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From: JJ on 20 Mar 2010 16:17
Hi all,

here is a question I wanted to ask about, regarding this concept of
the Universe being 4-dimensional.

The question is, how is this 4-dimensionality reflected in mathematic
equations used to solve physics problems?

For example, my understanding is (correct me if I am wrong) that
before "time" was added as one of the dimensions, some of the
calculations were done with just 3 dimensions and adding the dimension
of time helped in solving some of the calculations. Are the dimensions
in reflected in the equations as exponents of some kind?

Thanks in advance,

From: BURT on 20 Mar 2010 16:26
On Mar 20, 1:17 pm, JJ <sa...(a)temporaryinbox.com> wrote:
> Hi all,
>
> here is a question I wanted to ask about, regarding this concept of
> the Universe being 4-dimensional.
>
> The question is, how is this 4-dimensionality reflected in mathematic
> equations used to solve physics problems?
>
> For example, my understanding is (correct me if I am wrong) that
> before "time" was added as one of the dimensions, some of the
> calculations were done with just 3 dimensions and adding the dimension
> of time helped in solving some of the calculations. Are the dimensions
> in reflected in the equations as exponents of some kind?
>
> Thanks in advance,

The higher dimension is driving the expansion of the universe. But it
is a 4th dimension's space that we cannot place inside or outside the
univertse. But it is the interior oif the hypersphere model. The
universe is in the closed surface of a 4 dimensional round geometry.

Mitch Raemsch
From: Androcles on 20 Mar 2010 16:31

"JJ" <santa(a)temporaryinbox.com> wrote in message
news:19df96bb-07b8-4b5e-b277-6c7f5b006cf9(a)q21g2000yqm.googlegroups.com...
> Hi all,
>
> here is a question I wanted to ask about, regarding this concept of
> the Universe being 4-dimensional.
>
> The question is, how is this 4-dimensionality reflected in mathematic
> equations used to solve physics problems?
>
> For example, my understanding is (correct me if I am wrong) that
> before "time" was added as one of the dimensions, some of the
> calculations were done with just 3 dimensions and adding the dimension
> of time helped in solving some of the calculations. Are the dimensions
> in reflected in the equations as exponents of some kind?
>
> Thanks in advance,

Time is not a vector, it has no additive inverse.
http://mathworld.wolfram.com/VectorSpace.html

No physics problems are solved with sci-fi by failed mathematicians.
'Really, this is what is meant by the Fourth Dimension, though some people
who talk about the Fourth Dimension do not know they mean it. It is only
another way of looking at Time. There is no difference between Time and any
of the three dimensions of Space except that our consciousness moves along
with it.' -- Herbert George Wells - "The Time Machine" - 1895.

You're welcome in advance,

Androcles.





>

From: Darwin123 on 20 Mar 2010 18:01
On Mar 20, 4:17 pm, JJ <sa...(a)temporaryinbox.com> wrote:
> Hi all,
>
> here is a question I wanted to ask about, regarding this concept of
> the Universe being 4-dimensional.
>
> The question is, how is this 4-dimensionality reflected in mathematic
> equations used to solve physics problems?
I think the answer is in terms of matrix theory. A transformation
can be considered the change associated with a change in the observer.
If the change in a parameter is expressible in terms of a unitary
transformation, then the parameter can be called a dimension. Assuming
that the change in observer causes a unitary transformation simplifies
the problem a great deal, no matter what field of physics the problem
is in.
The time evolution operator is expressible as a unitary
transformation. The concept of dimension is generally used for
parameter that can be transformed as a tensor. This means that the
linear transformations are in some ways analogous to a rotation. This
means that discussion is usually confined to transformations that
preserve some quantity analogous to length. Thus, to be called a
dimension you need the linear transformations to be expressed as
unitary matrices.
Example: In 3-D Euclidean space, points are expressed as
vectors that can undergo a rotation. A rotation is defined as a linear
transformation that doesn't change the length of the vector. Thus, the
three geometric parameters are called dimensions because there is a
physically meaningful transformation (the rotation) that preserves
length.
Counter example: Temperature is usually not called a dimension
since there it does not transform with rotation, or with anything like
a rotation. There is no physically meaningful transformation that
includes temperature that can be expressed in terms of a unitary
matrix. Thus, you generally DON'T find temperature expressed as a
dimension. There is no "length" associated with temperature. Thus, the
concept of rotation has no meaning.
Gibbs managed to express some thermodynamics problems in terms of
geometry. So, I believe there are special cases where temperature is
called a dimension. However, these are exceptions that prove the rule.
Temperature is seldom called a dimension because there is seldom a
unitary transformation that shows how a change in observer results in
a change in temperature.
I think this concept extends through Galilean invariant physics
(Newtons laws), Lorentz invariant physics (special relativity),
covariant physics (general relativity) and quantum mechanics. A
parameter is called a dimension if it transforms as a tensor with a
unitary matrix.
In special relativity, the use of time as a "dimension" is tied
with the proper time being invariant to the Lorentz transformation.
The Lorentz transformation can be expressed in terms of a unitary
matrix. The proper time in SR is analogous to length in the Euclidean
case.
So if a parameter can be included in a unitary transformation,
then could be called a dimension. There has to be a length that is
unchanged. Time is a dimension because observers in different places
and different times can connect their observations through a unitary
matrix (i.e., the Lorentz transformation).
From: Sue... on 20 Mar 2010 18:39
On Mar 20, 4:17 pm, JJ <sa...(a)temporaryinbox.com> wrote:
> Hi all,
>
> here is a question I wanted to ask about, regarding this concept of
> the Universe being 4-dimensional.
>
> The question is, how is this 4-dimensionality reflected in mathematic
> equations used to solve physics problems?
>
> For example, my understanding is (correct me if I am wrong) that
> before "time" was added as one of the dimensions, some of the
> calculations were done with just 3 dimensions and adding the dimension
> of time helped in solving some of the calculations. Are the dimensions
> in reflected in the equations as exponents of some kind?
>
============

> Thanks in advance,

Does that violate causality ? :-)

<< the four-dimensional space-time continuum of the
theory of relativity, in its most essential formal
properties, shows a pronounced relationship to the
three-dimensional continuum of Euclidean geometrical space.
In order to give due prominence to this relationship,
however, we must replace the usual time co-ordinate t by
an imaginary magnitude

sqrt(-1)

ct proportional to it. Under these conditions, the
natural laws satisfying the demands of the (special)
theory of relativity assume mathematical forms, in which
the time co-ordinate plays exactly the same rôle as
the three space co-ordinates. >>
http://www.bartleby.com/173/17.html

<< if you know about complex numbers you will notice that
the space part enters as if it were imaginary

R2 = (ct)2 + (ix)2 + (iy)2 + (iz)2 = (ct)2 + (ir)2

where i^2 = -1 as usual. This turns out to be the essence
of the fabric (or metric) of spacetime geometry - that
space enters in with the imaginary factor i relative to
time. >>
http://www.aoc.nrao.edu/~smyers/courses/astro12/speedoflight.html

http://en.wikipedia.org/wiki/Complex_number

"Space-time"
http://farside.ph.utexas.edu/teaching/em/lectures/node113.html

Sue...


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