# Different metrics in different dimensions

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• Bob3141592
In summary: These two slices are at right angles to each other, and their distance is not defined by a metric (although a Riemannian metric can be induced on each slice). In summary, the conversation discusses the possibility of having different metrics for different subspaces in a space in R_n. The concept of direct products and phase spaces is mentioned, and the question is raised about the purpose of defining such spaces. An example of a four dimensional space-time with different metrics for different subspaces is also given.
Bob3141592
TL;DR Summary
Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?
I'm trying to get a handle on how general a space in R_n can be. Part of my motivation is the curled up dimensions physicists talk about. How does one dimension work differently than another dimension? Can one part of the dimensional structure follow one metric and another part follow a different metric?

I rather think it should be possible. That raises questions about the combinations of subspaces. Can R_1 X R_2 be different (say, taxicab geometry) from R_1 X R_3 (say, Euclidean) as long as R_1 X R_3 is consistent (um, somewhere in between maybe)?

Sorry if this is worded poorly, and if it's in an inappropriate folder. And how does one access the proper notation symbols?

Thanks.

Bob3141592 said:
Summary: Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?

I'm trying to get a handle on how general a space in R_n can be. Part of my motivation is the curled up dimensions physicists talk about. How does one dimension work differently than another dimension? Can one part of the dimensional structure follow one metric and another part follow a different metric?
Sure. You can build direct products of different topological spaces.
I rather think it should be possible. That raises questions about the combinations of subspaces. Can R_1 X R_2 be different (say, taxicab geometry) from R_1 X R_3 (say, Euclidean) as long as R_1 X R_3 is consistent (um, somewhere in between maybe)?

Sorry if this is worded poorly, and if it's in an inappropriate folder. And how does one access the proper notation symbols?

Thanks.
Phase spaces are considered In stochastic and physics which cover all possible states, i.e. their description. This leads to different dimensions in the components and thus different units and scales.

The actual question is not whether it can be defined rather what should it be good for, i.e. what do you want to do?

Bob3141592 said:
Summary: Given a space in R_n = R_1 X R_2 X R_3 X R_4 ... can the metric for the R_1 x R_2 subspace be different from the metric for the R_3 X R_4 subspace?
One example would be the four dimensional space-time we live in. You can pick out a two dimensional space-like slice using x and y coordinates and you can pick out an orthogonal two dimensional Minkowski slice using z and t coordinates.

## 1. What are metrics in different dimensions?

Metrics in different dimensions refer to the different ways in which data can be measured or evaluated. In science, metrics are used to quantify and compare various characteristics or properties of a system or phenomenon. These metrics can vary in terms of the units of measurement, the scale or range of values, and the dimensions or aspects being measured.

## 2. Why are different metrics used in different dimensions?

Different metrics are used in different dimensions because each dimension may require a specific type of measurement or evaluation. For example, in physics, distance is measured in meters while time is measured in seconds. In biology, the size of a cell may be measured in micrometers while the rate of a chemical reaction may be measured in moles per second. Using the appropriate metric for each dimension allows for more accurate and meaningful comparisons.

## 3. How do different metrics affect data analysis?

Different metrics can greatly affect data analysis as they can alter the interpretation of the data. For instance, using different units of measurement can lead to different numerical values, which can impact the conclusions drawn from the data. Additionally, using the wrong metric for a particular dimension can result in misleading or inaccurate conclusions.

## 4. What are some examples of different metrics in different dimensions?

Examples of different metrics in different dimensions include measuring length in meters, weight in kilograms, temperature in degrees Celsius, time in seconds, and volume in liters. In social sciences, metrics may include variables such as income, education level, or happiness, which can be measured in different units or scales.

## 5. How can scientists ensure the use of appropriate metrics in their research?

Scientists can ensure the use of appropriate metrics in their research by carefully selecting the units of measurement and dimensions to be evaluated. They should also consider the scale or range of values for each metric and ensure that it aligns with the data being collected. Additionally, it is important for scientists to clearly define and communicate the metrics used in their research to ensure the accuracy and reproducibility of their findings.

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