Negative dimensions?

[RossGr]

Originally posted by: ZeroNine8
I was using this definition of orthogonal:

N mutually orthogonal vectors span an N-dimensional
vector space, meaning that, any vector in the space can be
expressed as a linear combination of the vectors. This is
true of any set of N linearly independent vectors.

The term is used loosely to mean mutually independent or well
separated. It is used to describe sets of primitives or
capabilities that, like linearly independent vectors in
geometry, span the entire "capability space" and are in some
sense non-overlapping or mutually independent. For example,
in logic, the set of operators "not" and "or" is described as
orthogonal, but the set "nand", "or", and "not" is not
(because any one of these can be expressed in terms of the
others).

which basically means each coordinate is linearly independent of the others.

If you read your definition correctly you can see that orthogonal is a special case of linearly independent. The bold portion of your definition refers to the fact that what was said of orthogonal vectors is also true of linearly independent vectors. Any orthogonal set of vectors is linearly independent but not all sets of linearly Independent vectors are orthogonal. Linearly independent is the more general case, to from a basis for a space a set of vectors MUST be linearly Independent, but not necessarily orthogonal.

Vectors are linearly dependent or independent not coordinates. I take coordinate to mean a set of numbers which specify a location in the space. You can associate a vector with a coordinate if you assume the origin forms one end of the vector.

 

[ZeroNine8]

which basically means I'm just referring to a narrower set of conditions than I could be, correct?

 

[RossGr]

That is correct, a set of basis vectors can be orthogonal, they MUST be linearly independent. In the context of the dimension of a vector space being the number of linearly independent vectors required to span it, negitive dimensions do not make any sense.

 

[bwanaaa]

a dimension is a measurable distance.

Change in that dimension does n0t affect the value of that same object in another dimension.

So, i can understand three dimensions.

But time as a fourth dimension? I disagree. If time were a fourth dimension, you should be able to change an object in time WITHOUT affecting it in space. But time is DEFINED by space so it is just a shell game. Time is not a linearly independent variable that can be changed without affecting the spatial coordinates of an object (If there were no changes in space, you couldnt measure time)

And then they tell us that there are 11 dimensions in M theory-that strings can pass through these other dimensions. That I can believe.

Negative dimensions-I suppose they would be dimensions in antimatter. Although they might follow the rules of spacetime as we know them (volume=LxHxW), it might be possible that all 11 dimensions are equally weighted in antimatter--then Volume=product of eleven independent dimensional variables and if you dont specify the other 8 and just name three, it's like asking for the volume of a square---it's zero.

 

[silverpig]

Originally posted by: bwanaaa
a dimension is a measurable distance.

Change in that dimension does n0t affect the value of that same object in another dimension.

So, i can understand three dimensions.

But time as a fourth dimension? I disagree. If time were a fourth dimension, you should be able to change an object in time WITHOUT affecting it in space. But time is DEFINED by space so it is just a shell game. Time is not a linearly independent variable that can be changed without affecting the spatial coordinates of an object (If there were no changes in space, you couldnt measure time)

And then they tell us that there are 11 dimensions in M theory-that strings can pass through these other dimensions. That I can believe.

Negative dimensions-I suppose they would be dimensions in antimatter. Although they might follow the rules of spacetime as we know them (volume=LxHxW), it might be possible that all 11 dimensions are equally weighted in antimatter--then Volume=product of eleven independent dimensional variables and if you dont specify the other 8 and just name three, it's like asking for the volume of a square---it's zero.

Time is referred to as the fourth dimension, but it's really just the first dimension of time. The fourth spatial dimension is (as of now anyways) just a theoretical construct, but is completely different from time (or so it is thought anyways).

Another interesting idea is how about a second dimension of time? So you can go forward and backwards in time, and also "right" and "left" whoa.

 

[bwanaaa]

time is an artifact of the human mind. there is no such thing-there is only motion in space. if you ran a videotape backwards, you do not violate any law of physics. so time does not really exist as a dimension.

 

[f95toli]

Time IS a dimension. In physics the word "dimension" means the number of degrees of freedom. The best way to understand this is to ask how many numbers you have to use to define where in the space (not neccarily physical) something is happening.
In our world we need four numbers, you could for example give the position as latitude,longitude and height but you also need the to specify the time.

Physical problems with more than 3+1 dimensions are quite common. In dynamical problems you need 6+1 (3 for physical position, 3 for momentum and 1 for time), a volume in phase-space can therefore be for example a 6-dimensional sphere.

 

[bwanaaa]

you have to use to define where in the space (not neccarily physical) something is happening.

that is a contradiction-to define where something is in space IS physical, and can ONLY be physical. That you imagine a particular physical object or event occurring at a particular time is only adding a descriptor to reference your event/object to a particular 'keyframe' that is defined by other events/objects. Time is not an orthogonal variable. If it was orthogonal, then time travel would be possible-you would be able to independently specify time for an object relative to the other objects around it, which you cant.

 

[f95toli]

what I meant was the mathematical space in the problem. When you for example solve a differential equation the solution will span a solution space with a basis consisting of orthogonal functions (that is the basic idea in Fourier analysis).

I agree that time is a dimension with special properties, but it is a dimension from a mathematical point of view.
There is nothing that prevents you from going "back in time" in a problem since most equations (but not all) preserve time-reversal symmetry, so mathematically time is no different from any other coordinate.

And. as far as I know no one has been able to prove that time travel is impossible.

 

[rjain]

The "arrow of time" is defined by the second law of thermo. If you're interested in disproving that law, please show us how you will do it.

 

[f95toli]

I am not planning to

What I am saying is that the word "dimension" does not have to refer to only spatial dimensions, from a mathematical point of view time is also a perfectly good dimension, and so is momentum as long as you do not have TRSB.

My comment on time travel was that no one has been able to show that time travel is impossible. There are ways for the laws of thermodynamic to hold even if time-travel is allowed.
The possibility of time travel is an active area of research. There was a nice overview of the field in Scientific Amercan a few months ago, there was also a short comment in Science discussing a paper the recently appeared in Physical Review Letters that showed that "white holes", or Einstein-Rosen bridges as they are called, might not be impossible.

 

[rjain]

f95toli: shush. I wasn't talking to you.

Personally, I like Penrose's idea that black holes eat up entropy and quantum decoherence creates it, keeping entropy conserved overall.

 

[bwanaaa]

Well, i did some googling, the arrow of time is only proven by the asymmetric radioactive decay of the cobalt atom. Although entropy is constantly increasing in the aggregate, it is possible to decrease entropy with suitable investment of energy.

The reason that time is not a dimension is because you cannot measure the time frame of an object independent of other objects. At any given moment of time, all objects are 'doing something'. If time were a dimension, you should be able to 'age' an object independent of all other objects. Can you make time stand still?

This reminds me of the controversy surrounding Harrison's chronometer in the measurement of longitude. Having an accurate measure of time was key. Some people thought you could only measure time accurately using God's instruments-the rotation of the the 4 visible moons of Jupiter. That is was divinely ordained that circumnavigating the globe required the use of 'Godly' help. It was Harrison who figured out a way to keep a pendulum clock accurate on the high seas by using counterbalances. The analogy to the the current argument I am drawing is this---I say we cannot do time travel. I further say that the impossibility of time travel is evidence that time is not a dimension. But what if some lucky stiff figures out a way to do time travel? the 'time' variable then behave as a 'dimension'.

 

[rjain]

Investing energy to decrease energy locally requires entropy outside the system to increase by more than the entropy decrease in the system. This is basically how the second law is applied in real life situations.

The fact that we are moving through time doesn't make it any less of a dimension. We still have a position in that dimension and changing that position changes our state. If you are tied to your bed, that does not make your latitude, longitude, and altitude any less dimensions. If you'd like to dispute this matter, please show how time is not linearly independent from the dimensions of space and how it cannot be used to locate a point in space-time... Whether you think it's similar to space is irrelevant, as we already agree that it is not a dimension of space.

 

[f95toli]

Bwanaa: But you are missing the point. From a physical and a mathematical point of view the word "dimension" simply refers to the basis of a linear space (here I mean a mathematical "space").
If you need at least 4 numbers to define the postion in a space then that space has four dimensions by definition.
Phase space which is used in statistical physics has 6 dimensions, you could of course argue that 3 of those are not "real" since they are simply the momentum but that does not matter, you need 6 numbers to define a position in phase space and therefore it is 6-dimensional. The word "dimension" is not limited to spatial dimensions.

In GR universe is 4-dimensional, and in string theory there are at least 11 spatial dimensions+time .

 

[RossGr]

a dimension is a measurable distance.

This is not a complete definition of dimension. Read my posts about the mathematical meaning of the word, which has be repeated by f95toli and rjain. Dimension is simply not restricted to distance. That is specific application of the word, it is not general. Your logical fallacy is arguing from the specific to the general.