Minkowski metric tensor computation

[Advent]

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.

 

[atyy]

Did you sum over the repeated indices in the change of coordinates formula (summation convention)?

 

[Advent]

Did you sum over the repeated indices in the change of coordinates formula (summation convention)?

I think I did, in fact it explains the square of the derivatives.

 

[atyy]

I'm not sure about this, but I think in grr you should have terms like (dt/dr)^2.gtt+(dx/dr)^2.gxx+(dy/dr)^2.gyy+(dz/dr)^2+(dt/dr)(dx/dr)gtr + ...

where in your formula I've taken u=r,v=r and rho and sigma must be summed over all combinations of rho=t,x,y,x and sigma=t,x,y,z

 

[Advent]

I'm not sure about this, but I think in grr you should have terms like (dt/dr)^2.gtt+(dx/dr)^2.gxx+(dy/dr)^2.gyy+(dz/dr)^2+(dt/dr)(dx/dr)gtr + ...

Thanks for your posts.

I think now I understand your point. Do you mean that, in the equation of the line element should be 4*4=16 summands instead of only four, by varying the ro and sigma indices between all their range?

 

[Advent]

Ok, just answered too fast!

Then this was my error, thank you so much!

 

[atyy]

Ok, just answered too fast!

Then this was my error, thank you so much!

Wow, you sure compute fast! Good to know - I wasn't sure about this.