Random musings about mathematics

[Write4U]

No, we don't need fractals for GR or QM. The OP is about how space time can bend and that is described by GR which is Linear algebra, scalars, vectors, tensors, differential geometry and all the tricky notation that goes with it.

I have done some more thinking about this as the question of spacetime bending a warping is truly interesting. How and why does it do that?

We have discussed how it does that and from most accounts GR seems a satisfactory description.

But the "why" is still the question and that begs the question why there is "self-organization" of patterns at all.. Most philosophies , including religions assign a form of (quasi)-intelligence to the process that is scientifically still unexplained except as "fractality".

I ran across this interesting paper.

What Is Self-Organization?

Technological systems become organized by commands from outside, as when human intentions lead to the building of structures or machines. But many natural systems become structured by their own internal processes: these are the self-organizing systems, and the emergence of order within them is a complex phenomenon that intrigues scientists from all disciplines. —F. E. Yates et al., Self-Organizing Systems: The Emergence of Order

To express as clearly as possible what we mean by self-organization in the context of pattern formation in biological systems, we provide the following
8–WHAT IS SELF-ORGANIZATION? definition:
Self-organization is a process in which pattern at the global level of a system emerges solely from numerous interactions among the lower-level components of the system.

Moreover, the rules specifying interactions among the system’s components are executed using only local information, without reference to the global pattern. In short, the pattern is an emergent property of the system, rather than a property imposed on the system by an external ordering influence. Emergent properties will be defined in later chapters, but for now suffice to say that emergent properties are features of a system that arise unexpectedly from interactions among the system’s components.

An emergent property cannot be understood simply by examining in isolation the properties of the system’s components, but requires a consideration of the interactions among the system’s components. It is important to point out that system components do not necessarily have to interact directly. As described in Chapter 2, and Figure 2.4, individuals may interact indirectly if the behavior of one individual modifies the environmentand thus affects the behavior of other individuals.

more..... chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://assets.press.princeton.edu/chapters/s7104.pdf

IMHO, this is a fundamental question that, if answered, might reveal something new about the universe and spacetime.

What could the answer to this question reveal?

 

[Write4U]

And to answer a prior question;

Fractal curves in nature​

Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and lightning bolts.[3][4][5][6]
See also Romanesco broccoli, dendrite crystal, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality.

Dimensions of a fractal curve
Most of us are used to mathematical curves having dimension one, but as a general rule, fractal curves have different dimensions,[7] also see fractal dimension and list of fractals by Hausdorff dimension.

Fractal curve - Wikipedia

en.wikipedia.org

And a secondary reference:

Wayback Machine

 

[exchemist]

And to answer a prior question;

Fractal curve - Wikipedia

en.wikipedia.org

And a secondary reference:

Wayback Machine

Reported for attempted hijacking.

 

[Write4U]

Reported for attempted hijacking.

What the hell are you talking about?
Hijacking exactly what? GR?? I don't see the term GR in the Title. I do see the terms: bent, warped, contorted.

But all I see is the term General Relativity being cited as if that is the subject of the thread. Clearly, someone is ATTEMPTING TO HIJACK THIS THREAD AND IT ISN'T ME!

What is MISSING in the title is the term "curved", else the OP question is moot and useless for all intents and purposes. "Curved" is the only real scientific term used to indicate the bending, warping, or contorting of spatial geometry.

Curved space​

Curved space often refers to a spatial geometry which is not "flat", where a flat space has zero curvature, as described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry, though some simple cases can be described in other ways.

Curved space - Wikipedia

en.wikipedia.org

Reported for attempted hijacking.

 

[exchemist]

What the hell are you talking about?
Hijacking exactly what? GR?? I don't see the term GR in the Title. I do see the terms: bent, warped, contorted.

But all I see is the term General Relativity being cited as if that is the subject of the thread. Clearly, someone is ATTEMPTING TO HIJACK THIS THREAD AND IT ISN'T ME!

What is MISSING in the title is the term "curved", else the OP question is moot and useless for all intents and purposes. "Curved" is the only real scientific term used to indicate the bending, warping, or contorting of spatial geometry.


Reported for attempted hijacking.

This thread is not about fractals. It is about whether space can be "bent, warped, contorted", in other words
"distorted" from [otherwise presumed, ideal] straightness by being made to curve. That is implicitly about GR and related ideas.

We are not going to discuss fractals, just because it happens to have become your latest obsession.

 

[Write4U]

We are not going to discuss fractals, just because it happens to have become your latest obsession.

We are discussing space curvature and its causal implications, only as you want to address it, to the exclusion of all else.
We are not discussing GR either because it is your obsession. You can't have it both ways.

 

[exchemist]

We are discussing space curvature and its causal implications, as you want to address it, to the exclusion of all else.
We are not discussing GR either because it is your obsession. You can't have it both ways.

Really? GR is no obsession of mine. A brief glance at my posting history will show that I rarely make posts on GR, as I did not study it at university, not having learnt the maths of tensors, and have only the barest outline understanding of it. (SR I do sometimes post about, as I do think I understand how that works, but that is not relevant to this thread.)

 

[Write4U]

I do think I understand how that works, but that is not relevant to this thread.)

And you do understand fractality? I just presented evidence that many scientists think it is relevant to this thread.

That is implicitly about GR and related ideas.

Fractals are related ideas!

 

[exchemist]

And you do understand fractality? I just presented evidence that many scientists think it is relevant to this thread.

Fractals are related ideas!

There is fractal cosmology, about which you have already posted. About a month ago. That does not give you a licence to post any old unrelated crap about this particular branch of mathematics,

 

[Write4U]

There is fractal cosmology, about which you have already posted. About a month ago. That does not give you a licence to post any old unrelated crap about this particular branch of mathematics,

And you are the decider? There is Relativity about which everybody has already posted ad nauseam. But apparently, that gives you license to post this old "related" crap.

The point is that fractals are not old, they are new! And they are everywhere, just like general relativity, which has also become a branch of mathematics, surprise.

MATHEMATICAL GENERAL RELATIVITY: A SAMPLER

Mathematical general relativity is, by now, a well-established vibrant branch of mathematics. It ties fundamental problems of gravitational physics with beautiful questions in mathematics. The object is the study of manifolds equipped with a Lorentzian metric satisfying the Einstein field equations.

https://www.math.miami.edu/~galloway/papers/cgpbams.pdf

p.s. Define a "smooth universe" when we are speaking of a "chaotic curved" universe.

Shape of the universe: could it be curved, not flat?​

13 November, 2019

Now our new paper, published in Nature Astronomy, has come to a conclusion that may unleash a crisis in cosmology – if confirmed. We show that the shape of the universe may actually be curved rather than flat, as previously thought – with a probability larger than 99%. In a curved universe, no matter which direction you travel in, you will end up at the starting point – just like on a sphere. Though the universe has four dimensions, including time.

This looks smooth to you?

So while an anomaly in particle physics can be regarded as a hint that we may need to invent completely new physics, an anomaly in cosmology should be regarded as the only way we have to shed light on completely unknown physics.

Therefore, the most interesting result of our paper is not that the universe appears to be curved rather than flat, but the fact that it may force us to rearrange the pieces of the cosmic puzzle in a completely different way.

Dr Eleonora Di Valentino, Postdoctoral Researcher of Astrophysics, University of Manchester. This article is republished from The Conversation under a Creative Commons license. Read the original article.

Shape of the universe: could it be curved, not flat?

No matter how elegant your theory is, experimental data will have the last word. Observations of the retrograde motion of the planets were fundamental to the Copernican revolution, in which the sun replaced Earth at the centre of the solar system. And the unusual orbit of Mercury provided a...

www.manchester.ac.uk

 

[exchemist]

And you are the decider? There is Relativity about which everybody has already posted ad nauseam. But apparently, that gives you license to post this old "related" crap.

The point is that fractals are not old, they are new! And they are everywhere, just like general relativity, which has also become a branch of mathematics, surprise.

...[snip].......

I decline to have any further discussion with you about fractals on this thread. I've reported you again, and that's it from me.

 

[DaveC426913]

This looks smooth to you?

Yes.

That is map of a faint deviations in the CMBR that can only be detected by sensitive instruments, and those lumps are millions of light years across.

Imagine using a transmission electron microscope to detect the individual atoms on the surface of a billiard ball, and making a map of that, showing "angstroms of height over centimetres of surface area". Then add another dozen orders of magnitude onto it.

Yes, the CMBR is smooth.

 

[billvon]

We are not going to discuss fractals, just because it happens to have become your latest obsession.

Look at it this way. At least it's not microtubules.

 

[Write4U]

Yes, the CMBR is smooth.

And CMBR is all there is?

That is map of a faint deviations in the CMBR that can only be detected by sensitive instruments, and those lumps are millions of light years across.

are you saying that CMBR is fundamental spacetime or that it is a property of fundamental spacetime. There is nothing deeper underlying CMBR?

Look at it this way. At least it's not microtubules.

 

[exchemist]

And CMBR is all there is?

are you saying that CMBR is fundamental spacetime or that it is a property of fundamental spacetime. There is nothing deeper underlying CMBR?

Of course he is saying no such thing. You asked if it was smooth. And he says yes.

That's all.

Only an idiot (like you, perhaps?) might think radiation was the same thing as spacetime or that it was a property of it.

 

[Write4U]

Of course he is saying no such thing. You asked if it was smooth. And he says yes.

Well, I see granularity. But then, if CMBR is not representative of space, let's remove it and see what we do see as a more fundamental smoothness?

With the assumption of an original singularity (a granule) the only thing that is more fundamental and has no granularity, is absolute nothingness, no?

Is there a difference between a smooth surface and a smooth function?

A smooth function can refer to a function that is infinitely differentiable. More generally, it refers to a function having continuous derivatives of up to a certain order specified in the text. This is a much stronger condition than a continuous function which may not even be once differentiable. Aug 20, 2013

In Mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of the function define the rate of change of a function at a point. It is mainly used in fields such as physics, engineering, biology and so on. The primary purpose of the differential equation is the study of solutions that satisfy the equations and the properties of the solutions. Learn how to solve differential equations here.

https://byjus.com/maths/differential-equation/#

How does any dynamic action originate? Does it require a causal granularity, a differentiable?

 

[James R]

And you are the decider?

I'm the decider.

You're on thin ice, Write4U. Watch your step.

Reminder: this thread is about the contention that it is a fallacy that space can be bent, warped or contorted.

It is not about fractals. It is not about microwave background radiation. It is not about a wikipedia definition of what a differential equation is.

If you have nothing to say about the topic, don't post in this thread.

 

[Write4U]

Reminder: this thread is about the contention that it is a fallacy that space can be bent, warped or contorted.

And what does that mean?. Only arguments that space is not warped are allowed or can there be discussion what is causal to the warping of space to begin with?

As layman I wanted to establish what causes curvature to begin with. Apparently there has been great debate about this subject and is not something that can be posited without some background about what vectors and tensors are to engage in any discussion of falsification.

The Curious History of Vectors and Tensors​

September 03, 2024
By Ernest Davis

The backstory to tensor theory involves Carl Friedrich Gauss and Bernhard Riemann's creation of the theory of the intrinsic curvature of manifolds, as well as Augustin-Louis Cauchy's study of stress in continuum mechanics. Sep 3, 2024


tensors and vectors are mathematical objects

In the history of science and mathematics, a concept that currently seems straightforward and unproblematic often evolved in a way that appears weirdly indirect and convoluted in hindsight. Sometimes such ideas were met with a hostility that now seems incomprehensibly wrongheaded and pointless. One noteworthy example is the emergence of the concept of vectors in the 19th century, which is the subject of the first half of Robyn Arianrhod’s new book, Vector: A Surprising Story of Space, Time, and Mathematical Transformation.

The second half of Vector details the development of tensor calculus and its central role in Albert Einstein’s general theory of relativity. This historical thread is comparatively straightforward. The backstory to tensor theory involves Carl Friedrich Gauss and Bernhard Riemann’s creation of the theory of the intrinsic curvature of manifolds, as well as Augustin-Louis Cauchy’s study of stress in continuum mechanics. Gregorio Ricci-Curbastro—who was primarily responsible for the actual theory of tensors—worked with Tullio Levi-Civita to compose a classic exposition of the theory in their 1900 textbook, Méthodes de Calcul Différentiel Absolu et Leurs Applications.

In 1912, Einstein was looking for an existing mathematical theory to use for his own theory of spacetime and gravity that would accommodate his intuitions. His friend Marcel Grossmann searched the mathematical literature and told Einstein that Riemannian geometry and the theory of tensors might have what he needed. After years of intense study and labor, Einstein finally found success; in 1915, he published his theory of general relativity in the language of tensor analysis.

more.... https://www.siam.org/publications/siam-news/articles/the-curious-history-of-vectors-and-tensors/#

 

[James R]

And what does that mean?

It means that posts in this thread should be relevant to arguing for or against the contention that space can be bent, warped or contorted. It's right there in the thread title.

Fractals have no relevance to that question, as far as I am aware. Certainly, you have not said anything of relevance to the thread topic regarding fractals.

The same goes for your posts about microwave background radiation.

Differential equations are relevant in the mathematical description of curved spacetime, but your only post about them in this thread made no link between the equations and the application to the thread topic.

As layman I wanted to establish what causes curvature to begin with. Apparently there has been great debate about this subject ...

Not in the scientific community. The theoretical formulation of the general theory of relativity sets out precisely what causes curvature. The causes are right there in the stress-energy tensor of the theory, for all to see.

What "debate" are you aware of regarding the causes of curvature?

... and is not something that can be posited without some background about what vectors and tensors are to engage in any discussion of falsification.

It can certainly be "posited". But one can't talk with any authority about the general theory of relativity without having some understanding of the mathematics of the theory. Understanding the theory is not the same as just memorising some of the results predicted by it, by the way.

The vast majority of human beings lack the necessary mathematical training to be able to understand the formulation of the general theory of relativity. That is not a condemnation of the vast majority of human beings; it is just a statement of fact. There's no problem with not knowing. Problems more often arise when people who don't know pretend to know.

Einstein and his theories are, for some reason, a compelling magnet for cranks who can't do the math. Instead, they like to think that what they lack in mathematical knowledge and/or ability, they more than make up for with their boundless imaginations. That's delusional.

The Curious History of Vectors and Tensors
September 03, 2024
By Ernest Davis

This reference appears to support the contention that space can be bent, warped and "contorted".

What you have quoted from it is historically interesting, but it does little in terms of discussing the thread topic.

 

[Write4U]

This reference appears to support the contention that space can be bent, warped and "contorted".

What you have quoted from it is historically interesting, but it does little in terms of discussing the thread topic.

So all that ad hominem was used to affirm my position and rudimental understanding of what "bending" of space and its underlying "causality" means?
In that case I thank you.

I also find a certain irony in the fact that I always need to defend the concept of a mathematically ordered universe, which you constantly try to falsify.

Please note that by definition tensors are mathematical objects that are causal to the bending (warping) of space. By that logic, it is you who seems to have a limited understanding of the underlying "implicate order", as in Bohmian Mechanics.

I find little synopses like the following as interesting and pertinent to deeper understanding of the nature of spacetime and how Bohm's reality of the "explicate order" becomes manifest.

The Implicate Order in Bohmian mechanics is relevant to the definition of tensors in spacetime1. It is an ontological concept that describes a framework where everything is enfolded into everything, in contrast to the Explicate Order where things are unfolded2. Bohm's theory of the Implicate Order connects everything with everything else, allowing for the replication of past forms and the notion of temporal process345.

I call that discussion of a topic, rather than quoting textbook rote as is so desperately clung to in these "discussions", which does nothing to further support the contention that space can be bent, warped and "contorted" and WHY!