Are the three spatial dimension really two way? Spacetime is conventially described as consisting of three spatial dimensions and one time dimension. The time dimension is considered to be 'one way' in that we normally only go towards the future and can't go backwards to the past.Spatial dimensions are said to be bidirectional, and that it's possible to go one way or the other with equal freedom.For instance, the door to my apartment is directly opposite to the window in the living room. I can open the door, walk to the window and look out, then return to the door and close it. It seems that the door and the window maintain their relationship and I can go back and forth freely.But in reality the position of the door and the window are moving in space. The planet turns, the Earth moves along its orbit, the Sun moves around the galaxy, and the galaxy is moving through space.So when I walk from the door to the window, then from the window to the door, I am not retracing the same points in space. Everything has moved in space.So, is there any proof that it's possible to move from one point in space to another then back to the same point?
 A: It is always possible.  There is always some amount of momentum that can be imparted to you to get your back to a pre-selected point in space.  In your hypothetical example with the Earth moving and the Sun moving, it may require quite a lot of momentum (and if you're really unlucky, it may involve going straight through the center of the Earth, depending on the direction of the movement of the Earth and Sun).
It is trivially possible to show that you can define a spatial coordinate system which is a connected manifold that lets you travel this way or that.  It is also trivial to show that for a mere human with mere human skepticism, it works (as you demonstrated by walking through the door and back).  However, proving that the universe is well described by such spatial coordinates is a bit more daunting.  You start running into philosophical issues such as the Aggripan Trilemma (also known as the Münchhausen trilemma), which states that we can never truly know anything to be true because all justifications for anything we "know" eventually come down to:


*

*An axiomatic assumption that cannot be proven

*A circular argument

*A reduction to infinity


Needless to say, the assumption that space permits you to return to any point is an assumption which has been well accepted since long before Newton put forth his laws.  As you have noticed, it is an assumption, but it's one that people have found to be a generally useful assumption to assume.
The same goes for time.  We don't know that it's impossible to travel back in time.  We just have never seen a single instance of it ever occurring, and every single action we take appears to be along the march of time for all intents and purposes.  Many theories of timetravel are built around alternate structures which give the localized appearance of time traveling in a straight line.  Think of it similar to how the Earth appears to be flat locally, but we know that it's actually a spheroid... just a really big one.  Many timetravel theories presume structures along those lines.
A: Firtly you need to have a look at What is time, does it flow, and if so what defines its direction? where I discuss what the time dimension means in physics. The flow of time doesn't exist in physics. Exactly what the flow of time means is open to debate and possibly best left to the philosophers.
So the point is that the time axis is bidirectional in exactly the same way as the spatial axes. The equations of motion are time symmetric and apply to something moving to negative time just as they do to something moving to positive time. Admittedly I have never seen anything move to negative time (while sober) but as discussed in the linked question my guess is that this is the way human perception works rather than anything fundamental in physics.
But in any case how time behaves isn't relevant to the discussion of your main question. The reason you're getting confused about returning to the same point in space is because you think a point in space has some fundamental meaning i.e. there is some absolute position. But this isn't the case.
In physics all motion is relative. You are assuming we are moving relative to the Sun, the Sun is moving relative to the Milky Way and presumably the Milky way is moving relative to something else. But all motion is relative and there is no absolute motion. It is just as valid to treat ourselves as stationary and regard the Sun, Milky Way and everything else as moving relative to us. This may sound like a bit of sophistry, but I must emphasise that as far as physics is concerned this is a perfectly valid thing to do.
So I can construct a coordinate system with myself stationary at the origin, and in this coordinate system I can observe you to start at some point in space $\mathbf x$, move away then return to $\mathbf x$. In my coordinate system you have returned to the same point in space. In other coordinate systems that are moving relative to mine you are quite correct that you wouldn't have returned to the same point in space.
Physics tells us that all coordinate systems are equally valid, so the conclusion is that the same point in space is not meaningful when considering different observers/coordinate systems. It would only be universally meaningful if there was some absolute coordinate system that everyone could refer to. However relativity tells us no such absolute coordinate exists.
A: The two existing answers are good, with many good technical points. While respecting these answers I will try to give another answer that I wonder may perhaps be more satisfying to you.
The fundamental crux of your question comes when we look at your points:


*

*you give a good experimental definition of movement within your room

*you give another definition of movement within the solar system

*you allude towards the possibility of a future more fundamental definition


They can all give different kinds of answers. So the answer is clearly: it changes depending on what you mean by movement. There can be no answer at all until it is decided what precisely we mean. As physics research progresses, new ideas of movement are proposed. Currently there is no evidence to suggest that spatial dimensions are one way, and in fact all the evidence points to spatial dimensions being two way. However, maybe as a silver lining for your suggestion, the newest theory of space (relativity) already includes a way for this to be possible, because there is a maximum speed we can travel, the speed of light. So if the 'background movement' was more than the speed of light, we would indeed only be able to travel one way. So that means the discovery of 'one way only movement' would not even necessarily require us to rewrite our existing science from scratch.
Some comments have been made that your proposal that there exists an absolute kind of motion and absolutely fixed points in the universe is wrong. As far as I am aware, all current scientific evidence suggests these comments are correct in that there is no way to detect "absolute" motion. This is a kind of physical proof. However, I think your question suggests towards a more strict kind of proof. What we have not discovered is any evidence that there can be no possible deeper theory with absolute motion. Indeed I would consider it harder to go from a theory of fixed positions to a relative one than vice versa, and I would even venture to say there can never be an absolute proof of a lack of relative motion. So there is always the possibility we could discover 'one way spatial dimensions' in future, even if we have no reason to expect it now.
As a bonus, what would it look like if spatial dimensions really were 'one way' and how could we tell? We would have to be able to measure something, some kind of physical property or 'constant' (not the best word in this case!) that was constantly changing. But, it would change at a different speed if you travelled one way compared to the other way. That would show we were travelling relative to something else. But no matter how fast you travel, you would not be able to stop it changing completely. That would show however fast you are going, you would always be going one way. What kind of 'constant'? It could be something like the strength of the electric field, or maybe some physics we haven't discovered yet, perhaps a property of dark matter. But just to say again, nothing like this has been discovered yet.
To summarise:


*

*To get a deep and philosophical answer to this question, you need to deeply understand what you mean philosophically by "move", "the same point" and "back", as others pointed out.

*All the physical evidence currently suggests that it is possible to move back and forth, using what most professional scientists think is the most helpful yet fundamental definition of movement. However, this issue can never be absolutely proved one way or the other - and that's part of what makes physics so exciting.

