Looking at your properties of limits list, I never understood why they never just say that the function of limits commutes/distributes. It is after all knocked around as if it were an algebraic function.

4D Golf spoiler added for final 5-dimensional levels

Currently the only rigorous notion of 4-dimensionality I feel I understand well is the Linear Algebra notion of a vector space with 4 linearly independent vectors. Everytime I try looking into tensors, I just get confused, so I'd be interested in seeing what you mean by "a metric tensor with a negative eigenvalue." I know what a metric and an eigenvalue is, at least.

After seeing enough of these, visualizing in enough different ways, and thinking about it for too long, 4D space becomes so easy to understand that it's kind of a no-brainer. Not going to lie though, it took me years.

The negative eigenvalue in time has everything to do with entropy in a positive direction, especially in particle interactions. Spacetime is dynamic, which means that it is tending eventually to a point of rest. This played a large role alongside the universe's expansion leading to the assumption of heat death, that time itself would eventually breakdown, no events, even though space would continue to expand forever.

Don't beat me up if I got any of that wrong, by the way. There's been loads of observations in the last few years alone, especially with JWST, that have been telling us that our existing cosmological models are definitely fundamentally flawed and in need of rewriting.

To be honest I wouldn't know if you are right or wrong lol physics is by no means my area of expertise. I was hoping for more of a pure-math explanation. For me 4D or 5D or nD vector spaces already make sense (just add another coordinate), even if I don't know what they look like.

As far as purely mathematical, I don't know of any applications. I was trying to say in the intro while angrily shaking my fist that time has nothing to do with it. There's always one that's like "Time is the fourth!" All of these softwares are pure Euclidian space, because responsible adults made them. Lol

@xpaper - It sounds like you already got the gist of it. Most of this software seems to treat the concept of "four dimensions" as simply an extra coordinate, let's call it w. Changing that w coordinate gives us another space, while changing x, y, or z changes our place within that particular space.

In the context of Physics, a "scalar" is only a magnitude (e.g.: a speed given in miles per hour). A "vector" is a magnitude with direction (e.g.: 60MPH due North). Both are "tensors". A scalar is a "tensor of rank-0", a vector is a "tensor of rank-1", and so on.

More specifically, tensors are objects that can be described as points on a "manifold". A manifold is a space that may have different "local" and "global" properties. A common example is the Earth. Locally, its surface seems like a flat plane, but globally, it is a sphere. This is a concept from Topology.

The concept of tensors is not used as much when it comes to Linear Algebra. A rank-2 tensor can be described by a matrix, but tensors are not synonymous with matrices. In Physics, you might switch from using one branch of math to the other depending on what you are describing though. So, the term "four dimensions" is often referring to either one of two concepts when used in Physics:

1. The use of four coordinates where x, y, z describe where something is in regular Euclidean space, and a coordinate t that describes when it is there. Simple and intuitive. But when dealing with extremes of speed, like in the case of Relativity Theory, we would use "Minkowski spacetime".

2. In Minkowski spacetime, we still have the three axes of Euclidean space, but "time" is considered a fourth axis that objects travel along at the speed of light. This makes the speed of light constant for every observer and everything else is measured relative to that.

Personally, I feel that a lot of seemingly complex math has been thrown at what are essentially simple relationships, so I understand eastathenaeum's displeasure with how the idea of "four dimensions" is often presented.

Please tell me that you teach.

I have varying levels of familiarity with everything mentioned here. However, I feel like I still don't have an answer to the question of how someone would go about determining eigenvalues for the "metric tensor of a dimension" and what those objects are in a mathematically rigorous sense. But when I look into it that's precisely the part that remains mystifying.

It sounds like the confusion lies in assuming that spacetime is an object residing within dimensions as opposed to the dimensions being a part of spacetime itself. Basically the metric tensors are functions that translate how space looks to us with what is actually going on and visa versa, a feature of projection for mapping in differential geometry.

I'm thinking about either changing or removing the wording now, because on all levels, peoples' minds are already made up for whether or not time exists. To me that's kind of a dumb argument, because how do you experience something for yourself and claim it doesn't exist? It's like saying emotions don't exist because they are just biological chemical reactions, when in reality that's an argument for how they exist.

@eastathenaeum - As in teach professionally? You are too kind. No, I just love learning and sharing...While I cannot speak for xpaper, I think they were trying to grasp what "the negative eigenvalue of a metric tensor" means mathematically. In other words, how do we calculate that? Are we treating a metric tensor as a matrix and then finding its eigenvalue?

If I had to take a guess, from your descriptions on the dynamic nature of spacetime, I have a feeling that you are using the inverse of the resultant eigenvector(s) to define "time"...as if spacetime itself is expanding at the speed of light, something like Einstein's cosmological constant...

Don't quote me on that though. I could be entirely wrong. I'm just trying to understand exactly how you are looking at the situation. You obviously have a deep understanding of it.

@xpaper - The concept of spacetime is such an interesting amalgam of Differential Geometry, Topology, and Linear Algebra that it can be a challenge to understand it. Some of the best videos to get a general idea of the Math behind it that I have seen are: https://m.youtube.com/playlist?list=PL__fY7tXwodmfntSAAyBDxZ4_eE3ZwbFE

I usually approach the idea of "dimension" in terms of how many coordinates are needed to specify any location within a given space. So, a "1-dimensional line" only needs one coordinate to specify any point along it, a "2-dimensional plane" needs two coordinates to specify any location within it, and so on.

In the same way that we might think of a "2-dimensional square" as a slice of a "3-dimensonal cube", we might think of a 3D cube as a slice of a "4-dimensional hypercube". People might trip over visualizing such a thing, but again, all we are really thinking about is how many coordinates. When it comes to applying this idea to Physics, well...

To be honest, I think Einstein (and the theoretical physicists who followed him) go a little overboard with the Math sometimes, to the point of obscuring the actual physical phenomenon that they are describing...though I am not as harsh with my assessment of Relativity as Nikola Tesla. 😂

One has to keep in mind that Einstein formulated his idea of spacetime when scientists were arguing over the constitution of the aether. So, I think Einstein was like, "Here guys, let's just use this mathematical model to describe light and gravity for now. We'll figure out the composition of the aether later." Unfortunately, it seems to have led to a lot of confusion.

No, my understanding sucks, but thank you for the vote of confidence. I've only taken physics for engineering & science majors 1 & 2 in college as well as some others completely unrelated to cosmology. I've read about it a lot in Wikipedia over the decades and kept up with hypothetical models trying to make breakthroughs, but that hardly constitutes as studying.

The eigenvalue is determined from a metric tensor matrix, which is the scalar equivalent to the eigenvector, so yes, that is correct. I'm kind of falling down a rabbit hole trying to find pure mathematical context, and everything I'm looking up leads back to physical interpretation. In relativity, negative eigenvalues are pretty much a given rather than a consequence. I never gave it any thought.

I just called a friend over this after numerous carefully worded searches. He has two degrees, one in math, and he told me there is no purely mathematical context, that all it would do is flip the field and that it is never used. Now my brain hurts because this doesn't sound at all like the context laid out in relativity. Lol

P.S. I LOVE reading hate mail. I've wanted to create a hate mail page since the inception of this site, but so far have not received any. Please find something about my site that you hate, and then wait till you're drunk and pissed off to message me. Thanks in advance.

Hi Bill - I like what you did to the main page - looks good. And I just discovered you added bulltown to your neighbors page. I'll soon add you to the 'mutuals' sections of the main page and links page - really appreciate that! -joe

Thank you, it is mutually appreciated. Also, I have to mention that your appearance is iconic.