jdstokes
Aug23-08, 01:22 AM
Which of the following are true in curved spacetime?
\int d^4 x \delta^4(x - x_0) = 1 (1)
\int d^4 x \sqrt{-g} \delta^4(x - x_0) = 1 (2)
I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta function does not transform, whereas the fourth-order differential transforms in the opposite way to \sqrt{-g}, so the whole thing transforms as a scalar as it must.
I've also heard that \delta^4 is not a scalar, which suggests that (1) is the correct statement. However, this seems strange to me as I would think that (1) will fail to hold in curvilinear coordinates e.g.
\int d^4 x \delta^4(x - x_0) = 1 (1)
\int d^4 x \sqrt{-g} \delta^4(x - x_0) = 1 (2)
I think the first one is incorrect in curved spacetime, or in general when the metric is non-constant. I would argue this by saying that the delta function does not transform, whereas the fourth-order differential transforms in the opposite way to \sqrt{-g}, so the whole thing transforms as a scalar as it must.
I've also heard that \delta^4 is not a scalar, which suggests that (1) is the correct statement. However, this seems strange to me as I would think that (1) will fail to hold in curvilinear coordinates e.g.