Can an Alternating Metric Define a New Spacetime Topology?

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Discussion Overview

The discussion centers around the possibility of defining an alternative topology of spacetime based on a "metric" of alternating forms, as well as the relationship between topology and metrics. Participants explore whether a topology can be established without a traditional metric, focusing on the implications of using an alternating structure instead.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants inquire whether an alternative topology can be defined using a "metric" of alternating forms, questioning the necessity of a traditional metric.
  • There is a discussion about the definition of "metric," with some participants distinguishing between the topological-space sense and the metric tensor.
  • One participant asserts that any new metric must be a function of the existing metric, as the metric defines the topology of spacetime completely.
  • Another participant raises the question of whether it is possible to define a topology for purposes other than the metric topology.
  • Concerns are expressed regarding the nature of the metric tensor, with a participant noting that it does not satisfy the properties of a traditional metric (non-negativity and triangle inequality).

Areas of Agreement / Disagreement

Participants express differing views on the definitions and implications of metrics and topologies, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

There are limitations in the discussion regarding the assumptions made about the nature of metrics and topologies, as well as the dependencies on specific definitions that have not been fully clarified.

Phrak
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Can an alternative topology of spacetime be defined upon a "mertic" of alternating forms?

A less stringent question: Can a topology be defined, without first premising a metric, but premising an alternating structure instead?
 
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What do you mean by "alternating forms/structure"?
 
Phrak said:
Can an alternative topology of spacetime be defined upon a "mertic" of alternating forms?

A less stringent question: Can a topology be defined, without first premising a metric, but premising an alternating structure instead?

Yes ... but this new metric must be a function of the metric because the metric define the topology of spacetime completely.
 
Mueiz said:
Yes ... but this new metric must be a function of the metric because the metric define the topology of spacetime completely.

What is the metric?

And, yes a metric will define a topology via it's open balls, the metric topology. But can we define another topology perhaps for other purposes? Can't the space be given another topology other than the metric topology?
 
Phrak said:
Can an alternative topology of spacetime be defined upon a "mertic" of alternating forms?

A less stringent question: Can a topology be defined, without first premising a metric, but premising an alternating structure instead?
When you use the word "metric" do you mean in the topological-space sense or the metric tensor, which are two different concepts?
 
That is a key point, Dr. Greg.
Where does that other metric come from, the one that defines the topology? The metric tenser is not a metric -not non-neg and no triangle inequality.
Also see my 'Flat spatime topology-metric?' thread in math -> topology and geometry sections
 
DrGreg said:
When you use the word "metric" do you mean in the topological-space sense or the metric tensor, which are two different concepts?

I was thinking of the metric tensor. But now that you've brought it up, and I've had a chance to look at the Wiki page on the metric, I think the latter is what I want to know about. Thanks!
 

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