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5 hours ago, Mitcher said:

it was not an obligation for this constant to exist in the first place

The trouble with this is that the Einstein equations aren’t just invented out of thin air.  There are some fundamental principles of consistency and topology that greatly constrain the form these equations can take (See Meisner/Thorne/Wheeler for details). As it turns out, the equations including the constant are the most general form that fulfils all these conditions - so there needs to be a reason why the constant should be exactly zero.

Im not saying it can’t be zero, just that there would have to be a reason for it.

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19 minutes ago, Markus Hanke said:

The trouble with this is that the Einstein equations aren’t just invented out of thin air.  There are some fundamental principles of consistency and topology that greatly constrain the form these equations can take (See Meisner/Thorne/Wheeler for details). As it turns out, the equations including the constant are the most general form that fulfils all these conditions - so there needs to be a reason why the constant should be exactly zero.

Im not saying it can’t be zero, just that there would have to be a reason for it.

You are right, but Einstein invented his cosmological constant, suggested by Hilbert by the way, before the observed expansion which was discovered with surprise and even stupor. And even Hubble didn't believe in real expansion, he had another explanation for it.

Edited by Mitcher
messed up
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On 8/20/2022 at 8:09 AM, Mitcher said:

You are right, but Einstein invented his cosmological constant

Einstein, despite having come up with the equations initially, didn’t know about the full  set of principles underlying their form (the crucial topological concepts underpinning this were worked out by Ellie Cartan at a later date) - so he wouldn’t initially have been aware that the presence of the constant was the ‘normal’ state of affairs, hence it didn’t appear in his original formulation. So unfortunately any fine-tuning to precisely zero still lacks a physical mechanism or reason.

Again, I’m not saying it can’t be zero, just that this would be an example of unexplained fine-tuning.

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11 hours ago, Markus Hanke said:

Einstein, despite having come up with the equations initially, didn’t know about the full  set of principles underlying their form (the crucial topological concepts underpinning this were worked out by Ellie Cartan at a later date) - so he wouldn’t initially have been aware that the presence of the constant was the ‘normal’ state of affairs, hence it didn’t appear in his original formulation. So unfortunately any fine-tuning to precisely zero still lacks a physical mechanism or reason.

Again, I’m not saying it can’t be zero, just that this would be an example of unexplained fine-tuning.

ok, but this cosmological constant, as well with the Hubble parameter, are prone to varrying with very large scale of time isn't ?

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On 8/22/2022 at 5:11 AM, Mitcher said:

ok, but this cosmological constant, as well with the Hubble parameter, are prone to varrying with very large scale of time isn't ?

That’s a really good question! It is indeed possible to vary lambda with time, location, or both - the resulting models are called “agegraphic dark energy models”. There are both advantages and problems associated with these, but I must admit that this isn’t something I’ve been following, so I don’t know where things stand on this. It hasn’t caught on in the mainstream though.

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23 hours ago, Markus Hanke said:

It is indeed possible to vary lambda with time, location, or both - the resulting models are called “agegraphic dark energy models”.

That's very interesting.
As both Big Bang theory, and GR, assume isotropy and homogeneity of the universe, how do they reconcile with some fundamental aspects, such as lambda being non-isotropic and non-homogenous in these  agegraphic models ?

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12 hours ago, MigL said:

That's very interesting.
As both Big Bang theory, and GR, assume isotropy and homogeneity of the universe, how do they reconcile with some fundamental aspects, such as lambda being non-isotropic and non-homogenous in these  agegraphic models ?

I don’t know, to be honest - I’ve never looked into these models. Over the last few years my focus has been elsewhere, so I haven’t been keeping up with latest developments as much as I would have liked to.

Another one for the future to-do list 👍

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On 8/23/2022 at 9:25 PM, Markus Hanke said:

That’s a really good question! It is indeed possible to vary lambda with time, location, or both - the resulting models are called “agegraphic dark energy models”. There are both advantages and problems associated with these, but I must admit that this isn’t something I’ve been following, so I don’t know where things stand on this. It hasn’t caught on in the mainstream though.

One can imagine the acceleration is isotropic for a given age of the Universe but again, how the mainstream is not looking at this, and how it keeps speaking about the Hubble constant ? If it accelerates it is not constant, right ? What am I missing there ?

On 8/23/2022 at 9:25 PM, Markus Hanke said:

That’s a really good question! It is indeed possible to vary lambda with time, location, or both - the resulting models are called “agegraphic dark energy models”. There are both advantages and problems associated with these, but I must admit that this isn’t something I’ve been following, so I don’t know where things stand on this. It hasn’t caught on in the mainstream though.

I mean, instead of looking at the Hubble slope I want to see a curve.

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12 hours ago, Mitcher said:

how the mainstream is not looking at this

They are looking at it - if you search for this on arXiv, you’ll find quite a number of papers on this subject. I would imagine it hasn’t become part of the general consensus, because there are also problems and issues with these models.

12 hours ago, Mitcher said:

If it accelerates it is not constant, right ?

You are right, yes. I think the problem here is that the error bars for the precise value of the Hubble constant are far bigger than the effects of the acceleration, so at the moment we aren’t yet in a position to draw a meaningful graph for this. This is a work in progress.

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On 8/19/2022 at 11:51 PM, Markus Hanke said:

As it turns out, the equations including the constant are the most general form that fulfils all these conditions - so there needs to be a reason why the constant should be exactly zero.

Im not saying it can’t be zero, just that there would have to be a reason for it.

Penrose once said that the probabilities for the Universe to exist were almost zero. However, there it is.

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16 hours ago, Mitcher said:

Penrose once said that the probabilities for the Universe to exist were almost zero. However, there it is.

The probability that the electron (or any quantum particle) exists at a specific point on a space axis is exactly zero.

Yet all those zero probabilities can be summed to exactly 1 over the entire space axis.

 

There are many counter intuitive results in Maths like this.

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On 8/29/2022 at 1:49 PM, studiot said:

The probability that the electron (or any quantum particle) exists at a specific point on a space axis is exactly zero.

Yet all those zero probabilities can be summed to exactly 1 over the entire space axis.

 

There are many counter intuitive results in Maths like this.

Yes, I forgot the name of that function which is null everywhere excepted on 0 where it is infinite. Yet the integral of that function is infinite. Brilliant.

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10 minutes ago, Mitcher said:

Yes, I forgot the name of that function which is null everywhere excepted on 0 where it is infinite. Yet the integral of that function is infinite. Brilliant.

It’s the Dirac delta function, and you have

\[\int_{-\infty}^{\infty}\delta(x)dx=1\]

Edited by Markus Hanke
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The Big Bang Theory, how then did matter (energy & form) initially become organized, progress, and develop into the current reality.  Rather than Big Bang, the origin of the Universe might better be described as a “Sudden and Orderly Development”, i.e., involving the known laws of physics in conjunction with natural geometry. So, the question is, where did they come from?

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16 hours ago, Mitcher said:

Yes, I forgot the name of that function which is null everywhere excepted on 0 where it is infinite. Yet the integral of that function is infinite. Brilliant.

Other strange mathematical constructions, of potential interest to cosmologists are Gabriel's Horn and Peano curves.

Gabriel's horn is part of a family of infinite n dimensional objects that enclose our bound a finite (n+1) dimensional object.

Peano curves are n dimensional objects that fill (cover) spaces of greater (n+1, n+2 etc) dimension.
These completely cut though and violate conventional metrics such as Euclidian, Riemannian etc.

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6 hours ago, studiot said:

Other strange mathematical constructions, of potential interest to cosmologists are Gabriel's Horn and Peano curves.

Gabriel's horn is part of a family of infinite n dimensional objects that enclose our bound a finite (n+1) dimensional object.

Peano curves are n dimensional objects that fill (cover) spaces of greater (n+1, n+2 etc) dimension.
These completely cut though and violate conventional metrics such as Euclidian, Riemannian etc.

Very interesting +1

This brings to mind the Sierpinski cube - infinite surface area enclosing zero volume :)

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On 8/29/2022 at 1:49 PM, studiot said:

The probability that the electron (or any quantum particle) exists at a specific point on a space axis is exactly zero.

Yet all those zero probabilities can be summed to exactly 1 over the entire space axis.

 

There are many counter intuitive results in Maths like this.

Good point, zero probability does not imply impossibility if the continuum is a reality. In maths it's to do with the measure problem, as I'm sure you know. The measure of an interval can be zero or not, but the measure of a point is always zero. The measure of any denumerable set in a continuum is zero. Lo and behold!

On 8/31/2022 at 10:35 PM, Mitcher said:

Yes, I forgot the name of that function which is null everywhere excepted on 0 where it is infinite. Yet the integral of that function is infinite. Brilliant.

(My emphasis.)

The integral of Dirac's delta distribution is finite. If you want the probability of a denumerable set to be non-zero, you must construct a combination of delta distributions. For example p(x)=p1delta(x-x1)+p2delta(x-x2) with p1+p2=1.

The derivative of delta makes sense, but the square doesn't... Think of that.

It's a subtle business.

8 hours ago, studiot said:

Other strange mathematical constructions, of potential interest to cosmologists are Gabriel's Horn and Peano curves.

Gabriel's horn is part of a family of infinite n dimensional objects that enclose our bound a finite (n+1) dimensional object.

Peano curves are n dimensional objects that fill (cover) spaces of greater (n+1, n+2 etc) dimension.
These completely cut though and violate conventional metrics such as Euclidian, Riemannian etc.

Very interesting. Fractality makes me wonder so much...

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Singularity functions are not new in Mathematics.

Here are some ways of handling them and indeed making good use of them.

https://en.wikipedia.org/wiki/Singularity_function

note this can be downloaded as a pdf.

 

11 hours ago, joigus said:

Very interesting. Fractality makes me wonder so much...

All these oddities and fractals us rethink our ideas of 'points' and point set topology.

This is because the classical view of a point is as a 'static' identifiable object.

But if we look the other way for a fractal, (expanding to larger and larger scales,  rather than contracting to smaller and smaller ones) our 'static' points (I would rather use the word fixed, but Banach has already bagged that in the fixed point theorem) are not static at all but change as the scale increases.

So if we take the traditional isotropic and homogeneous n dimensional ball about a point,  ie as a neighborhood including that point as an interior point, we find the properties of that point depend to some extent on the other points in the ball, which in turn depend upon the scale of the ball.

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On 9/1/2022 at 3:20 PM, studiot said:

Other strange mathematical constructions, of potential interest to cosmologists are Gabriel's Horn and Peano curves.

Gabriel's horn is part of a family of infinite n dimensional objects that enclose our bound a finite (n+1) dimensional object.

Peano curves are n dimensional objects that fill (cover) spaces of greater (n+1, n+2 etc) dimension.
These completely cut though and violate conventional metrics such as Euclidian, Riemannian etc.

I knew about Peano curves but had never heard of Gabriel's horn, thanks for pointing them to me.

On 9/2/2022 at 12:12 AM, joigus said:

The integral of Dirac's delta distribution is finite.

Correct, it's equal to 1, not infinity, my mistake.

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