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\begin{align}
  & d{{s}^{2}}={{g}_{\mu \nu }}dx{{'}^{\mu }}dx{{'}^{\nu }} \\
 & d\tau =\frac{1}{c}\sqrt{{{g}_{\mu \nu }}dx{{'}^{\mu }}dx{{'}^{\nu }}} \\
 & =\frac{1}{c}\sqrt{{{g}_{\mu \nu }}\frac{dx{{'}^{\mu }}}{dt'}\frac{dx{{'}^{\nu }}}{dt'}}dt=\sqrt{{{g}_{00}}}dt'
\end{align}

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d s 2 = g μ ⁢ ν d x d ′ μ x ν ′ d ⁢ τ = 1 c ⁢ g μ ⁢ ν d x d ′ μ x ν ′ = 1 c ⁢ g μ ⁢ ν ⁢ d x μ ′ d ⁢ t ′ ⁢ d x ν ′ d ⁢ t ′ ⁢ d ⁢ t = g 00 ⁢ d ⁢ t ′ missing-subexpression fragments d superscript 𝑠 2 subscript 𝑔 𝜇 𝜈 d x superscript superscript 𝑑 𝜇 ′ x superscript superscript absent 𝜈 ′ missing-subexpression 𝑑 𝜏 1 𝑐 fragments subscript 𝑔 𝜇 𝜈 d x superscript superscript 𝑑 𝜇 ′ x superscript superscript absent 𝜈 ′ missing-subexpression absent 1 𝑐 subscript 𝑔 𝜇 𝜈 fragments d x superscript superscript absent 𝜇 ′ 𝑑 superscript 𝑡 ′ fragments d x superscript superscript absent 𝜈 ′ 𝑑 superscript 𝑡 ′ 𝑑 𝑡 subscript 𝑔 00 𝑑 superscript 𝑡 ′ {\displaystyle{\displaystyle\begin{aligned} &\displaystyle d{{s}^{2}}={{g}_{% \mu\nu}}dx{{{}^{\prime}}^{\mu}}dx{{{}^{\prime}}^{\nu}}\\ &\displaystyle d\tau=\frac{1}{c}\sqrt{{{g}_{\mu\nu}}dx{{{}^{\prime}}^{\mu}}dx{% {{}^{\prime}}^{\nu}}}\\ &\displaystyle=\frac{1}{c}\sqrt{{{g}_{\mu\nu}}\frac{dx{{{}^{\prime}}^{\mu}}}{% dt^{\prime}}\frac{dx{{{}^{\prime}}^{\nu}}}{dt^{\prime}}}dt=\sqrt{{{g}_{00}}}dt% ^{\prime}\end{aligned}}}

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ds2=gμνdx'μdx'νdτ=1cgμνdx'μdx'ν=1cgμνdx'μdt′dx'νdt′dt=g00dt′

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