I was wondering what are the distances, speeds, energies etc. were relativity can cause notable differences.

I am asking because I heard in a TV that for a distance from here to Jupiter for example, euclid geometry calculations start to deviate from reality.

I don't know if he reffered to distances in the universe general, or for distances within a certain gravitational field (in our case, the Sun's)

I am not sure if this is correct or wrong.

In case this is valid, we should have orbit modifications for spacecrafts sent to Jupiter, for example. Do we have any such corrections?

I believe that's only true at or near the speed of light. There's nothing about space that would cause geometry to fail in that distance. Not that I know of, at least.

Euclidian geomtry has no problem with the universe so long as travel time is instanteous and we see all objects exactly where they really are. We don't, of course. Light travels at a finite speed and so we see objects as they were in the past. Even close objects such as Jupiter. In fact calculating where Jupiter 'should be' compared to where we see it was one of the ways in which the speed of light was first calculated.

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Depends on the precision you need in your controls synchronization. Relativity is extremely relevant to GPS satellites, which are only 150 miles away and traveling at 17,000 mph.

GPS orbits are about 20,000 km (12,400 miles), which implies 14,000 km/hr (8700 mph)

I think you're thinking of the ISS/shuttle or something else in LEO.

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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.

It's definitely GPS.
http://www-astronomy.mps.ohio-state....Unit5/gps.html

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?

No, swansont is right about the height of the orbit--from your link: "Each satellite in the GPS constellation orbits at an altitude of about 20,000 km"

Warped spacetime is not Euclidean so I guess this could be what they meant.

If the geometry of the universe was closed then the angles of a triangle would add up to more than 180 degrees and if open, less than 180 degrees. Only if the universe is flat (Euclidean) would they add up to 180 degrees. I think this is still an open question (if you'll excuse the pun). Maybe this is what they were referring to.

Huh - I assumed GPS satellites were in LEO. I stand corrected.

I was meaning that he probably was thinking about the GPS constellation, as it is one of the very few "real-world" applications of SR and GR. (I'm not ambitious enough to claim to understand the mind of God, only a half-cocked attempt to understand the mind of a fellow board member. ) It even gets a mention in Hawking's "A brief history of time", IIRC.

Didn't spot the incorrect altitudes though. Oops. ops:

It is astonishing though, that to provide the current level of accuracy, you have to correct for what was once considered a pretty exotic effect.

What about Dirac's relativistic corrections to the quantum equations?

True, but here we're talking about a bit of consumer electronics calculating a correction factor derived from General Relativity. (It'll be interesting to see how Galileo does.)