On page 161 of Carroll's(adsbygoogle = window.adsbygoogle || []).push({}); Spacetime and Geometry, Carroll writes that

[tex]\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\sigma}\delta g^{\rho\sigma}.[/tex]

##\delta g_{\alpha \beta}## denotes an arbitrary, infinitesimal variation of the metric.

Why is there a minus sign? By the regular rules of raising and lowering indices, shouldn't it just be

[tex]\delta g_{\mu\nu}=g_{\mu\rho}g_{\nu\sigma}\delta g^{\rho\sigma}?[/tex]

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# Error in Carroll?

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