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I understand that in general relativity all observers agree on what it is they see (looking at the same object or event) when they account for the effects on their observations of the gravitational field they sit in, on their distance and motion relative to the observed.

Does this mean that in general relativity there is a universe-wide present all objects and observers everywhere live in (so they are contemporaries, living at the same moment in cosmic time, X years after the big bang), but that it only for practical reasons—for example, time dilation and the (finite) speed of light which makes that we see a distant galaxy as it was in a distant past—that it cannot actually be observed?

Níck Aguiar Alves
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Anton
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Your first paragraph contains some physics misunderstandings. Gravitational time dilation depends on the gravitational potential, not the gravitational field, and in any case we can't define a gravitational potential for a cosmological spacetime. Also, we can't define the motion of one object relative to another object if they are separated by cosmological distances.

Does this mean that in general relativity there is a universe-wide present all objects and observers everywhere live in (so they are contemporaries, living at the same moment in cosmic time, X years after the big bang) but that it only for practical reasons -time dilation and the (finite) speed of light which makes that we see a distant galaxy as it was in a distant past- cannot actually be observed?

This is not a very well-defined physics question, since physics doesn't make philosophical claims like "there is a universe-wide present." We can define a preferred cosmological time coordinate which is the time reading on a clock that has been at rest relative to the Hubble flow since the big bang. If you want to say that this implies a "universe-wide present," you can, but it's not a meaningful scientific statement.