A Is the variation of the metric ##\delta g_{\mu\nu}## a tensor?

[Kostik]

TL;DR Summary

Two contradictory equations are shown. Where is the flaw?

From $$0 = \delta(g^{\alpha\mu}g_{\mu\nu}) = g^{\alpha\mu} \delta g_{\mu\nu} + g_{\mu\nu} \delta g^{\alpha\mu}$$ we have $$g^{\alpha\mu} \delta g_{\mu\nu} = -g_{\mu\nu} \delta g^{\alpha\mu} \,\, . $$ Multiply both sides by $g_{\alpha\beta}$ to get $$\delta g_{\beta\nu} = -g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \qquad(*)$$ (This is Dirac's eq. (26.9) in "GTR".) On the other hand, the variation $\delta g^{\alpha\mu} = \bar{g}^{\alpha\mu} - g^{\alpha\mu}$ should be a tensor; therefore, $$\delta g_{\beta\nu} = g_{\alpha\beta} g_{\mu\nu} \delta g^{\alpha\mu} \quad\qquad(**)$$ Which equation is correct, $(*)$ or $(**)$?

 

[Kostik]

I think I found the (or an) answer. The relations $$\delta g^{\mu\nu} = -g^{\mu\rho}g^{\nu\sigma}\delta g_{\rho\sigma} \quad , \qquad \delta g_{\mu\nu} = -g_{\mu\rho} g_{\nu\sigma} \delta g^{\rho\sigma} \qquad(*)$$ appear to indicate that $\delta g_{\mu\nu}$ and $\delta g^{\mu\nu}$ are not tensors, since their indices are not raised and lowered in the usual way by the metric tensor. But this is not true; they are tensors, as can be seen by calculating $\delta g'_{\mu\nu}$ using $$\delta g_{\mu\nu} = \bar{g}_{\mu\nu} - g_{\mu\nu} \,\, .$$ $\qquad$ Given the variation $\delta V_\mu$, to find the variation of the contravariant vector $\delta V^\mu$, we must calculate $$\delta V^\mu = \delta (g^{\mu\alpha}V_\alpha) = g^{\mu\alpha}\delta V_\alpha + V_\alpha \delta g^{\mu\alpha} \,\, .$$ Notice that the last term "spoils" the usual rule for raising and lowering indices; the Leibniz rule requires that we account for the variation in $V_\alpha$ and the metric tensor. In the case of the metric tensor $g_{\alpha\beta}$, the Leibnitz rule gives $$\delta g^{\mu\nu} = g^{\mu\alpha} g^{\nu\beta} g_{\alpha\beta} + 2\delta g^{\mu\nu}$$ which gives $(*)$.

$\qquad$ In other words, the Leibniz rule is paramount; the “raising and lowering” rule does not apply across the “$\delta$”.

 

[PeterDonis]

the screenshot below

What reference is this screenshot from?

 

[Kostik]

@PeterDonis These are my own notes.

If I have made an error, please critique it!

 

[PeterDonis]

These are my own notes.

Then you should post them directly using LaTeX for equations, in accordance with the forum rules.

If I have made an error, please critique it!

We can't if all we can quote is an image. We need to be able to quote individual equations and words. That's why we need things posted directly using LaTeX for equations.

 

[Kostik]

That makes sense, I will endeavor to use LaTex in the future. Obviously it’s much faster for me to take a snapshot of my own notes (made with MSWord).

 

[PeterDonis]

I will endeavor to use LaTex in the future.

If you're not going to re-post your notes using LaTeX now, I'm going to close this thread. You asked for feedback; that can only be done if you post your content the way I described.

 

[PeterDonis]

Obviously it’s much faster for me to take a snapshot of my own notes (made with MSWord).

If you can't take the time to post using direct text and LaTeX, why should we take the time to read and answer?

Sorry to be blunt, but that's one of the reasons behind the PF rules on this: you need to be respectful of other posters' time, not just your own.

 

[Kostik]

The narrative in #2 has now been added in LaTex.