speeding out of 3-space

[Adam Lore]

I'm presuming that this has come up before, but:

Consider a 2-d being living on the surface of a sphere. She is trapped by supergravity (or whatever) on the surface, and cannot access the third dimension.
But, if she moves fast enough she could move in a straight line, zooming off of the sphere into outer (3) space.

This raises the question, could WE theoretically move something fast enough to get it into a 4th dimension?

(I don't necessarily mean to suggest that we could actually just go out and do this, or that we could move that fast, but does it work "on paper"?)

[Keiji]

In such a situation, yes.

However, it's pretty obvious that what you describe isn't the case for our universe; if it was, gravity would not work as we know it.

[Adam Lore]

Is there any way to modify the scenario so that something like this applies to our physical universe?

[Keiji]

I wouldn't know. The only thing I can think of is if it used quantum forces instead of gravity, but that's not really my best subject.

[papernuke]

ehh.. i haven't been here in a while but

I'm presuming that this has come up before, but:
Consider a 2-d being living on the surface of a sphere.

Her being on a on a sphere already introduces her to the third dimension because as we all know, a sphere isn't flat, and if she were on the sphere, her 2D body would be warped in 3D already.

[Keiji]

The surface of a sphere is 2D, which is where the 2D being would be. They'd have no idea they were on a sphere, except for observing how things moved (curvature).

[papernuke]

Wouldn't they be able to feel that they are not on a flat surface? That they are being bent in another direction?

[zero]

I don't know, but it seems doubtful. Does a fish feel wet?

[papernuke]

I don't know, but it seems doubtful. Does a fish feel wet?

Ehh, probably not, then I guess the only way they would be able to tell would be by

The surface of except for observing how things moved (curvature).

[quickfur]

Is there any way to modify the scenario so that something like this applies to our physical universe?

Dunno, first it has the big assumption that the universe is the surface of a 3-sphere, and then it assumes that momentum is not restricted to 3D, which, as far as we can tell, isn't the case (since otherwise fast moving particles, like light, would simply vanish off the universe instantly).

OTOH, this insane idea just occurred to me that, perhaps the reason nothing can exceed light-speed is because it is precisely the "escape velocity" into 4-space. Relativistic effects associated with near light-speed travel are a side-effect of whatever force it is that confines us to the 3D surface of the 3-sphere (sorta like a centrifugal force that expends some of the kinetic energy to keep the particle in the universe, but we observe that as time dilation/space contraction, increased expenditure of energy to accelerate, etc.), but anything that breaks the lightspeed barrier will instantly escape out of the universe, and so cannot be observed. (Yes, it's a totally wacked idea, but it's food for thought.)

[PWrong]

OTOH, this insane idea just occurred to me that, perhaps the reason nothing can exceed light-speed is because it is precisely the "escape velocity" into 4-space. Relativistic effects associated with near light-speed travel are a side-effect of whatever force it is that confines us to the 3D surface of the 3-sphere (sorta like a centrifugal force that expends some of the kinetic energy to keep the particle in the universe, but we observe that as time dilation/space contraction, increased expenditure of energy to accelerate, etc.), but anything that breaks the lightspeed barrier will instantly escape out of the universe, and so cannot be observed. (Yes, it's a totally wacked idea, but it's food for thought.)

Sounds interesting. You could try looking at the equations for escape velocity and those for relativity, and see if you can find some connection or reformulate one in terms of the other.

[Eric B]

Never thought of it that way before. I do know that when you approach the speed of light, the dimension you are heading in in effect shrinks, as time and space sort of become "reversed". The same thing happens inside of the Schwarzchild radius of a black hole. The direction you are being pulled in now becomes like time, as you can only move forward; it appears to be flattened into nothing, because you're freefalling at the speed of light, and then time supposedly opens up into a dimension you are free to travel in. They never do explain much about this.
An analogy used is that there are four dimensions, and when you move at the speed of light in any one of them, it collapses down to 0 (in spatial length), and there are three others you have freedom in. Your total velocity in all dimensions has to be c, so if you begin accelerating in one of the other dimensions, your velocity in the one you had been traveling in must decrease.

We are currently moving at the speed of light in the direction we call "future". If we start moving in one of the other three directions, our velocity in the time dimensions decreases (time for us slows down). Since this transformation begins occuring under the speed of light, I wonder if that means you can start to somehow experience something like a fourth space dimension. Like if your at half the speed of light, then you see two partly collapsed dimensions (besides right/left and up/down) that you're traveling at c/2 in. (This dimensions would contain, in one direction, images of your past. would the other direction somehow contain the future, or at least different options?) Maybe that phenomenon where the vision is warped and everything is concentrated in front of you (including stuff behind you) has something to do with it.

So hypothetically; going any faster would be basically fully opening up time as a sort of space dimension (though once you're off the 3-brane, would the collapsed dimension open back up?) Since the coordinate length, time, and mass would be "imaginary", paerhaps that woul dbecome quite literal, and you would cease to exist. I tend to believe that three dimensional matter cannot exist in 4D, because its hypervolume would be 0, by virtue of being h×w×d×0. (unless there's some sort of hyperthickness, such as perhaps the strings which may have more, tiny dimensions, but then we would be so "thin" as to still not really exist anyway or easily fall apart). Of course, we have no way of accelerating anything past the light barrier, so that is all not even hypothetical, really.

[Keiji]

Welcome back, Eric

If we consider only one direction of motion, we can graph the concept you're describing on a plane, with time on the x axis and space on the y axis. Then a stationary object would be a horizontal line and an object moving at the speed of light (thus stationary in time) would be a vertical line.

Now, let us consider an object M which has some very small time-thickness δ (which for us would be a fourth dimensional thickness, though in this analogy it is second dimensional), so that the curve traced out has a non-zero area approximately equal to δt (where t is the total time observed by M).

We can now think of the mass of M being the area of this curve intersected with a vertical bar of width δ, divided by δ (hi differentiation). Therefore if M travels through space at a constant speed which is a significant fraction of the speed of light, the mass becomes the area of a parallelogram whose perpendicular height (actually horizontal) is δ and width (actually vertical) is greater than δ, so the mass increases above the rest mass. If M travels through space at the speed of light, its mass becomes infinite as it traces out a vertical bar. This is consistent with the usual "increasing mass" phenomenon.

However, you can now see that the mass of M only increases as observed by an external observer, as the bar being used to measure mass would be perpendicular to the observer's motion curve. To M, M's mass remains constant... but everyone else's increases instead.

While travelling at the speed of light in space, you would not be able to see your past or future because light does not travel in time (because it travels at the speed of light in space) so it would always be parallel to you. In addition you would not be able to travel through time yourself, as you have no muscles or equipment which can push into the time dimension.

[Eric B]

Is this illustrated anywhere?
I'm trying to picture what you're saying, and I'm so used to spacetime diagrams having time as vertical, space as horizontal and the speed of light at 45°.
I take it that at rest M would have a slight thickness in "height", and when moving at the fraction of the speed of light, it would then consist of two diagonal parallel lines.

[Keiji]

Oh right, I completely forgot about relativistic simultaneity.

Forget everything I said

[TheBetterGamer]

Such a scenerio could exist, concluding to what you're saying, but unfortunately little Sally wouldn't be able to move off the sphere.
She is stuck in that dimension of 2d and cannot move any other direction.

[PWrong]

Such a scenerio could exist, concluding to what you're saying, but unfortunately little Sally wouldn't be able to move off the sphere.

You're right, but you could have a hypothetical universe where surfaces have an "escape velocity". So if Sally moves fast enough on the sphere she'd escape. She'd probably immediately fall apart, being a one-atom-thick layer of atoms in a 3D universe.

[wendy]

I really don't follow

You take a photograph, and you crumple it up on the ground, but it's still whole: no one has escaped. Pictures are variations of ink on plain paper: it takes consciousness to be aware that the varying hues are different things.

We are variations of space, and it takes our consciousness to become aware of what is and what is not ours. We sometimes get this right, too. Unfortunately, we are not things floating on top of space, but are etched into the very nature of space itself.

And for all it is space being curved, it isn't curved in any higher space: curvature is a property of space itself. And what, would ye imagine ye become, if nothing more than a squirt of ink, like the man who escapes from the photograph above?

[PWrong]

That's almost certainly the way the universe works. To "escape" the 3rd dimension simply by going really fast is silly and obviously impossible. However a hypothetical magical universe in which this kind of talk does make sense is reasonable.

In particular, in string theory there's talk about branes within branes within branes, and how they stick together. Atoms are made of tiny branes (probably strings) but are stuck to a larger brane, which could vibrate in a higher dimensional brane. If that's the case you could have a spherical brane embedded in a higher dimensional brane, with atoms stuck to it, and for some reason if they go fast enough they might escape.

[TheBetterGamer]

Technically you can't have one dimension in another.
And even if Sally COULD move faster than speed of light, she wouldn't fall/fly off.
Unless her world was in a 3D world[Which is this scenerio], then all is possible.

[PWrong]

Technically you can have anything you want. We're talking about maths, not the real world.

What you have is an equivalence relation gluing a bit of the tangent space of the sphere to bits of the tangent space of R³. The glue takes effect at a certain speed.

[TheBetterGamer]

So we're glued to our world?
AAAAAAAAHHHHHHHH!!!!
*starts knawing at glue*
*people stare at TBG*

O_O
Umm......

Yeah.......

[holomanga]

Photons can't escape, but (theoretically) tachyons could. Thats how they might travel through time.