Help With Raising and Lowering Indices

[dm4b]

Well, this isn't so much for general raising and lowering of indices. It's a specific step within the formation of the Ricci Tensor in the Linearized Gravity problem.

I trying to get from 7.5 to 7.6 in Sean Carrol Spacetime and Geometry, page 275.

I'm not matching with the 2nd term in 7.6.

I'm pretty sure I'm having a problem when an index is raised on a derivative. 7.6 seems to imply this is somehow equivalent to raising an index on the perturbation.

can anybody provide clarification on this or fill in the steps?

I've been stuck on this once before, figured out .. now years later, I'm stuck on it again ... frustrating!

Any help would be much appreciated!

 

[dm4b]

Or, to put it another way, is the following valid?

$$\partial^{\sigma}h_{\sigma}_{\mu}=\eta^{\sigma}^{\epsilon}\partial_{\epsilon}h_{\sigma}_{\mu}=\partial_{\epsilon}h_{\mu}^{\epsilon}=\partial_{\sigma}h_{\mu}^{\sigma}$$

As you can see on the third term, I use the neta to raise an index on h instead of the partial now. is that valid?

Since the metric is full of constants in the Minkowski metric, seems like it would be valid to move it inside the partial and operate on h. BUT, seems like this would not be true in general, maybe?

 

[bcrowell]

You can't raise and lower indices on the partial derivative $\partial$, but you can on the covariant derivative $\nabla$. I don't have a copy of Carroll, so I'm not sure what the context is. The manipulation shown in your #2 is valid in SR, where $\partial$ and $\nabla$ are the same thing. It would be valid in GR if you changed $\partial\rightarrow\nabla$ and $\eta\rightarrow g$. I think the worry you're expressing about the variability of the metric in a derivative boils down to exactly what the covariant derivative is designed to take care of.

 

[qbert]

it's valid in this context.

he's using the constant SR metric $\eta$ to raise and lower indices. since
the metric is constant you can "pull it through" derivatives. what you should pay
attention to is WHY in this context is he raising and lowering indices with $\eta$ instead of g.

 

[Rasalhague]

The manipulation shown in your #2 is valid in SR

Some of this is above my head but... when you say valid in SR, do you mean it would only be valid in an inertial coordinate system in flat spacetime?

 

[dm4b]

You can't raise and lower indices on the partial derivative $\partial$, but you can on the covariant derivative $\nabla$. I don't have a copy of Carroll, so I'm not sure what the context is. The manipulation shown in your #2 is valid in SR, where $\partial$ and $\nabla$ are the same thing. It would be valid in GR if you changed $\partial\rightarrow\nabla$ and $\eta\rightarrow g$. I think the worry you're expressing about the variability of the metric in a derivative boils down to exactly what the covariant derivative is designed to take care of.

Those are partials in that equation, which can be raised and lowered, with the effect of changing the sign on the 0-compenent, or the partial with respect to time. But, you're right, thanks to metric-compatibility (i.e. the covariant derivative of the metric is zero) a similar operation would be okay in all of GR.

So, I think you and gbert are right, that I'm okay in this context. I just always get tripped up on this - probably will again in a month from now ;-)

But, I still don't think it is true in general (outside GR and SR), because metric compatibility isn't always guaranteed.

 

[bcrowell]

Some of this is above my head but... when you say valid in SR, do you mean it would only be valid in an inertial coordinate system in flat spacetime?

Yeah, I guess so. It's valid when $\partial$ and $\nabla$ are the same thing, which would not be the case in a flat spacetime described in non-Minkowski coordinates.

 

[Rasalhague]

Yeah, I guess so. It's valid when $\partial$ and $\nabla$ are the same thing, which would not be the case in a flat spacetime described in non-Minkowski coordinates.

Thanks. Just a little quibble to check I understood. Incidentally, I like the fact that some people call flat spacetime "Minkowski space" and inertial coordinates a "Lorentz frame", whereas others give Minkowski's name to inertial coordinates, and Lorentz's name to flat spacetime ;-)

 

[George Jones]

Or, to put it another way, is the following valid?

$$\partial^{\sigma}h_{\sigma}_{\mu}=\eta^{\sigma}^{\epsilon}\partial_{\epsilon}h_{\sigma}_{\mu}=\partial_{\epsilon}h_{\mu}^{\epsilon}=\partial_{\sigma}h_{\mu}^{\sigma}$$

As you can see on the third term, I use the neta to raise an index on h instead of the partial now. is that valid?

Since the metric is full of constants in the Minkowski metric, seems like it would be valid to move it inside the partial and operate on h. BUT, seems like this would not be true in general, maybe?

On page 274, Carroll explians why this is done.

As before, we can raise and lower indices using $\eta^{\mu \nu}$ and $\eta_{\mu \nu}$,since the corrections would be of higher order in the perturbation.

To first order,

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

gives

$$g^{\mu \nu} = h^{\mu \nu} - \eta^{\mu \nu}.$$

Consequently,

$$\begin{equation*} \begin{split} \partial_\sigma h_{\mu}^{\sigma} &= \partial_\sigma \left( g^{\sigma \nu} h_{\nu \mu} \right) \\ &= \partial_\sigma \left[ \left( \eta^{\sigma \nu} - h^{\sigma \nu} \left) h_{\nu \mu} \right] \\ &= \eta^{\sigma \nu} \partial_\sigma h_{\nu \mu} - \left( \partial_\sigma h^{\sigma \nu} \right) h_{\nu \mu} - h^{\sigma \nu} \partial_\sigma h_{\nu \mu} \end{equation*} \end{split}$$

I don't think Carroll states explicitly that the derivatives of $h$ are assumed to be small, but some books do. Assuming this gives, to first order of "smallness",

$$\partial_\sigma h_{\mu}^{\sigma} = \eta^{\sigma \nu} \partial_\sigma h_{\nu \mu}.$$

I think this is what qbert meant.