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Dark Energy Force


Thought-experiment provides a formula for (new) dark energy force.


Author: Dan Visser (independent cosmologist and Art-painter), Almere, the Netherlands.

Publication-dates: Version-1 was submitted in October 7 2010, in retrospective to the first paper of Dan Visser and Christopher Forbes in the Vixra-archive about a possible new cosmology. Then version-2 needed a textual revision after the publication of several new papers, wherein I gave a better explanation without altering the mathematical content of the original ‘dark energy force-formula’. Version-3 again needed a better textual explanation for the benefit of entropy-gravity, of course without altering the mathematics.

Date: Version-3 November 25 2013.


A (new) ‘dark energy force-formula’ was introduced on April 10 2004 by Dan Visser, Almere, the Netherlands on his website (reference 1a). The (new) formula was picked up by a PhD-mathematician and -Physics, Christopher Forbes (UK), in the summer of 2009, leading to email-contact among them, and resulting in a publication of a general mathematical expression, whereof Dan’s (new) ‘dark energy force-formula’ indeed resulted from the general expressing (reference 2). Additional an amount of (new) dark energy resulted too. Afterwards Dan published his derivation of the (new) force in the Vixra-archive on October 7 2010 (in retrospective). He called his mathematical exercises a ‘thought-experiment’. That had as result the (new) ‘dark energy force-formula’. In the versions following (version-2 and -3) only textual changes were made for servicing a better understanding of the thought-experiment, of course without altering the original mathematical content. Further development was focussed on working–out the (new) dynamics in the Double Torus hypothesis. This is a hypothesis that claims the Big Bang cosmology can no longer be maintained as the model for the universe. The main issue in ‘Dan’s-thought-experiment’ is a scaling-away-principle’,which is characterized by ‘scaling-away’ two black holes from each other (small and large), as well as ‘melting them together’ and making them ‘temperature connected’. Then an observer would receive the same evaporation-radiation from both black holes at the same time. These three exercises (scaling-away, melting-together and temperature-connection) had the aim to keep entropy conserved in the universe. However, ‘scaling-away’ causes a ‘change of dark information’ at the surface of one or the other black hole. Such a change had to correspond to a ‘new force’. That was further exercised, analyzed, synthesized, combined and translated as a mathematical thought-experiment. The result: Dan’s (new) ‘dark energy force-formula’.


My thought-experiment was described in April 4 2004 and put on my website on April 10 2004. The revision of version-1 was October 7 2010 and was aimed on a better textual-explanation, without altering the mathematical ‘thought-experiment’. Still the fundamental mathematical derivations are unchanged and still actual.

I started with Hawking's prediction of 1974, who claimed a large black hole could evaporate very slowly and a small black hole, called a Planck hole - which is the smallest possible black hole in Big Bang cosmology - would evaporate in a flash. I had a faint idea that such a process might introduce 'extra time’, so I started writing my simple mathematical derivations, which  became ‘my thought-experiment’.

I combined the ‘scaling-away’ of a small- and large black hole ‘from each other’, while an observer had to be capable of receiving equal amounts of evaporation-radiation from both black holes at the same time. Although the evaporation-radiation was assumed to be thermic-radiation, I had a the idea it should affect 'dark information' probably by ‘extra time’.

In my vision the ‘scaling-away principle’ changed the 'dark-information' at the surface of these black holes. So ‘such a scaling-away movement’ should cause the ‘same change’ of dark information’. In order to get two equations with one unknown ‘dark information-change’ parameter, and ‘melting them together’, while connecting them in temperature, could achieve that ‘conservation of entropy’. So, I translated this in a mathematical exercise, which I called my ‘thought-experiment’. The result was my (new) ‘dark energy force-formula’.

It was a different force from forces known in the Big bang cosmology, because a  ‘time-extension’ was introduced. That implied the introduction of a wider universe. Then my )new) force was picked-up from the internet (from my website) by Christopher Forbes (UK) in the summer of 2009. He identified himself as a British PhD mathematician and physicist (also Fellow of the Royal Astronomical Society). Soon he came forward with a general mathematical equation, whereof my (new) ‘dark energy force formula’ appeared to be a solution! But also an amount of (new) dark energy came out of the equation. Besides,  my dark energy force formula appeared to have a ‘+’ and (‘-‘) sign. My ‘scaling-away-principle’ could be associated with the ‘-‘ sign, so the ‘+’ sign must have the meaning of ‘scaling-getting-closer’. I had not recognized that during the exercise of my thought-experiment. So, Forbes and I decided to publish the results in the Vixra-archive in co-authorship. That publication was of September 1 2009, followed by two other publications in co-authorship. The rest of the publications afterwards, I did on my own in a solitary way without any connection to any institute. The publications convinced me of additional theoretical evidence for the existence of a wider universe than the Big Bang. However, although my website already comprehended the derivation of my (new) ‘dark energy force formula’, I decided to publish my ‘dark energy force formula’ in the Vixra-archive afterwards (or so to say: in retrospective) in order to synchronize our actions and future developments: Reference: http://vixra.org/abs/1010.0013). All the papers published afterwards were of my hand only, because my contact with Christopher Forbes 'faded away slowly' (since 2010, and ending in the beginning of 2011). One of his last messages to me gave me still hope he would be the one to formulate an extensive mathematical framework, called  the Triple Torus Topology ('triple' because mathematical boundaries should determine the physical boundaries), or the Forbes-Visser Model (F-V model). However, until now I have never heard of him since.

So, I ‘scaled away’ from each other two differently sized black holes, a small and a large one, in order to obtain an equal amount of evaporation-radiation from both, in order to make possible a simultaneous detection by an observer. The evaporation-radiation exists of Hawking-radiation, which is related to S=¼ A; S is the entropy and A the amount of Planck-surfaces. Hawking claimed his radiation is thermic.

However, quantumdynamics demands that information can never been lost in the universe, even after having been disappeared in black holes. Although this is acknowledged by theoretical analysis, it is a paradox to some scientists. However, new mathematics in loop-quantum-gravity also postulates that the forces in the atom forbid the forming of singularities (a singularity is just an infinite small point with an infinite large energy).

That means that ‘fundamental information’ should be maintained in the universe. However, whether this is a paradox or not, it is not relevant for this ‘thought-experiment’. The purpose is to “scale-away” black holes from each other for finding a new dynamical action, which I named: "dark energy force". The ‘dark energy force’ might reveal sub-quantum-information. Maybe it could be ‘fundamental information by ‘time affecting quantum-gravity’. ‘Sub’ in this context means information from below the Planck scale.

The only fundamental way to measure ‘scaled away black holes from each other’, is by their ‘temperature’! We are also used to do this with light. Every light-wave has its own typical temperature. However, Hawking-radiation is rather difficult to observe. So, I used a ‘refined chance-principle’ connected to ‘temperature, which I implemented in the rest of the exercise.

The issue here is thus: An observer sees the same entropy for both black holes. However, not without excitation of a ‘new force’. In fact it is the conservation of entropy and that cannot be reach without a ‘new force’. The force is ‘dark’. So: A new dark energy force.

One can ask: Why would you do that? Well, In the beginning of a Big bang the entropy must have been maximal, but at the end of an ever expanding space-time after the Big Bang the entropy is maximal again. So it looks as if the entropy in the beginning was the end of the entropy of a Big Bang that happened before. In some way entropy seemed to be conserved. Anyhow, that was my motive. But I couldn’t express this in version-1.

Whatever, from another historical perspective movement of energy was observed as ‘going from warm to cold areas’, but new insights learned that ‘lost information’ was defined at the surfaces of event-horizons of black holes. More ‘lost-information’ inevitably would lead to a higher entropy (S). Hawking derived this as S=4(pi)m², where (m) is the mass of the black hole. This can be rewritten, as follows:

The surface of a globe is A=4(pi)r², with r=2m, where r is the radius of the event-horizon. The result becomes A=4(pi).(2m)²=16(pi)m². Comparing this with S=4(pi)m², we find S=1/4 A (where A is the amount of Planck surfaces). However, entropy must be without dimensions, so (S) must be divided by the elementary Planck-surface Oe.

Meanwhile the event-horizon of a black hole becomes smaller, due to the evaporation of its surface. Small black holes evaporate faster and more intensively than large ones. The temperature of the black hole (Ts) is proportional to the gravity of a black hole: Ts ~ Fz ~ m/r² ~ m/m² ~ (1/m).       

Starting my “thought-experiment”.

I took the following product:

[a light-way (ct) from myself up to the light-horizon of a black hole] x [the distance (s = 0,5 rs) from the light-horizon up to the event-horizon of the black hole in order to “observe” the evaporation of two black holes simultaneously (large and small) through a kind of entanglement within the observer].

It is not possible to look beyond the light-horizon of a black hole, but still there must be an unknown chance to observe this and even more deeper. This “chance” starts at (ct).(s) / (ct) + (s). However, within a black hole the total of comparable chances is (ct) + (s) = 1, so the chance will be (ct).(s). This could carry out “more detailed chances” than is known from quantum mechanics.

Therefore I relate the “temperature” to this chance (ct).(s):

Ts ~ ct . 0.5 rs                                                         (1)

However, this chance must be also combined with Ts ~ 1/m to connect with the temperature of a black hole as a complete physical system. The result is:

Ts ~ (ct. 0.5 rs). 1/m

From this follows:

ct ~ (2 m/rs). Ts

From this follows:

ct ~ 2. (1/2.c²/G) . Ts

ct ~ c²/G . Ts                                                          (2)

According to S=4(pi)m², the entropy S at the surface of a black hole is proportional to m². This means as soon as two equally sized black holes form one black hole, the event-surface becomes 2x larger, while the mass only increases with a factor 21/2 . I call this effect 1, which propagates (2 - 21/2 ) m = 1.4 m. This affects (ct) to the observer.



On the other hand this effect leads to a specific analysis of dark energy and dark matter as follows. I call this effect 2:

I take the ratio of the black hole surface A and the black hole mass m, defined as A/m. This ratio A/m is constant for as well a single black hole as for two black holes put together. According to the afore effect 1, the ratio becomes larger with a factor 1.4, only if the two black holes are put together. Compared to the original ratio then follows: 1.4 (A/m) - A/m = 0.4 =40%. This is 40% had to escape via the black hole surface, leaving behind 60% in the larger black hole. The escaping energy must be dark energy with an anti-gravitational property. So, an anti-gravitational dark energy of 40 / 60 = 2/3 stays connected to the combined black holes. Consequently 1/3 must be identified as dark matter with a gravitational property.

Conclusion: The basic ratio of darke energy / dark matter is defined as 2 : 1. This means basicly 66% is dark energy and 33 % is dark matter. The fact that nowadays 73% dark energy is observed (calculated) and 23% dark matter (observed and calculated) is due to an unkwown dynamic in the big bang. This includes that the big bang also might be part of an other cosmological model.
Back to the effect 1, this results in:
ct~m.(2 - 21/2)                                                     (3)          
Now both sides in the proportional-formula (3) are divided by rs (the schwarzschild-radius):
ct/ rs ~ (m/rs ).(2-21/2) and because m/rs can be rewritten in 1/2 (c²/G), the result is:
ct/ rs ~ 1/2 (c²/G) . (2-21/2)
ct/ rs ~ c²/G - (0.5 .21/2). c²/G
c²/G ~ ct/ rs + (0.5 . 21/2).c²/G
substitution in (2) results in:
         ct       (0,5 .21/2).c²
ct ~ { _   +   _________ } . Ts
          rs               G
        Ts ctG + rs Ts (0,5 .21/2 ).c²
ct ~ _____________________
                         rs G
         2Ts ctG + rs Ts c² 21/2
ct ~ ___________________
                       2rs G
2 rs G ct ~ 2Ts ctG + rs Ts c² 21/2
2 rs G ct - 2Ts ctG ~ rs Ts c² 21/2
2t (rs G c - Ts c G) ~ rs Ts c² 21/2
        rs Ts c² 21/2
2t ~ ________
      rs G c - Ts c G
        rs Ts c² 21/2
t~  -__________                                                 (4)                                                     
       2 G c (rs - Ts )
This time (t) to observe radiation from both black holes. Whether this is a large or small black hole depends on rs and Ts. For rs >> Ts (which is a large black hole) follows:
       0.5 c 21/2 .Ts
t ~ ____________
The restriction means: rs =2mG/c² >> Ts , so, m >> 0.5 . (c²/G) . Ts.
But because Ts ~ 1/m than follows m >> 0,5 . (c²/G) . 1/m.
So, than the restriction changes in :
m² >> 0.5 . (c²/G) which means m² >> 0.5 . 1,36 . 1027 >> 0,068 . 1028
This means one sun-mass of 2.1030 [kg] imaginated as a blackhole, is a large black hole. The time (t),  with the restriction of Ts ~ 1/m, results in:
      0.5 c 21/2
t ~________                                                       (5)
The dimension is [m/s] /{[kg].[m³/kg.s²]) = [s/m²].
So to translate time in seconds (this means to enable it , to observe it really), a multiplication is necessary with the unity of a blackhole-surface, which is an elementary surface quantum Oe [m²].
From this follows:
      0.5 c 21/2
t = ________ . Oe [s]
Oe can be replaced by (Lplanck)² = hG/c³
       0.5 c 21/2   hG
t = _____ .      ____ [s]
        mG             c³
From this follows:
t = 0.5 .21/2  . ____   [s]                                   (6)
So here is the time to have a unknown chance of observing radiation of a large blackhole. This is determined by Planck’s constant (h) and Einsteins energy E= mc². This was expected.
But now firts the other restriction rs << Ts .
Now I define time as (t'), because in principle, it is different from time (t).
Then starting again from formula (4):
       0,5 c 21/2 rs Ts
t' ~ ____________
           G (rs - Ts )
Now the case is: rs is neglectable to Ts (a small black hole):
         0,5 c 21/2 rs .Ts
t' ~ ______________
              G . (- Ts )
                     c 21/2
t' ~ - 0,5 rs . _____
In this I substitute rs = 2mG/c² .
               2mG   c 21/2
t' ~ -0,5 . ____ . _____
                 c²          G
           m 21/2 
t' ~ - _______   [kg]/[m/s] =[(kg/m).s]
Again (t’) must be expressed in seconds, but now for a small black hole.
So, there must be divided by the dimension of massdensity [kg/m]. But by what value?
The is this: a small black hole exists, when a Planckmass and light, are both present at the same time, so (hc/G)1/2 [kg] . c [m/s] is actual. Moreover the Planckmass is defined at the Plancklength, so also (1/c). (hG/c)1/2 .c [m.(m/s)] = (hG/c)1/2 [(m/s).m] must be actual.
To get a volume [m3] of a small blackhole per second, (hG/c)1/2 [(m/s).m] must be taken per 1 m/s, or multiplied by 1 [s/m].
The reult is (hc/G)1/2 [kg] / {(hG/c)1/2 [(m/s).m] .1 s/m}]= (hc/G)1/2 . (hG/c)-1/2 = c/G [kg/m].
So to get the time (t') for a small black hole, there must be divided by c/G [kg/m], or multiplied by G/c [m/kg]. This will express (t’) in seconds. The result is:
          m 21/2
t' = - _______ . G/c [s]
           m G  21/2
t' = - ________ [s]                                            (7)
Now the time to observe a small black hole is determined by G and c² , while m must be negative to get positive time .
This defines my information-subpoint-particles in my "darkfield",where -m is the returned-information of small black holes.
After having found two time-durations for observing a small and large blackhole, I introduce the duo-time factor, called DQT-factor, which means both time-durations will be connected. The 'Q' stands for a detailed chance below Quantum
I have found two times (t) en (t'), which I connect to  E x t' working in the opposite to E x t .
The result is:
              G 21/2                                 h
DQT = - ______ . m. 0.5 .21/2 .          ____
                c²                                      mc²
This can be rewritten:
              G c 21/2                              h
DQT= - ______   . m. 0.5 21/2 .          ____
                   c³                                   mc²
This makes possible to replace (hG/c³) [m²] in Oe [m²]. So than follows:
DQT= - m . Oe . c. 1/mc²
DQT= - Oe /c [m.s]                                             (8)
Intermezzo: Could both times ever be equal to each other? No ! Accept in an empty universe, or a universe which hasn’t started yet. I will show this:
     G 21/2                                  h
- _______ . m  =  0.5 21/2  .        ___
     c²                                          E
                     h               G 21/2
0.5 21/2  .      ___ +     _______ . m = 0
                     mc²              c² 
0.5 21/2  . h . c² + Gmc². 21/2
________________________  = 0
                  c² mc²
c²(0.5 21/2  . h + Gm.) 21/2
________________________ = 0
                 c² mc²
21/2  (0.5h + Gm²)
________________ = 0
With the restriction of Gm² >> 0,5h, or let us say Gm² >> hc, or m²>>hc/G, or m²>> m²Planck this is giving the following derivation:
  21/2 Gm²
__________ = 0
 21/2 Gm²
__________ = 0
  G 21/2
_______ . m = 0
This can only be for m = 0. Thus only both times can be equal if there are no masses. This means both times are only equal for small black holes, which loose all their radiation.  Under these circumstances there are no blackholes to give radiation.
And in the other case:
21/2 (0.5h + Gm²)
______________ = 0 , with Gm² << 0.5h, follows:
0.5 .21/2 . h [J. s]
_______________ = 0 [s]
      E [J]
In this expression there is energy E in the dimension [J], so in the expression 0.5 .21/2. h [J. s] = 0 only the time can be 0. If E=0 and the time is finite, than the result would be infinite, but that is not the case, it is 0. So “the time dimension must be = 0”, and that means the universe had not yet started. Anyway this also proves that the universe had a finite energy before it started.
Now I continue:
Both times can not be equal to each other. It always demands a DQT-factor to be a product, which gives an unknown chance to observe the radiation after a time (t') and (t), for a small and large blackhole.
I subsitute the DQT-factor in the product of energy and time:
U = (E x t) . (E x –t’) = E² . t.-t’ = E². DQT = E² . - Oe /c [J².m.s] =[(J.s)².m/s]
Those two forms of "energy x time" , symbolize the 100 % unknown chance of observing a large and small blackhole simultaneously. This co-existence of two different blackholes in one moment, means: You obtain an energy (U) for one black hole, for which (U) has to be divided by 2.
U = 0,5 . E² . - Oe /c = E² . - Oe /2c [J².m.s] =[(J.s)².m/s]

The energy (U) is a temporal energy from below quantumscale, because the source of the energy is orinormally from inside the black hole. Therefore (U) has anti-gravitational features.

Thus, to get a real presentation of the new energy, I have to accept the existence of E and U together at "the same time".
In other words: The Cosmos exists of having a chance to be involved with Einstein’s energy and the returned information from an unknown energy force. This is resulting in the next equation:
Uu = E . U = mc² [J]. E². - Oe /2c [J².m.s]
This introduces:
My dark energy force formula:
Uu = - 0.5 . E² . mcOe [J³. m.s] = [(J.s)³.m/s²] = [(kg)³.m7/s5]                     (9)
In this formula E²= E²kin + E²0 is embedded. There is also a dark matter impuls (mc) as part of a dark matter flow (1/2 mc Oe) [kg.(m3/s)]. In total the sign is “-“, which means there is a repulsive gravitational property: dark energy. .
The dimension [(J.s)³.m/s²] shows a three dimensional spin (J.s), which accelerates (m/s²).

This represents a force in a torus geometry of dark energy and dark matter.

Moreover, my formula can be be rewritten furthermore in:
Uu = - 0,5 . E² . mcOe [(kg)³.m7/s5]
Uu = -0,5 m³c5Oe [(kg³.(m³/s)] . 1 [m4 /s4]                                                        (10)
Uu = -0,5 m³c5Oe [(kg³.(m³/s)] . 1/G [N]
From this follows:
dark energy force formula:
Uu = - (c5Oe /2G) . (m)3 [(kg.m)³.N/s]               (11)
Here c is the lightspeed, G is the Newton-constant, Oe = (L planck )2  and m is mass.
Control of the dimensions.
{[m5/s5] . [m2]}  /  {[m3] / [kg.s2]} [kg3 ] = {[kg3 ]. [m7/s5]} . {[kg.s2] / [m3]} = [kg3 ]. {[kg] . [m4/s3]} = [kg3 ]. {[kg] . [m/s2]. [m3/s] = [kg3 ]. N . [m3/s] = [kg3 . m3] . [N/s] = [(kg . m)3] . (N/s)] 
"Thougt-experiment and formulas" are designed and owned by Dan Visser, Almere, Netherlands, First published on April 10, 2004
Text modifications were made on July 18 2004, September 18, 2005 and March 31 2008, April 5 2008 and April 19 2008, September 28 2008, September 2010 and October 7 2010.
This page and all the other pages of this website are juridical protected.  Comment to Dan Visser, email dan.visser@planet.nl ;
You are not permitted to copy, broadcast, download, store (in any medium), transmit, show or play in public, adapt or change in any way the content of Dan Visser 's web pages for any other purpose what so ever without the prior written permission of Dan visser.
[2} My dark energy force-formula was used by Christopher Forbes (UK), PhD-math and Physics, and his colleague, in three “pre-papers” published in the viXra-archive. The “papers“ are viXra:0909.0005 [pdf], “Short Article On A Newly Proposed Model Of Cosmology“, submitted on Sep 1, 2009 in the category “Relativity & Cosmology“; viXra:0910.0016 [pdf], “Mathematical and Phenomenological Elements of the Twin-Tori Model of Physics and Cosmology“, submitted on October 11 2009 in the category “Mathematical Physics“; viXra:0911.0061 [pdf], “A New Quantum Gravity Framework Based on the Twintori Model of Cosmology. (Part 1), submitted on November 28 2009 in the category “Atrophysics“.
Author: Dan Visser, Almere, The Netherlands (ingE, independent cosmologist and art-painter).