Is the 'distance function' the same as the 'metric' in physics texts?

Thread starter pmb_phy
Start date Mar 16, 2007
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Function Metric

In summary: They do mention the triangle inequality and show how it can be violated, but they don't go into quite as much detail as in MTW.In summary, Schut'z Geometrical methods of mathematical physics and Walds General Relativity use a distance function which is not the same as the metric. Schutz refers to a "distance function" between two points on page 1 and as one exampe this function is the usual Euclidean distance function, and other examples are similar. In Wald he speaks of an open ball in Rn in which the Euclidean function is used as the "open ball" function about a particular point. The purpose of the distance function

Apr 5, 2007

Haelfix

Science Advisor

"Anybody disagree with the above assertion by Ohanian? If so please post your thoughts"

The source of the confusion it seems to me is the observation that locally spacetime is not R^4 in the most naive way. The isometry group of a local neighberhood in spacetime is not SO(4), but rather SO(3,1)! So we require an additional physical map from R^4 --> R^3 * R^1, and this induces a new distance function.

<h2>1. What is the difference between the 'distance function' and the 'metric' in physics texts?</h2><p>The 'distance function' and the 'metric' are related concepts, but they are not exactly the same. The distance function is a mathematical function that calculates the distance between two points in a space. The metric, on the other hand, is a mathematical object that defines the distance between any two points in a space. In physics texts, the metric is often used to describe the geometry of space and time, while the distance function is used to calculate specific distances between points.</p><h2>2. Can the terms 'distance function' and 'metric' be used interchangeably in physics texts?</h2><p>No, the terms 'distance function' and 'metric' cannot be used interchangeably in physics texts. As mentioned before, they refer to different mathematical concepts and serve different purposes. While the metric is used to define the distance between any two points in a space, the distance function is used to calculate specific distances between points.</p><h2>3. How are the 'distance function' and the 'metric' related in physics?</h2><p>The 'distance function' and the 'metric' are related in that the metric is used to define the distance function. The metric provides the mathematical framework for calculating distances between points in a space, while the distance function uses this framework to calculate specific distances.</p><h2>4. Are there any other terms used in physics texts that are similar to the 'distance function' and the 'metric'?</h2><p>Yes, there are other terms used in physics texts that are similar to the 'distance function' and the 'metric'. For example, the term 'distance measure' is often used instead of 'distance function', and the term 'distance metric' is used instead of 'metric'.</p><h2>5. Can the 'distance function' and the 'metric' be applied to any type of space?</h2><p>Yes, the 'distance function' and the 'metric' can be applied to any type of space, including Euclidean spaces, non-Euclidean spaces, and even abstract mathematical spaces. As long as there is a well-defined notion of distance between points, the concepts of distance function and metric can be used to describe it.</p>

1. What is the difference between the 'distance function' and the 'metric' in physics texts?

The 'distance function' and the 'metric' are related concepts, but they are not exactly the same. The distance function is a mathematical function that calculates the distance between two points in a space. The metric, on the other hand, is a mathematical object that defines the distance between any two points in a space. In physics texts, the metric is often used to describe the geometry of space and time, while the distance function is used to calculate specific distances between points.

2. Can the terms 'distance function' and 'metric' be used interchangeably in physics texts?

No, the terms 'distance function' and 'metric' cannot be used interchangeably in physics texts. As mentioned before, they refer to different mathematical concepts and serve different purposes. While the metric is used to define the distance between any two points in a space, the distance function is used to calculate specific distances between points.

3. How are the 'distance function' and the 'metric' related in physics?

The 'distance function' and the 'metric' are related in that the metric is used to define the distance function. The metric provides the mathematical framework for calculating distances between points in a space, while the distance function uses this framework to calculate specific distances.

4. Are there any other terms used in physics texts that are similar to the 'distance function' and the 'metric'?

Yes, there are other terms used in physics texts that are similar to the 'distance function' and the 'metric'. For example, the term 'distance measure' is often used instead of 'distance function', and the term 'distance metric' is used instead of 'metric'.

5. Can the 'distance function' and the 'metric' be applied to any type of space?

Yes, the 'distance function' and the 'metric' can be applied to any type of space, including Euclidean spaces, non-Euclidean spaces, and even abstract mathematical spaces. As long as there is a well-defined notion of distance between points, the concepts of distance function and metric can be used to describe it.