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- In what coordinate system is set up and solved the Einstein's Field Equation and how does it relate to our continuum?

Hello all,

I have a question on a pivotal concept of GR that I've never managed to fully grasp.

In what coordinate system is the Einstein's Field Equation set up and solved?

I've always assumed it's an Euclidean 4D space, whose metric is irrelevant because we are dealing with scalar functions, solutions of the ten independent differential equations.

But if it's so, how does it map to our 4D curved continuum? I've always seen our curved spacetime only mapped to a "local" flat space in the infinitesimal region around a given point, but never "globally".

What if a want to solve the EFE in an extended spacetime region, i.e. our solar system for several days?

From a procedural standpoint, should I have to take a "snapshot" of the masses in my curved spacetime to express the stress-energy tensor as a function of my curvilinear coordinates, map it to a 4D flat space, solve the EFE to obtain the metric tensor

What boundary conditions should I use i.e. for velocities, expressed in which coordinate system?

And what would be the use for such a

I have a question on a pivotal concept of GR that I've never managed to fully grasp.

In what coordinate system is the Einstein's Field Equation set up and solved?

I've always assumed it's an Euclidean 4D space, whose metric is irrelevant because we are dealing with scalar functions, solutions of the ten independent differential equations.

But if it's so, how does it map to our 4D curved continuum? I've always seen our curved spacetime only mapped to a "local" flat space in the infinitesimal region around a given point, but never "globally".

What if a want to solve the EFE in an extended spacetime region, i.e. our solar system for several days?

From a procedural standpoint, should I have to take a "snapshot" of the masses in my curved spacetime to express the stress-energy tensor as a function of my curvilinear coordinates, map it to a 4D flat space, solve the EFE to obtain the metric tensor

*g*as a function of Euclidean coordinates, then map it back to my curved spacetime?What boundary conditions should I use i.e. for velocities, expressed in which coordinate system?

And what would be the use for such a

*g*? To compute the motion equations of the masses in my curved spacetime?