A Why Can't I Derive Eq.(7) from Phys.Lett.B Vol.755?

PRB147
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I encountered a problem in reading Phys.Lett.B Vol.755, 367-370 (2016).
I cannot derive Eq.(7), the following snapshot is the paper and my oen derivation,
I cannot repeat Eq.(7) in the paper.
Filho.png


##g^{\mu\nu}## is diagonal metric tensor and##g^{\mu\mu}## is the function of ##\mu## only, ##\mu=x,y,z##.
My derivation is as follows, but I cannot repeat their result and my result contains the cross term differential ##\partial_x \partial_y##.

Filho2.png

(The first line is their result), while the last line is mine.
Would anyone can help me to elucidate this problem?
 
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What does the three horizontal lines after ##\mathcal{D}^2## mean in eq 7? It sometimes means that something is true by definition (identical to) https://en.wikipedia.org/wiki/Triple_bar

If this is the case, you can not derive eq. 7.

PRB147 said:
andgμμ is the function of μ only
indices are not variables, thus the metric is not a function of µ.
 
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PRB147 said:
I cannot derive Eq.(7)
That's because the equation that precedes (7), namely ##{\cal D}=\sum_{\nu}g_{\nu\nu}^{-1/2}\partial_{\nu}##, is a nonsense. And it seems that the authors of the paper are not well versed in tensor calculus. To see how Eq. (7) should be correctly written and derived, google Laplace-Beltrami operator.
 
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PRB147 said:
My derivation is as follows
Please use the PF LaTeX feature to post equations directly, not as images. There is a LaTeX Guide link at the lower left of each post window.
 
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malawi_glenn said:
What does the three horizontal lines after ##\mathcal{D}^2## mean in eq 7? It sometimes means that something is true by definition (identical to) https://en.wikipedia.org/wiki/Triple_bar

If this is the case, you can not derive eq. 7.indices are not variables, thus the metric is not a function of µ.
thank you for your comment, I thought the author's meaning is ##g_{xx}## depends only on x; ##g_{yy}## depends only on y; etc.
 
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Demystifier said:
That's because the equation that precedes (7), namely ##{\cal D}=\sum_{\nu}g_{\nu\nu}^{-1/2}\partial_{\nu}##, is a nonsense. And it seems that the authors of the paper are not well versed in tensor calculus. To see how Eq. (7) should be correctly written and derived, google Laplace-Beltrami operator.
Thank you very much, I will read the relevant references according to your guidance.
 
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@PRB147, your posts #5 and #6 are a mess. Please pay more attention to proper quoting and formatting.
 
PeterDonis said:
@PRB147, your posts #5 and #6 are a mess.
Well, they aren't now because I have used magic moderator powers to edit them and clean them up. But my advice still stands.
 
PRB147 said:
thank you for your comment, I thought the author's meaning is ##g_{xx}## depends only on x; ##g_{yy}## depends only on y; etc.
No it does not...
 
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