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As far as time dilation effects go you can figure out at what velocity the gamma factor (1/sqrt(1-v^2/c^2)) becomes significant. I guess it depends on what you consider significant too.
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I think the gamma factor explains quite well some limits, but I think its primaily related with special relativity.
I think the comment:
"for a distance from here to Jupiter for example, euclid geometry calculations start to deviate from reality.
", might be based on the time-space curvature that occurs in gravity fields, as predicted by General Relativity.
The Sun's gravity field is not that strong to cause big differences for small paths, but if we integrate/add these differences for a path from the Sun to Jupiter, these differences might start to become notable. Could this be the explanation of what I heard in TV, or is this just nonsense?