Re: Epistemology 201: The Science of Science

From: Daryl McCullough (stevendaryl3016_at_yahoo.com)
Date: 03/03/05 

Date: 3 Mar 2005 12:35:46 -0800

Lester Zick says...

>I've read many over the years on SR and GR. None deal with the actual
>mechanics. They're just descriptive.

That's all that science *ever* is. Science consists of coming up
with models (or descriptions) for the way the world works. The
model is not the world any more than a map of a city is the city.
But hopefully it is close enough that we can use it for guidance,
for prediction.

>I'm simply referring to our location is space relative to the origin
>of BB or whatever temporal origin we imagine there is. If these are
>concentric or radially opposed we have an isotropic isomorphic
>relation between the two metrics, temporal and spatial.

I don't know what you mean by "two metrics"? As I said, the word "metric"
has a very specific meaning in special and general relativity, but that
notion of metric is a *spacetime* metric.

>But this requires the special assumption of a privileged place for
>us in the universe.

I don't know why you think that. As I explained to Albert, one way
to understand this is to look at a simpler model with only one spatial
dimension and one temporal dimension. Think of the surface of the
Earth as 2D spacetime. Longitude (East-West) represents spatial location,
and lattitude (North-South) represents temporal location. The beginning
of time (the Big Bang) is the South Pole, and the end of time (the Big
Crunch) is at the North Pole. At the beginning of time (the South Pole)
there is only one spatial point, and all the matter that exists at that
time is in one point. Then at later times (that is, farther north), space
expands till it reaches its maximum expansion at the equator. Then it
starts contracting again, until all of space is contracted at a single
point again at the North Pole.

Asking where (spatially) the Big Bang took place is like asking what
is the longitude of the South Pole. The South Pole doesn't have a longitude,
or it has all possible values of longitude, because the South Pole is a
single point.

>Let me see if I can explain this anisotropic isomorphism by means of
>an analogy another poster used. Let's suppose for the sake of argument
>that BB occurs at (0,0) and we are located at (5,5) and we're looking
>in the direction of (10,10) and see the same red shift as at (0,0).

I'm not sure what those coordinates are supposed to mean. In terms
of the simplified model of 1 space dimension and 1 time dimension,
a point in a closed spacetime is characterized by a lattitude (from
90 degrees South to 90 degrees North) and a longitude (from -180
degrees to +180 degrees). Lattitude represents time and longitude
represents spatial location. To say that space is
homogeneous is to say that for any circle of constant lattitude
(representing the universe at a specific time), all points are
the same. Each point on a line of lattitude is as much "at the
center" as any other point. To say that space is isotropic is
to say that looking East, things look exactly the same as when
looking West.

To say that space is expanding is to say that the circumference
at 60 degrees South lattitude is greater than the circumference
at 59 degrees South lattitude.

>Now we understand we are looking back in time as we look in the
>direction of (10,10) but the difficulty is we are looking at (10,10)
>and not (0,0) where time actually began, the temporal metric origin.

You can't "look" in the direction of the past or the future. Any
direction you can turn is pointing in a spatial direction.

>Okay. But there is one part of Euclidean geometry that is accurately
>reflected in Cartesian coordinates and that is the idea that straight
>lines are the shortest distance between points. Non Euclidean
>geometries just ignore this problem by claiming things cannot or
>simply don't travel in straight lines.

That's not quite right. In non-Euclidean geometry, a "line"
is *defined* to be the shortest path between two points,
but the path must be *in* the space. So considering the
surface of the Earth to be a non-Euclidean 2D geometry,
the shortest distance between two points on the surface
of the Earth is a "great circle" connecting those points.
Great circles are the lines for the Earth.

>But that doesn't alter the fact straight lines remain the shortest
>distance between points whatever other assumptions are made.
>Euclidean geometry thus remains the limiting notion implicit in
>every non Eucidean geometry possible in this respect and must be
>accounted for whatever special assumptions are employed in addition.

That's pretty much right. Every non-Euclidean geometry can be
built up out lots of little pieces, each of which is approximately
Euclidean. Any manifold, Euclidean or non-Euclidean, looks Euclidean
if you zoom in on a small enough piece. The surface of the Earth is
a non-Euclidean 2D surface, so there are inevitable distortions if
you try to draw a proportionally correct map of the Earth on a flat
piece of paper. But if you aren't mapping the entire Earth, but just
a single small town, then the non-Euclidean nature of the Earth
becomes unimportant.

>As long as lines in any form are used in any geometry of any order,
>straight lines remain the shortest distance between points in geometry
>claims to the contrary notwithstanding.

Basically, what you are asserting is that space *must* be Euclidean.
But why must it?

>>Sure, in the same way that saying that the northern border of the US
>>is the same as the southern border of Canada doesn't make it so. Geometry
>>is *descriptive* of the universe, and my point is that everything you
>>need to know about the geometry of the universe is conveyed by a
>>complete collection of maps for parts of the universe, together with
>>a specification of which maps are overlapping. There is no need to
>>postulate a higher dimension in order to describe a non-Euclidean
>>universe.
>
>Unless a higher order space is required to accommodate objects
>postulated to explain the effects described.

Why is a higher order space required? It's true that if you want
to embed the non-Euclidean geometry inside Euclidean geometry,
you have to use a higher-dimensional Euclidean geometry to do
the embedding. But why must it be embedded in Euclidean geometry?

>Your explanation for
>contiguous shapes seems to require a higher dimension than the one
>postulated for the shapes themselves.

Only if you want to put together the object inside Euclidean
space.

>>>A Kline bottle is a 3D object.
>>
>>I don't know why you would say that. Its intrinsic geometry is
>>2D---it is a curved surface. A Kline bottle cannot
>>be embedded in Euclidean 3D, so I don't know in what sense you
>>would call it a 3D object.
>
>My bad. I thought it was a kind of ellipsoid with a hole through it.

The hole is only necessary if you try to embed a Kline bottle in
Euclidean 3D space. It's kind of like trying to draw a knotted rope
on a 2D piece of paper. You can't actually draw it without cutting
holes in the rope for it to pass through itself. But in Euclidean
3D you can have a knotted rope that doesn't have any holes. In
Euclidean 4D, you can have a Kline bottle that doesn't require any
holes.

>>Maybe you don't know what a Kline bottle. One way to describe it
>>is to take two Mobius strips, and glue their edges together. It
>>can't be done in Euclidean 3D.
>
>So it's 4D? The mobius strip is 3D I hope?

The Kline bottle is 2D, but it requires 4D if you want to construct
it in Euclidean space. Similarly, a Mobius strip is also 2D, but
it requires 3D if you want to construct it in Euclidean space.

>>Euclidean geometry obeys Euclid's axioms for geometry. In particular:
>>
>> 1. Given two points, there is exactly one line connecting them.
>> 2. Given a line, and a point not on that line, there is a second
>> line that passes through that point that is parallel to the first.
>> 3. The sum of the interior angles of a triangle is 180 degrees.
>>
>>These are not true of non-Euclidean geometry.
>
>Well, Daryl, I think as described above that non Euclidean geometries
>have to honor the idea of a straight line as the shortest distance
>between points as long as they use lines in any form or shapes defined
>in terms of lines of any kind.

A line being the shortest distance between two points is true by
definition. However, the idea that lines that are parallel at
one point are always parallel is not true for non-Euclidean geometries.
For example, on the surface of the Earth, the "lines" are the
great circle routes, and any two intersect, so there are no parallel
lines.

>>>As far as I am concerned Euclidean space is just the three dimensional
>>>manifold,
>>
>>No, it's a three-dimensional manifold with specific connectivity
>>and geometry. The surface of a sphere is a 2D surface, but it
>>isn't a Euclidean 2D surface.
>
>Which makes the surface of a sphere a 3D object even though
>traversible along two mutually orthogonal coordinates.

No, it doesn't. The dimensionality of a space is the number
of independent directions in the space. On the surface of a
sphere, there are two independent directions: East-West, and
North-South. So it's two dimensional. 3D comes into play only
if you want to consider the sphere embedded in a higher-dimensionality
Euclidean space.

In other words, *if* you insist that everything is embedded in
Euclidean space, *then* what you say follows. 

--
Daryl McCullough
Ithaca, NY

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