Nobel Prize for Physics 2011 is awarded to Saul Perlmutter, Brian Schmidt, Adam Riess

The Nobel Prize for Physics 2011 has been awarded half to Saul Perlmutter and the second half jointly to Brian Schmidt, Adam Riess, for the discovery the the acceleration of the expansion of the universe using supernovae at high Z. This discovery had been widely predicted as a candidate for a Nobel prize. See for example my predictions for last year.

Saul Perlmutter

Brian Schmidt

Adam Riess

In the late 1990s these astronomers upset the prevalent belief that the universe’s acceleration must be slowing down due to the pull of gravity. They observed the brightness of distant supernovae in the universe, using them as standard candles to gauge distance. A comparison of the redshift and the brightness was found to be more consistent with the view that the rate of expansion is increasing.

The simplest way to model the expansion is to add a cosmological constant term to Einstein’s gravitational field equations. Such a term means that energy is added to space as it expands which must be compensated by negative gravitational energy from the increasing rate of expansion. The popular term dark energy has been used to refer to this mysterious feature of space and time but its origin remains a mystery and the Nobel prize has only been awarded for the discovery of the acceleration, not for dark energy.

Physicists can naively calculate a theoretical value for dark energy from vacuum effects due to known particles. Sadly the result they get is 120 orders of magnitude greater than the value that can account for the accelerating expansion. A smaller value could be obtained if the negative contribution from fermions were to partly cancel the positive contribution from bosons, but to get the observed result would require an imperfect cancellation of two numbers to 120 decimal places and this seems hardly likely to happen by coincidence. It is thought that only a full quantum theory of gravity can resolve this problem.

As a side note it is worth mentioning that Erwin Hubble himself was never awarded a Nobel prize for his discovery of the expanding universe. At the time work in astronomy and cosmology was not considered eligible and Hubble campaigned to have this changed. The Nobel committee relented after his death. Since then there have been a few prizes given for work in astronomy but mostly in relation to fundamental physics, including the discovery of cosmic rays, cosmic neutrinos, the microwave background radiation etc.

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15 Responses to Nobel Prize for Physics 2011 is awarded to Saul Perlmutter, Brian Schmidt, Adam Riess

And, “Physicists can naively calculate a theoretical value for dark energy from vacuum effects due to known particles. Sadly the result they get is 120 orders of magnitude greater than the value that can account for the accelerating expansion.”. Wouldn’t this technically be 60 orders of magnitude? We don’t have any _known_ particles heavier than the top quark.

Good that you watch your quotas, Phil. The URL of this article ends with “risss” so “s” is overrepresented. To compensate for this privilege, the first paragraph contains the words “laat year” where “s” is underrepresented. 😉

This has been anticipated for some time. It is nice to see cosmological work get the prize.

One little point that I have is with the statement about negative energy. The de Sitter spacetime has the equation of state w = -1, which means the pressure p = -ρ is negative, where ρ is the vacuum energy which propels this cosmic expansion. It is not really “negative energy.”

You can express it terms of pressure if you like, but I prefer to use energy conservation simply because a lot of people think that dark energy defies the conservation law. It doesn’t.

The de Sitter spacetime is such that in a patch chosen by the analyst there is conservation of energy. From a first law of thermodynamics perspective dE = dQ – dW the system is adiabatic dQ = 0 and dE = ρdV and dW = pdV and so

dQ = 0 = (ρ + p)dV,

which corroborates with the equation of state w = -1 or p = -ρ. We might think of the pressure as doing “negative work” which cancels out the apparent creation of energy ρdV as the volume of some region increases. So far the data is close to w = -1 — within error bars.

“Wouldn’t this technically be 60 orders of magnitude?”

The de Sitter radius comes out too small by about 60 orders of magnitude, 10^{-35} metres instead of 10^{26} metres. The de Sitter radius is proportional to the reciprocal of the square root of the cosmological constant.

The notion of negative pressures is of course mathematical and physical non-sense. Magnetic fields carry tension along flux lines and this gives to the energy momentum tensor effective negative pressure term. This plus some good principles is a good starting point if one wants to understand situation at deeper level.

TGD actually leads to three different views about dark energy corresponding to articulations in different length scales.

a) The mere imbeddability of critical cosmology to M^4xcP_2 implies negative “pressure term” in energy momentum tensor: this is essentially constraint force induced by imbeddability. This cosmology is fixed apart from a parameter fixing its duration: transformation to sub-critical cosmology is forced by imbeddability.

b) A more microscopic vision is based on the study of solutions of field equations leads to identification of magnetic flux tubes as basic objects: at primordial times they were string like objects X^2xY^2 in M^4xCP_2 and later thicken during cosmic evolution respecting conservation of magnetic flux and total magnetic energy. Magnetic tension gives effective negative pressure and dark energy as magnetic energy.

c) Still more microscopic vision is based on the observation that TGD predicts space-time regions with both Minkowskian and Euclidian signatures o the f induced metric. The latter includes space-time counterparts of generalized Feynman diagrams. In GRT limit the Euclidian regions must have non-vanishing cosmological constant with correct sign in order to obtain CP_2 as as extremal of Einstein-Maxwell action. The space-time average of this very large cosmological constant is small and of correct order of magnitude.

Second and third visions can be consistent because ordinary matter is created as dark energy transforms to visible and dark matter inside magnetic flux tubes and leads to formation of galaxies, stars, etc…

By the way, p-adic physics have been now independently discovered: Susskind did it;-)! Congratulations for the happy discoverer;-). I did it already for almost two decades ago and have written 15 books applying p-adic physics in different contexts (see my blog posting).

P-adic physics is fundamentally linked to Feigenbaum’s route to chaos of nonlinear dynamical systems. QFT is based on nonlinear Yang-Mills fields and the relevance of the Feigenbaum’s universality for hep-th has been discussed in many papers, for example,

“A comparison of the redshift and the brightness was found to be more consistent with the view that the rate of expansion is increasing.”

I have to question whether the observations show the rate of expansion is increasing now. The supernovas may appear dimmer than they should be for their redshift in some interpretation, but could it not be that there was an acceleration in the past, which may or may not be continuing now? So if for example the universe expands in cycles (for example due to cyclical matter creation which offsets the expansion in the matter creation region), then there would be a period of acceleration and a period of deceleration, etc., and depending on where we are in the cycle, we would see distant supernovas farther away, but the acceleration period may be over. So the increasing rate of expansion is subject to interpretation, not an observation.

Re the post on negative gravitational energy, you appear to be saying that the gravitational potential energy increases with expansion, cancelling out the expansion pressure. But if gravity is expressed in general relativity as a “distortion of the space-time continuum” due to the presence of energy, and the universe is filled with energy, then how could any potential be measured? It is easy to measure ten feet on a ladder at the surface of Earth, but how do you measure it when ten feet is becoming eleven feet? I assume the cosmological energy conservation equation is mainly for the purpose of keeping the books straight, to square with the conservation law.

According to that equation, the (net) energy of the universe is zero. Is that right? Then the question of where the energy of the universe comes from is not answered, but made more complicated, since it appears there are now several kinds of energy whose origin is not explained by science.

As a more general speculation, it seems to me there is something redundant in the “spacetime-energy complex”, especially when it has to describe a set of universal initial conditions, which could be eliminated if matter is modelled as a particular motion of space, in which case spacetime and energy are the same thing, and matter is a particular form of spacetime. This is because to say that “there is a distortion of the space-time continuum” is the same as saying that “space moves”. (This allows a model in which space rotates in a particular way, to absorb pressure created in an expanding spatial field which expands at a constant rate at its limit.) Then the universal initial condition is reduced to spacetime (or energetic space).

With these comments I hope not to offend anyone whose education is many powers greater than mine (as well as awesome in its complexity). But the universe may be idiotically simple at its root.

It is surprised to hear that the three U.S. physicists had won the Nobel Prize in physics for discovering through the study of supernovae that the universe is accelerating expansion.
The Hubble’s redshift and the Doppler Effect are the facts. It doesn’t mean the universe is expansion or universe is accelerating expansion.
The expansion of the universe is based on the true of the Big Bang theory. If the Big Bang theory is not true. Even though the Hubble’s redshift, and the Doppler Effect is the facts. The expansion or accelerating expansion of the universe is not supported.
The Big Bang theory and “dark energy” are not just a crazy idea, it is nonsense. Is anyone believe the “dark energy” able to create the real energy to push the supernovae accelerating expansion? If you do believe it. You may believe “dark human” can turn spirit to a real person.
In Scientific community , the Big Bang and ”dark energy” issue had been argue for so many year, And now act rashly to award the Nobel Prize in physics for the universe is accelerating expansion .I believe that someday will be proved the 2011 of the Nobel Prize for physics turn out to be a joke.
The expansion of Universe should not true. If it is true, the Big Bang theory and “dark energy” will be true.
Now raise a big question. If he Big Bang theory and “dark energy” are not true. Also the Hubble redshift, and the Doppler Effect is the facts, How could be proved the universe is not expansion or not accelerating expansion?
The space of the universe has only three kind of possibility. One is Euclidean space, one is elliptical space and the other is hyperbolic space. These three kinds of space can only be hypnosis to be one of the three only, And, it cannot be identify by proved.
1. If the universe is a Euclidean space, due to the Hubble redshift, and the facts of the Doppler Effect, There must be the expansion of the universe and the Big Bang also true. This is contradiction to the Big Bang is not true. So, the space of the universe may not be a Euclidean space
2. The space of the universe will never be an Elliptical space. If the space of the Universe is an Elliptical space. When we see anything from the east, can be seen from the west too. Obviously, it is not so
3. Remaining space is The Hyperbolic space.

Let analyze, if supernovae happen in Hyperbolic space.

Hubble’s laws are derived from Euclidean rules and Euclidean formulas. However, assume the Universe is in Hyperbolic space. Very logically, we must derive its rules and formulas from Hyperbolic rules and non-Euclidean formulas.

The rules and formulas of Hyperbolic space are quite different from Euclidean space. Hence, the results derived from utilizing these two systems must be different. These differences may be the keys to unveil the mystery of the Universe.
“Now we try to prove, the universe is not expansion or not accelerating expansion. Even though Hubble’s redshift and the Doppler effect are the facts”

A. LIGHT SPHERICAL WAVE FRONTS

When photon travel a distance of r. The equation of a light spherical front in Euclidean space is

x2 + y2 + z2 = r2 ————– (1)

From Hyperbolic geometry, the equation of the light spherical front is

Where k is the constant of the space curvature. （cosmological constant）
(From page 298 of non-Euclidean Geometry by Allen Liou, 1964.)

Comparing equations (1) and (2), we can see very obviously that the area of the light spherical fronts is very much different. Even though they have the same radius. Therefore, the Doppler Effect should not be the same between Euclidean space and Hyperbolic space.

The area of the Light Spherical Front in Euclidean space is 4πr2.
What is the area of the Light Spherical Front in Hyperbolic space?

Let us determine the circumference of a circle in Hyperbolic space first:

Let PQ be the chord of a circle of radius r, which subtends an angleθ, M be the midpoint of the chord, and O be the center of the circle.
See pic 1.

From the formula of the right-angle in Hyperbolic trigonometry, we have (page 143 of non-Euclidean Geometry by Allen Liou, 1964.)

sinh PQ/2k = sinh r/k sin∠POQ/2

If angle θ 0
We have ds/2k = sinh r/k dθ/2
or ds = k sinh r/k dθ

Integrating both sides, we have

Circumference = 2πk sinh r/k

Then, let ds are the length of the arc of the spherical circle, and r be the radius.
By same formula, we have, see pic 2.

ds = k sinh r/k dθ

The area of the circle strip is
d (area of circle strip) = 2πk sinh AM/k ds

But
sinh AM/k = sinh r/k sinθ

Therefore
d(area of circle strip)=2πk[sinh r/k sinθ][k sinh r/k dθ]
= 2πk 2 sinh2 r/k dθ

Integrating both sides, we have
area of sphere = 4πk 2 sinh2 r/k

B. DOPPLER EFFECT OR “LIOU’S STRETCH EFFECT”

When a photon travels a distance r, the area of the Light Spherical Front in Euclidean space is 4πr2.

area of sphere = 4πr 2

But the area of the Light Spherical Front in Hperbolic space is

area of sphere = 4πk 2 sinh2 r/k

Compare the two Spherical Areas in the two different spaces with the same r. We easily to see that, if we are in Hyperbolic universe, Light Spherical Front stretch from 4πr 2 to 4πk 2 sinh2 r/k. We temporary called this stretch by “Liou’s stretch effect”.

The Spherical Front of photon may only travel a distance r in Hyperbolic space. But in Euclidean space, it appears to travel a distance of k sinh r/k.

From the difference of r and k sinh r/k in Euclidean space, it looks like the object moves from point r to point k sinh r/k, but the object actually stays still in Hyperbolic universe.

When we use Redshift of Doppler Effect in Euclidean space to calculate velocity of galaxy from point r to point k sinh r/k. Actually, there is no movement from point r to point k sinh r/k. It only cause by the stretch of curvature of Hyperbolic space (“Liou’s stretch effect”).

I will use the velocity to calculate the space constant (cosmological constant). Use redshift of frequency the result is the same.

D. CALCULATION OF SPACE CURVATURE (OR COSMOLOGICAL CONSTANT) IN HYPERBOLIC UNIVERSE

Let s = k sinh r/k – r

Where s is the distant of galaxy moving from position r to the position k sinh r/k.
Taking the derivative of both sides, we have

ds/dt = cosh r/k dr/dt – dr/dt

where ds/dt =v (the velocity of galaxies at the remote distance of r), and dr/dt is the speed of light c.

Therefore
v = cosh r/k c – c

v = c[cosh r/k – 1] = 2c sinh2 r/2k ———- (3)

There are several versions of the Hubble’s constant. We will select the one most popular one. In which, the velocity of galaxies at a distance of six billion light-years move away at a velocity of roughly 90,000 kilometers/sec.
Hence v=90,000 kilometers/sec and r=6 bly.

Hence we have

v = 90,000 kilometers/sec and r = 6 bly.

Therefore
90,000 = 2c sinh2（6bly/2k ）

∴ 0.3c = 2c sinh2（3bly/k ）

∴ k = 3bly/ sinh-1√0.12

∴ k = 7.931965828 bly

Where bly is billion light-years.

C. DISCUSIONS

1. Hubble’s constant was not constant.

From equation (3), v = 2c sinh2 r/2k , the velocity of galaxies and the remote distance of r were not exactly linear proportions in Euclidean universe. The velocity is more likely in slightly acceleration observed in Euclidean universe.

Here, the cosmological constant, k = 7.931965828 bly was based on the Hubble’s Law at 6 bly. If we based on a difference distance, like one on a distance of 1 or 2 bly, the k value should be slightly different. If we use different versions of Hubble’s Law, the cosmological constant k will be even more different. We really need an accurate data to determine the constant k

Assuming k = 7.931965828 bly is correct; the Hubble’s diagram in Euclidean space should look like the following diagram.

From this chart, we should call Hubble’s accelerator instead of Hubble’s constant.

2. Is Universe’s redshift cause by DOPPLER EFFECT or “LIOU’S STRETCH EFFECT”?

From Hubble’s Law, the speed by which a galaxy moves away is proportional to the distance to the galaxy. A galaxy with distance of 6 bly has a velocity of 90,000 km/s. For a galaxy 30 bly away, its speed will be 450,000 km/s. This is beyond the speed of light a lot. It is contradict to the fact of the speed of light is constant.
In recent year, astronomer’s observed that Hubble’s constant is not constant. The galaxies actually moving away accelerated, like supernovae.

From this two facts, the Universe’s redshift is more likely to be caused by the “LIOU’S STRETCH EFFECT”.
. And Doppler Effect caused by the STRETCH of the Hyperbolic space (“LIOU’S STRETCH EFFECT”) not by the speed of velocity in Euclidean space.

So, the universe is not expansion nor accelerating expansion. Even though Hubble’s redshift and the Doppler effect are the facts.

In other words,The universe is still in Hyperbolic space

Lets think critically about the analysis of this effect. The data indicates at most a 5% departure from a flat and empty universe. The Friedmann coordinates are artificially curved, and still noninertial even when normalized. Fictitious gravity and fictitious sources of gravity dominate a naive analysis using Friedmann coordinates.

Congratulations, well deserved!

Lubos Motl wrote a post with some more details on the quantum contributions to the CC in February:

http://motls.blogspot.com/2011/02/enigmatic-cosmological-constant.html

And, “Physicists can naively calculate a theoretical value for dark energy from vacuum effects due to known particles. Sadly the result they get is 120 orders of magnitude greater than the value that can account for the accelerating expansion.”. Wouldn’t this technically be 60 orders of magnitude? We don’t have any _known_ particles heavier than the top quark.

Good that you watch your quotas, Phil. The URL of this article ends with “risss” so “s” is overrepresented. To compensate for this privilege, the first paragraph contains the words “laat year” where “s” is underrepresented. 😉

I was in a rush to be the first to blog the result, 🙂

This has been anticipated for some time. It is nice to see cosmological work get the prize.

One little point that I have is with the statement about negative energy. The de Sitter spacetime has the equation of state w = -1, which means the pressure p = -ρ is negative, where ρ is the vacuum energy which propels this cosmic expansion. It is not really “negative energy.”

You can express it terms of pressure if you like, but I prefer to use energy conservation simply because a lot of people think that dark energy defies the conservation law. It doesn’t.

The de Sitter spacetime is such that in a patch chosen by the analyst there is conservation of energy. From a first law of thermodynamics perspective dE = dQ – dW the system is adiabatic dQ = 0 and dE = ρdV and dW = pdV and so

dQ = 0 = (ρ + p)dV,

which corroborates with the equation of state w = -1 or p = -ρ. We might think of the pressure as doing “negative work” which cancels out the apparent creation of energy ρdV as the volume of some region increases. So far the data is close to w = -1 — within error bars.

“Wouldn’t this technically be 60 orders of magnitude?”

The de Sitter radius comes out too small by about 60 orders of magnitude, 10^{-35} metres instead of 10^{26} metres. The de Sitter radius is proportional to the reciprocal of the square root of the cosmological constant.

The notion of negative pressures is of course mathematical and physical non-sense. Magnetic fields carry tension along flux lines and this gives to the energy momentum tensor effective negative pressure term. This plus some good principles is a good starting point if one wants to understand situation at deeper level.

TGD actually leads to three different views about dark energy corresponding to articulations in different length scales.

a) The mere imbeddability of critical cosmology to M^4xcP_2 implies negative “pressure term” in energy momentum tensor: this is essentially constraint force induced by imbeddability. This cosmology is fixed apart from a parameter fixing its duration: transformation to sub-critical cosmology is forced by imbeddability.

b) A more microscopic vision is based on the study of solutions of field equations leads to identification of magnetic flux tubes as basic objects: at primordial times they were string like objects X^2xY^2 in M^4xCP_2 and later thicken during cosmic evolution respecting conservation of magnetic flux and total magnetic energy. Magnetic tension gives effective negative pressure and dark energy as magnetic energy.

c) Still more microscopic vision is based on the observation that TGD predicts space-time regions with both Minkowskian and Euclidian signatures o the f induced metric. The latter includes space-time counterparts of generalized Feynman diagrams. In GRT limit the Euclidian regions must have non-vanishing cosmological constant with correct sign in order to obtain CP_2 as as extremal of Einstein-Maxwell action. The space-time average of this very large cosmological constant is small and of correct order of magnitude.

Second and third visions can be consistent because ordinary matter is created as dark energy transforms to visible and dark matter inside magnetic flux tubes and leads to formation of galaxies, stars, etc…

For a short summary see this article.

See also the blog postings Do we really understand solar system

and

Cosmic evolution as transformation of dark energy to matter.

By the way, p-adic physics have been now independently discovered: Susskind did it;-)! Congratulations for the happy discoverer;-). I did it already for almost two decades ago and have written 15 books applying p-adic physics in different contexts (see my blog posting).

P-adic physics is fundamentally linked to Feigenbaum’s route to chaos of nonlinear dynamical systems. QFT is based on nonlinear Yang-Mills fields and the relevance of the Feigenbaum’s universality for hep-th has been discussed in many papers, for example,

http://iopscience.iop.org/0295-5075/82/1/11001

http://www.ejtp.com/articles/ejtpv7i23p75.pdf

http://www.worldscinet.com/ijbc/18/1803/S0218127408020756.html

These are comments of a non-scientist.

“A comparison of the redshift and the brightness was found to be more consistent with the view that the rate of expansion is increasing.”

I have to question whether the observations show the rate of expansion is increasing now. The supernovas may appear dimmer than they should be for their redshift in some interpretation, but could it not be that there was an acceleration in the past, which may or may not be continuing now? So if for example the universe expands in cycles (for example due to cyclical matter creation which offsets the expansion in the matter creation region), then there would be a period of acceleration and a period of deceleration, etc., and depending on where we are in the cycle, we would see distant supernovas farther away, but the acceleration period may be over. So the increasing rate of expansion is subject to interpretation, not an observation.

Re the post on negative gravitational energy, you appear to be saying that the gravitational potential energy increases with expansion, cancelling out the expansion pressure. But if gravity is expressed in general relativity as a “distortion of the space-time continuum” due to the presence of energy, and the universe is filled with energy, then how could any potential be measured? It is easy to measure ten feet on a ladder at the surface of Earth, but how do you measure it when ten feet is becoming eleven feet? I assume the cosmological energy conservation equation is mainly for the purpose of keeping the books straight, to square with the conservation law.

According to that equation, the (net) energy of the universe is zero. Is that right? Then the question of where the energy of the universe comes from is not answered, but made more complicated, since it appears there are now several kinds of energy whose origin is not explained by science.

As a more general speculation, it seems to me there is something redundant in the “spacetime-energy complex”, especially when it has to describe a set of universal initial conditions, which could be eliminated if matter is modelled as a particular motion of space, in which case spacetime and energy are the same thing, and matter is a particular form of spacetime. This is because to say that “there is a distortion of the space-time continuum” is the same as saying that “space moves”. (This allows a model in which space rotates in a particular way, to absorb pressure created in an expanding spatial field which expands at a constant rate at its limit.) Then the universal initial condition is reduced to spacetime (or energetic space).

With these comments I hope not to offend anyone whose education is many powers greater than mine (as well as awesome in its complexity). But the universe may be idiotically simple at its root.

Congratuation to awarded to Saul Perlmutter, Brian Schmidt, Adam Riess!

It is surprised to hear that the three U.S. physicists had won the Nobel Prize in physics for discovering through the study of supernovae that the universe is accelerating expansion.

The Hubble’s redshift and the Doppler Effect are the facts. It doesn’t mean the universe is expansion or universe is accelerating expansion.

The expansion of the universe is based on the true of the Big Bang theory. If the Big Bang theory is not true. Even though the Hubble’s redshift, and the Doppler Effect is the facts. The expansion or accelerating expansion of the universe is not supported.

The Big Bang theory and “dark energy” are not just a crazy idea, it is nonsense. Is anyone believe the “dark energy” able to create the real energy to push the supernovae accelerating expansion? If you do believe it. You may believe “dark human” can turn spirit to a real person.

In Scientific community , the Big Bang and ”dark energy” issue had been argue for so many year, And now act rashly to award the Nobel Prize in physics for the universe is accelerating expansion .I believe that someday will be proved the 2011 of the Nobel Prize for physics turn out to be a joke.

The expansion of Universe should not true. If it is true, the Big Bang theory and “dark energy” will be true.

Now raise a big question. If he Big Bang theory and “dark energy” are not true. Also the Hubble redshift, and the Doppler Effect is the facts, How could be proved the universe is not expansion or not accelerating expansion?

The space of the universe has only three kind of possibility. One is Euclidean space, one is elliptical space and the other is hyperbolic space. These three kinds of space can only be hypnosis to be one of the three only, And, it cannot be identify by proved.

1. If the universe is a Euclidean space, due to the Hubble redshift, and the facts of the Doppler Effect, There must be the expansion of the universe and the Big Bang also true. This is contradiction to the Big Bang is not true. So, the space of the universe may not be a Euclidean space

2. The space of the universe will never be an Elliptical space. If the space of the Universe is an Elliptical space. When we see anything from the east, can be seen from the west too. Obviously, it is not so

3. Remaining space is The Hyperbolic space.

Let analyze, if supernovae happen in Hyperbolic space.

Hubble’s laws are derived from Euclidean rules and Euclidean formulas. However, assume the Universe is in Hyperbolic space. Very logically, we must derive its rules and formulas from Hyperbolic rules and non-Euclidean formulas.

The rules and formulas of Hyperbolic space are quite different from Euclidean space. Hence, the results derived from utilizing these two systems must be different. These differences may be the keys to unveil the mystery of the Universe.

“Now we try to prove, the universe is not expansion or not accelerating expansion. Even though Hubble’s redshift and the Doppler effect are the facts”

A. LIGHT SPHERICAL WAVE FRONTS

When photon travel a distance of r. The equation of a light spherical front in Euclidean space is

x2 + y2 + z2 = r2 ————– (1)

From Hyperbolic geometry, the equation of the light spherical front is

tanh2 x/k + tanh2 y/k + tanh2 z/k = tanh2 r/k —— (2)

Where k is the constant of the space curvature. （cosmological constant）

(From page 298 of non-Euclidean Geometry by Allen Liou, 1964.)

Comparing equations (1) and (2), we can see very obviously that the area of the light spherical fronts is very much different. Even though they have the same radius. Therefore, the Doppler Effect should not be the same between Euclidean space and Hyperbolic space.

The area of the Light Spherical Front in Euclidean space is 4πr2.

What is the area of the Light Spherical Front in Hyperbolic space?

Let us determine the circumference of a circle in Hyperbolic space first:

Let PQ be the chord of a circle of radius r, which subtends an angleθ, M be the midpoint of the chord, and O be the center of the circle.

See pic 1.

From the formula of the right-angle in Hyperbolic trigonometry, we have (page 143 of non-Euclidean Geometry by Allen Liou, 1964.)

sinh PQ/2k = sinh r/k sin∠POQ/2

If angle θ 0

We have ds/2k = sinh r/k dθ/2

or ds = k sinh r/k dθ

Integrating both sides, we have

Circumference = 2πk sinh r/k

Then, let ds are the length of the arc of the spherical circle, and r be the radius.

By same formula, we have, see pic 2.

ds = k sinh r/k dθ

The area of the circle strip is

d (area of circle strip) = 2πk sinh AM/k ds

But

sinh AM/k = sinh r/k sinθ

Therefore

d(area of circle strip)=2πk[sinh r/k sinθ][k sinh r/k dθ]

= 2πk 2 sinh2 r/k dθ

Integrating both sides, we have

area of sphere = 4πk 2 sinh2 r/k

B. DOPPLER EFFECT OR “LIOU’S STRETCH EFFECT”

When a photon travels a distance r, the area of the Light Spherical Front in Euclidean space is 4πr2.

area of sphere = 4πr 2

But the area of the Light Spherical Front in Hperbolic space is

area of sphere = 4πk 2 sinh2 r/k

Compare the two Spherical Areas in the two different spaces with the same r. We easily to see that, if we are in Hyperbolic universe, Light Spherical Front stretch from 4πr 2 to 4πk 2 sinh2 r/k. We temporary called this stretch by “Liou’s stretch effect”.

The Spherical Front of photon may only travel a distance r in Hyperbolic space. But in Euclidean space, it appears to travel a distance of k sinh r/k.

From the difference of r and k sinh r/k in Euclidean space, it looks like the object moves from point r to point k sinh r/k, but the object actually stays still in Hyperbolic universe.

When we use Redshift of Doppler Effect in Euclidean space to calculate velocity of galaxy from point r to point k sinh r/k. Actually, there is no movement from point r to point k sinh r/k. It only cause by the stretch of curvature of Hyperbolic space (“Liou’s stretch effect”).

I will use the velocity to calculate the space constant (cosmological constant). Use redshift of frequency the result is the same.

D. CALCULATION OF SPACE CURVATURE (OR COSMOLOGICAL CONSTANT) IN HYPERBOLIC UNIVERSE

Let s = k sinh r/k – r

Where s is the distant of galaxy moving from position r to the position k sinh r/k.

Taking the derivative of both sides, we have

ds/dt = cosh r/k dr/dt – dr/dt

where ds/dt =v (the velocity of galaxies at the remote distance of r), and dr/dt is the speed of light c.

Therefore

v = cosh r/k c – c

v = c[cosh r/k – 1] = 2c sinh2 r/2k ———- (3)

There are several versions of the Hubble’s constant. We will select the one most popular one. In which, the velocity of galaxies at a distance of six billion light-years move away at a velocity of roughly 90,000 kilometers/sec.

Hence v=90,000 kilometers/sec and r=6 bly.

Hence we have

v = 90,000 kilometers/sec and r = 6 bly.

Therefore

90,000 = 2c sinh2（6bly/2k ）

∴ 0.3c = 2c sinh2（3bly/k ）

∴ k = 3bly/ sinh-1√0.12

∴ k = 7.931965828 bly

Where bly is billion light-years.

C. DISCUSIONS

1. Hubble’s constant was not constant.

From equation (3), v = 2c sinh2 r/2k , the velocity of galaxies and the remote distance of r were not exactly linear proportions in Euclidean universe. The velocity is more likely in slightly acceleration observed in Euclidean universe.

Here, the cosmological constant, k = 7.931965828 bly was based on the Hubble’s Law at 6 bly. If we based on a difference distance, like one on a distance of 1 or 2 bly, the k value should be slightly different. If we use different versions of Hubble’s Law, the cosmological constant k will be even more different. We really need an accurate data to determine the constant k

Assuming k = 7.931965828 bly is correct; the Hubble’s diagram in Euclidean space should look like the following diagram.

From this chart, we should call Hubble’s accelerator instead of Hubble’s constant.

2. Is Universe’s redshift cause by DOPPLER EFFECT or “LIOU’S STRETCH EFFECT”?

From Hubble’s Law, the speed by which a galaxy moves away is proportional to the distance to the galaxy. A galaxy with distance of 6 bly has a velocity of 90,000 km/s. For a galaxy 30 bly away, its speed will be 450,000 km/s. This is beyond the speed of light a lot. It is contradict to the fact of the speed of light is constant.

In recent year, astronomer’s observed that Hubble’s constant is not constant. The galaxies actually moving away accelerated, like supernovae.

From this two facts, the Universe’s redshift is more likely to be caused by the “LIOU’S STRETCH EFFECT”.

. And Doppler Effect caused by the STRETCH of the Hyperbolic space (“LIOU’S STRETCH EFFECT”) not by the speed of velocity in Euclidean space.

So, the universe is not expansion nor accelerating expansion. Even though Hubble’s redshift and the Doppler effect are the facts.

In other words,The universe is still in Hyperbolic space

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Lets think critically about the analysis of this effect. The data indicates at most a 5% departure from a flat and empty universe. The Friedmann coordinates are artificially curved, and still noninertial even when normalized. Fictitious gravity and fictitious sources of gravity dominate a naive analysis using Friedmann coordinates.