Einstein Field equation

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Is EFE an equation to help us find the curvature cause by mass? Like the schwarzchild metric? In addition, is it a must to use polar coordinate for EFE to work since it contains dr? Can we used the ordinary Euclidean coordinate in minkoswki space for EFE? If we can't is it because there is no radius(dr) component in it? Cause curvature results in excess radius, right?

Lastly, geodesic equation is the one that allows us to determine the path of an object in gravitational field by integrating it, right?
 

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Is EFE an equation to help us find the curvature cause by mass? Like the schwarzchild metric?
Yes, the EFE allows us to calculate the curvature caused by the presence of mass (and energy). The Schwarzchild metric is one particular solution to the EFE, the one that you get if you apply the EFE to the vacuum outside of a spherical mass.

In addition, is it a must to use polar coordinate for EFE to work since it contains dr? Can we used the ordinary Euclidean coordinate in minkoswki space for EFE? If we can't is it because there is no radius(dr) component in it?
You can use any coordinates you wish in solving the EFE - it holds true in all coordinate systems. However, in any given problem some coordinates will be much easier less hard to use than others. We use spherical instead of Minkowski coordinates to derive the Scwarzschild solution for the same reason that we use spherical instead of Euclidean coordinates in the Newtonian solution for planetary orbits - for these particular problems the equations are a lot simpler written in spherical coordinates.


If you haven't already tried working through http://preposterousuniverse.com/grnotes/grtinypdf.pdf [Broken], give it a try. This is about as simple of an introduction as you will find (some will say that it's too simple); if you can't follow the math there, that's a pretty good hint that you'll need more math background before you're ready to go up against the EFE.
 
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