Re: metric or measure?

Seán O'Leathlóbhair wrote:
> jerry_friedman@xxxxxxxxx wrote:
> > Seán O'Leathlóbhair wrote:
> > > jerry_friedman@xxxxxxxxx wrote:
....

> > > > But the most interesting metric--that of space-time in the theory of
> > > > relativity, which appears to describe the real world--can be negative.
> > > > Maybe mathematicians call that a pseudometric or something.
> > > >
> > > > --
> > > > Jerry Friedman
> > >
> > > I think that this is an example of a technical word having different
> > > meanings in different disciplines.
> >
> > Yep.
>
> This phenomenon is worth remembering. It is common to hear people
> attempting to impose a precise technical meaning on a common word
> without realising that their precise meaning is only one of many and
> not necessarily the most appropriate for the context.
>
> In this case, both your meaning and mine are probably inappropriate
> since the original context was neither serious physics nor serious
> maths.
....

> > > In
> > > theoretical computing, the notion may be laxer and allow zero. An even
> > > looser definition is required to allow your metric.
> >
> > Not mine! Minkowski's, Einstein's, etc.
>
> I only mean the one that you referred to. I did not mean to suggest
> that you devised it. I am familiar with it.

Got it, and I'm glad to hear you and maybe your teachers didn't
consider it *that* uninteresting.

> If my memory is correct, positive distances are "space like" and it
> would be possible to travel from one to the other without exceeding the
> speed of line. Negative distances are "time like" and it is not
> possible to travel between them without exceeding the speed of light.
> So, the generalisation is attractive, the _silly_ negative distances
> represent inaccessible points (events).

That's right, except...

> We are getting very off topic for an English newsgroup.
....

Even so, I have to correct what I got wrong. The thing that can be
negative is the analogue of the distance *squared* (-t^2 + x^2 + y^2 +
z^2). If people really took this seriously as telling you a distance,
the "distance" could be imaginary--but I don't remember anybody ever
taking the square root of that thing. Sorry if I misled anybody.

And I hope I haven't misled anybody into thinking I understand general
relativity.

> > > I am not aware of a standard pure maths term that includes your
> > > function.
> >
> > I tried to look, and it seems that pure math and relativity are on
> > different Riemann sheets (to use one of my graduate advisor's
> > expressions). Both Wikipedia and Mathworld
> > <http://mathworld.wolfram.com/> define the metric as positive definite,
> > but in their articles on Minkowski space and relativity, both blithely
> > use "metric" for the thing that can be negative, as theoretical
> > physicists do. The Wikipedia article is helpful enough to say that you
> > drop the requirement that d(x,y) = 0 iff x = y, you get a pseudometric;
> > if you drop the requirement of symmetry, you get a quasimetric; and if
> > you drop the triangle inequality, you get a semimetric (whereas if you
> > strengthen it in a certain way, you get an ultrametric, which I'd
> > thought was called a hypermetric--and you can combine these, as in
> > "quasi-pseudo-ultrametric").
>
> A nice expression. Pseudo metric sounds vaguely familiar but the rest
> of those terms don't. However, it is a couple of decades since I did
> this stuff seriously.
....

> > > Pure mathematicians are usually uninterested in the actual
> > > values of the metric and are mostly interested in the topology that it
> > > implies. They are also likely to disagree that your "metric" is the
> > > most interesting. Real world things are generally less interesting
> > > than the abstract.
> >
> > To get back off topic, you can't change what people are interested in,
> > but I think it would be nice for math professors and courses to at
> > least mention that a generalization of this idea is central to
> > relativity and hence quite possibly to physical reality. Besides, the
> > metric tensor as a solution to a partial differential equation
> > involving the distribution of mass and energy--that's pretty slick.
>
> Pure mathematicians are weird creatures. I remember often hearing PhD
> students describe their research to each other. Commonly one would
> spend ages explaining to the other what he was doing. A common
> response was: "Now I understand but why do you find that interesting?".
> Failing to find obscure pure maths interesting may not be too hard to
> understand but the funny bit is that the conversation would often be
> repeated the other way around. Despite being interested in one very
> specialised area they could not understand that another equally
> specialised area could also be interesting. For example, I find number
> theory rather dull but many others find it fascinating.
....

Even I like it, the little I know of it. But you're right that I
shouldn't have said the Minkowski metric or the metric tensor of GR is
the most interesting. It's a matter of taste.

--
Jerry Friedman

.

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