- #1
Advent
- 29
- 0
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 \eta with the signature (-+++). Under some coordinate change, say x_{\mu} \rightarrow x_{\overline{\mu}}, then the metric changes as 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}}. So, If I change the coordinate system from Cartesian (t,x,y,z) to spherical (t,r, \theta, \varphi) with the following equations
x = r \cos(\varphi) \cos (\theta), y = r \cos(\varphi) \sin (\theta), z = r \sin(\varphi), t = t. The four non-zero componentes of the metric \eta in spherical coordinates should be:
g_{11} = (\frac{ \partial t }{ \partial t })^2 g_{1'1'}=-1
g_{22} =(\frac{ \partial x }{ \partial r })^2 g_{2'2'}=\cos^2(\varphi) \cos^2 (\theta)
g_{33} =(\frac{ \partial y }{ \partial \theta })^2 g_{3'3'}=r^2 \cos^2(\varphi) \cos^2 (\theta)
g_{44} =(\frac{ \partial z }{ \partial \varphi })^2 g_{4'4'}=r^2 \cos^2(\varphi)
And finally, the line element 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 which is incorrect.
Thanks for your time, any help will be appreciated.
In flat spacetime we have \eta with the signature (-+++). Under some coordinate change, say x_{\mu} \rightarrow x_{\overline{\mu}}, then the metric changes as 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}}. So, If I change the coordinate system from Cartesian (t,x,y,z) to spherical (t,r, \theta, \varphi) with the following equations
x = r \cos(\varphi) \cos (\theta), y = r \cos(\varphi) \sin (\theta), z = r \sin(\varphi), t = t. The four non-zero componentes of the metric \eta in spherical coordinates should be:
g_{11} = (\frac{ \partial t }{ \partial t })^2 g_{1'1'}=-1
g_{22} =(\frac{ \partial x }{ \partial r })^2 g_{2'2'}=\cos^2(\varphi) \cos^2 (\theta)
g_{33} =(\frac{ \partial y }{ \partial \theta })^2 g_{3'3'}=r^2 \cos^2(\varphi) \cos^2 (\theta)
g_{44} =(\frac{ \partial z }{ \partial \varphi })^2 g_{4'4'}=r^2 \cos^2(\varphi)
And finally, the line element 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 which is incorrect.
Thanks for your time, any help will be appreciated.
