Weak Field Approximation - Quick Sign Question

[binbagsss]

http://www.mth.uct.ac.za/omei/gr/chap7/node3.html

Shouldn't eq 45 have a minus sign, looking at eq 29.
Although I'm confused because the positive sign makes sense when comparing with the Newton-Poisson equation.
I can't see a sign error in eq 29.

(I believe the metric signature here is (-,+,+,+))

Anyone?
thanks..

 

[Mentz114]

When you lower the derivative index of ${h_{00}}^{,i}$ in (29) it changes the sign. I think what is written on that page is correct.

 

[binbagsss]

When you lower the derivative index of ${h_{00}}^{,i}$ in (29) it changes the sign. I think what is written on that page is correct.

Ahh thanks,
but these are not covariant/contravariant derivatives they are just partials , is this correct?

Also if equation (29) has an upper index then I thought (45) would have one upper and lower?

 

[Mentz114]

Ahh thanks,
but these are not covariant/contravariant derivatives they are just partials , is this correct?

Also if equation (29) has an upper index then I thought (45) would have one upper and lower?

Yes, the comma usually means partials, the semi-colon is used for covariant derivatives.

We have (29)

${\Gamma^i}_{00} \approx -\frac{1}{2}\epsilon {h_{00}}^{,i}$

which has one upper index only because $h_{00}$ is a number ( a component).

In (45) ${\Gamma^i}_{00,i}$ has no indexes because $i$ is summed over. This is a scalar which it must be to give us Poisson's equation.

 

[binbagsss]

Yes, the comma usually means partials, the semi-colon is used for covariant derivatives.

We have (29)

${\Gamma^i}_{00} \approx -\frac{1}{2}\epsilon {h_{00}}^{,i}$

which has one upper index only because $h_{00}$ is a number ( a component).

In (45) ${\Gamma^i}_{00,i}$ has no indexes because $i$ is summed over. This is a scalar which it must be to give us Poisson's equation.

I mean the $h_{00,ii}$ in [45], Taking a lower derivative of (29)-${\Gamma^i}_{00,i}$ I thought would give $h_{00,i}^{,i}$