# Metric Nomenclature: Lorentz & Minkowski

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• kent davidge
In summary, the terms "Lorentz metric" and "Minkowski metric" are often used interchangeably, but can refer to different things depending on the context. The Lorentz metric is a specific form with a signature of -1,1,1,1 or +1,-1,-1,-1, while the Minkowski metric can refer to the geometry independently of coordinates or to the specific line element in Cartesian coordinates. However, it may also be applied to other coordinate systems, such as cylindrical or spherical. The term "Lorentzian" is more commonly used to refer to a manifold with a pseudo-metric of signature (1,3) or (3,1), while "Minkowski space
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?

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.

Martin Scholtz
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."

## 1. What is the significance of Lorentz & Minkowski in metric nomenclature?

Lorentz and Minkowski are two important figures in the development of metric nomenclature. They both made significant contributions to the understanding of space and time, which led to the development of the metric system and the concept of spacetime.

## 2. How does the Lorentz transformation relate to metric nomenclature?

The Lorentz transformation is a mathematical formula that describes how measurements of space and time change between two observers in different frames of reference. This is an important concept in metric nomenclature, as it allows for the measurement of spacetime intervals and the understanding of how they are affected by motion.

## 3. What is the difference between Lorentzian and Minkowskian metrics?

Lorentzian and Minkowskian metrics are two types of metrics used in relativity. The Lorentzian metric is used in special relativity, which describes the behavior of objects in uniform motion. The Minkowskian metric is used in general relativity, which describes the behavior of objects in non-uniform motion and in the presence of gravitational fields.

## 4. How does the Minkowski diagram represent metric nomenclature?

The Minkowski diagram is a graphical representation of the Minkowskian metric. It is a two-dimensional diagram that shows the relationship between space and time in special relativity. This diagram is useful for visualizing concepts such as time dilation and length contraction, which are important in metric nomenclature.

## 5. What is the role of metric tensors in Lorentz & Minkowski?

Metric tensors are mathematical objects that describe the relationship between coordinates in different frames of reference. In Lorentz & Minkowski, metric tensors are used to calculate spacetime intervals and to determine the curvature of spacetime in general relativity. They play a crucial role in understanding the geometry of spacetime and are essential in metric nomenclature.

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