- #61
anuttarasammyak
Gold Member
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1. We can derive the equation
$$ \delta g^{\mu\nu}=-g^{\mu\alpha}g^{\nu\beta}\delta g_{\alpha\beta} $$
by indeces up down operation. In the world whose metric tensor is g,
$$ g^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}g_{\alpha\beta} $$
In another new world whose metric tensor is ##\bar{g}##,
$$ \bar{g}^{\mu\nu}=\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta}\bar{g}_{\alpha\beta} $$
Say the difference of metric tensors is small
$$ \bar{g}_{\mu\nu}=g_{\mu\nu}+\delta g_{\mu\nu} $$ we can get this equation in first order of ##\delta g##.
2. In the new world where ##\delta g \neq 0##, in first order of ##\delta g##.
$$ g^{\mu\nu} \neq \bar{g}^{\mu\alpha}\bar{g}^{\nu\beta} g_{\alpha\beta} $$
$$ \delta g^{\mu\nu} \neq \bar{g}^{\mu\alpha}\bar{g}^{\nu\beta} \delta g_{\alpha\beta} $$
Neither old metric nor the differece of new-old satisfies indecies up-down relation but their sum satisfies it. Reciprocally, neither new metric nor the difference of old-new satisfies indecies up-down relation but their sum does it.
3. $$(\delta g)^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}(\delta g)_{\alpha\beta}$$
satisfies the indeces up down relation. I want concilliation of 2 and 3 for my deeper understanding.
$$ \delta g^{\mu\nu}=-g^{\mu\alpha}g^{\nu\beta}\delta g_{\alpha\beta} $$
by indeces up down operation. In the world whose metric tensor is g,
$$ g^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}g_{\alpha\beta} $$
In another new world whose metric tensor is ##\bar{g}##,
$$ \bar{g}^{\mu\nu}=\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta}\bar{g}_{\alpha\beta} $$
Say the difference of metric tensors is small
$$ \bar{g}_{\mu\nu}=g_{\mu\nu}+\delta g_{\mu\nu} $$ we can get this equation in first order of ##\delta g##.
2. In the new world where ##\delta g \neq 0##, in first order of ##\delta g##.
$$ g^{\mu\nu} \neq \bar{g}^{\mu\alpha}\bar{g}^{\nu\beta} g_{\alpha\beta} $$
$$ \delta g^{\mu\nu} \neq \bar{g}^{\mu\alpha}\bar{g}^{\nu\beta} \delta g_{\alpha\beta} $$
Neither old metric nor the differece of new-old satisfies indecies up-down relation but their sum satisfies it. Reciprocally, neither new metric nor the difference of old-new satisfies indecies up-down relation but their sum does it.
3. $$(\delta g)^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}(\delta g)_{\alpha\beta}$$
satisfies the indeces up down relation. I want concilliation of 2 and 3 for my deeper understanding.
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