Disformal transformal transformation using xact

[NUR JAMAN]

Hi,
I am trying to do disformal transformation, the standard one ,f μν = g μν + B(φ)φ _μ φ_ ν where the latter( _ ) is partial derivative of scalar function Phi. I have to calculate the ricci Scalar of this new disformal metric f in terms of original metric g. I am doing so by replacing the christoffel by the new one(Calculated by hand). Now the problem I am facing is in simplification of the terms. Also Total derivative issue may applicable since R will appear in the action of the theory. Any kind help is welcome either in simplification or is there are any other way of doing this.


Thanks

[Miguel Zumalacárregui Pérez]

Hi Nur,

You can find the computation in https://arxiv.org/pdf/1308.4685.pdf My experience is that rather than working with the Christoffel (which is not a tensor and for which xAct has very specific rules) it's better to define the difference between the normal and the disformal christoffel, which is a well defined tensor (cf eq 35 of the aforementioned paper). From that you can compute the Ricci, the Einstein-Hilbert action and easily isolate the total derivatives (eqs 36, 38)

Another trick you can use is to redefine the field to pi(phi) = \int_0^phi \sqrt{B(x)} dx. That way you remove all the derivatives of B, and you can always undo the transformation at the end (see eq 27 of see also https://arxiv.org/pdf/1210.8016.pdf). With this trick the computation is simple and can be done/checked by hand.

If that doesn't work I have some old code to compute (more complicated) disformal transformations that worked reasonably well (although it might be for an older version of xAct).

[NUR JAMAN]

Hi Miguel,
Thank you for the kind reply , I have checked with the trick as you described in the second portion, It is really excellent and circulations are made simple. If you can help me in any kind for the xact implementation of the process described in first part it will be very help full. I am getting no idea in software implementation since I am very new in xact. Also let me  thank you for the reference it helps me a lot both for physical understanding