perturbations of the metric determinant

[David Pirtskhalava]

HI,

First of all I'd like to thank you a lot for such a helpful software.
I have a small question:

It seems to me that when I try to expand the sqrt of metric
determinant on up to the second order

Perturbation[Sqrt[Detg[]], 2] // ExpandPerturbation // Simplification

I'm getting the output, which differs by the standard expression by a
factor of 2 in the second order. Namely, instead of the standard

Sqrt[g] = 1+h/2+ h^2 / 8 - h^2_{\mu\nu}/ 4

Im getting something like

1+h/2+ h^2 / 4 - h^2_{\mu\nu}/ 2.

why do you think it happens?

Thanks,

david

[JMM]

Hi David,

Thank you for using xAct.

I think the problem could be coming from the fact that for a tensor T
we have

perturbedT = T + eps Perturbation[T] + eps^2 Perturbation[T] / 2! +
eps^3 Perturbation[T] / 3! + ...

You get the full expression you want with xPert using (I assume eps is
the perturbation parameter)

Perturbed[ Sqrt[ Detg[] ], 2 ] / Sqrt[ Detg[] ] // ExpandPerturbation

Collect[ NoScalar[%], eps, ToCanonical ]

Note the use of NoScalar. The result is

1 + eps trace(h1) / 2 + eps^2 ( trace(h2) / 4 + trace(h1)^2 / 8 -
trace(h1.h1) / 4 )

where h1 is the first-order perturbation of the metric and h2 is its
second-order perturbation. I guess the last term is this formula is
the term you are worried about.

If you still think this result is incorrect, please let us know.

Cheers,
Jose.

[JMM]

On Jul 10, 10:18 am, JMM <Jose.Martin-Gar...@obspm.fr> wrote:
> Hi David,
>
> Thank you for using xAct.
>
> I think the problem could be coming from the fact that for a tensor T
> we have
>
> perturbedT = T + eps Perturbation[T] + eps^2 Perturbation[T] / 2! +
> eps^3 Perturbation[T] / 3! + ...

Sorry, the previous formula was wrong. It must be:

perturbedT = T + eps Perturbation[T] + eps^2 Perturbation[T, 2] / 2!
+eps^3 Perturbation[T, 3] / 3! + ...

This sort of expansion is precisely what you get with the command
Perturbed.

Jose.

[David Pirtskhalava]

Hi Jose,

Thank you very much for the reply. It works perfectly!

Cheers,

david

[magma]

David's question is typical of new xPert users.
At heart the problem lies in the way one represents the perturbed
metric.
Classic GR reference books either do not deal with higher order
pertubations at all, or use various/different conventions.
Just compare:
MTW's Gravitation pag. 966 (exercise), where the
"$PerturbationParameter" is implicit in the "h"
with

Wald's General Relativity page 184 where the same parameter is
explicit and the mathematcal setting is similar to xPert (but only
linear perturbations are considered).

And in the literature one finds anything in between, so to speak, with
the typical

g[eps]= g0 + eps h1[T] + eps^2 h2[T]
+eps^3 h3[T] + ...

where the eps is factored out , but the 1/n! is not.

David Brizuela, José M. Martín-García and Guillermo A. Mena Marugán,
the authors of xPert have chosen the mathematically most sensible
representation of the perturbed metric, using the h's as the
coefficients of a Taylor expansion and you will discover that xPert
code is a little jewel of efficiency and cleanliness, if you take a
minute to look at it.

The documentation however, could - in my opinion- be easily improved.
The documentation starts right away by describing and showing the
function Perturbation and this function is prominent in all sections.
Nowhere - unfortunately - is explained what is it meant ,
mathematically, by "perturbation".
The real mathematical definition of Perturbation in xPert can be
inferred only by the function Perturbed, which however is relegated to
a supporting role in the documentation.
By looking at Perturbed one sees that the epsilon^n/n! is factored out
and the h's are really the epsilon derivatives for epsilon ->0

Therefore I suggest that xPert documentation start with a section
describing Perturbed and Perturbation (in this order).
This would immediately set the mathematical playground for the rest of
the documentation, thus putting the new and in-xPert user quickly at
ease.

[magma]

errata:

>
> g[eps]= g0 + eps h1[T] + eps^2 h2[T]
> +eps^3 h3[T] + ...
>

the previous formula should read:

g[eps]= g0 + eps h1 + eps^2 h2
+eps^3 h3 + ...