Ask Ethan #104: What if we grew a fourth spatial dimension? (Synopsis)

If the dimension grew as large as the Earth-Sun distance, in fact, everything in our Solar System would become unbound. If this lasted for more than a few days out of the year — even if gravity went back to normal for the other 50 weeks or so — our Solar System would dissociate completely in only a century.

Yeah but if it was only a few hundred meters deep, think of how your rent/mortgage would go down! With a hundredfold increase in living space per square foot of land, even us regular schmucks could afford a (3-D) apartment in lower Manhattan. Or wherever else you want to live. :)

interesting post, but why did it include a picture of Ghostbusters committing sex acts on each other. Did you really not notice that?

In an infinite universe chances are there is a duplicate of our exact situation elsewhere, right? Also a copy 1 atom different ? Ad infinitum

In his essay; "Just the Right Size" Isaac Asimov relates his experiences dealing with the producers of "Fantastic Voyage", when they asked him to novelize the screenplay.
He explained that if you were going to shrink anything to a microscopic level you would have two choices;
Either get rid of enough atoms to reduce a human to the size of a microbe, in which case there would be insufficient atomic structure for human thought;
Or;
Compact the atoms together as you shrink them in which case you would have a 180lb microbe that would be on the verge of reaching critical mass.
He then threw in the fact that Brownian Motion would shake your miniature submarine to pieces.
The producers told him to just "Mix it up".

@Ethan wrote:

It is relatively straightforward, in the context of general relativity, to construct spacetimes where the number of “large” (i.e., macroscopic) dimensions changes with time. Not only could you have had a large extra dimension in the past, you can have one in the future, or you could even construct a spacetime where that number oscillates, or gets larger-and-smaller-and-larger again over time.

Then General Relativity is wrong. If something is mathematically possible, then it happens. I'm fully aware of how thoroughly tested GR has been, but if predictions don't match observations then you need to check the math. GR is one of the most successful theories of all time but because it does have known shortcomings such as on the singularity side of an Event Horizon, GR should be treated as a rule of thumb rather than the holy grail of universal mechanics.

"Then General Relativity is wrong. "

How do you mean "wrong"? Wrong like "My aunt's name is Ethel" or "profit is the money left over after all expenses are paid" wrong?

"If something is mathematically possible, then it happens."

Ah, well THAT is wrong.

It is POSSIBLE that there is a teapot orbiting Jupiter. It doesn't mean there MUST be a teapot orbiting Jupiter.

"but if predictions don’t match observations then you need to check the math"

It is known you will die, denier. However, we have not OBSERVED this yet. Therefore you're immortal???

"GR should be treated as a rule of thumb"

Well, yes. And it is.

Come up with something better. Until then, we'll use the best one we have.

"In an infinite universe chances are there is a duplicate of our exact situation elsewhere, right?"

No. The possibilities may be a higher order of infinity, so that even an infinite universe cannot contain more space than there are possible ways of it to occur.

E.g. (and this may be wrong with integers, but some mathematician can give you a real set that does this if this set doesn't) imagine a six sided dice rolled an INFINITE number of times. You can claim "Well, this set of dice rolls will happen more than once!" except that it won't be preceded by the same infinite preceding list of numbers from that dice roll. So it won't be EXACTLY the same series of numbers, since the last time that series occurred, there was a different set of numbers before it.

@ Denier #5

your comment seems a bit radical, yet I feel is due to lack of understanding, it's geometry and not that hard. In GR you can start with one dimension (and I think in fact this is how you learn in in calculation terms, and it's how the formula is set-up, since it's the simplest form). Then you can pick one more coordinate and an angle at which it is oriented to the fist coordinates/dimension... then a 3rd.. and so on. and the formula works for any arbitrary number.. you can put i.e. 7, .. and you will have 7 precisely defined terms in that part of your field equation. In fact, I would call that a success of the theory, because it formulation is consistent and allows for that. it's not "finely tuned" by hand to work in 3d and nothing else. it gives a result for 2d space, and it can give a result for 25d space if it needs.

@Sinisa Lazarek #8

I don't doubt the math in the paper was done correctly or that GR allows for extra dimensions. I'm saying that a theory allowing for such a thing is dubious due to the observations that in 4.6 Billion Years our solar system has not dissociated. In the article @Ethan went on to say there are QM theories that limit our universe to the number of dimensions we have. The existence of other viable theories combined with the total lack of any observational evidence to support that aspect of GR should cast doubt on GR.

@ Denier

I have to admit I don't understand your reasoning and objections about GR in this respect.

you say.." I’m saying that a theory allowing for such a thing is dubious due to the observations that in 4.6 Billion Years our solar system has not dissociated... " I don't understand what does dissociation of our solar system have to do with the fact that a coordinate system can have as many coordinates as you need. On the other hand the statement that any such theory is dubious is ridiculus, since any vector even in classical mechanics can have as many coordinates.. Is classical mechanics dubious because I can calculate a trajectory in 2d space if I only choose x and y coordinates.. even though I live in 3d+1 reality? This is what you are saying... and I think you'll agree how nonsensical this sounds.

it changes nothing.. and you have it backwards for GR.. it's not an aspect of GR that allows this.. it's geometry. The metric in geometry can be calculated for any number of coordinates.. period. GR didn't invent this.

Denier, I agree with Sinisa in this though i would generalize it. Mathematicians and scientists often model different aspects of nature using N-dimensional models - with varying Ns depending on the system. Really it can be done for any number (N) as long as they are independent variables. This doesn't mean reality must have N dimensions. In fact it couldn't mean that since different models use different Ns.

Sinisa #10 & eric #11

As I've taken some time to digest what you're saying, I think I understand your point. The extra dimensions do not spring from the GR mathematics, but rather GR provides a canvas for which this kooky idea could possibly be painted. GR neither spawns nor prohibits the idea, but is instead agnostic to its existence.

If that is the case then I am.......er......less right.

I've got a bigger fear than merely planets getting messed up.

I never thought about the fundamental physical forces "leaking out" like that, but if they are confined to 3, would that mean my OTHER fear wouldn't happen? Namely, I am terrified that if we existed in 4 spacial dimensions, everyone would leak apart at every scale. The contents of your stomach would slosh out into the 4th dimension, and the contents of your blood stream, and the contents of your cells for that matter, all would just spill over since they aren't being held in on the 2 new sides.

"Namely, I am terrified that if we existed in 4 spacial dimensions, everyone would leak apart at every scale"

Are you not afraid of the Jabberwokky? The jaws that bite, the claws that catch!

"all would just spill over since they aren’t being held in on the 2 new sides."

How do you know? Blood cells are no different from skin cells or endothelial cells or any other cell. Why would opening a dimension not allow a four-body to be the result? Why should our extent be zero?

Wow,

I think a good example of the type of mathematics you are looking for is the random walk. If you aren't familiar with the concept, an object is located at the origin in a discrete space. The object moves one unit for each unit of time. The direction in which the object moves is randomly determined. It is along one of the axes of one of the dimiensions of the space in all cases. Thus, for a 1-D random walk, the object is limited to 2 choices of direction with probability 50% of either. For the 2-D case there are 4 possible directions (either direction along either axis) and in the general N-d case, there are 2N choices.

I will leave the details to one who's more mathematical than I am, but the whole question of the random walk is whether it is inevitable that the object will eventually return to the origin. For the 1-D case, the answer is yes; given enough time, the object inevitably returns to the origin. For the 3-D case, the answer is no, even in infinite time, the object does not inevitably return to the origin (I believe that the more mathematically precise formulation deals with the limit of the probability of return to the origin as the time goes to infinity. It either approaches one or zero, depending on the dimensionality of the random walk).

"I think a good example of the type of mathematics you are looking for is the random walk"

No, that's nothing like what I'm either asking or saying.

Your body, your cells, whatever, occupy their full volume in three dimensions.

When an extra dimension turns up, why would it be empty when the three dimensions are?

We CANNOT measure the fourth space dimension.

How do you know your body doesn't have a 4-d shape that still remains whole and solid?

WE DON'T.

When the apple in flatland is introduced into the 2D world, everyone "sees" it as 2D. Mr Apple has sensory perception into the third dimension, he knows he's a whole 3D object.

But none of the flatlanders, their perceptions limited to 2D, would EVER know this by perceiving.

Anyone asked Mr Apple what shape the flatlanders are? Why must their shape be 2D? All we know is that they only perceive 2 dimensions.

...when the three dimensions aren't...

"I will leave the details to one who’s more mathematical than I am, but the whole question of the random walk is whether it is inevitable that the object will eventually return to the origin. For the 1-D case, the answer is yes; given enough time, the object inevitably returns to the origin. For the 3-D case, the answer is no"

In none of the cases is a return to the origin inevitable unless the realm to wander is finite but unbounded.

Keep adding random numbers between +1 and -1 and you may ON AVERAGE expect 0, but any actual series will never get back to 0.

2D random walk definitely doesn't go back, with a divergence equal to Sqrt(N).

And I could probably get round to seeing that the same proof proves that the 1D random walk ALSO diverges similarly.

You forget, each random step changes the origin for the next. It is HARDER to get back to the origin, even in 1D, on the second step than it is to move further away. So on average, "further away" is the rule.

@Wow #19: You wrote "In none of the cases is a return to the origin inevitable unless the realm to wander is finite but unbounded."

That's not true. In one dimension (even an INFINITE one dimension), an unbiased random walk is guaranteed to eventually cross any designated boundary. "Returning to the origin" is one specific instance of such a designated boundary. Reaching the origin from any other designed starting point is another such instance (see the "gambler's ruin" problem).

In two dimensions, the level-crossing problem is not guaranteed, but is "highly probable." In three or more dimensions, the probability of an random walk crossing any specified boundary falls rapidly with increasing dimension.

Wow,

I'm sorry if you don't see the applicability, but what you were discussing was the question of whether in an infinite universe with an infinite amount of time, the universe will reach a configuration identical to the starting one. A simplification to that question would be in an infinite universe containing a single particle, given an infinite amount of time will that particle return to its starting location. This simpler question ignores the existence of other particles and characteristics other than location.

As Michael points out, returning to the origin is a special case of the more general "crossing any designated boundary" requirement. From the Wikipedia article, the probability of a single particle returning to its starting location in a 3-D universe is about 34%. For a large number of particles, each with only a probability of 34% of returning to their original locations, it's clear that the probability of ALL particles in the universe returning to their original locations is vanishingly small indeed. Factor in other attributes of particles such as their velocities, and the probability is even lower.

https://en.wikipedia.org/wiki/Random_walk

"That’s not true. In one dimension (even an INFINITE one dimension), an unbiased random walk is guaranteed to eventually cross any designated boundary. "

Cross, not return.

I wasn't certain if it would also be true it would tend to fly off. I'd have to dig up my shcoolbooks for the proof of the random walk.

There doesn't seem to be any need for a random number walk where you don't have the same number to be crossing, however. A *unit* distance, however, in a single line, yes It's possible because there isn't the infinite variation of any angle from the original

Without SOME random in there, it can't really be random

"Ibut what you were discussing was the question of whether in an infinite universe with an infinite amount of time, the universe will reach a configuration identical to the starting one. "

Oh, right.

No, I may be (by chance if nothing else) the same collection of atoms in the same order as some time before.

However, It's not ME there. It's my identical twin.

"From the Wikipedia article, the probability of a single particle returning to its starting location in a 3-D universe is about 34%."

That's not the random walk I know of.

I was talking about the original post, where we "suddenly" grew an extra dimension. If the universe already HAS a massive 4th dimension (unlikely as it may be), we're still alive, so clearly something is keeping all our juices in, but I was talking more about the original idea.

"I was talking about the original post, where we “suddenly” grew an extra dimension."

This isn't the same as developing an extra hole, though.

You're presupposing that any such dimension would not have anything in it at all and that in creating this dimension, nothing that currently exists in 3 dimensions has any 4thD extent.

However, this presupposition is pure conjecture.

Take a Circle. Draw it through the third dimension. You have a cylinder. There is no hole in it.

At all.

Indeed, please show me a 2D cylinder here in three dimensions.

Presupposing that if we developed an extra dimension we would necessarily have no extent in it is presupposing the conclusion you wish to draw: That our juices would flood out like a Tetley tea bag.

Your argument boils down to "If our juices could fall out through this 4th dimension, our juices would fall out through this 4th dimension."

Can you see the logical problem here?

You are ALSO presupposing that there are no higher dimensions.

This is not the case. See String Theory. These higher dimensions are necessary in string theory, but there's no need for its precepts to be nonexistent unless there are strings as theorised.

These dimensions merely have no extent in the available space we have now. "Creating" such a dimension is merely unrolling that dimension, similarly to unrolling a thin string from its rolled up 0 dimensional shape of a dot to its 1 dimensional shape of a line.

Your problem resides on the presence of two suppositions. Neither at all supported.