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MeJennifer
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What is the hypervolume of a hypercube in a Minkowski space?
I understand that that is the case for a Euclidean space.Hurkyl said:4-Volume = Duration * Length * Width * Height.
To be honest, I really dislike the notion of a tensor density. I much prefer thinking about the differential 4-formrobphy said:Technically speaking, I think the notion of a "tensor density" arises here.
But I think Hurkyl's response is correct.
Just to make sure we're on the same page -- the unit 4-sphere is not the set of all points a unit (Minkowski) distance away from the origin. That object is... well, in Minkowski 2-space it would be a hyperbola. I'm not sure what it's called in Minkowski 4-space.MeJennifer said:How about the volume of a unit 4-sphere and the 4-volume of a unit 4-ball in Minkowski space?
These questions seem so basic, surely I am not the first person who asks such questions.
Anybody who can provide some numbers?
Well Hurkyl you seem to be much better in visualizing what a sphere is in Minkowski space, I already have enough trouble visualizing Euclidean 4-space let alone being able to visualize a sphere in Minkowski space, but whatever you want to call it, that is what I am asking for.Hurkyl said:Just to make sure we're on the same page -- the unit 4-sphere is not the set of all points a unit (Minkowski) distance away from the origin. That object is... well, in Minkowski 2-space it would be a hyperbola. I'm not sure what it's called in Minkowski 4-space.
I think you are right.quasar987 said:I don't think the unit sphere has a finite volume.
The hypervolume is a mathematical concept that measures the space occupied by a multi-dimensional object. It is a generalization of the concept of volume in three dimensions to higher dimensions.
The hypervolume is calculated by multiplying the lengths of all the sides of the multi-dimensional object. For example, in two dimensions (a square), the hypervolume would be calculated by multiplying the length and width. In three dimensions (a cube), the hypervolume would be calculated by multiplying the length, width, and height.
The hypervolume has various applications in science, including in physics, biology, and computer science. It is used to measure the size and shape of complex objects, model and analyze high-dimensional data, and to understand the behavior of systems with multiple variables.
The concept of hypervolume is closely related to the concept of dimensionality. As the number of dimensions increases, the hypervolume of an object also increases. Furthermore, the hypervolume of a multi-dimensional object can be thought of as the amount of space it occupies in that particular dimension.
No, the hypervolume of an object cannot be negative. It is a measure of the space occupied by an object, and space cannot have a negative value. In some cases, the hypervolume of an object can be zero if all the dimensions of the object have a length of zero.