Disclaimer - I am not a physicist, nor do I have any deep understanding of space-time, relativity or orbital mechanics. This thread was started simply out of a place of curiosity.
The thought experiment goes like this: - Since the universe is expanding everywhere at the same time, is it possible to stay completely and relativistically still? - Since our earth is orbiting around the sun, and the sun is affected by the gravitational effects of the larger galactic arm, and the galaxy has some relativistic vector through the universe, what is the vector of the Earth? More concisely, if we want to ignore direction, at what speed is Earth moving relative to absolute zero? - Would we observe time dilation of natural physical objects in the universe that are experiencing vast velocities relative to us?
Kind of a mouthful of questions, but I just wonder how physicists take into account these variables when calculating trajectories within our own solar system.
I'm not a qualified physicist either, but did study the subject at university. My take on it is:
Yes. When you say 'completely and relativistically' you mean, "in comparison to my own frame of reference".
There is no 'absolute zero' as everything is 'relative' to everywhere else. This is one of the presumptions of Special Relativity - an objects speed can only be measured in terms of another object. There is no 'absolute frame of reference'. The 'Vector of the Earth' only has meaning relative to another object. eg. if Earth is moving at a speed of 1000 km/hr, this figure only has meaning if you add 'relative to the Sun'.
Yes, all objects with a different velocity relative to 'us' will be experiencing a slightly different rate of time to us depending on the magnitude of the difference. Although there is a caveat on this - expansion of the universe does not affect the equations of Special Relativity, as space is expanding and additional space is inserted between us. Although the 'coordinates in space' may be moving apart, it is important to note that an objects velocity is different to its position even if the position changes. A distant galaxy that is moving away from us due to additional space being created between us is not experiencing a different velocity, only a changing position, so there is no different frame of reference in which a different rate of time is experienced.
You would find that distant galaxies, although appearing to move away from us at great speed, actually don't have much of a different velocity to us. Therefore their rate of time is not dilated as much as you would expect from our point of view.
You can largely ignore time dilation due to Special Relativity in our own solar system, as velocities for common objects like probes are usually not really that fast. However General Relativity time dilation is a factor for communications, satellites and observations of objects close to mass like planets or the Sun (for example, observations of Mercury or clocks on GPS satellites).