According to our senses, space and time seem to be absolute, not varying according to our mass / speed, but is there a way we could physically experience the effects of Relativity ?
I mean be conscious of its effects, distortion of space/time for ex., without any device or instruments, just our normal senses.I’m talking theoretically, considering the limitations of the human body but not the technical requirements.
I am not a physicist or a strong math jockey, but it was my understanding that the lack of observable differences was exactly what “relativity” was designed to explain.
I don’t know much about physics, but isn’t the primary concept of General Relativity the idea that in the observer’s frame of reference everything stays constant? Any experience of the effects would violate this principle.
Or, what DrFidelius said.
If we could make a spacecraft travel fast enough you could look outside and see relativistic distortion of the starlight and anything else in sight.
If you could get close to a large black hole that wasn’t absorbing anything (and therefore not putting out fatal radiation) you ought to be able to see it warp the images of the stars behind it. I’m not sure if you could see the light redshift with the naked eye.
One of the axioms of relativity is that there’s no experiment you can do that tells you that you’re moving at a constant velocity (as opposed to your being fixed in place and the Universe moving around you.) You don’t physically experience time as passing more slowly when you move close to the speed of light (whatever that might mean); clocks that are at rest with respect to you tick away at their normal rate, and you see clocks that are moving relative to you ticking off the seconds more slowly.
That said, there’s nothing that theoretically bars you from accelerating yourself up to a substantial fraction of the speed of light relative to the Earth and comparing notes with people on Earth. We can imagine an interstellar ship that (through science or magic) accelerates continuously at 1 g for two years, and then turns off the rocket engines. At this point, the ship will be moving at about 90% of the speed of light, and for every further year that passes on Earth, only about 160 days will elapse on the spaceship. There’s nothing barring us from doing this other than the phenomenonal, colossal, mind-boggling expense of building such a ship.
I recall one of the Apollo missions tested this. They took along an atomic clock which had an earthbound partner, and measured an observable timekeeping difference upon return. My memory is my cite; I suggest you exercise your Google-fu for details.
If you can move close to c, and look out the window, you should see all kinds of strange visual effects. Here is a primer on some of the effects. This youtube video does the opposite - it slows down light to bring out relativistic aberrations.
Tried out my own Google-fu after posting, and I could not find specifics on the experiment I reported. One site even said no such experiment ever happened. Oh well…
Even with modern technology we humans live and move far, far below the threshold of noticing any relativistic phenomena with our senses.
Barring any technology, and since we’re talking Special or General Relativity, the only thing to work with here is your sense of time and sight. We’re notoriously bad at noticing the subtleties of the former and with the latter, there’s no way to get up to any speed relative to anything else to witness red-shifting or blue-shifting behind or in front of us.
Of course, with technology there’s all the cool and interesting stuff like time dilation, physically shrinking along the direction of your velocity, etc. There’s even the more passive stuff like Einstein Rings which require modern telescopes and image processing to resolve—But for now, being able to physically sense this stuff is far into the future and dependent on technology.
Despite all of the "if"s in the answers (which go against the point of the OP), I’m pretty sure the answer is no.
I like this video of A Slower Speed of Light - well, except for the commentary. It’s a game where you collect orbs and slow down light.
MikeS is completely correct but perhaps not forceful enough. Let me put it like this:
Time always moves at 1 sec per sec in your reference frame. Nothing can ever change that.
Your mass is always constant in your reference frame. Nothing can ever change that.
The world in your reference frame always looks normal to you. Nothing can ever change that. The things you do, the signals you send, the aging you see, every process runs identically to before you started moving at speed.
The idea that the world appears different to you simply because you are moving quickly is perhaps the biggest misunderstanding of relativity. Many of the physics threads here begin with this as an assumption, sometimes stated, sometimes unstated, but always leading to confusion.
The proof of this is that we here on the earth are - right now and always - moving at 99.999~% of the speed of light in some other object’s reference frame. Just as we see the objects that are at the edge of what is to us the observable universe moving at 99.999~% of the speed of light, those objects see us, at the edge of what is to them the observable universe, moving at 99.999~% of the speed of light. You can’t argue that it would be different if we could speed ourselves up: the point of relativity is that no such difference can ever be seen.
We can look out of our frame and see the effects of such apparent speed, but we can never experience them inside of our frame, whether that is the earth or a spaceship.
I await the inevitable nitpicking.
Well, it’s called “Relativity” for a reason. 
Well, it needs a “device” but it is pretty simple. And very cool.
You might not consider this a demonstration of relativity, but you would be hard pressed to explain it without.
I was going to suggest a compass and a wire hooked up to a battery, but yeah, that too. Magnetism is inherently relativistic. The current-carrying wire is perhaps a better example, since it’s easy to calculate the speed of the electrons causing the relativistic effects, and it’s a literal snail’s pace. Also because the apparatus is a lot easier to get ahold of than huge chunks of copper and rare-earth magnets.
True. But you can’t ignore just how brilliantly cool the magnet and the copper pipe is 
It does have the advantage that you don’t need an electrochemical device to drive the electrons along. Two simple and passive components. (And I want one.)
It’s probably not as direct an experiment as you were asking for, but the detection of high-energy particles with very small half-lives is a good confirmation of relavistic predictions (as was historically used as such). If they didn’t experience time dilation (or length contraction, depending on your frame), we would see a tiny fraction of what we actually detect. It’s not quite something you can conduct with your bare hands, but certain experiments along those lines are within the scope of an undergrad physics lab.
Here’s a link to a story about a guy who did this during a family camping vacation.
He took three cesium clocks to a campsite located thousands of feet higher than his house. The predicted time gain (due to the reduced gravitational field) was 22 nanoseconds; measured time gain was 23 nanoseconds.
Unfortunately, this doesn’t meet the OP’s “without any device or instruments” requirement.
Correct me if I am wrong, but my understanding is that the theory of relativity corrected some observed anomalies in the orbit of Mercury. Given that Mercury can be observed and that some features of its orbit can be observed, (eg, length of time of transit), that might be as close as you can get to the OP.