Accelerating Universe

[naysay]

[Tensor]

What is this supposed to be and what are your claiming?

[naysay]

What is this supposed to be and what are your claiming?

It lays out the underlying mechanics of a sphere undergoing a dynamical accelerating expansion. If the expansion of the universe is in fact a composite of two component expansions, each being driven by two time scales, and if those time scales have a squared relationship, then the composite expansion along the resultant hypotenuse should accelerate.

[John Jones]

What are you stating?

[Swift]

naysay,

That is not an acceptable way to present your idea. For one thing, those attachments are illegible. You need to actually type the text in a post and attach diagrams, as necessary.

I also hope you have read the rules on ATM, as you will be held to them.

[naysay]

naysay,

You need to actually type the text in a post and attach diagrams, as necessary.

No problem. Will post soon...

[naysay]

We will make this simple. The first thing we notice is a sphere that grows over time.



Physics tells us that the smallest unit of time is Planck, t_p. So we assume that at the first Planck moment of existence, the sphere had some surface area S_p. But it grows over time and so is dynamical in nature. We label the surface area A_c. It doesn’t matter what we call the time scale. Let’s call it tau. How does the sphere surface grow? There is only one way. To evaluate A_c we must multiply the size of the sphere at its beginning point S_p by the sliding ratio tau/t_p:

A_c = S_p(tau/t_p). (1)

Now let’s write Eq. (1) in terms of Planck length l_p and the light-speed constant:

A_c = S_p(c*tau/l_p). (2)

Now let’s assert something bold. Let’s say that the sphere contains a literal interior with a literal radius.



This allows us to evaluate the surface area A_c using a completely different equation. For now, light speed is the only speed we know. Using it in our equation also helps us stay consistent with Eq. (2) where the light-speed constant was multiplied by the time unit. Here we must use a different unit of time, however. Remember tau clocked the surface area growth of the sphere. This new time, t, will clock the radial growth of the sphere. Here is our new equation for surface area

A_c = 4pi(c*t)^2. (3)

Thus far we have identified three times: 1) Planck time, a constant, 2) tau which clocks the surface area expansion of A_c, and 3) t, which clocks the radial expansion of the sphere.

Now we set the r.s. of Eq. (2) equal to the r.s. of Eq. (3),

S_p(c*tau/l_p) = 4pi(c*t)^2, (4)

and easily see the relationship between the two timescales:



Now let’s find the two dynamical scale factors involved.

We move 4pi to the left hand side of Eq. (4) and take the square root of both sides. This gives us r_tau and r_t



Both scale factors grew to the same size but according to two completely different time scales.

Now let’s find the metric that explains this growth. According to coordinate systems and metrics, these two time scales and their corresponding scale factors must be oriented at right angles and the hypotenuse evaluated to find the true distance measurement. But the hypotenuse they create is actually a brand new, bigger radius with a completely new time scale and scale factor!



We conclude that the expansion of the sphere is actually accelerating and we did not know it until now. Let’s call the new time scale T and the new scale factor a(T). Here’s the metric:



Does it look familiar? Yes, it is the Friedmann metric. Only now, it accelerates! And we know why – because beneath its surface, two hidden time scales are driving two hidden expansion factors.

[naysay]

What are you stating?

It's out there now. See post #7.

[John Mendenhall]

What sphere? Where? When? Why is it growing? You cannot 'assume' something is true and then build a theory (or an ATM idea) on that assumption. Remember, observation, theory, predictions, results, repeatable by anyone, and peer reviewed by folks with legitimate credentials.

Good luck. Please find some observations to support your claims. As an alternative, you could become a string theorist . . . .

[caveman1917]

and easily see the relationship between the two timescales:

In other words, surface area goes by radius squared. Yes we know.

In other words, the square root of the surface area goes like the radius. Yes we know.

According to coordinate systems and metrics, these two time scales and their corresponding scale factors must be oriented at right angles and the hypotenuse evaluated to find the true distance measurement. But the hypotenuse they create is actually a brand new, bigger radius with a completely new time scale and scale factor!

I don't see why you're doing this, but let's go along and calculate .

which is not accelerating.

This actually describes a completely empty universe btw, there is no energy density to slow down the expansion and no dark energy to speed it up (the second time derivative of the scale factor is zero).

[naysay]

which is not accelerating.

In cosmology we do a substitution and rewrite a dynamical scale factor in terms of redshift; r_tau becomes (1+z)^-1 and r_t becomes (1+z)^-2 meaning your math is incorrect. If you take the square root of the sum of squares you should arrive at a completely new scale factor which does accelerate.

[caveman1917]

In cosmology we do a substitution and rewrite a dynamical scale factor in terms of redshift

There is absolutely no reason to rewrite the scale factor in terms of redshift, especially when calculating its long-term behaviour.

; r_tau becomes (1+z)^-1 and r_t becomes (1+z)^-2 meaning your math is incorrect.

You said

which when put in your triangle gives the scale factor i calculated.

You don't even have to calculate it out, Thales' theorem immediately tells you that will grow like if

[naysay]

You said

which when put in your triangle gives the scale factor i calculated.
will grow like if

You are thinking in terms of a 3-sphere only and performing mathematical evaluations based on that geometry. In cosmology, we think in terms of a 3-sphere embedded in 4-D spacetime. Your calculations are correct only in a 3 manifold, but not in a 4. I specifically pointed out that the two scale factors are dynamical in nature. They move according to clocks. They can either inflate the system or contract the system, but they must be moving – always. It is the presence of clocks in the calculations which begins to separate cosmology from other studies.

The language which all cosmologists understand is that models always “clock” their respective scale factors using the redshift expansion factor; namely, (1+z)^-1, and conventional cosmology attaches this “clock” to the expanding surface area of the universe sphere. The balloon analogy, probably the simplest version of modern models (and still in use today), clocks the growing universe sphere in this manner. The model at hand derives a second dynamical scale factor in the system based on the presence of a second clock. This second clock incorporates an additional factor of (1+z)^-1 and enlarges its respective scale factor at (1+z)^-2 (remember Eq. (5) showing the two clocks’ proportional relationship).

Something else you must understand – the Friedmann metric considers that coordinate points remain fixed in space while only the dynamical scale factor enlarges the system. This was a natural outgrowth of Friedmann’s (and others’) assumption of isotropy and an unbounded, no-center universe. Friedmann’s scale factor, a(t), is all that is moving. Yes, it is an idealized (possibly even simplistic) approach, but it is nonetheless elegant and allows the universe to be predictable and understood in a few simple equations. Cosmology by nature builds simple models; astrophysics puts them to the test to see if their predictions hold to the limits of the physical system.

For all the reasons stated above, you cannot simply evaluate the square root of the sum of r_tau^2 and r_t^2 and expect any applause from the world of cosmology. Those are dynamical scale factors and must be evaluated as the square root of [(1+z)^-1]^2 + [(1+z)^-2]^2. Your answer will be the brand new dynamical scale factor of

(1+z)^-1*sqrt[1+(1+z)^-2].

Now start plugging in z values, like 0, .2, .4, .6, .8, 1, 1.2, 1.4, 1.6, 1.8, 2. Drop those answers in one column in an Excel spreadsheet, then use the same z values in the usually understood Friedmann dynamical scale factor of (1+z)^-1 and drop those answers in the next column in your spreadsheet. Remember that z = 0 in our current cosmological era. As you look back in time, z increases. Now, starting at z = 2, begin a side-by-side comparison of the two columns as you move forward in time toward the current era and you will see a gentle and naturally occurring acceleration of the new scale factor.

[WayneFrancis]

ummm. Lost a whole post... forgive me while I redo it.

[caveman1917]

You are thinking in terms of a 3-sphere only and performing mathematical evaluations based on that geometry.

I wasn't doing any such thing, i was merely completing what you specified, ie that and that is the hypotenuse of the triangle with and as sides.

In cosmology, we think in terms of a 3-sphere embedded in 4-D spacetime.

Notwithstanding that the spatial hypersurfaces may just as well be hyperbolic or flat, even if they are spherical then it is still not a "3-sphere embedded in 4-D spacetime".

The language which all cosmologists understand is that models always “clock” their respective scale factors using the redshift expansion factor

That's quite a red herring, as i said earlier there is no reason whatsoever to put it in terms of redshift. Experimental cosmologists will put it in terms of redshift to relate observations to a specific value of the scale factor, and thus to calculate the time the light was emitted that they're observing. However once they have related the redshift of their observation to a specific value of the scale factor they still have to use the actual formulation of the scale factor in terms of cosmological time to get the time of emittance.

When you want to calculate whether your universe is accelerating or not, you need to check the scale factor as a function of time (ie the second step experimental cosmologists do). Putting it in terms of redshift is a step away from what you need to do, and completely irrelevant.

and conventional cosmology attaches this “clock” to the expanding surface area of the universe sphere. The balloon analogy, probably the simplest version of modern models (and still in use today), clocks the growing universe sphere in this manner.

That's just plain incorrect. If anything the balloon analogy "clocks" the expansion of the universe to the expansion of the arc-length between stationary points on the sphere, not the surface area.

For all the reasons stated above, you cannot simply evaluate the square root of the sum of r_tau^2 and r_t^2

So you changed your mind about your previous statement? :

these two time scales and their corresponding scale factors must be oriented at right angles and the hypotenuse evaluated to find the true distance measurement

Specifically note that one doesn't even have to evaluate any square root, this is a purely geometrical result already known to Thales of Miletus.

and expect any applause from the world of cosmology.

I don't expect any applause from the world of cosmology. Do you?

Those are dynamical scale factors and must be evaluated as the square root of [(1+z)^-1]^2 + [(1+z)^-2]^2. Your answer will be the brand new dynamical scale factor of

(1+z)^-1*sqrt[1+(1+z)^-2].

And with this you just made things a lot worse than they already were. At least in the previous incarnation you had a valid FLRW universe, granted it was a completely empty universe, but at least it was a valid solution.

If you're now going to claim that

you don't even have a valid solution to the EFE (with isotropy and homogeneity) anymore.

You see, the equation

is valid for every FLRW universe irrespective of the form of the scale factor (accelerating, decelerating,..).

You might want to try deriving the above equation from the general FLRW metric and you'll see that it doesn't depend on the form of

[naysay]

If you're now going to claim that

you don't even have a valid solution to the EFE (with isotropy and homogeneity) anymore.

I did not claim this.

You see, the equation

is valid for every FLRW universe irrespective of the form of the scale factor (accelerating, decelerating,..).

This is something we already know.

You might want to try deriving the above equation from the general FLRW metric and you'll see that it doesn't depend on the form of

I am not aware of any derivations based on the FLW metric.

[naysay]

I wasn't doing any such thing, i was merely completing what you specified, ie that and that is the hypotenuse of the triangle with and as sides.



Notwithstanding that the spatial hypersurfaces may just as well be hyperbolic or flat, even if they are spherical then it is still not a "3-sphere embedded in 4-D spacetime".



That's quite a red herring, as i said earlier there is no reason whatsoever to put it in terms of redshift. Experimental cosmologists will put it in terms of redshift to relate observations to a specific value of the scale factor, and thus to calculate the time the light was emitted that they're observing. However once they have related the redshift of their observation to a specific value of the scale factor they still have to use the actual formulation of the scale factor in terms of cosmological time to get the time of emittance.

When you want to calculate whether your universe is accelerating or not, you need to check the scale factor as a function of time (ie the second step experimental cosmologists do). Putting it in terms of redshift is a step away from what you need to do, and completely irrelevant.



That's just plain incorrect. If anything the balloon analogy "clocks" the expansion of the universe to the expansion of the arc-length between stationary points on the sphere, not the surface area.



So you changed your mind about your previous statement? :


Specifically note that one doesn't even have to evaluate any square root, this is a purely geometrical result already known to Thales of Miletus.



I don't expect any applause from the world of cosmology. Do you?



And with this you just made things a lot worse than they already were. At least in the previous incarnation you had a valid FLRW universe, granted it was a completely empty universe, but at least it was a valid solution.

If you're now going to claim that

you don't even have a valid solution to the EFE (with isotropy and homogeneity) anymore.

You see, the equation

is valid for every FLRW universe irrespective of the form of the scale factor (accelerating, decelerating,..).

You might want to try deriving the above equation from the general FLRW metric and you'll see that it doesn't depend on the form of

I'm sorry to post again so soon but I need to find out if you and I are going to be spinning our wheels "teaching each other" concepts we should already know. Specifically, here is where I am headed in bringing this up. Where do you think (1+z)^-1 came from? Do you think Friedmann derived this factor? Your post above seems to indicate "yes". Is so, you will be incorrect. By the way, do you actually know what Friedmann derived? Can you post it back to me? Do you think he derived the FRW metric? Do you think that we use the FRW metric to derive other things? What part of the metric is actually Friedmann in nature? And here is where you seem to be way off base. You seem to think that if a dynamical scale factor accelerates slightly as approaching the present era, and thus deviates slightly from (1+z)^-1, that it is eliminated somehow from Friedmann's derivation. In that, you are also incorrect. But do you know why? Can you tell me why the dynamical scale factor can deviate and still fully satisfy the true Friedmann equation? (That is, if you even know what the true Friedmann equation is.) I am trying to determine if you and I can even have a discussion. You are already so far off course with the fundamental model I am proposing. Are we going to spend our time teaching one another?

[caveman1917]

I did not claim this.

You clearly said:

Your answer will be the brand new dynamical scale factor of

(1+z)^-1*sqrt[1+(1+z)^-2].

and

This is something we already know.
{...}
I am not aware of any derivations based on the FLW metric.

If you already know that then how can you not be aware of the derivation?

You seem to be under the mistaken impression that changing the scale factor involves changing the expression. However that expression is true for any valid solution of the field equations under isotropy and homogeneity, if you were aware of its derivation you'd see that those are the only assumptions necessary. And that's why putting the scale factor in terms of redshift is a red herring.

Change that expression and at least one of the following must be false:
1. isotropy
2. homogeneity
3. light follows null geodesics

If however you do decide to keep the scale factor at (and really, how could you not?) then what does

Your answer will be the brand new dynamical scale factor of

(1+z)^-1*sqrt[1+(1+z)^-2].

mean?

[caveman1917]

Where do you think (1+z)^-1 came from?

I know very well where it comes from. In fact i'm sure that anybody who realizes that it equals the ratio of the wavelength when light was emitted versus when it is received[*] can immediately see why it must equal the scale factor (when we, as is convention, set the scale factor at the present time to be 1), since it is perfectly clear that the ratio of wavelengths of light at emittance and reception equals the ratio of the scale factors at those times[**]. All irrespective of how the scale factor actually evolves, wether accelerating or decelerating or even the universe being a huge harmonic oscillator.

You seem to have another opinion on that, so i'm going to ask you, where do you think comes from?

* that's how is defined
** under the 3 conditions given in my previous post

[fcunnane]

I'm just a math yodel but explain to me with two different time scales how anything is "actually" accelerating and not just "appearing" to accelerate due to your addition of non-symetric timing?

[John Mendenhall]

OP, would you please define the location of a point in your 4D universe? x, y, . . . . whatever.

Thank you, John M. This is a very simple question, and I would appreciate a prompt answer.

[PetersCreek]

30-day time limit. Thread closed.