I Recast of a conformal line element

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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
 
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silverwhale said:
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.
silverwhale said:
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.
 
martinbn said:
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
 
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)##.
 
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..
 
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