The Illusion of the 3rd Dimension: A Mathematical Perspective on Space and Time

[evanghellidis]

Holographic principle aside, I believe that there is no theoretical basis for a 3rd spatial dimension, only experiential, since we can see and measure depth, as well as height and length. From what I've read, however, all forms of depth perception require movement, or otherwise don't allow measurements to be made(perspective). That lead me to the conclusion that the 3rd dimension of space may be just the illusion created by several 2-dimensional "frames" super-imposing over an infinitesimal amount of time in our memory.

In other words, there'd be 2 space dimensions, one time dimension and one "memory" dimension, or imaginary time. If "the present moment" is a volume in this 2+1 spacetime with an infinitesimal depth(=time), therefore mathematically real, "memory", or "past+future" are the rest of the volume and mathematically defined as real+imaginary(or, we always relate the past/future to the present). Technically, I suppose this would be a 2+2 dimensional universe, what with time being complex.

So, am I making an error in judgement, or can I commence bedazzling my drinking buddies with this little revelation of mine? And if I'm really on to something, who else thought of this before me? I just recently found out that complex time has already been proposed by Hawking.

 

[Chalnoth]

Holographic principle aside, I believe that there is no theoretical basis for a 3rd spatial dimension, only experiential, since we can see and measure depth, as well as height and length. From what I've read, however, all forms of depth perception require movement, or otherwise don't allow measurements to be made(perspective). That lead me to the conclusion that the 3rd dimension of space may be just the illusion created by several 2-dimensional "frames" super-imposing over an infinitesimal amount of time in our memory.

In other words, there'd be 2 space dimensions, one time dimension and one "memory" dimension, or imaginary time. If "the present moment" is a volume in this 2+1 spacetime with an infinitesimal depth(=time), therefore mathematically real, "memory", or "past+future" are the rest of the volume and mathematically defined as real+imaginary(or, we always relate the past/future to the present). Technically, I suppose this would be a 2+2 dimensional universe, what with time being complex.

So, am I making an error in judgement, or can I commence bedazzling my drinking buddies with this little revelation of mine? And if I'm really on to something, who else thought of this before me? I just recently found out that complex time has already been proposed by Hawking.

It wouldn't work because if there were only 2+1 dimensions, then forces like gravity and electricity would fall off with the inverse of the distance, instead of the inverse of the square of the distance as we observe.

 

[Chronos]

3D looks good because we see no wrap around effects that would be obvious in a 2D universe.

 

[evanghellidis]

forces like gravity and electricity would fall off with the inverse of the distance, instead of the inverse of the square of the distance as we observe.

The idea being that those forces disperse and lose energy in that extra dimension, if I understood that correctly. Hmm...Well, that sort of gives this 2+1 model a driving principle and explains why time "passes". Movement through time would be explained as the result of various forces dispersing in the imaginary time dimension.

Time here being understood as a medium in which that 2D matter moves, so essentially a space-like dimension, although one made up of the individual threads of each particle. Basically, each point on the timeline of a particle is that same particle, except it's being affected by different external forces in the space dimension, since other particles would have moved closer/further. As such, the force that acts between the two versions of the same particle would be oriented towards an arbitrary "future", where the spatial configuration of a particle system allows more force to spill out in the time dimension.

I really get the feeling I'm somehow rephrasing string theory...

 

[Chalnoth]

The idea being that those forces disperse and lose energy in that extra dimension, if I understood that correctly. Hmm...Well, that sort of gives this 2+1 model a driving principle and explains why time "passes". Movement through time would be explained as the result of various forces dispersing in the imaginary time dimension.

It doesn't work like that. You need 3 spatial dimensions to get the right falloff, plus one time dimension.

 

[George Jones]

An elaboration on what Chalnoth has said:

All of the the spacetimes for general relativity in 2+1 dimensions have constant spacetime (not just spatial) curvature (zero curvature, if there is no cosmological constant). 2+1 general relativity has no local gravitational degrees of freedom.

 

[debra]

It doesn't work like that. You need 3 spatial dimensions to get the right falloff, plus one time dimension.

We have 3 dimensions to contain objects. The objects could not move around without time, so time is just another addition to allow objects to move around in 3D. Its not a dimension as such, just a quantity needed to allow spatial movement.

The whole thing must be (IMO) an intelligent design. i.e. it would not just happen by itself for no reason. I suppose some type of intelligence though it was a good idea to build somewhere to live - hence 3D space and time. I suppose even mathematics by itself could come up with the whole idea. (but I do not really know for sure!)

 

[evanghellidis]

mathematics by itself could come up with the whole idea

That's actually how I arrived at this, although my reasoning wasn't entirely mathematical. The motion of two massive bodies defines a plane and things at really big scales tend to be planar. The natural progression of dimension starts with 0, which would be that really dense initial point. Why must we assume that the first event in the universe exploded straight into 3 dimensions, besides the anthropic principle? What is the basis of this, other than what we observe with our limited senses?

That's really the essence of my argument. Imagine that time suddenly froze, yet you could still see(the photons would bounce back and forth on their now finite trajectory, for instance). What would the universe look like? Isn't that what a photograph is, in fact, a moment in time? Since our perception of 3-dimensionality is so deeply tied with the passage of time, I believe we can equate time with the 3rd space dimension.

Still, that's not the entire gist of it. There's also the notion of complex time, so the math still has 4D(2+2). Would that agree with GR?