Metric Nomenclature: Lorentz & Minkowski

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Discussion Overview

The discussion revolves around the terminology and definitions related to the Lorentz and Minkowski metrics in the context of spacetime geometry. Participants explore the distinctions and similarities between these metrics, their forms in different coordinate systems, and the implications of their nomenclature within the framework of general relativity.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that the Lorentz metric is a specific form, while the Minkowski metric can represent a broader class of metrics, including those in spherical coordinates.
  • Others argue that the terms "Lorentz metric" and "Minkowski metric" are often used interchangeably, with different interpretations based on coordinate choices.
  • A participant interprets the Minkowski metric specifically as the form ##-c^2 dt^2 + dx^2 + dy^2 + dz^2## but acknowledges that it could apply to other coordinate systems as well.
  • There is a suggestion that a "Lorentz metric" could refer to any metric with a -1,1,1,1 or +1,-1,-1,-1 signature, not necessarily limited to flat metrics.
  • Another participant emphasizes the term "Lorentzian" for metrics with the specified signature, suggesting that "Lorentz" alone may refer to a specific flat metric.
  • One participant critiques the term "metric" as misleading, proposing "pseudo-metric" as a more accurate descriptor for the non-degenerate bilinear form used in relativity.
  • There is a mention of the distinction in signature notation between different regional conventions, referring to the pseudo-metric as having either (1,3) or (3,1) signatures.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and applications of the Lorentz and Minkowski metrics, with no consensus reached on their interchangeability or specific meanings. The discussion remains unresolved regarding the precise terminology and implications of these metrics.

Contextual Notes

Participants note the potential confusion surrounding the term "metric," highlighting its non-positive definite nature and the need for precision in terminology when discussing spacetime models.

kent davidge
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Can I say that the Lorentz metric is the specific form ##-c^2dt^2 + dx^2 + dy^2 + dz^2## whereas the Minkowski metric is the metric of Minkowski space which can take the Lorentz form I just gave, but can also, e.g., be written in spherical coordinates?
 
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AFAIK the terms "Lorentz metric" and "Minkowski metric" are used interchangeably, and there are at least two usages of both terms, one to just refer to the geometry independently of any choice of coordinates, and the other to refer specifically to the line element in Cartesian coordinates.
 
I usually interpret "Minkowskii metric" to be the specific form ##-c^2 dt^2 + dx^2 + dy^2 + dz^2##. I couldn't say, though, that it might not be applied to a cylindrical flat line element like ##-c^2\,dt^2 + dr^2 + r^2\,d\phi^2 + dz^2## or a spherical flat line element. The difference is that in one case, one assumes that it singles out a specific metric, in the other case one assumes it singles out any of a class of equivalent metrics.

I would assume that a "lorentz metric" was any metric with a -1,1,1,1 or a +1,-1,-1,-1 signature, and not even necessarily flat.

But I could be wrong, I don't have a reference to back that up.
 
pervect said:
I would assume that a "lorentz metric" was any metric with a -1,1,1,1 or a +1,-1,-1,-1 signature, and not even necessarily flat.

I think the usual term for this is "Lorentzian", or if one wants more precision, "locally Lorentzian". "Lorentz" without the "ian" seems to me to be a specific reference to the flat metric with this signature.
 
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The term "metric" is highly misleading to begin with. Since it's not a positive definite bilinear form but just a non-deggenerate one, it's a "funcamental form" rather than a metric of relativistic space-time models. Another good term, I like is "pseudo-metric" since formally it behaves in many ways just like a metric.

In GR the space-time model is a pseudo-Riemannian manifold with a pseudo-metric of dignature ##(1,3)## if you are a west-coast guy (or equivalently ##(3,1)## if you are an east-coast guy). This is sometimes also called a Lorentzian manifold.

Minkowski space is the special case of a flag (affine) Lorentzian manifold.
 
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Thumbs up for "dignature."
 

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