Is there such a thing as absolute velocity, and if so, can we negate the effects of time dilation by traveling in the opposite direction?
Assuming there is absolute velocity, if we were to move an object with a velocity equal to in value and opposite in direction, of our own velocity in reference to an extragalactic frame, would that object experience less time dilation than what we do?
This would also assume all influences from general relativistic gravitational time dilation aside.
There are a few key points you must bear in mind when thinking about special relativity.
The first is that motion is not absolute, but relative. That means you do not have 'a velocity' in the sense of a single velocity in an absolute sense. At a given moment you have an infinite number of velocities, each of which is relative to a different reference frame. You are moving at one velocity relative to the Sun, at other velocities relatives to cars and people passing on the street, and so on.
The next, which is a consequence of the lack of absolute motion, is that the effects of special relativity are symmetrical. So a passing object will seem to be time-dilated compared with time in your reference frame, while you will seem equally time dilated compared with time in the object's frame.
Thirdly, you yourself do not experience the time dilation or length contraction that others might attribute to you when they observe you from another reference frame. It is rather like perspective. If you stand far away from me, you will appear to me to be smaller, and I will appear to you to be smaller- that is not because either of us has shrunk!
If you apply those ideas to the question you have put, you will realise that the first part of your question is rather meaningless. When you say an object has a velocity equal and opposite to our own, you are assuming we have an absolute velocity, which we don't. In any case, what counts is our velocity relative to the moving object. In our frame, the object will appear time dilated by some amount, and in the frame of the object we will appear time dilated by exactly the same amount.
