Recast of a conformal line element

[silverwhale]

TL;DR Summary

In Birrell an dDavies QFT on CS a rewrite of a conformal line element is done. But this recasting seems to me not to be correct.

Hello PhysicsForums-Readers,

On page 59 of Birrells and Davies QFT on CS, the line element $ds^2 = dt^2 - a(t)^2 dx^2$, where $a(t)$ is some conformal factor defined as $a({\eta}) = dt/d{\eta}$.
Then in 3.83 the equation is rewritten to $ds^2 = a(\eta)^2 (d^2 \eta - dx^2)$. IMHO this cannot be true.
But how can the author recast eqaution 3.81 (mentioned above) to this one? maybe because the map is a conformal map??

Can anyone enlighten me on this rewrite? Thank you!
Silverwhale

 

[martinbn]

On page 59 of Birrells and Davies QFT on CS, the line element $ds^2 = dt^2 - a(t)^2 dx^2$, where $a(t)$ is some conformal factor defined as $a({\eta}) = dt/d{\eta}$.

No, $\eta$ is defined by this relation.

Then in 3.83 the equation is rewritten to $ds^2 = a(\eta)^2 (d^2 \eta - dx^2)$. IMHO this cannot be true.

You just make a change of variables, instead of $t$ use $\eta$. Substituting $dt = a(\eta)d\eta$ in the first equation, gives you this.

 

[silverwhale]

No, $\eta$ is defined by this relation.

You just make a change of variables, instead of $t$ use $\eta$. Substituting $dt = a(\eta)d\eta$ in the first equation, gives you this.

Thank you martinbn for your answer.

In page 59, the definition is $d \eta = dt/a$, that I do know; from which $a(\eta) * d\eta = dt$ follows (which I wrote), right?

Before I start explaining my problem (I hope this time better), We should not forget that the factor $a(t)$ depends on the variable $t$ as does $dt^2$.

Now, If we change the variable $t$ by $\eta$ in the line element, then we should get: $$ds^2 = d\eta^2 - a^2(\eta) dx^2.$$
That is not 3.83..

Next, If we subsitute in 3.81 $dt$ by $a(\eta) d\eta$, then $$ ds^2 = a^2(\eta) d\eta^2 - a^2(t) dx^2.$$ the problematic factor $a^2(t)$ still appears.

Last, if we take each term by itself in 3.81 and make a change of variables just in the second term, and substitute in the first, then yes we get 3.83, but that contradicts IMHO the definition 3.81 of the conformal line element $ds^2$ where $a(t)$ changes, when $dt$ changes in the coordinate axis..
Finally, saying $a(t)$ is the same as $a(\eta)$ does not make sense to me as $a$ should note the same map..
Silverwhale

 

[martinbn]

No, i am not saying replace the letter $t$ with the letter $\eta$, that would be usleless. The relation $d\eta=\frac{dt}a$ gives you, if you integrate it, each of the $t$ and $\eta$ as a function of the other, say $t=f(\eta)$. Then you make this change of variables. You keep the $x$ and you change $t$ to $\eta$ using $t=f(\eta)$.

 

[silverwhale]

Yes, I do get your point.
But then, I get $a(f(\eta))$ which ist not equivalent (as a function) to $a(\eta)$ That is my problem. Both are called $a$, but they are two different functions..