Re: The Absurd Claim of the Metric as a Tensor


Daryl McCullough wrote:
Koobee Wublee says...

The definition is absurd. ds^2 is an invariant scalar. The metric is
a matrix.

You are extremely confused about this subject. ds^2 is a *function*.
Look at the equation for ds^2 in, say, good-old 2D Euclidean space
in Cartesian coordinates:

ds^2 = dx^2 + dy^2

ds^2 depends on dx and dy. If you make dx twice as large, and
you make dy twice as large, then ds^2 will be four times as
large.

The whole point of this equation is to describe a function;
a quadratic function from vectors to reals. ds^2 is not a
fixed number, it is a function of the displacement vector.

When people talk about the metric, they mean the mapping
from vectors to reals, and ds^2 specifies that mapping.

It's a definition. Your definition doesn't model real world correctly
because you think of "metric" as a matrix which means that - for
example - in the 2D plane the following two "metrics" are different:

[1 0]
[0 1] in rectangular coordinates

and

[1 0 ]
[0 r^2] in polar coordinates.

Since these two are the same 2D plane, one cannot sensibly define
"metric" as a "matrix".

These are matrices, are they not?

Yes, but they are *not* metrics. That's the point---a metric
is not a matrix.

[g], the metric or a matrix, is not ds^2 period.

Sure, a tensor is not a matrix.

This matrix [g] is observer dependent.

Yes, and [g] is not the metric.

Let me rephrase what I said. The Minkowski metric does not necessarily
imply spacetime is flat.

Yes, it certainly does.

Coordinate systems OTOH are a means to turn certain pre-existing
physical manifestations into numbers or some other quantification
means.

The matrix [g] is useless without knowing what coordinate system one
chooses.

That's why [g] is not the metric. The metric does not depend
on coordinate systems.

The metric along
can never tell you about the geometry it is describing. You also need
the choice of coordinate system to complete the observation.

Yes but that's different than saying that metric is
coordinate-dependent.

Why not? The metric is [g] which is a matrix.

The metric is *not* [g]! Why do you keep saying that?

But, the metric is not an abstract mathematical quantity. It is the
matrix [g].

No, it's not. The metric is the mapping from pairs of vectors to reals.
[g] is the representation of the metric in a particular coordinate
system.

[g] is also called the metric.

http://mathworld.wolfram.com/EuclideanMetric.html

The POINT is that GRists have developed their own terminology which
within the context of what they mean is correct but it is confusing to
innocent people passing by. They also use the bastardized terminology
to attack people and divert attention from the REAL ISSSUES.

and you do a good job Daryl.

Ask anyone, the metric is [g]. ds^2 is the line element.

Mike









--
Daryl McCullough
Ithaca, NY

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