- #1
dEdt
- 288
- 2
On page 161 of Carroll's 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]
[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]