::LaTeXForm("\delta")

d:=g_{a b} g^{b c};
eliminate_metric(_);

And it doesn't work. What am I doing wrong?

---

There is something to be said for that, but in general Cadabra takes a 'conservative' approach in the sense of not pushing things down the users' throats.

Keep in mind that even though you define

 g_{a b}::Metric;

this does not attach any property to the g symbol with indices in different places. Yes, for a metric it probably would make sense to automatically attach meaning to the object with mixed and the object with upper indices, but at the moment it does not do that.

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This works:

{a,b,c}::Indices(position=fixed).
g_{a b}::Metric.
g_{a}^{b}::LaTeXForm("\delta").

d:=g_{a b} g^{b c}; 
eliminate_metric(_);

Note that I added the position=fixed argument to the Indices property (to indicate that upper indices mean something different than lower indices) and I also added a full stop after the LaTeXForm property.

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It doesn't work for me, it gives this:

g_{a b}*g^{b c}
g_{a}^{c}

I'm using cadabra 2.1.7.

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LaTeXForm only works in the notebook interface, not on the command line.

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OK, so I guess the only way to do that is by substituting in the post_process function. :_(

Thanks a lot for all your time and patience.