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## \Delta P_1=\int dt \sqrt{-g} P^r_{\ \ x^1} ##

where g is the induced metric on the worldsheet and r and t are the coordinates on the worldsheet which are also two of the background spacetime coordinates along with ## x^1 ##.

My problem is, I'm not sure I understand the reason for the presence of ## \sqrt{-g} ## in the integrand instead of integrating only ## P^r_{\ \ x^1} ##. Could anyone explain?

Thanks