How can I calculate changes in amplitude using gauge transformations in GR?

In summary: The complete line element isds^2 = a^2(\eta) \left( -(1+2\psi) d\eta^2 + 2\omega_idx^id\eta + \left[ (1+2\psi)\gamma_{ij} + 2\chi_{ij} \right] dx^idx^j \right)
  • #1
Logarythmic
281
0
I have been told that using a metric

[tex]g_{00} = -a^2(\eta)(1+2\psi)[/tex]
[tex]g_{oi} = g_{i0} = a^2(\eta)\omega_i[/tex]
[tex]g_{ij} = a^2(\eta) \left[(1+2\phi)\gamma_{ij} + 2\chi_{ij} \right][/tex]

and a gauge transformation

[tex]x^{\bar{\mu}} = x^{\mu} + \xi^{\mu}[/tex]

with

[tex]\xi^0 = \alpha[/tex]
[tex]\xi^i = \beta^j[/tex]

gives the changes in the amplitude as

[tex]\delta \psi = \alpha' + \frac{a'}{a} \alpha[/tex]

and so on.

But how do I calculate these changes? How do I start?
 
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  • #2
The 'gauge' transformation is a coordinate transformation. You know how the metric changes in a coordinate transformation. That should certainly be enough info to start.
 
  • #3
Yes like

[tex]g_{\bar{\mu} \bar{\nu}} = \frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}} \frac{\partial x^{\nu}}{\partial x^{\bar{\nu}}} g_{\mu \nu}[/tex]

but I can't get it right. I get that the perturbations change like

[tex]\delta g = -\partial_{\bar{\nu}} \xi^{\nu} -\partial_{\bar{\mu}} \xi^{\mu}[/tex]

but then what?
 
  • #4
If

[tex]g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}[/tex]

and

[tex]g_{00} = -(1+2\psi)[/tex]

then

[tex]h_{\bar{\mu} \bar{\nu}} = h_{\mu \nu} -\partial_{\mu} \xi_{\nu} -\partial_{\nu} \xi_{\mu}[/tex]

and

[tex]h_{00} = -2\psi[/tex].

But how do I get that

[tex]\psi \rightarrow \psi + \alpha' + \frac{a'}{a}\alpha[/tex]??

If I use the above equation for [itex]h_{\bar{\mu} \bar{\nu}}[/itex] I just get that

[tex]\psi \rightarrow \psi + \alpha'[/tex].

Please help someone!
 
  • #5
As usual, I'm pounding my head over the tex. The extra term comes from the effect of the transformation on the overall scale factor a. a(eta) -> a(eta+alpha) -> a(eta)+alpha*a'(eta) -> a(eta)(1+(a'/a)*alpha). (In case I never get the tex straightened out).
 
Last edited:
  • #6
I don't get it. Where does this come from?
 
  • #7
Logarythmic said:
I don't get it. Where does this come from?

The scale factor 'a' changes under the transformation.
 
  • #8
That I got, but how do I get your transformation?
 
  • #9
a(eta) goes to a(eta+alpha). I just took the first term of the taylor series expansion of a(eta).
 
  • #10
I don't follow. How does this couple with my metric transformation?
 
  • #11
I have to confess, I've only had to deal with this metric perturbation formalism once. And I found it pretty confusing myself. So I'm not sure I can clearly answer your question. But I do know that that is where your extra term comes from. It seems to me there is a review paper around by Brandenberger and Muhkanov that was pretty handy. But I don't have access to it right now.
 
  • #12
Ok. I think have all papers ever written about this here, but all they say is that "one can easily see that..."
 
  • #13
Logarythmic said:
Ok. I think have all papers ever written about this here, but all they say is that "one can easily see that..."

Annoying, isn't it?
 
  • #14
Yes. Very.
 
  • #15
Logarythmic said:
Yes. Very.

I guess the point is what is [itex] \eta [/itex] ??
 
  • #16
In problems like this eta is usually the conformal time. Just a specific parametrization of the time coordinate.
 
  • #17
Correct...
 
  • #18
Dick said:
In problems like this eta is usually the conformal time. Just a specific parametrization of the time coordinate.


Ok. Thanks. What is the definition? What is the relation with [itex] x_0 [/itex]?
 
  • #19
Conformal time is defined as

[tex]\eta = \int_0^{x_0} \frac{dx_0'}{a(x_0')}[/tex]
 
  • #20
Logarythmic said:
Conformal time is defined as

[tex]\eta = \int_0^{x_0} \frac{dx_0'}{a(x_0')}[/tex]

Ok. Then why not simply do the change of coordinates in that expression?
 
  • #21
?

I have an expression for a gauge transformation for scalars:

\bar{Q}(x^{\mu}) = q(x^{\mu}) - \xi^{\nu}\partial_{\nu}Q[/tex].

This gives the transformation for the scale factor as above. Then I have the transformation for the matric which gives the above expression for [itex]h_{\mu\nu}[/itex], but how can I get these together to yield

[tex]\psi \rightarrow \psi + \alpha' + \frac{a'}{a}\alpha[/tex]??

The next problem is to find a transformation

[tex]\omega_i \rightarrow \omega_i - \partial_i\alpha + \beta_i'[/tex]

but that was easy.

The complete line element is

[tex]ds^2 = a^2(\eta) \left( -(1+2\psi) d\eta^2 + 2\omega_idx^id\eta + \left[ (1+2\psi)\gamma_{ij} + 2\chi_{ij} \right] dx^idx^j \right)[/tex]
 

1. What is a gauge transformation in general relativity?

A gauge transformation in general relativity is a mathematical concept that involves changing the coordinates used to describe a physical system. It does not change any physical properties of the system, but rather just the way it is described.

2. Why are gauge transformations important in general relativity?

Gauge transformations are important in general relativity because they allow us to choose different coordinate systems to describe the same physical system. This can often simplify calculations and make it easier to understand the underlying physics.

3. How do gauge transformations relate to the principle of general covariance?

The principle of general covariance states that the laws of physics should be the same in all coordinate systems. Gauge transformations are a mathematical representation of this principle, as they allow us to change coordinate systems without changing the underlying physical laws.

4. Can gauge transformations affect the predictions of general relativity?

No, gauge transformations do not affect the predictions of general relativity. They only change the way the physical system is described, but the underlying physics remains the same. This is similar to changing the units used to measure a physical quantity - it doesn't change the value, just the way it is expressed.

5. Are gauge transformations unique in general relativity?

No, there are many different gauge transformations that can be applied in general relativity. Some may be more useful or convenient for certain problems, but they are all valid and do not change the physical predictions of the theory.

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