So I got this matrix of the sphere times an "essential constant" $\alpha$: $$g=\alpha\begin{bmatrix}1 & 0\\0 & \sin^{2}(\theta)\end{bmatrix}$$ I'm asked to prove that $\alpha$ is a an essential constant using two different methods. I figured that one way would be to calculate the Lie derivative $\mathcal{L}_{\xi}$ and setting that equal to $\frac{\partial g_{\mu\nu}}{\partial\alpha}$ and then to show that there's no solution to that; which I did successfully. But I don't know what a second way to prove it would be. Any help is greatly appreciated.
Essential constant
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differential-geometry
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2what is an "essential constant"? – 2017-01-11
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0@AccidentalFourierTransform A non-absorbable constant. – 2017-01-11
1 Answers
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Here's what I've found: One considers twice the metric tensor field: one for a given value of the constant and one for another value of it. Then the two metrics are compared to check whether they are equivalent or not. If they are, this means that the constant is spurious. (It's from a paper by G.O. Papadopoulos,2006)
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0how do you compare two metrics? e.g., they are conformally equivalent; is this enough? – 2017-01-11
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0[Here](https://arxiv.org/pdf/gr-qc/0503096v3.pdf)'s a link to the paper. – 2017-01-11