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## Main Question or Discussion Point

I am a little bit confused about the metric tensor and would like some feedback before I proceed with my learning of GR.

So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come into existence only when the coordinate system is selected. However, consider the equations for the components of the metric tensor, viz.,

$$

g_{ij} =

\frac{ \partial{x'^1}}{\partial{x^i}} \frac {\partial{x'^1}}{\partial{x^j}} +

\frac{ \partial{x'^2}}{\partial{x^i}} \frac {\partial{x'^2}}{\partial{x^j}} +

\frac{ \partial{x'^3}}{\partial{x^i}} \frac {\partial{x'^3}}{\partial{x^j}}

$$

This equation implies that the metric tensor is not just about the space itself, not even about ##a## chosen coordinate system alone, but is a function of ##two## coordinate systems, the primed and the unprimed. In particular, if we choose, say, the spherical polar coordinate system, typically we choose the Cartesian coordinate system as the primed one, and derive the metric tensor. But the above equation ##-## I think ##-## holds for any two coordinate systems and there is nothing stopping us from choosing the cylindrical coordinates instead of Cartesian as the primed system. So the metric tensor is really a function of ##two## coordinate system, in addition to describing the geometry of the space itself?

I understand the reasoning for choosing the Cartesian coordinate system as the primed one, namely that the differential length in it is evaluated as ##ds^2=dx^2+dy^2+dz^2##, but it appears that the Cartesian coordinates are being given a special status. Can someone clarify to help me understand this better?

Thanks.

So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come into existence only when the coordinate system is selected. However, consider the equations for the components of the metric tensor, viz.,

$$

g_{ij} =

\frac{ \partial{x'^1}}{\partial{x^i}} \frac {\partial{x'^1}}{\partial{x^j}} +

\frac{ \partial{x'^2}}{\partial{x^i}} \frac {\partial{x'^2}}{\partial{x^j}} +

\frac{ \partial{x'^3}}{\partial{x^i}} \frac {\partial{x'^3}}{\partial{x^j}}

$$

This equation implies that the metric tensor is not just about the space itself, not even about ##a## chosen coordinate system alone, but is a function of ##two## coordinate systems, the primed and the unprimed. In particular, if we choose, say, the spherical polar coordinate system, typically we choose the Cartesian coordinate system as the primed one, and derive the metric tensor. But the above equation ##-## I think ##-## holds for any two coordinate systems and there is nothing stopping us from choosing the cylindrical coordinates instead of Cartesian as the primed system. So the metric tensor is really a function of ##two## coordinate system, in addition to describing the geometry of the space itself?

I understand the reasoning for choosing the Cartesian coordinate system as the primed one, namely that the differential length in it is evaluated as ##ds^2=dx^2+dy^2+dz^2##, but it appears that the Cartesian coordinates are being given a special status. Can someone clarify to help me understand this better?

Thanks.