I Recast of a conformal line element

  • I
  • Thread starter Thread starter silverwhale
  • Start date Start date
silverwhale
Messages
78
Reaction score
2
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
 
Physics news on Phys.org
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..
 
In this video I can see a person walking around lines of curvature on a sphere with an arrow strapped to his waist. His task is to keep the arrow pointed in the same direction How does he do this ? Does he use a reference point like the stars? (that only move very slowly) If that is how he keeps the arrow pointing in the same direction, is that equivalent to saying that he orients the arrow wrt the 3d space that the sphere is embedded in? So ,although one refers to intrinsic curvature...
ASSUMPTIONS 1. Two identical clocks A and B in the same inertial frame are stationary relative to each other a fixed distance L apart. Time passes at the same rate for both. 2. Both clocks are able to send/receive light signals and to write/read the send/receive times into signals. 3. The speed of light is anisotropic. METHOD 1. At time t[A1] and time t[B1], clock A sends a light signal to clock B. The clock B time is unknown to A. 2. Clock B receives the signal from A at time t[B2] and...
So, to calculate a proper time of a worldline in SR using an inertial frame is quite easy. But I struggled a bit using a "rotating frame metric" and now I'm not sure whether I'll do it right. Couls someone point me in the right direction? "What have you tried?" Well, trying to help truly absolute layppl with some variation of a "Circular Twin Paradox" not using an inertial frame of reference for whatevere reason. I thought it would be a bit of a challenge so I made a derivation or...
Back
Top