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From: eric gisse on 5 Dec 2008 16:22
On 5 Dec 2008 12:38:12 -0800, stevendaryl3016(a)yahoo.com (Daryl
McCullough) wrote:

>eric gisse says...
>
>>As I recall, the only consistent way to define Newtonian space-time is
>>on constant time slices. Hard to have temporal ordering when time
>>never changes.
>
>No, that's not true. I think what you might be thinking is that
>there is no *metric* except for events that are on the same
>time-slice. But you can have a perfectly good manifold without
>having a metric.

How does one define distances between events without a metric that
describes both eents?
From: Daryl McCullough on 5 Dec 2008 18:06
eric gisse says...
>
>On 5 Dec 2008 12:38:12 -0800, stevendaryl3016(a)yahoo.com (Daryl
>McCullough) wrote:
>
>>eric gisse says...
>>
>>>As I recall, the only consistent way to define Newtonian space-time is
>>>on constant time slices. Hard to have temporal ordering when time
>>>never changes.
>>
>>No, that's not true. I think what you might be thinking is that
>>there is no *metric* except for events that are on the same
>>time-slice. But you can have a perfectly good manifold without
>>having a metric.
>
>How does one define distances between events without a metric that
>describes both eents?

One doesn't.

Without a metric, there is no notion of distance between events. But
you can do a lot of physics without a metric. For instance, Newtonian
physics (including Newtonian gravity) can be formulated in a
coordinate-independent way as a theory on a 4-D manifold without
a metric.

What you *do* need is a notion of parallel transport (which
allows you to do covariant derivatives). In a metric theory
such as GR, parallel transport can be defined in terms of the
metric. In a theory without a metric, it has to be introduced
as a primitive.

--
Daryl McCullough
Ithaca, NY

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