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As I found in this Wikipedia Article about Schwarzschild metric $$g_{00}=(1-\frac{r_s}{r})$$ where g - metric tensor.

I understand what is $r_s$. It is a Schwarzschild radius. For the Earth, it will be 8.9 mm. But what is $r$ it is not clear? For example, what will be $r$ on the Earth surface (6371 km from the Earth Center) or at 10000km from the Earth Center?

Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field (of course I neglect the Earth rotation. I hope the influence of the Earth rotation is very small).

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  • $\begingroup$ $r$ is just a coordinate, the radial coordinate. Have you looked at en.wikipedia.org/wiki/Schwarzschild_coordinates $\endgroup$
    – Eletie
    Feb 27, 2021 at 21:17
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    $\begingroup$ If you assume the Earth is spherically symmetric and ignore the fact that it is rotating, then the Schwarzschild metric describes its exterior gravitational field but not its interior field. $\endgroup$
    – G. Smith
    Feb 27, 2021 at 21:51
  • $\begingroup$ @Eletie In the article "Schwarzschild metric" it is written "$r$ is the radial coordinate (measured as the circumference, divided by 2π, of a sphere centered around the massive body).". This is where my concern is. I need and example how to calculate it for the numbers I mentioned in the question. $\endgroup$
    – Zlelik
    Feb 28, 2021 at 11:21

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Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field (of course I neglect the Earth rotation. I hope the influence of the Earth rotation is very small

The Earth is not spherically symmetric so that's a problem straight away. It is also rotating, so that's another issue.

That said the effects of GR are measurable and can be calculated. In particular there has been success measuring gravitational time dilation and the effect of frame dragging. Note that frame dragging is only relevant in the case of a rotating body and is not modeled by the Schwarzchild solution. To model frame dragging you need e.g. the Kerr metric. Frame dragging is a very small effect for Earth.

Also, is it applicable to use the Schwarzschild metric to calculate the Earth Gravitational field

This may seem like a pointless statement, but it's perfectly reasonable to try any model as long as you understand the deficiencies of the model, which you seem to. These GR effects are all very small for Earth. Newtonian gravity works extremely well for Earth's gravitational field. It's worth noting that the famous general relativistic correction for Mercury's orbit is smaller than perturbations from the gravitational fields of other planets on Mercury's orbit. GR effects are, simply put, very small in the solar system.

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  • $\begingroup$ I understand that Earth is not symmetric and rotating, but I assume that the deviation is small and can be neglected. I would like to calculate Electromagnetic field difference in gravitational field. I found some equations like $\Large B=\frac{H}{\sqrt{g_{00}}}$. this is why I am asking which metric is better for the Earth gravity. $\endgroup$
    – Zlelik
    Feb 28, 2021 at 8:25
  • $\begingroup$ Your questions makes no mention of the EM field and I do not understand how using the metric to calculate the gravitational field could give you an EM field ? I have never seen the relationship for $B$ and $H$ you mention and the relationship I am familiar with is $\mathbf{H}=\frac{1}{\mu_0}\mathbf{B}-\mathbf{M}$. I would think B field variations due to GR around Earth would be extremely small compared to other variations in Earth's B field. $\endgroup$
    – StephenG - Help Ukraine
    Feb 28, 2021 at 16:42
  • $\begingroup$ I found it in this book books.google.nl/books?id=HudbAwAAQBAJ chapter 10 para 90. If I can understand how to calculate $g_{00}$ in Schwarzschild metric, it will be good enough for me to start. $\endgroup$
    – Zlelik
    Feb 28, 2021 at 21:09

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