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Shortened a bit for clarity (and added some punctuation while at it).
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Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as  : $$g_{\mu\nu}+h_{\mu\nu}$$ or or equivalently as  : $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that: $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$$$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu,$$ and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: whatWhat is the relationship between $$\delta(g^{\mu\nu})$$ and$\delta g^{\mu\nu}$ and $$h^{\mu\nu}$$$h^{\mu\nu}$? This is actually proving that for arbitrary tensors,: $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}$$$$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}.$$ How to prove this?

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as  $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as  $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: what is the relationship between $$\delta(g^{\mu\nu})$$ and $$h^{\mu\nu}$$ This is actually proving that for arbitrary tensors, $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}$$ How to prove this?

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as: $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as: $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that: $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu,$$ and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: What is the relationship between $\delta g^{\mu\nu}$ and $h^{\mu\nu}$? This is actually proving that for arbitrary tensors: $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}.$$ How to prove this?

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Qmechanic
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Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: what is the relationship between $$\delta(g^{\mu\nu})$$ and $$h^{\mu\nu}$$ This is actually proving that for arbitrary tensors, $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}$$ How to prove this?

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: what is the relationship between $$\delta(g^{\mu\nu})$$ and $$h^{\mu\nu}$$

Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu$$and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: what is the relationship between $$\delta(g^{\mu\nu})$$ and $$h^{\mu\nu}$$ This is actually proving that for arbitrary tensors, $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}$$ How to prove this?

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Rescy_
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