On 12/18/2017 11:44 PM, [email protected] wrote:

    Invariants are always the important things in physics because they
    are what we can have intersubjective agreement on.

    Brent


*IIUC, the field equations are covariant, which means coordinate system independent. *

Right.  Covariant means that something that changes in such a way that invariant things remain the same.  So vectors components transform covariantly so that they keep the vector physically the same.

*Isn't Newton's Law of Gravitation also coordinate independent? That is, if we use Newton to calculate the planetary orbits, won't we get the same results in different coordinate systems? *
Right.

*... Is there a distinction in GR between frame independence and coordinate independence? AG*

I'd say frame independence is a special case of coordinate independence.  It refers only to relative motion of coordinate frames.

Brent

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