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Hi, I'm having problem with understanding tensors and the Einsteins summation convention, so I decided to start doing explicit calculations, and I'm doing it in the wrong way. Hope someone could help me to clarify the concepts.
In flat spacetime we have [tex]\eta[/tex] with the signature (-+++). Under some coordinate change, say [tex]x_{\mu} \rightarrow x_{\overline{\mu}}[/tex], then the metric changes as [tex]g_{ \overline{\mu} \overline{\nu}}=\frac{ \partial x^{\rho}}{ \partial x_{\overline{\mu}}} \frac{ \partial x^{\sigma}}{ \partial x_{\overline{\nu}}}g_{ \overline{\rho} \overline{\sigma}}[/tex]. So, If I change the coordinate system from Cartesian [tex](t,x,y,z)[/tex] to spherical [tex](t,r, \theta, \varphi)[/tex] with the following equations
[tex] x = r \cos(\varphi) \cos (\theta)[/tex], [tex] y = r \cos(\varphi) \sin (\theta)[/tex], [tex] z = r \sin(\varphi) [/tex], [tex] t = t [/tex]. The four non-zero componentes of the metric [tex]\eta[/tex] in spherical coordinates should be:
[tex]g_{11} = (\frac{ \partial t }{ \partial t })^2 g_{1'1'}=-1[/tex]
[tex]g_{22} =(\frac{ \partial x }{ \partial r })^2 g_{2'2'}=\cos^2(\varphi) \cos^2 (\theta)[/tex]
[tex]g_{33} =(\frac{ \partial y }{ \partial \theta })^2 g_{3'3'}=r^2 \cos^2(\varphi) \cos^2 (\theta)[/tex]
[tex]g_{44} =(\frac{ \partial z }{ \partial \varphi })^2 g_{4'4'}=r^2 \cos^2(\varphi)[/tex]
And finally, the line element [tex]ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}=dt^2+\cos^2(\varphi) \cos^2 dr^2+ r^2 \cos^2(\varphi) \cos^2 d^2 \theta + r^2 \cos^2(\varphi) d^2 \varphi[/tex] which is incorrect.
Thanks for your time, any help will be appreciated.
In flat spacetime we have [tex]\eta[/tex] with the signature (-+++). Under some coordinate change, say [tex]x_{\mu} \rightarrow x_{\overline{\mu}}[/tex], then the metric changes as [tex]g_{ \overline{\mu} \overline{\nu}}=\frac{ \partial x^{\rho}}{ \partial x_{\overline{\mu}}} \frac{ \partial x^{\sigma}}{ \partial x_{\overline{\nu}}}g_{ \overline{\rho} \overline{\sigma}}[/tex]. So, If I change the coordinate system from Cartesian [tex](t,x,y,z)[/tex] to spherical [tex](t,r, \theta, \varphi)[/tex] with the following equations
[tex] x = r \cos(\varphi) \cos (\theta)[/tex], [tex] y = r \cos(\varphi) \sin (\theta)[/tex], [tex] z = r \sin(\varphi) [/tex], [tex] t = t [/tex]. The four non-zero componentes of the metric [tex]\eta[/tex] in spherical coordinates should be:
[tex]g_{11} = (\frac{ \partial t }{ \partial t })^2 g_{1'1'}=-1[/tex]
[tex]g_{22} =(\frac{ \partial x }{ \partial r })^2 g_{2'2'}=\cos^2(\varphi) \cos^2 (\theta)[/tex]
[tex]g_{33} =(\frac{ \partial y }{ \partial \theta })^2 g_{3'3'}=r^2 \cos^2(\varphi) \cos^2 (\theta)[/tex]
[tex]g_{44} =(\frac{ \partial z }{ \partial \varphi })^2 g_{4'4'}=r^2 \cos^2(\varphi)[/tex]
And finally, the line element [tex]ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}=dt^2+\cos^2(\varphi) \cos^2 dr^2+ r^2 \cos^2(\varphi) \cos^2 d^2 \theta + r^2 \cos^2(\varphi) d^2 \varphi[/tex] which is incorrect.
Thanks for your time, any help will be appreciated.