I Metric Nomenclature: Lorentz & Minkowski

[kent davidge]

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?

 

[PeterDonis]

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.

 

[pervect]

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.

 

[PeterDonis]

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.

 

[vanhees71]

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.

 

[DEvens]

Thumbs up for "dignature."