Negative Dimensions: attn Mathematicians!

Lets say you have xyz*t…a 4d field which, when distorted by
mass, describes gravity.

Furthermore, 4 imaginary dimensions, ijk*l…describe a
quarterion…a mathematical short-cut to rotate an object on its
axes.

What about negative dimensions?
Does a -x*-y*-z*-t coordinate system exist? if so, can a construct
be computed in such a coordinate system?

Well, ya didn’t need all that hooha at the beginning to ask simply if there could be negative dimensions.

I would contend that there is no such thing.

First off, I assume you mean spacial dimensions, i.e. the familiar 2d and 3d.

Well, in our normal 3 (maybe 4) dimensional world, we only describe distances in positive terms. Nothing can ever be -3 feet away from something. That would be meaningless.

Likewise, on a 2d piece of paper, all distances are measured as positive values.

So, what the heck would a negative dimension mean?

If you can answer that, maybe you’re onto something.

Well, Looking through an older issue of Scientific American, the
article descrived negative energy and exotic (negative) matter.

I thought if they could theoretically exist, why not negative dimensions?

That, and I wanted to provide a solid foundation describing the
continuum known as subspace in the Star Trek fanfic I’m writing.

The number of dimensions is a cardinal (counting) number. In 3d space, you need 3 coordinates, for example. Since cardinal numbers are be non-negative, I don’t see how you would have a negative number of coordinates.

(Fractals have non integer dimensions, but they use a different definition of dimension. I think they are still non-negative)

Nope, negative dimensions are right out the window.

In the vector space usage, what DrMatrix said applies. Fractal dimension is based on the concept of self-similarity, and also need to be non-negative. Here’s an article about the first sense; that should give you a clear sense that there are no negative dimensions.

Nitpick: the quaternions do not have an imaginary number of dimensions–they have 4, with unit vectors 1, i, j, and k.

Negative distances make sense in a one-dimensional if you have a direction associated with your “distance”, although I’d be hard-pressed to call it a distance. But that’s off to the side.

How would you define the position in inverted space past the singularity of a black hole? Wouldn’t you use negative dimensions?

The singularity is a zero dimensional point. There is no space past the singularity inverted or otherwise.

I think that’s just assumed, you have to take into account the space that it bends into itself.

Sorry… I was thinking of Naked Singularities… just took 2 posts to remember the term I was thinking of. NS bend space time to cause inverted space, which would be negative dimensions.

You talking about the space inside the Event Horizon,
or do Naked Singularities not have EHs?

In Ian Stewart’s recent book Flatterland, there’s some explanation of what living in spaces with many different dimensions would mean. He further goes on to show what fractional dimensions would mean and what some odd variations on dimensions would be like. He may attempt to define “negative dimensions” there. I only glanced at the book so I’m not sure. I just did a search on Google and I find some references to negative dimensions. It’s not that difficult to come up with clever, consistent definitions of various generalizations of mathematical terms. Whether these are useful definitions are another matter. For instance, after explaining how it’s possible to have the base for number representations be not just 10, but also 2, or 8, or 13, or any positive integer, we could go on to show that it’s possible to define a way in which to use negative bases for number representations. It may not be useful, but it’s a clear, consistent definition.

Illuvatar may be thinking of the extended spacetime in Kruskal coordinates, which doesn’t correspond to any real values of the usual spacetime coordinates. These would not be considered as “negative coordinates”, however.

The Cosmic Censorship principle says that there are no Naked Singularities. That is, every singularity (except the Big Bang) must be clothed by an event horizon. I like the principle, but it is something that is just assumed. I have no idea what a naked singularity would be like. And since dimensionality of a space is the number of coordinates necessary to specify a point, I still don’t understand how any space could have a negative number of dimensions.

The phrase doesn’t have to mean anything.

What would the square root of a negative number mean?

These constructs don’t need meanings, as long as you have a useful way of working with it. If I’d design a consistent set of formulae which describe how to use negative dimensions, that’s all you’d need.

I figured, if you use the rules to derive gravity from four positive
dimensions, maybe you could get anti-gravity from four negative
dimensions.

Well yes, but it is desirable that a new improved definition for a construction be an extension of the old definition. That is, when both definitions can be applied, they yield the same result. The fractal dimension is an extension of the usual definition of dimension, because a space that has three dimensions by the usual definition of dimension has a fractal dimension equal to three. If you can come up with a useful definition of dimension that is an extension of the usual definition and includes negative values for dimensions, then I have no problem with that.

What DrMatrix said in his last post is a very good point. If someone comes up with a useful extension of the concept of dimension that is consistent with the current usage, and includes negative dimensions, then we have negative dimensions. Until then, there’s no such thing. I don’t believe that anyone has done so at this point, but I don’t keep up with the research literature.

No… NS are theoretically possible. One example is if you have a star that is spinning fast enough that when it goes supernova and forms the singularity it forms into a ring rather than a single point.

Something like this is possible but scientist say that it is highly unlikely that any actually exist. A NS inverts space time, I think I heard the analogy of an hour-glass where the center was the NS, the top half is space-time and the bottom half is inverted space-time that exists underneath/outside normal space-time.

Last I heard, the consensus was that NS were barely possible, but highly unstable: Anything you could possibly do to observe one would clothe it. Which is almost as good as saying that they don’t exist.

Could the 6 or 7 tightly curled up (Plank sized) dimensions called for in String Theory and M Theory be called negitive dimensions,sense they are everywherebut nothing can be located with them because they are smaller than even a black hole.
But sense your writeing for a Star Trek story you can just make up negitive dimensions and use them however you wish. It couldn’t possibly be any worse than any of the other made up physics in the Star Trek universe.
By the way I love the new show Enterprise, but that’s one for Cafe`Society.