# Gauge transformations in GR

1. May 17, 2007

### Logarythmic

I have been told that using a metric

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

and a gauge transformation

$$x^{\bar{\mu}} = x^{\mu} + \xi^{\mu}$$

with

$$\xi^0 = \alpha$$
$$\xi^i = \beta^j$$

gives the changes in the amplitude as

$$\delta \psi = \alpha' + \frac{a'}{a} \alpha$$

and so on.

But how do I calculate these changes? How do I start?

2. May 17, 2007

### Dick

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. May 17, 2007

### Logarythmic

Yes like

$$g_{\bar{\mu} \bar{\nu}} = \frac{\partial x^{\mu}}{\partial x^{\bar{\mu}}} \frac{\partial x^{\nu}}{\partial x^{\bar{\nu}}} g_{\mu \nu}$$

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

$$\delta g = -\partial_{\bar{\nu}} \xi^{\nu} -\partial_{\bar{\mu}} \xi^{\mu}$$

but then what?

4. May 17, 2007

### Logarythmic

If

$$g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}$$

and

$$g_{00} = -(1+2\psi)$$

then

$$h_{\bar{\mu} \bar{\nu}} = h_{\mu \nu} -\partial_{\mu} \xi_{\nu} -\partial_{\nu} \xi_{\mu}$$

and

$$h_{00} = -2\psi$$.

But how do I get that

$$\psi \rightarrow \psi + \alpha' + \frac{a'}{a}\alpha$$??

If I use the above equation for $h_{\bar{\mu} \bar{\nu}}$ I just get that

$$\psi \rightarrow \psi + \alpha'$$.

5. May 17, 2007

### Dick

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: May 17, 2007
6. May 17, 2007

### Logarythmic

I don't get it. Where does this come from?

7. May 17, 2007

### Dick

The scale factor 'a' changes under the transformation.

8. May 17, 2007

### Logarythmic

That I got, but how do I get your transformation?

9. May 17, 2007

### Dick

a(eta) goes to a(eta+alpha). I just took the first term of the taylor series expansion of a(eta).

10. May 17, 2007

### Logarythmic

I don't follow. How does this couple with my metric transformation?

11. May 17, 2007

### Dick

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. May 17, 2007

### Logarythmic

Ok. I think have all papers ever written about this here, but all they say is that "one can easily see that..."

13. May 17, 2007

### Dick

Annoying, isn't it?

14. May 17, 2007

### Logarythmic

Yes. Very.

15. May 17, 2007

### nrqed

I guess the point is what is $\eta$ ??

16. May 17, 2007

### Dick

In problems like this eta is usually the conformal time. Just a specific parametrization of the time coordinate.

17. May 17, 2007

### Logarythmic

Correct...

18. May 17, 2007

### nrqed

Ok. Thanks. What is the definition? What is the relation with $x_0$?

19. May 17, 2007

### Logarythmic

Conformal time is defined as

$$\eta = \int_0^{x_0} \frac{dx_0'}{a(x_0')}$$

20. May 17, 2007

### nrqed

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