Comoving-physical coord. problem

[sanctus]

I have question but it is hard to explain, I'll give my best.

 

consider a spatially isotropic and homogenuos universe (ie FRW-universe), with a distribution function:

 

[math]f(x,\tilde{p}^i)[/math] where the tilde denotes the comoving momenta and there is no 0-component of the enrgy-momentum 4-vector because one expresses it via g(x)(p,p)=-m^2 (ie. there are only 7 degrees of freedom).

 

Now, since one is in a spatially homogenous and isotropic universe, the distribution function can only depend the norm of the momentum which I note for clarity P.

Sot far everything is clear to me, but now for the definition of P the book says:

[math]P=\sqrt{\delta_{ij}p^ip^j}=\sqrt{a^2\gamma_{ij} \tilde{p}^i \tilde{p}^j}[/math]

where [math]\gamma_{ij}[/math] is the tensor in the FRW metric, ie:

[math]g_{\mu\nu}dx^\mu dx^\nu =a^2\left(-dt^2+\gamma_{ij}dx^i dx^j\right)[/math]

Hence a is the expansion factor.

 

Also the p's without tilde are the physical coordinates. Now, why is P given by a minkowski metric if defined in physical coordinaties and by FRW metric in comoving coordinates?

 

Hope someone understands my problem.

[sanctus]

I eventually asked my prof...I should have read more carefully the book, one chose an orthonormal basis there...

 

I write this just in case someone wondered about this but never posted...