[Halc]

However, do you agree that if the space is represented as a "surface of a balloon" - somehow, we must monitor a curvature in space?

Not if the curvature is below the sensitivity of the instruments to measure it.

So, you agree that based on our instruments we do not see any curvature in our Universe (as far as we can see and monitor - At least 13 Bly for any direction)

I don't see how looking a long distance would help.  You need to propose a way to measure it.  They've done it over a distance of say 50 km, but no curvature was measured at that scale.  Nobody has tried something significant like billions of light years.  We're not capable of putting instruments that far away.

How can we claim in one hand that based on our mathematical theory the Universe should have a curvature,

Who said that?

Maybe it's finite but not curved.  Read my other post.   Maybe the finite/infinite distinction depends on how the size is measured.

In other words - as we see no curvature up to a minimal radius of 13 Billion light year, we must find what is the minimal circumference of  "surface of a balloon" that is needed to support that none curvature at that radius.

No measurement at 13 BLY has been taken.

I fully accept/agree with the following answer from evan_au:

There is an experiment proposed for the 2030s that would search for gravitational waves using lasers bounced of satellites, perhaps 2.5 million km apart....

If I understand it correctly, our scientists try to find a breakthrough technology/ideas in order to give an answer to some of key questions including the idea of curvature.[/quote]
That setup would indeed measure curvature of space, but we already know that local space is curved due to there being nearby mass.  A decent measurement of the curvature of the universe probably needs to be done away from local mass.

Let's use the circumference of the Earth as an example-
At the early time, our scientists didn't really measure the circumference of the Earth.

You're trying to measure the curvature of a 3D ball, not a 2D surface.  Light on a 2D surface follows the surface,  Techniques used to find the size of a solid will not work on a surface geometry.
It can be done, but it requires non-locality.  Even Earth measurement used non-locality.  I can think of some local ways to do it, but they're not obvious, and they won't work for a surface since they require looking in directions other than on the surface.

Sort of a challenge:  How to measure the size of the planet without moving.

[Dave Lev (OP)]

How can we claim in one hand that based on our mathematical theory the Universe should have a curvature,

Who said that?
Maybe it's finite but not curved.  Read my other post.   Maybe the finite/infinite distinction depends on how the size is measured.

Wow
After all of our discussion, do you agree that there is a possibility that there is no curvature in our Universe and maybe it is also infinite?
Please be aware that those ideas are key elements in Theory D.

Therefore, at the meeting point of that hypersphere, we might have one galaxy from the left which is moving at a velocity of n * c, while from the other side (right) another galaxy is coming at a velocity of n * c.

They're the same galaxy, and receding in all directions.  Nothing that distant is coming at us.  The universe is expanding, remember?

I see a severe contradiction between the idea of expansion and the 4D module.
In order to get the 4D module Minkowski had placed the time as orthogonal to the 3D of space.
So, if we set an expansion in 4D module, we must set an expansion in each dimension of this module.
If we only set an expansion in the 3D space dimensions and ignore the time dimension, than we set a severe violation in Minkowski formula.
So, in order to validate Minkowski concept under the idea of space expansion, we also must set an expansion in the time.
But, if I understand it correctly, there is no way to set an expansion in time.
Therefore, do you agree that Minkowski and expansion can't work together?

This discusses how a 3D creatures would measure the curvature of a 3D ball, not the analogous 2D creatures measuring the curvature of the 2D surface of Earth.

I also do not agree to that analogy.
We are living in a 3D space.
Minkowski had added mathematically one more dimension - time.
If it is correct - than we are living in a 4D Universe.
This should be correct everywhere. So, you can't go back and eliminate one dimension. Hence your analogy to 2D is not realistic.
However, if we are not living in a 4D Universe - than there is no value for Minkowski calculation.
It is just an unrealistic mathematical idea.
My conclusion is as follow:
1. In our space there is only 3D, so we are not living in a 4D Universe. In our real universe the time isn't orthogonal to the 3D space.
2. Hence, Minkowski concept for 4D Universe is just unrealistic mathematical concept of Space-time.
3. As there is no way for setting the expansion in time, Minkowski formula break down under the expansion idea.
4. If the 4D concept of space-time is unrealistic, than its outcome as a curvature in space is also unrealistic.
5. If our universe is infinite (or might be infinite - based on your answer), we must find a theory that gives a clear explanation for infinite Universe!
6. I can't see how the BBT gives explanation for an infinite Universe in a very limited time frame of 13.8 Billion years
7.  Thermodynamics , Atom creation & BBT

Lack of mass creation is not a BBT thing.  It is a law of thermodynamics, which is not based at all on BBT.

Any real theory must show the source for all the matter in our Universe.
If our universe is infinite, than the total matter in the universe is also infinite.
If I understand it correctly - in order to bypass the thermodynamic obstacle, our scientists doesn't give an explanation for the source of particles before the BBT. They just show the transformation process from particles to Atoms after the Big bang: From Particles to Atoms, From Atoms to stars, and from stars to galaxies.
However, we know that particles are not stable over time. So, if there were particles before the big bang - their life time was quite short (few seconds?). Therefore, I can't see any option to accumulate infinite quantity of particles, just in order to wait for the mighty Big bang to come and start the process of converting all of them into real atoms.
By that converting process, our scientists believe that they have overcome the thermodynamics obstacle.
However, thermodynamics is always there. It was there after the Big bang and also before the Big bang.
They can't bypass that thermodynamics law!
They must show how all particles had been created before the Big bang!!!
Without it, the whole BBT is useless.
In any case, if I understand it correctly - it took over 300,000 years after the big bang just to set the first Atom. So, even if there was some quantity of particles before the big bang, they had to stay with us for at least 300,000 years before being converted into real Atom. Is it real? How can we hold a particle for so long time? What is the expected life time of a average particle?
How can we hold those particles there for unlimited time before the Big bang and for the minimum 300,000 years after the big bang just in order to be transformed into Hydrogen Atom which will be used as a basic element for the whole Universe?
I have proved that Atom is a cell of energy.
So, even if there was unlimited quantity of quarks before the big bang, a significant energy must be added in order to set the transformation into Atom. So, how the Big bang could get the energy which was requested for that transformation of Atom creation?
How even one atom can be created without real magnetic/electric field and without acceleration?
How the Big bang could set the requested magnetic/electric field and acceleration in order to accomplish the mission of only atom creation?
What is the source of the energy of the Big bang? Why it had suddenly happened?
How the thermodynamics can supports the Big bang requested power?
If it is so difficult to create/transform even one atom, how the Big bang could set the whole matter in our infinite Universe - if it is infinite)
Sorry - we can't hide behind the idea that it isn't our job to find the source for all the particles before the Big bang and the source of power which is needed for the Big bang to take place. We have to offer a valid theory how all of that particles/matter had been created (before and after the Big bang) - especially if our universe is infinite!!!
We must find a way how the thermodynamics is part of the creation particles/Atoms process - before and/or after the big bang!

[Halc]

Wow
After all of our discussion, do you agree that there is a possibility that there is no curvature in our Universe and maybe it is also infinite?

Since I never said it cannot be infinite, I don't know where 'all this discussion' about it comes from.  My actual personal stance is that the size of the universe is dependent on how you measure it.

I see a severe contradiction between the idea of expansion and the 4D module.

It's no different than a 2D model (one of space, the other time, so an expanding circle), which is easier to envision if you find 4D too much for you.  The balloon analogy is 2D of space, but even that seems beyond your ability to envision.

In order to get the 4D module Minkowski had placed the time as orthogonal to the 3D of space.
So, if we set an expansion in 4D module, we must set an expansion in each dimension of this module.
If we only set an expansion in the 3D space dimensions and ignore the time dimension, than we set a severe violation in Minkowski formula.

All the formulas work fine locally.  If you look up the balloon analogy on the web, it is stated that it is only intended to be used locally.  Yes, your point about Minkowski spacetime is relevant on the large scale.  If space was a hypersphere holding to Minkowski's work on the large scale, then the size of the universe would be known.  It would take only about 87 billion light years to get back to here.  The universe is probably not that small.  My point is that in a hypersphere, the 4 dimensions are orthogonal at every point, but they look more like a 4d version of a polar coordinate system, which looks like a normal Cartesian coordinate system locally.

So, in order to validate Minkowski concept under the idea of space expansion, we also must set an expansion in the time.
But, if I understand it correctly, there is no way to set an expansion in time.

You must not understand it correctly.  If time expanded with space, there would be no apparent expansion.  We'd not see distances to far objects increasing.  At best it would look like everything staying put but getting smaller, with gravity dropping steadily as the mass of everything decreases.  I suppose that is a valid, if not strange, way to describe physics.  All the physical constants would become time dependent.

Therefore, do you agree that Minkowski and expansion can't work together?

Minkowski's model is a local one, and never was a cosmological scale model. The two models are not contradictory, but Minkowski does not paint a picture of the universe. Mind you it could:

The thread I have going is trying to describe a flat universe according to the Minkowski description.  While not a hypersphere, it turns out that the universe is finite in size and very much has an edge. The universe is still isotropic from any point, despite having an edge. This is the standard model but using Cartesian coordinates, not comoving coordinates.  I said the size depends on how you measure it.  I'm measuring it along straight lines, not bent ones as is usually done.  I'm still working acceleration of expansion into the picture, but that won't change the size of it, just curve a lot of lines that were straight without that acceleration.
I suppose this topic inspired me to explore that view of the universe.

This discusses how a 3D creatures would measure the curvature of a 3D ball, not the analogous 2D creatures measuring the curvature of the 2D surface of Earth.

I also do not agree to that analogy.
We are living in a 3D space.

So use the hypersphere model then, which is for 3D creatures.  I was trying to simplify it for somebody obviously incapable of envisioning a hypersphere.

1. In our space there is only 3D, so we are not living in a 4D Universe.

Doesn't follow.  Logic fail.

2. Hence, Minkowski concept for 4D Universe is just unrealistic mathematical concept of Space-time.

It works, predicting exactly what we see.  If it doesn't, show me how it would make an empirical difference.  Your words are empty without that.

4. If the 4D concept of space-time is unrealistic, than its outcome as a curvature in space is also unrealistic.

True.  Curvature of space makes no sense in 3D, yet it has very much been measured, so it must be curvature in spacetime.  You just demonstrated the 4D concept by a valid indirect argument.

5. If our universe is infinite (or might be infinite - based on your answer), we must find a theory that gives a clear explanation for infinite Universe!

Or at least a coordinate system that yields infinite distances.

6. I can't see how the BBT gives explanation for an infinite Universe in a very limited time frame of 13.8 Billion years

Good argument.  You're using a coordinate system (like Minkowski's) with a finite light speed.  Yes, under that system, the size is finite.  See my other thread.  See the standard picture in post 464 which clearly shows faster than light speed of objects, not possible in the coordinates that Minkowski and special relativity uses.  It is a different coordinate system (I know not the name of it), but the one used when discussing very distant things.

Any real theory must show the source for all the matter in our Universe.

Any theory of the origin of the universe needs to, yes.  The BBT is not such a theory.  It is only a model of what happened subsequently.

If our universe is infinite, than the total matter in the universe is also infinite.

You need to add homogeneity postulate to make that statement stick, but that is typically assumed, so yes.

If I understand it correctly - in order to bypass the thermodynamic obstacle, our scientists doesn't give an explanation for the source of particles before the BBT.

Since the particles formed subsequent to the big bang, I don't see why they would want to do that.  All that matter you talk about formed after the big bang.

They just show the transformation process from particles to Atoms after the Big bang: From Particles to Atoms, From Atoms to stars, and from stars to galaxies.

You forgot energy to particles.

However, we know that particles are not stable over time. So, if there were particles before the big bang - their life time was quite short (few seconds?). Therefore, I can't see any option to accumulate infinite quantity of particles, just in order to wait for the mighty Big bang to come and start the process of converting all of them into real atoms.

You're right.  I can't see that either.

By that converting process, our scientists believe that they have overcome the thermodynamics obstacle.

By attributing the statement above to 'our scientists', you seem to be formulating a strawman argument.

How can we hold a particle for so long time? What is the expected life time of a average particle?

Some are quite stable.  Some are not and thus come and go. Electrons and protons are quite stable, and neutrons quickly bound with the latter.  Together they could easily last that 379000 years before they turned into the first atoms.

What is the source of the energy of the Big bang? Why it had suddenly happened?

There are actually hypotheses concerning the dynamics beyond the big bang singularity. Look them up for some interesting reading, but I have little patience for your naive concepts such as talking about time before time.  The total energy of the universe may well be zero, so a theory of the origin of the singularity need not posit any sort of energy anomaly to prevent violation of thermodynamic law.

[Dave Lev (OP)]

Thanks Halc

2. Hence, Minkowski concept for 4D Universe is just unrealistic mathematical concept of Space-time.

It works, predicting exactly what we see.

Why do you claim that it is predicting exactly what we see?
Do we see any special verification/evidence for that confirms space - time concept? Do we really see a curvature in our universe?

4. If the 4D concept of space-time is unrealistic, than its outcome as a curvature in space is also unrealistic.

True.  Curvature of space makes no sense in 3D, yet it has very much been measured, so it must be curvature in space-time.  You just demonstrated the 4D concept by a valid indirect argument.

Did we really measure that Curvature of space?
It was stated before that so far we couldn't find any curvature in our universe.
Please explain this answer.

6. I can't see how the BBT gives explanation for an infinite Universe in a very limited time frame of 13.8 Billion years

Good argument.  You're using a coordinate system (like Minkowski's) with a finite light speed.  Yes, under that system, the size is finite.  See my other thread.  See the standard picture in post 464 which clearly shows faster than light speed of objects, not possible in the coordinates that Minkowski and special relativity uses.  It is a different coordinate system (I know not the name of it), but the one used when discussing very distant things.

So do you agree that the BBT can't support an infinite Universe (without curvature)? Only a finite curvature Universe?

After all of our discussion, do you agree that there is a possibility that there is no curvature in our Universe and maybe it is also infinite?

Since I never said it cannot be infinite, I don't know where 'all this discussion' about it comes from.  My actual personal stance is that the size of the universe is dependent on how you measure it.

Once you accept the idea that the universe is/could be infinite (without curvature), than do you accept the idea that for this infinite Universe the BBT is useless and therefore we must look for more updated theory?

We are living in a 3D space.

So use the hypersphere model then, which is for 3D creatures.  I was trying to simplify it for somebody obviously incapable of envisioning a hypersphere.

https://en.wikipedia.org/wiki/Hypersphere
"The term hypersphere was introduced by Duncan Sommerville in his discussion of models for non-Euclidean geometry.
The first one mentioned is a 3-sphere in four dimensions."
The main problem with "hypersphere" is that it doesn't represent our real 3-sphere Universe.
Therefore, as our universe isn't hypersphere, there is no room for curvature in our real universe.

I see a severe contradiction between the idea of expansion and the 4D module.

It's no different than a 2D model (one of space, the other time, so an expanding circle), which is easier to envision if you find 4D too much for you.  The balloon analogy is 2D of space, but even that seems beyond your ability to envision.

I fully understand your explanation about the 2D.
However, I don't agree with this analogy.
It is really strange to me that in one hand you add the time as it an extra dimension to our real 3D space (call it 4D space-time) and then eliminate one real dimension from that 4D space-time just to show that our real 3D space is like a 2D in that unrealistic 4D space-time.
Do you agree that we are still living in a 3D space?
We can add as many other dimensions as we like to our real 3D space.
For example - we could add velocity as another dimension and call it: space -time-velocity space.
However, I don't agree that we can now say that we are living in a 1D of that space -time-velocity Universe.
It is clear to me that in any kind of universe that we might offer - It is forbidden to eliminate one real space dimension (x,y,z). Therefore the analogy of 2D dimension is just not relevant.
I also can't agree with the idea that in one hand we offer an idea of curvature, while on the other hand we show that we won't be able to verify if this curvature idea is correct or incorrect (as our space is considered as a surface of a balloon in an unrealistic space-time concept) .
If there was a curvature in our Universe - why don't we see it?
I have full confidence that even if we could install our measurements tools at 13 Billion years from each other we won't find even one centimeter of curvature in our Universe.
Therefore, it seems to me that the main success of the Curvature idea is just an extra living time for the BBT.
Therefore, I still wonder why our scientists do whatever it takes to keep the BBT in life?
Why is it so important for them?
Why they don't start looking on all the evidences without the BBT glass and try to find a more accurate theory to our real Universe?

[Kryptid]

Sorry - If there was a curvature in our Universe we should see it!

Not if it's too subtle for us to measure with current technology.

I have full confidence that even if we could install our measurements tools at 13 Billion years from each other we won't find even one centimeter of curvature in our Universe.

Your confidence is not evidence. Even if we didn't observe curvature under such circumstances, that still wouldn't eliminate the possibility of curvature. The curvature could still be so subtle that it cannot be detected with any possible technology.

Therefore, I still wonder why our scientists do whatever it takes to keep the BBT in life?

Because it matches the evidence.

[Halc]

Why do you claim that it is predicting exactly what we see?

To be more precise, it doesn't introduce any predictions that contradict empirical measurements.

Do we see any special verification/evidence for that confirms space - time concept? Do we really see a curvature in our universe?

The theory does not predict that these things can be distinguished.

Did we really measure that Curvature of space?

Sure.  In principle, you can measure the circumference C of Earth with a tape measure and also its diameter D through a hole passing through.  In flat spaceitme, the ratio would be π.  In reality, the diameter would be larger than C/π.  No, they've not put a tape measure through the center of Earth, but there any theory that predicts a ratio of π has been falsified.

It was stated before that so far we couldn't find any curvature in our universe.
Please explain this answer.

I meant on a large scale, not a small scale like a planet or star.  Black holes cannot exist without curved space.  If your idea denies curved space, then it cannot have black holes.  They're space bent to the point of breaking.

So do you agree that the BBT can't support an infinite Universe (without curvature)? Only a finite curvature Universe?

It can.  It just depends on how you measure distance. I got a finite universe by using Minkowski coordinates (a defined inertial reference frame, constant light speed, etc).  The other coordinates (the standard one you see in all the books) do away with all that.  Distances are not measured along inertial lines.  Objects move faster than light due to this different way of measuring their distance.  A completely different set of events is labeled as 'now' and this events are not simultaneous with ours in any inertial sense.

So same universe, different picture of it.  One is finite, and one is not.

Once you accept the idea that the universe is/could be infinite (without curvature),

Depends what you mean by curvature.  I said the standard picture measures the size of the universe along a line that would be curved in a Minkowski style coordinate system.  The straightness of the lines all depends on how you map the events.

than do you accept the idea that for this infinite Universe the BBT is useless and therefore we must look for more updated theory?

The BBT is not inconsistent with either view.  In the standard coordinates used, things move arbitrarily faster than light, so a limit to the size is not concluded.  In the Minkowski coordinates, light speed is fixed in any frame and the size of the universe is frame dependent, but always finite.

The main problem with "hypersphere" is that it doesn't represent our real 3-sphere Universe.

How do you know that? 

Do you agree that we are still living in a 3D space?

I was planning to move next Tuesday, but so far I'm still here.

If there was a curvature in our Universe - why don't we see it?
I have full confidence that even if we could install our measurements tools at 13 Billion years from each other we won't find even one centimeter of curvature in our Universe.

There you go.  You know the answer, so why bother putting the instruments out there?

Therefore, it seems to me that the main success of the Curvature idea is just an extra living time for the BBT.

Who said the hypersphere model was a particularly successful one?  It's just a possibility that hasn't been falsified.  My Minkowski model is another finite one that is not curved. It has an edge like the hypersphere doesn't.
I drew a picture of it yesterday here:
https://www.thenakedscientists.com/forum/index.php?topic=76976.msg577095#msg577095
The purple line is the edge of the universe, growing further away from 'here' as time progresses.  There's another one on the other side not depicted.

The upper dark blue line represents comoving 'now', and distance is measured along that line.  It is curved, meaning the distance is measured with a bunch of meter sticks all moving at different velocities relative to each other.
The horizontal lighter blue line is the inertial now, with all the meter sticks having the same velocity.  They are stationary in the selected frame.

The light blue line intersects the purple line. Thus the univserse is finite in size when measured that way.  The dark blue line never hits the purple line, and thus the universe is infinite when measured that way.
Same (standard) model, but different ways of measuring things.

I had not added dark energy to the picture.  Still working on that.  Adding it will not change what I say here in this thread, but it very much affects that red light cone line.

[Dave Lev (OP)]

Thanks Halc & Kryptid

1. With regards to curvature:

Sorry - If there was a curvature in our Universe we should see it!

Not if it's too subtle for us to measure with current technology.

Yes, I agree
With the current technology we can't see any curvature.
In the future, we might use LISA as advised by Evan_au:

There is an experiment proposed for the 2030s that would search for gravitational waves using lasers bounced of satellites, perhaps 2.5 million km apart.
The goal was to look for tiny variations in the phase of a laser beam over this distance - looking for variations of around 20 picometers with periods ranging from 1 second to 10 hours.
It is outside the mission goals, but the same equipment could be used in the same way as a surveyor's laser to measure the distance between the satellites with an accuracy of perhaps 1nm over a period of a year. That may reveal some deviation from flat space due to the nearby Sun (superimposed on orbital disturbances from Earth, the Moon, Jupiter, Venus and Mars - and the Milky Way itself).

As a bonus, it is expected that this device could get a measure of the expansion of the universe (the Hubble constant) which is independent of the two current (conflicting) answers based on various methods.
However, measuring deviation of intergalactic space from perfectly flat space of the universe would be beyond such a device.
See: https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna

However, Even LISA might not discover any curvature.
Let's assume that in 2200 we will develop a supper advanced technology that sets the satellites at 2.5 Billion Km apart (instead of only 2.5Million Km in Lisa). However,  even in that range we won't find any curvature.
So, what is needed to us to understand that there is no curvature in our Universe?
In any case, we know the accuracy of our current technology. Therefore, why can't we say that based on the current accuracy there is no curvature in space?
Why instead of assuming that one day we should find a curvature in our Universe, we don't say today that up to the current level of accuracy our universe is flat and we don't see any curvature?
Don't you agree that based on the statement of zero curvature (at the current accuracy level) we must solve the enigma of our Universe?
How long do we have to wait in order to understand that there is no curvature in our Universe?
Just an example -
If I will tell you that an elephant is hiding behind the tree, would you believe that there is an elephant there?
If you will be very positive, how long are you going to wait for that elephant to move away from the tree in order to show himself?
In the same token:
Our scientists want to believe that there is a curvature in our Universe but it is hiding and therefore we can see it with the current technology. That is perfectly clear to me.
However, I would like to ask - How long do we have to wait until we all understand that the curvature is just in our imagination?


2. With Regards to Hypersphere:

Who said the hypersphere model was a particularly successful one?  It's just a possibility that hasn't been falsified. 

Thanks
So, the hypersphere model is a possibility that hasn't been falsified yet.
Therefore, do you agree that as long as there is no solid prove for that hypersphere possibility - we can't really use it as an evidence for the curvature in the Universe?

3. With regards to Minkowski model:

My Minkowski model is another finite one that is not curved. It has an edge like the hypersphere doesn't.
I drew a picture of it yesterday here:
https://www.thenakedscientists.com/forum/index.php?topic=76976.msg577095#msg577095
The purple line is the edge of the universe, growing further away from 'here' as time progresses.  There's another one on the other side not depicted.

I have looked at your modeling but I'm not sure that I fully understand how it really works.
In one hand you claim that: "Minkowski model is another finite one that is not curved"
On the other hand you claim that: "The purple line is the edge of the universe, growing further away from 'here' as time progresses"
So, if the purple line is the edge of the universe, then how could it be that there is no curvature in the Universe?
How could it be that in a finite Universe with an edge there is no curvature?
Don't you agree that a finite space/universe must have a curvature?
In any case, please advice if I understand it correctly as follow:
The main idea with the 4D space-time of Minkowski is that even if we see that the Universe is completely flat (without any sort of evidence for curvature) the Universe/space must be a finite universe.
However, what is the size of that finite Universe?
As you claim that Minkowski model has an edge like the hypersphere doesn't, then would you kindly calculate the size/edge of our current space/universe based on this model?
If the 4D space-time of Minkowski model is correct, than why based on the flat Universe that we see (and measured), and the Picture that you drew, we can't extract the total size of our real finite Universe?
Why don't we try to calculate the minimum size of the finite Universe as a direct outcome of Minkowski 4D space-time model?
How can we claim that: The edge of the universe is growing further away from 'here' as time progresses, while we have no clue about the where is 'here' and what is the edge/size of the Universe?
You also add:

The light blue line intersects the purple line. Thus the univserse is finite in size when measured that way.  The dark blue line never hits the purple line, and thus the universe is infinite when measured that way.
Same (standard) model, but different ways of measuring things.

So, we can get different outcomes as we use the Minkowski model differently.
Therefore, do you agree that theoretically if we move along the opposite direction of the lines, than we might find that the Universe is shrinking?

The main problem with "hypersphere" is that it doesn't represent our real 3-sphere Universe.

How do you know that? 

In our real 3D space all the three dimensions are orthogonal to each other.
That is very clear to all of us.
However, I still don't see any prove for the idea that the time is orthogonal for those 3 Dimension in space.
It might be a great mathematical concept. But if in our real life, the time isn't orthogonal to the space, than the outcome of Minkowski space-time model is an imaginary mathematical Universe.
In this imaginary Universe, there might be a curvature/edge in the space or even in the time.
Therefore, we get that magneficent image as you have drawn.
However, in order to use it in our real Universe, we must first prove that the time is orthogonal to the 3D space.
So, can you please prove that our time is orthogonal to the 3D space?
I wonder what Einstein would say about that modeling...
In any case, I see it as follow:
If we can prove that the time is orthogonal to the 3D space, than your image represents our real Universe and it proves that it can be finite with edge - as a direct outcome from the image.
However, if we can't prove it - than why don't we set this model as a mathematical concept?
Why do we use it as a real model for our universe?

Therefore, I still wonder why our scientists do whatever it takes to keep the BBT in life?

Because it matches the evidence.

Which evidence?
Do we see any curvature in our Universe?
Can we prove that our Universe is Hypersphere?
Can we prove that in our real universe the time is orthogonal to the 3D space?
If we can't prove the above and our universe is infinite, how the BBT could fit in this Universe?

[Kryptid]

Which evidence?

The abundance of the chemical elements matches Big Bang predictions, the metric expansion of space, the existence of the cosmic microwave background, the morphology and distribution of the galaxies and quasars, the fact that the Big Bang model agrees with the age of some of the oldest stars in the universe as measured using other methods, the fact that we now know that the CMBR temperature was higher in the past among other things.

Do we see any curvature in our Universe?

Not so far. But we don't need to.

Can we prove that our Universe is Hypersphere?

It might not be a hypersphere, so we don't know.

Can we prove that in our real universe the time is orthogonal to the 3D space?

I don't think that's necessary.

If we can't prove the above and our universe is infinite, how the BBT could fit in this Universe?

The inability to prove any of the above things does not rule out the Big Bang. Science is not about proof.

[Halc]

Therefore, why can't we say that based on the current accuracy there is no curvature in space?

If current technology lacks the precision to distinguish a thing from the absence of the thing, no conclusion about the thing can be drawn from the technology in question.

Just an example -
If I will tell you that an elephant is hiding behind the tree, would you believe that there is an elephant there?

That's a positive claim being compared to one that is not.  We're only claiming the tree is large enough to hide an elephant (by like 20 orders of magnitude), so we can't be sure.

Our scientists want to believe that there is a curvature in our Universe but it is hiding and therefore we can see it with the current technology. That is perfectly clear to me.

Got any evidence for this claim, or it just another item on the long list of things that are clear to you?


Therefore, do you agree that as long as there is no solid prove for that hypersphere possibility - we can't really use it as an evidence for the curvature in the Universe?

3. With regards to Minkowski model:

My Minkowski model is another finite one that is not curved. It has an edge like the hypersphere doesn't.
I drew a picture of it yesterday here:
https://www.thenakedscientists.com/forum/index.php?topic=76976.msg577095#msg577095
The purple line is the edge of the universe, growing further away from 'here' as time progresses.  There's another one on the other side not depicted.

I have looked at your modeling but I'm not sure that I fully understand how it really works.

It's the standard model (the accepted one  * ), so it works like that one.  All I did was draw a picture of it using a standard inertial coordinate system where measurements are taken with meter sticks that are all moving at the same speed instead of different speeds like they usually depict.  Minkowski space is inertial, so he uses a coordinate system like I drew there, but rarely uses it at that scale.  I figured out that it is quite possible to scale it up all the way.

* The picture differs from the actual accepted model in that I didn't include effects of dark energy into it.  Haven't got that far yet.

In one hand you claim that: "Minkowski model is another finite one that is not curved"
On the other hand you claim that: "The purple line is the edge of the universe, growing further away from 'here' as time progresses"
So, if the purple line is the edge of the universe, then how could it be that there is no curvature in the Universe?

By not curved, I mean it is Euclidean space.  Big triangles drawn anywhere have 3 angles that add up to 180°, which is not true of the curved hypersphere model.

How could it be that in a finite Universe with an edge there is no curvature?

Kind of like a Euclidean ball is finite in size and has an edge.

Don't you agree that a finite space/universe must have a curvature?

Locally it does. No large scale curvature has ever been measured.  You just posted all this stuff about denial of curvature, and here you are insisting on it.  My coordinate system doesn't demand flat space, but it works with flat space, so that's what I drew.

In any case, please advice if I understand it correctly as follow:
The main idea with the 4D space-time of Minkowski is that even if we see that the Universe is completely flat (without any sort of evidence for curvature) the Universe/space must be a finite universe.

Measured that way, it would need infinite time to grow to infinite size.  The model has finite time (it is a big bang model), so it hasn't had time to get bigger than I've drawn.  Nothing can move faster than light in Minkowski spacetime.
It is actually an infinite model.  There is no upper edge (time goes up), so after arbitrary time X years, the universe will grow to X light years in radius.

However, what is the size of that finite Universe?

Currently about 1.09e31 cubic light years, assuming a comoving frame.  It is only slightly larger as measured in Earth's frame.
The thread topic was about the limit of the size of a stationary object.  I had to draw a picture of the universe to see it.  The max size of the object is about 2.75e10 light years if it is stationary relative to Earth, but it can be arbitrarily larger if it is moving fast enough.  See below for more on that.

As you claim that Minkowski model has an edge like the hypersphere doesn't, then would you kindly calculate the size/edge of our current space/universe based on this model?

13.75 BLY away of course.  The universe is that old and nothing can move faster than light in that coordinate system, so that's how large it is.
Again, this isn't a different model, just the same model in Minkowski coordinates.

If the 4D space-time of Minkowski model is correct, than why based on the flat Universe that we see (and measured), and the Picture that you drew, we can't extract the total size of our real finite Universe?
Why don't we try to calculate the minimum size of the finite Universe as a direct outcome of Minkowski 4D space-time model?

We can.  I did it above.  The size they typically quote measures distance a different way (with the meter sticks all moving at different speeds).

How can we claim that: The edge of the universe is growing further away from 'here' as time progresses, while we have no clue about the where is 'here' and what is the edge/size of the Universe?

Here is here.  Relativity says there are no absolute coordinates.  The Minkowski coordinates do not contradict that.  In drawing the picture, I selected a frame where Earth is at the exact center of the universe.  See the dark blue line labeled 'now'?  I can choose a frame that puts us anywhere on that line.  By doing so, the universe gets larger, which is why the selection of reference frame allows me to posit larger objects than the one you get in Earth's frame.  The choice of frame similarly gets us closer to one of the purple lines, so we can be arbitrarily close to the edge if you like.

You also add:

The light blue line intersects the purple line. Thus the universe is finite in size when measured that way.  The dark blue line never hits the purple line, and thus the universe is infinite when measured that way.
Same (standard) model, but different ways of measuring things.

So, we can get different outcomes as we use the Minkowski model differently. [/quote]It's not a different model, just different coordinates.  We get different answers by measuring via a different method.

Therefore, do you agree that theoretically if we move along the opposite direction of the lines, than we might find that the Universe is shrinking?

One cannot move along any line except a worldline.  The brown lines show some worldlines, but they can angle upwards at any angle up to 45°.

The main problem with "hypersphere" is that it doesn't represent our real 3-sphere Universe.

How do you know that? 

In our real 3D space all the three dimensions are orthogonal to each other.
That is very clear to all of us.[/quote]Oohh  you can see them?  No wonder stuff is clear to you and nobody else.  Anyway, coordinate lines are orthogonal on a hypersphere as well.

However, I still don't see any prove for the idea that the time is orthogonal for those 3 Dimension in space.

Newton assumed that, and worked out a speed of light that was measurably different depending on your speed. If you moved the time axis, the space axes should have stayed the same. It wasn't, so time is orthogonal, per relativity theory.  If you assign a different direction to the time axis, the space dimensions must move with it, which is mathematically what relativity of simultaneity is.  The theory followed from constant light speed and showed that the two are the same thing (spacetime).  Relativity of simultaneity (four orthogonal axes) has been repeatedly demonstrated.  Some deny it and assume a 3D universe and no time dimension at all, but their inability to orient the x,y,z axes without assuming an orientation of the nonexistent t axis seems kind of hypocritical.  The math is much more complex if done in a 3D universe.

It might be a great mathematical concept. But if in our real life, the time isn't orthogonal to the space, than the outcome of Minkowski space-time model is an imaginary mathematical Universe.

It isn't a universe at all.  It's a model, and one that corresponds exactly to what we see.  If you disagree, find the empirical contradiction, but please stop it with the 'I don't like it, so it is clear that it is wrong' nonsense.  You not liking something is probably good evidence that something is correct.

In this imaginary Universe, there might be a curvature/edge in the space or even in the time.
Therefore, we get that magneficent image as you have drawn.

What I drew is Euclidean everywhere, even at the edges.

However, in order to use it in our real Universe, we must first prove that the time is orthogonal to the 3D space.
So, can you please prove that our time is orthogonal to the 3D space?
I wonder what Einstein would say about that modeling...[/quote]Einstein (together with the other contributers like Minkowski) showed it as described above.  If you don't understand that theory, then you can't assert that it doesn't show this.  I can't help you.  You imagine your idea being taught in textbooks someday, and yet you're incapable of understanding a theory where somebody else has done all the work, let along working out your own.

In any case, I see it as follow:
If we can prove that the time is orthogonal to the 3D space, than your image represents our real Universe and it proves that it can be finite with edge - as a direct outcome from the image.
However, if we can't prove it - than why don't we set this model as a mathematical concept?
Why do we use it as a real model for our universe?

Can we prove that in our real universe the time is orthogonal to the 3D space?

I don't think that's necessary.

Einstein showed it.  You get a Newtonian universe without it, and that has been falsified.  Even those that deny the existence of the time dimension (presentists) find that they need to posit a fixed yet unmeasurable orientation for it.  The relativity people are free to move it around of course.

[Dave Lev (OP)]

However, I still don't see any prove for the idea that the time is orthogonal for those 3 Dimension in space.

Newton assumed that, and worked out a speed of light that was measurably different depending on your speed.

How can we call it a proof?
Actually, Newton sees it quite differently:
https://astarmathsandphysics.com/a-level-physics-notes/special-and-general-relativity/3013-newton-s-views-on-space-and-time.html
"Newton's Views on Space and Time"
"Newton founded classical mechanics on the view that space is something distinct from matter and that time passes uniformly at every point in space, without regard to whatever happens in the world. For this reason he spoke of absolute space and absolute time, so as to distinguish these entities from the various ways by which we measure them.
It is stated clearly:
"Space had three dimensions – the normal directions of three dimensional space, and time threaded the whole of space and moved uniformly at the same rate throughout space."
So, where do you see Newton approval for the idea that the time should be orthogonal to space?

It might be a great mathematical concept. But if in our real life, the time isn't orthogonal to the space, than the outcome of Minkowski space-time model is an imaginary mathematical Universe.

It isn't a universe at all.  It's a model, and one that corresponds exactly to what we see.

On the contrary.
Based on the 4D module, the math shows that the Universe has a curvature.
Do we really see ant curvature in our Universe?
Why can't we assume that if so far we don't see any curvature in the Universe, than there is high chance that there is no curvature in the Universe?
So, why do you claim that the model corresponds exactly to what we see?
Where is the curvature?

Can we prove that in our real universe the time is orthogonal to the 3D space?

I don't think that's necessary. Einstein showed it. 

Is it?
Please see the following info about:
Einstein's Spacetime
https://einstein.stanford.edu/SPACETIME/spacetime2.html
By 1905 he had shown that FitzGerald and Lorentz's results followed from one simple but radical assumption: the laws of physics and the speed of light must be the same for all uniformly moving observers, regardless of their state of relative motion. For this to be true, space and time can no longer be independent. Rather, they are "converted" into each other in such a way as to keep the speed of light constant for all observers. (This is why moving objects appear to shrink, as suspected by FitzGerald and Lorentz, and why moving observers may measure time differently, as speculated by Poincaré.) Space and time are relative (i.e., they depend on the motion of the observer who measures them) — and light is more fundamental than either. This is the basis of Einstein's theory of special relativity ("special" refers to the restriction to uniform motion)
It is also stated:
"Einstein did not quite finish the job, however. Contrary to popular belief, he did not draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric."
So, Einstein proved that "Space and time are relative" but he didn't claim that they are orthogonal to each other.
The Fourth Dimension is Minkowski idea:
That insight came from Hermann Minkowski (1864-1909), who announced it in a 1908 colloquium with the dramatic words: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".
Please be aware that Einstein lived long after Hermann Minkowski.
Therefore, he was fully aware about Minkowski 4D dimension.
He could accept that idea. But based on the above explanation it seems clearly that Einstein didn't agree with the idea that the Time is orthogonal to space.
He claimed that "Space and time are relative".
So, If Einstein didn't accept the idea of 4D Dimension, who am I to disagree with him?
Speaking about Einstein:
If I understand it correctly - He actually disagree with the idea of BBT.
https://guardianlv.com/2014/03/albert-einstein-debunked-the-big-bang-theory/
"Albert Einstein’s theory of relativity walked hand in hand with the Big Bang theory, but recently resurfaced manuscripts show that the physicist debunked this idea and believed that the universe expanded steadily and eternally."
He also disagree with his personal idea about the cosmological constant:
https://www.space.com/9593-einstein-biggest-blunder-turns.html
"In 1917, Albert Einstein inserted a term called the cosmological constant into his theory of general relativity to force the equations  to predict a stationary universe in keeping with physicists' thinking at the time. When it became clear that the universe wasn't actually static, but was expanding instead, Einstein abandoned the constant, calling it the '"biggest blunder" of his life"
So, Einstein called the "cosmological constant"  as the '"biggest blunder" of his life.
"But lately scientists have revived Einstein's cosmological constant (denoted by the Greek capital letter lambda) to explain a mysterious force called dark energy that seems to be counteracting gravity -- causing the universe to expand at an accelerating pace."
Therefore, we must respect this scientist and NOT use his "biggest blunder" to prove the unrealistic idea of "dark energy", as there is no dark energy in our Universe.
How our scientists dare to use the "cosmological constant" in Einstein formula against his clear wish and then claim that it is based on Einstein?
Don't you agree that once we add this "cosmological constant" we clearly contradict Einstein formula?
So, our scientists use in one hand the 4D as Einstein Idea - which is incorrect, while on the other hand the also use the  "cosmological constant" in Einstein formula against his clear wish just to prove the BBT which Einstein did not accept.
So, how our scientists could use "Einstein" name in order to prove the BBT which he didn't agree with?

[Halc]

Actually, Newton sees it quite differently:
https://astarmathsandphysics.com/a-level-physics-notes/special-and-general-relativity/3013-newton-s-views-on-space-and-time.html
"Newton's Views on Space and Time"
"Newton founded classical mechanics on the view that space is something distinct from matter and that time passes uniformly at every point in space, without regard to whatever happens in the world. For this reason he spoke of absolute space and absolute time, so as to distinguish these entities from the various ways by which we measure them.
It is stated clearly:
"Space had three dimensions – the normal directions of three dimensional space, and time threaded the whole of space and moved uniformly at the same rate throughout space."

Yes, Newton viewed it as 3D.  Time in that interpretation (presentism) is something that flows at a uniform rate everywhere.  If that model is done in 4D, there is one direction that time can be oriented that is orthogonal.  Newton didn't see things in 4D like that, but some did.  The block universe concept is quite old actually, but spacetime (them being the same thing in the same units) is something different, introduced in the 20th century.

So, where do you see Newton approval for the idea that the time should be orthogonal to space?

If you're going to have a coordinate system, the coordinates (like x y z) need not be orthogonal to work, but things sure work a lot easier if you do it.  They just can't be parallel.  Then it ceases to function as a coordianate system.
Since Newton posited absolute space and time, there is really only one way to orient the time axis orthogonally if you're going to represent it as an axis.  It works at other orientations, but only one is orthogonal.

It isn't a universe at all.  It's a model, and one that corresponds exactly to what we see.

On the contrary.
Based on the 4D module, the math shows that the Universe has a curvature.

Not sure which model you are referring to.  The 4D thing is an interpretation of time (block universe) and it doesn't assert curvature or lack of it, or even relativity.  The Newtonian model was falsified whether it is interpreted as 3D or 4D.  It predicts frame dependent measurement of light speed.

So, why do you claim that the model corresponds exactly to what we see?
Where is the curvature?

The Minkowski model is a local model, and if expanded beyond locality, corresponds to a non-curved space.

Please see the following info about:
Einstein's Spacetime
https://einstein.stanford.edu/SPACETIME/spacetime2.html
By 1905 he had shown that FitzGerald and Lorentz's results followed from one simple but radical assumption: the laws of physics and the speed of light must be the same for all uniformly moving observers, regardless of their state of relative motion. For this to be true, space and time can no longer be independent. Rather, they are "converted" into each other in such a way as to keep the speed of light constant for all observers. (This is why moving objects appear to shrink, as suspected by FitzGerald and Lorentz, and why moving observers may measure time differently, as speculated by Poincaré.) Space and time are relative (i.e., they depend on the motion of the observer who measures them) — and light is more fundamental than either. This is the basis of Einstein's theory of special relativity ("special" refers to the restriction to uniform motion)
It is also stated:
"Einstein did not quite finish the job, however. Contrary to popular belief, he did not draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric."
So, Einstein proved that "Space and time are relative" but he didn't claim that they are orthogonal to each other.

The stuff you quote does not imply that.
As I said, a coordinate system is abstract, so if you don't mind totally complicating the math, you can put the axes in any non-orthogonal orientation that turns you on.

The Fourth Dimension is Minkowski idea:

It was proposed centuries earlier than that.  A 4 dimensional universe was suggested for instance by Anselm of Canterbury somewhere around year 1100.  It is probably older than that.  Minkowski may have first suggested that time and space are ontologically the same thing.

If I understand it correctly - He actually disagree with the idea of BBT.
https://guardianlv.com/2014/03/albert-einstein-debunked-the-big-bang-theory/
"Albert Einstein’s theory of relativity walked hand in hand with the Big Bang theory, but recently resurfaced manuscripts show that the physicist debunked this idea and believed that the universe expanded steadily and eternally."

Expansion and BBT was unknown at the time when relativity theory was published. 

He also disagree with his personal idea about the cosmological constant:
https://www.space.com/9593-einstein-biggest-blunder-turns.html
"In 1917, Albert Einstein inserted a term called the cosmological constant into his theory of general relativity to force the equations  to predict a stationary universe in keeping with physicists' thinking at the time. When it became clear that the universe wasn't actually static, but was expanding instead, Einstein abandoned the constant, calling it the '"biggest blunder" of his life"

Just so.  That was him accepting the BBT as a better explanation of things than his cosmological constant.

So, how our scientists could use "Einstein" name in order to prove the BBT which he didn't agree with?

I don't think Einstein's name comes up much in any evidence of BBT.  It isn't his theory.  Try Hubble and others.

[Dave Lev (OP)]

That was him accepting the BBT as a better explanation of things than his cosmological constant.

It seems to me that you have missed the key point about Einstein formula.
So the question is as follow:
In order to prove the BBT, is it correct that our scientists have used Einstein formula?
If so, could they prove the BBT without adding the cosmological constant to Einstein formula?
Do you agree that Einstein have stated that adding this cosmological constant is the biggest mistake of his life?
Therefore, Do you agree that Einstein will not accept the idea that our scientists have used that cosmological constant in his formula?
Hence, do you agree that it is forbidden to add that cosmological constant to Einstein formula and still call it Einstein formula?
In other words do you agree that the statement that the BBT had been mathematically proved based on Einstein formula - is totally incorrect?

[Dave Lev (OP)]

Which formula is this?
And I don't think any cosmological theory has ever been claimed as proved.

In the following article it is stated:
https://en.wikipedia.org/wiki/Cosmological_constant
From the 1930s until the late 1990s, most physicists assumed the "cosmological constant to be equal to zero.[4] That changed with the surprising discovery in 1998 that the expansion of the universe is accelerating, implying the possibility of a positive nonzero value for the cosmological constant.[5]"
The question is as follow:
Do we need to use the cosmological constant in Einstein formula in order to find an explanation for "the discovery in 1998 that the expansion of the universe is accelerating"?
Can we get an explanation for that discovery without the cosmological constant?

[Dave Lev (OP)]

OK. That quote suggests that the cosmological constant and dark energy are the same explanation, not alternate explanations for the same effect.

In order to prove the idea of the dark energy, our scientists were obliged to set a positive cosmological constant value in Einstein formula.
Therefore, do you agree that without this positive value, they would not be able to add the dark energy into Einstein formula?

Yet your statement here seems to suggest that this one explanation is unrealistic, leaving no alternative.  So your statement certainly doesn't follow from the quote you gave

As I have proved, Einstein had claimed that it is forbidden to use this cosmological constant in his formula.
Hence, without using the cosmological constant, how our scientists could prove the existence of that dark energy in Einstein formula?
If they can't use the cosmological constant, than they have a severe problem with the idea of dark energy.
Without the dark energy, they can't explain the acceleration in the expansion process.
Without this explanation, they have a severe problem with the BBT.
So, do you see the contradiction?

[Dave Lev (OP)]

You claim to have proved that Einstein made a claim about the forbidden usage of some constant in some unspecified formula.  That sounds pretty inconsistent if the formula makes any mention of this constant.

I'm not sure that I fully understand the meaning of this answer.
However:
1. Do you agree that in order to prove mathematically the BBT, our scientists are Using Friedmann equations?
2. Do you agree that Friedmann equations are based on Einstein' general relativity formulas?
https://en.wikipedia.org/wiki/Friedmann_equations
"The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity."
"They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density {\displaystyle \rho } \rho  and pressure {\displaystyle p} p"
3. Do you agree that in that Einstein general relativity formula the cosmological constant is a key element that is needed to validate the existence of the dark matter? So it must have a positive value?

In order to prove the idea of the dark energy, our scientists were obliged to set a positive cosmological constant value in Einstein formula.
Therefore, do you agree that without this positive value, they would not be able to add the dark energy into Einstein formula?

See post 497 for my reply to this repeated question.

Your message at 497 was:

Which formula is this?
And I don't think any cosmological theory has ever been claimed as proved.

and my reply at 498 was:
In the following article it is stated:
https://en.wikipedia.org/wiki/Cosmological_constant
From the 1930s until the late 1990s, most physicists assumed the "cosmological constant to be equal to zero.[4] That changed with the surprising discovery in 1998 that the expansion of the universe is accelerating, implying the possibility of a positive nonzero value for the cosmological constant.[5]"
In that article it is stated clearly:
Since the 1990s, studies have shown that around 68% of the mass–energy density of the universe can be attributed to so-called dark energy.[6] The cosmological constant Λ is the simplest possible explanation for dark energy, and is used in the current standard model of cosmology known as the ΛCDM model. While dark energy is poorly understood at a fundamental level, the main required properties of dark energy are that it functions as a type of anti-gravity, it dilutes much more slowly than matter as the universe expands, and it clusters much more weakly than matter, or perhaps not at all."
So, do you agree that dark matter is a key element in our understanding of the Universe?
It is stated clearly: "The cosmological constant Λ is the simplest possible explanation for dark energy, and is used in the current standard model of cosmology known as the ΛCDM model"
Therefore, the cosmological constant Λ in Einstein formula is used as a possible explanation for the existence of dark matter.
4. Do you agree that Einstein had rejected the idea of using the cosmological constant in his formula?
5. Therefore, do you agree that without that cosmological constant we can't confirm mathematically the existence of dark matter?
6. If so, do you agree that without the dark matter we can't explain the acceleration in the expansion?
7. If we can't explain that acceleration in the expansion, how can we prove the BBT?

[Halc]

1. Do you agree that in order to prove mathematically the BBT, our scientists are Using Friedmann equations?

Nobody proves anything, and I've said that repeatedly.  So I will not agree to such a statement.

2. Do you agree that Friedmann equations are based on Einstein' general relativity formulas?

wiki Friedmann_equations
"The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity."
"They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density and pressure"

It says "within the context of GR" and "derived ... from [GR equations for this specific metric]", so yes, the expansion of the universe was first worked out directly from GR theory, even before it was first observed.  Einstein's equations predict expansion.

3. Do you agree that in that Einstein general relativity formula the cosmological constant is a key element that is needed to validate the existence of the dark matter? So it must have a positive value?

The GR formulas that Friedmann used did not contain a cosmological constant.  Einstein added it to attempt a model of a static universe, and abandoned it when Hubble falsified the static model.
None of this has anything to do with dark matter.  I think you mean dark energy, and none of this has to do with that either.  All the above discussion concerns work done in the 20's and 30's.  We're still discussing an early model with steady expansion.

So, do you agree that dark matter is a key element in our understanding of the Universe?
It is stated clearly: "The cosmological constant Λ is the simplest possible explanation for dark energy, and is used in the current standard model of cosmology known as the ΛCDM model"
Therefore, the cosmological constant Λ in Einstein formula is used as a possible explanation for the existence of dark matter.

I assume you mean dark energy, and the universe does not behave in a way a model without it (or an equivalent) predicts.

4. Do you agree that Einstein had rejected the idea of using the cosmological constant in his formula?

History records him abandoning it, yes, since he was using it for a static model (a non-big-bang one).  Nobody is resurrecting such a static model.

5. Therefore, do you agree that without that cosmological constant we can't confirm mathematically the existence of dark matter?

This statement makes no sense.  If you mean dark energy, the two are the same thing.  If you're asking if it needs to be a constant, I don't think that has been demonstrated.

6. If so, do you agree that without the dark matter we can't explain the acceleration in the expansion?

I am unfamiliar with other suggested models (than say λCDM) and the terminologies they use, so I don't know if the term 'dark energy' is used in them.  You really need to stop saying dark-matter in all these statements. 

7. If we can't explain that acceleration in the expansion, how can we prove the BBT?

Yet again, science isn't involved in proofs.  They have explained it, so the if-we-don't part of your question is irrelevant, like asking why apples fall if we've no explanation for the effect?

[Dave Lev (OP)]

Thanks Halc

None of this has anything to do with dark matter.  I think you mean dark energy,

Yes, it is all about dark Energy.

1. Do you agree that in order to prove mathematically the BBT, our scientists are Using Friedmann equations?

Nobody proves anything, and I've said that repeatedly.  So I will not agree to such a statement.

So, what is the added value of Friedmann equations?
Do you mean that there is no mathematical prove for the BBT?
If so, why do you so positively support this theory?

[Kryptid]

Do you mean that there is no mathematical prove for the BBT?

There is no mathematical proof for any theory. That isn't how theories work.

[Dave Lev (OP)]

Do you mean that there is no mathematical prove for the BBT?

here is no mathematical proof for any theory. That isn't how theories work.

If there is no mathematical proof for the BBT, than do you agree that there is a chance that it is incorrect?
During this discussion, I have set several obstacles for the feasibility of the BBT.
One of the main obstacles is time frame.
I really can't see how can we fit all of this magnificent Universe in only 13.8 Billion years.
If the Universe is infinite in its size and there is no curvature - than why do you insisted that the 13.8 BY is still ok to set it?
Few days ago I have watched two amazing programs of "How the universe works" at the national geographic channel TV.
In one program they have discussed about the SMBH.
They have asked how the SMBH had been created.
They have set a calculation that if we take a BH and give it the time to eat as much as it wants, it can't technically be converted to SMBH after 13.8 BY.
Much more time is needed.
However, we see massive spiral galaxies (with embedded SMBH) with estimated age of about 12 BY.
So, they have wondered how could it be that the SMBH had been created in so short time after the BBT and could also add around it all the Billion stars in order to set the mighty spiral galaxy?
They have tried to come with some ideas, but they were not sure about those hypothetical ideas.
They also wondered why we see mainly find BH and SMBH.
In SMBH there are over than Billion BH.
Why don't we see as many mid size BH with only few Millions of BH?
In any case, they were positively sure that the SMBH can't evolve from a BH due to time limitation.
 
In the other program they have discussed about the heavy metal as Iridium and gold.
They have found that those metals could not been created by a supernova. The maximum power of that supernova is Setting Iron.
They have stated that one hypothetical idea for the creation of those two heavy metals is an explosion in twin neutron stars.
So, in order to calculate the requested time for the gold in our Planet, we need to add all the requested time as follow:
1. Time for the first Hydrogen
2. Time for the first Hydrogen cloud.
3. Time to set the first star. However, we have already found that in order to set a star in Hydrogen cloud there must be a nearby SMBH). so if there is no SMBH after the Big bang, how the gas cloud could be set any sort of star?
In any case, let's assume that somehow the hydrogen cloud had set the first Hydrogen star.
3. Time for the first hydrogen star to set the first Supernova explosion. Due to this explosion we get the iron. However, it is very rare and it is stes the iron as dust in space.
4. So, how long it might take to gather all the dust in a first iron dust cloud.
5. How long it might take to this first iron dust could to set the first twin neutron star systems.
6. How long it might take to this first twin Neutrom star system to set the explosion in order to create the first iridium and Gold in the whole Universe?
7. How long it might take to those heavy metal to arrive to our planet?
If they are too far away, do you agree that due to the expansion in space, they have no chance to get by Asteroids to our planet?
So, how do we have got Gold and iridium in only 13.8 Billion years?

In this program they didn't even try to explain how heavier Atoms as plutonium had been created.
It seems to me that even an explosion in a twin Neutron star is not good enough.
So, how do the the plutonium had been created and how it had been arrived to our planet in only 13.8 BY?

Based on theory D, all the atoms had been created at the excretion disc of a SMBH (Yes, including the plutonium).
There is no need to any chain of explosions to set any sort of heavy Atom.
Our planet had got all its matter in his first day. No needed for any special delivery ( as water, gold, plutonium...) from any kind of asteroid.
So, if the BBT has no real approval, why don't you open your mind to a breakthrough theory?
A theory which is so simple and gives a perfect explanation to any activity in our infinite universe?

[evan_au]

in order to set a star in Hydrogen cloud there must be a nearby SMBH

I don't see this dependency.

You need a hydrogen cloud, and it must be fairly dense to collapse under it's own gravity. But the early universe was fairly dense.

Time for the first hydrogen star to set the first Supernova explosion.

This is thought to be fairly quick (in astronomical terms): probably less than a million years.

The reason is that the early stars were formed from Hydrogen and Helium, and these have difficulty radiating away energy. To collapse under their own gravity, the first "Population III" stars must have been very massive, perhaps 100 times the mass of the Sun. Since the rate of burning their nuclear fuel increases as something like the 4th power of the mass, these stars would have had short lives.

When these first stars exploded, they would have distributed a lot of elements up to iron, which made the formation of the next generation of "Population II" stars much easier.
See: https://en.wikipedia.org/wiki/Stellar_population#Population_III_stars

one hypothetical idea for the creation of those two heavy metals is an explosion in twin neutron stars.

I think the phrase you want is "merger of twin neutron stars"; but I agree that certainly produces a dramatic explosion!

It is no longer hypothetical - in 2017 such an event was observed in gamma rays and visible light; the imminent event was heralded by gravitational waves arriving more than a minute before the gamma rays.

Spectroscopic analysis suggested that gold (and other heavy elements) were formed in this event. It is likely that Plutonium and heavier nuclei were also formed, but these radioactive elements have fairly short half-lives compared to Uranium, Thorium and Potassium-40 that we do find in reasonable concentrations on Earth.

Conceivably, a similar mechanism could occur if a neutron star were disrupted by a black hole - some of the neutron-rich material may be sprayed out into space, beyond the escape velocity of the black hole.
See: https://en.wikipedia.org/wiki/Neutron_star_merger#Observed_mergers

how do the the plutonium had been created and how it had been arrived to our planet in only 13.8 BY?

There is no measurable amount of natural plutonium that arrived on Earth during its formation almost 5 billion years ago. The longest-lived isotope of Plutonium (Pu-244) has a half-life of 80 million years, so it will have decayed by now.

There is some natural Plutonium generated continually on Earth, as a neutron occasionally strikes a Uranium atom in Uranium ores; this unstable Uranium nucleus decays into Plutonium (and then into Americium). The balance between creation of Plutonium and decay of Plutonium is biased heavily on the decay side, so the long-term average level of Plutonium in Uranium ores is extremely low.

A considerable amount of Plutonium is generated in nuclear reactors; the operation of some nuclear reactors is optimised to maximise production of Plutonium.
See: https://en.wikipedia.org/wiki/Isotopes_of_plutonium

There are signs that elements with half-lives of millions of years have arrived on Earth - sedimentary bands with traces of Fe-60 (half-life 2.7 million years) have been found in ocean basins. But as you say, Iron is likely to be produced in supernova explosions, which are expected to be far more frequent than neutron star mergers.

After all, any isolated massive star can go supernova - but it takes two very closely orbiting neutron stars to merge (and this implies two closely orbiting massive stars that had already gone supernova).
 
See: https://en.wikipedia.org/wiki/Near-Earth_supernova#Past_events