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## Main Question or Discussion Point

Many text books use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity.

Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass) whose coordinate radii are 10km and 11km, i.e., 1km apart. Since coordinate radius is being used, the gravitational bending effect is being avoided by measuring

But my question is, isn’t the components of the metric tensor expected to take care of the geometry of the space? In particular, isn’t the component ##g_{11} = -\frac{1}{A(r)}## where ##A(r) = 1-\frac{2k}{r} ## supposed to account for the curvature of space due to gravity? If the space was Euclidean, in spherical coordinates, we would have used ## ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2##, but in Schwarzschild space, we are using ## ds^2 = \frac{1}{A(r)}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2 ##. So, shouldn't the equation for ##\Delta r_{shell}##, which is derived by setting ##d\theta = d\phi = 0## given us the same value of 1km? Is the metric tensor for the Schwarzschild space incorrect as it does not match the observed/differently measured distance of 1km? What is the

What am I missing?

Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass) whose coordinate radii are 10km and 11km, i.e., 1km apart. Since coordinate radius is being used, the gravitational bending effect is being avoided by measuring

*around*object. Then they calculate the radial distance using the formula $$ \Delta r_{shell} = \int_{r1}^{r2} \frac{dr}{\sqrt{1 - \frac{2k}{r}}} $$ where ##k=GM/c^2## and show this to be more than 1km, 1.18km in this example.But my question is, isn’t the components of the metric tensor expected to take care of the geometry of the space? In particular, isn’t the component ##g_{11} = -\frac{1}{A(r)}## where ##A(r) = 1-\frac{2k}{r} ## supposed to account for the curvature of space due to gravity? If the space was Euclidean, in spherical coordinates, we would have used ## ds^2 = dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2##, but in Schwarzschild space, we are using ## ds^2 = \frac{1}{A(r)}dr^2 + r^2 d\theta^2 + r^2 \sin^2\theta d\phi^2 ##. So, shouldn't the equation for ##\Delta r_{shell}##, which is derived by setting ##d\theta = d\phi = 0## given us the same value of 1km? Is the metric tensor for the Schwarzschild space incorrect as it does not match the observed/differently measured distance of 1km? What is the

*real*separation distance between the 2 shells? 1km or 1.18km?What am I missing?