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I was hoping that some genius could explain what we would observe in the universe around us in the following hypothetical scenario in which the earth itself were traveling at or near the speed of light. I am confident that somebody can tell me that my hypothetical situation cannot exist, but they are not really the point here, though I don't mind being proven dumb.

Scenario: Say that our solar system, jointly with the Orion Spur in which it resides, is traveling at near the speed of light as it prepares to merge with the adjacent Perseus Arm of our galaxy (some 5,000-10,000 ly away). How does that change how we see the Perseus Arm? What do we see as far as the rest of the Milky Way galaxy? How does our speed change what we see in distant galaxies? Or in other words, does the Milky Way galaxy as a whole seem to move faster or slower due to our speed? Do other galaxies seem to be moving faster or slower than their actual motion?

Hope that makes sense and hope somebody besides me finds it fascinating to consider light speed's impact on our perception.

EDIT1: I guess a component of this question is regarding red shifts and blue shifts, as well as how our observation of star evolution and galactic evolution would be impacted. Such as, would distant galaxies appear to be evolving faster or slower than actual and whether we would think they were moving toward or away from us, just due to our own near c motion?

EDIT2: I have reduced my scenarios to just one, as all responders are consistently telling me that such is better. Hope that the change aids the dialogue. My apology as a newby to this forum.

Thomas
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In very very short: if the rest of the Universe travels at a velocity close to the speed of light relatively to us, everything will happen slower in the Universe, every length (or distance) in the direction of movement will be contracted (but not perpendicularly to it, hence a deformation), and electromagnetic waves will be heavily doppler shifted. Everything will be totally normal in the solar system, of course, as speed is not a thing in itself but has only a relative meaning.

This only addresses scenario 1. If our galaxy was spinning such as the outer limit travel at a speed close to c in an frame having its origin at the galactic centre and axes not spinning relatively to the average distribution of matter in the Universe, first you need quite some forces to prevent it to dislocate, and then some effects could be detected in our frame if those forces were not gravitational interaction.

Matt
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Massive objects cannot travel at the speed of light, so I will assume you are asking about speeds close to c. And I will assume this all is relative to the CMB.

I will address scenario 1, because there are too many questions in one.

As our solar system travels near speed c relative to the CMB, time in the rest of the universe will seem to speed up (relative to us). Now if you have a clock here on earth, it will still seem to tick normally.

It is when you try to compare this clock to another clock somewhere else (outside the Solar system) in the universe, that you will see our clock tick way slower. In fact, as you go arbitrarily close to the speed of light, our clock might never tick during the whole 13.8 billion years since our universe existed (assuming the clock always existed).