Area & Volume Naming Conventions: 4 Questions

[Robert Friz]

TL;DR Summary

Area and Volume in Four Dimensions

1. Area is the naming convention assigned to that which is within a closed diagram in the x-y dimensions.
2. Area is also the naming convention used in simplified Lorentzian diagrams in the x-t dimensions.
3. Volume is the naming convention used to that which is within a closed vessel in the x-y-z dimensions.
4. What, if not volume, is the naming convention for that which is within a closed vessel in the t-x-y-z dimensions?

Should not all four of these be designated with a different name?

 

[Mark44]

Summary:: Area and Volume in Four Dimensions

1. Area is the naming convention assigned to that which is within a closed diagram in the x-y dimensions.
2. Area is also the naming convention used in simplified Lorentzian diagrams in the x-t dimensions.
3. Volume is the naming convention used to that which is within a closed vessel in the x-y-z dimensions.
4. What, if not volume, is the naming convention for that which is within a closed vessel in the t-x-y-z dimensions?

Should not all four of these be designated with a different name?

Length, area, and volume are geometric terms that relate to measures of one-dimensional, two-dimensional, and three-dimensional regions, respectively. To my knowledge, there are no terms for measures in regions of four or higher dimensions. Since we are creatures of three dimensions and aren't generally capable of visualizing spaces of higher dimension, there's probably not a need for a term that describes the measure of a higher-dimension region.

 

[SiennaTheGr8]

4-volume, no?

 

[Robert Friz]

I might volunteer "frame". It is sort of in common use for 4-dimension curvature is it not?

 

[HallsofIvy]

More generally, "hyper-volume".

 

[vanhees71]

Length, area, and volume are geometric terms that relate to measures of one-dimensional, two-dimensional, and three-dimensional regions, respectively. To my knowledge, there are no terms for measures in regions of four or higher dimensions. Since we are creatures of three dimensions and aren't generally capable of visualizing spaces of higher dimension, there's probably not a need for a term that describes the measure of a higher-dimension region.

Of course in 4D you have 4-volume, 3-hypersurface, 2-hypersurface (area), 1-hypersurface (line) (though I've never seen anybody a 2D submanifold of a 4D manifold a 2-hypersurface, but it would make sense for a systematic naming scheme).

 

[Robert Friz]

Perhaps:

I just realized that the "area" described by x-t might best be described by "vector". I have to go have a nap and think about Vanhees71's suggestion, which certainly has merit. Interesting discussion -- the nomenclature will be different for physicists and for lay people if we pursue this too far...

 

[Mark44]

I just realized that the "area" described by x-t might best be described by "vector".

I don't think so. If you have a graph in the t-x plane, that would describe the position of an object moving along the x-axis at various times t. The area of the region between the graph and the t-axis would not have any physical meaning. However, if you had a graph of the velocity of the object at various times, the area under that graph represents the total displacement of that object.

A (t, x) coordinate could be thought of as a vector, but a region in the t-x plane represents an area. These are separate concepts.

 

[PeterDonis]

"frame". It is sort of in common use for 4-dimension curvature is it not?

No. "Frame" is short for "reference frame", which is a different thing.