Local isotropic length transformation - hypothesis

[caracal]

I made some corrections.

 

IT seems that the Friedman equation in co-contracting coordinates without lambda seems to be similar than in expanding universe with lambda.

 

The important result seems to be that if L is exponential in co-contracting coordinate system, then the apparent expansion gives same kind of term to 1st Friedman equation ( k^2 = constant ) than cosmological constant does.

 

There is also one new density term in the model: compact matter (the last section in this post, which is unfinished)

 

 

 

There has been several mistakes in the latest post above.

 

Mistakes in last post:

----------------------------------

-There may not be any delay term in gravitation after all -when the contraction changes in m1 and m2 happen simultaneously.

 

in rigid coordinate system-section:

-the units of radiation are wrong. the factor L is L at the moment of emission, not L=1. But it is right that L=constant for radiation

-the density dependencies is wrong, they should be 1/La^3 and 1/La^4

-The newtons gravitation law is right, but i forgot to take into account two things: r depends on scale factor r=constant in static universe only and - the forces remain same, yes, but accelarations gets slower (newtons 2nd law), since the mass of the matter increases.

-The friedman equation is wrong since the density formulas are wrong

 

in co-contracting coordinate system section:

-the units of radiation are wrong, the factor L should be L at the moment of emission. But it is right that L=constant for radiation

 

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i hope it is now without mistakes:

 

 

Corrected model:

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Rigid/fixed coordinate system (s,t,E)
------------------------------------------------------------------------------------------------------------------------------------------------

Length, time and energy units:

Length : [latex] ds= constant [/latex]
Time : [latex] dt= constant [/latex]
energy : [latex] dE = constant[/latex]

 

No lambda

Active Gravity

 

Scale factor, if no gravity and no expansion:

static universe : [latex] a_{nullG}(t) = 1 = constant [/latex]

dynamic universe: [latex] a_{nullG}(t) \neq constant [/latex]

 

Length,time and energy units of matter:

[latex] ds'= L(t)ds [/latex]
[latex] dt'= L(t)dt [/latex]
[latex] dE' = \frac{dE}{L(t)}[/latex]

 

Convention for L(t): [latex] L(t_{0}) = 1 [/latex]

 

Particle contraction Law: [latex] \frac{dL(t)}{dt}< 0 , L(t) -> 0 ,when t -> infinity [/latex]

 

Length, time and energy units of radiation:
.............................................
[latex] ds'= L(t_{em})ds = constant [/latex]
[latex] dt'= L(t_{em})dt = constant [/latex]
[latex] dE' = \frac{dE}{L(t_{em})} = constant[/latex]

(Units of radiation does not change after emission due to infinite time dilation)

 

Changes in Gravity and Newton's 2nd Law in static a=1 universe:

 

Newton's Gravitation Law:

[latex] G' = G_{0} * L(t)^{2} [/latex]
[latex] m' = m_{0} / L(t) [/latex]
[latex] r' = r_{0} [/latex] (in static universe a=1=constant)

[latex] => F' = G' \frac{m_{1}'m_{2}'}{r^{2}} = F [/latex]

=> Gravitation forces do not contract universe in static universe

Newton's 2nd Law:

[latex] F' = m'a' = F => a'= F/m' = (F/m) * L(t) [/latex]

(In static universe, All accelerations slow down since the mass of matter has increased by factor 1/L )


changes in gravitation In dynamic universe,when a is not constant:

[latex] G' = G_{0} * L(t)^{2} [/latex]
[latex] m' = m_{0} / L(t) [/latex]
[latex] r' = r_{0}a(t) [/latex]

[latex] => F' = G' \frac{m_{1}'m_{2}'}{r^{2}a^{a}} = \frac{F}{a^{2}} [/latex]

=> in dynamic universe Rigid coordinate system if a<1 the effect of gravitation becomes stronger, and if a>1 the effect of gravity becomes weaker.

 

Density of matter and radiation:

Density of matter :[latex] \rho_{matter}=\frac{\rho_{matter,0}}{L(t)a(t)^3} [/latex]

Density of radiation: [latex] \rho_{rad}=\frac{\rho_{rad,0}}{L(t)a(t)^4} [/latex]

=>

1st Friedman Equation in rigid coordinates:

[latex] \left ( \frac{\dot{a}}{a} \right )^{2}= \left ( \frac{8 \pi G}{3}\right) \left( \frac{\rho_{mass,0}}{L(t)a(t)^3} + \frac{\rho_{rad,0}}{L(t)a(t)^4}\right ) [/latex]

 

(is this right?)

 

 

Comparing Friedman equations between rigid and co-contracting system

 

 

when comparing friedman equations between rigid and co-contracting coordinates, remember transformation equations between the two systems:

 

density : [latex] \rho' = \frac{\rho}{L^{4}} [/latex]
pressure [latex] p' = \frac{p}{L^{4}} [/latex]

gravitation constant : [latex] G' = G * L^{2} [/latex]

scale factor [latex] a' = \frac{a}{L} [/latex]

time derivative of a : [latex] (da/dt)' = 1/L^{2} [/latex]

 

Equal infinitesimal time interval: [latex] dT' = \frac{dT}{L(t)} [/latex]

 

Equal time interval: [latex] T' =\int_{0}^{T} \frac{dT}{L(t)} [/latex]

 

(this section is not finished)

 

 

--------------------------------------------------------------------------------------------------------------------------------------------------------
Co-contracting coordinate system: (s' , t' , E')
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Length,time and energy units:

[latex] ds'= L(t)ds [/latex]
[latex] dt'= L(t)dt [/latex]
[latex] dE' = \frac{dE}{L(t)}[/latex]

Convention for L(t'): [latex] L(t'= now) = 1 [/latex]


Transformation equations between rigid coordinate system (s,t,E) and co-contracting coordinate system (s',t',E'):

density : [latex] \rho' = \frac{\rho}{L^{4}} [/latex]
pressure [latex] p' = \frac{p}{L^{4}} [/latex]
mass [latex] m' = \frac{m}{L}[/latex]
gravitation constant : [latex] G' = G * L^{2} [/latex]
forces: [latex] F' = \frac{F}{L^{2}}[/latex]
acceleration: [latex] a' = \frac{a}{L}[/latex]
velocity: [latex] v' = v [/latex]
scale factor [latex] a' = \frac{a}{L} [/latex]

 

Equal infinitesimal time interval: [latex] dT' = \frac{dT}{L(t)} [/latex]

 

Equal time interval: [latex] T' =\int_{0}^{T} \frac{dt}{L(t)} [/latex]

(Co-contracting time t' is accelerating relative to rigid time t)

 

Apparent expansion in universe that is static in rigid coordinate system:

IF [latex] a_{nullG}(t) = 1 [/latex]

=> [latex] a_{nullG}(t') = \frac{1}{L(t')} [/latex]

(There is apparent expansion in co-contracting coordinate system even if the universe is static in rigid coordinate system)

 

Length, time and energy unit of matter:

[latex] ds_{matter} = ds_{0} = constant [/latex]
[latex] dt_{matter} = st_{0} = constant [/latex]
[latex] dE_{matter} = dE_{0} = constant [/latex]

 

Length, time and energy unit of radiation:

[latex] ds_{rad} = \frac{ds'}{L(t_{em}')/L(t_{obs}')}[/latex]

 

[latex] dt_{rad} = \frac{dt'}{L(t_{em}')/L(t_{obs}')}[/latex]

 

[latex] dE_{rad} = dE' * \frac{L(t_{em}')}{L(t_{obs}')} [/latex]

(Radiation units does not change after emission event due to time dilation.)

(note that [latex] t_{em} < t_{obs} [/latex] )

 

Density of matter and radiation:

[latex]\rho{rad}(t') = \frac{\rho_{rad,0}}{a(t')^{4}} [/latex]

[latex]\rho_{matter}(t') = \frac{\rho_{matter,0}}{a(t')^{3}} [/latex]

-Densities changes similarly as in expanding universe


Apparent expansion only universe:

if no active Gravity, there is only apparent expansion:

[latex] a_{null G} = \frac {1}{L(t')} [/latex]

This is solution from 1st friedman equation in empty space:

[latex] \left ( \frac{\dot{a}(t')}{a(t)} \right )^{2} = f(t') [/latex]

[latex] => f(t') = \left ( \frac{\dot{L}(t')}{L(t')} \right )^{2}[/latex]

 

1st Friedman equation:

 

Putting these 3 terms above together in Friedman equation:
=>
1.st Friedman equation:

[latex] \left ( \frac{\dot{a}(t')}{a(t)} \right)^{2} = (\frac{8 \pi G }{3})(\frac{\rho_{matter,0}}{{a(t')^3}} + \frac{\rho_{rad,0}}{a(t')^{4}}) + \left(\frac{\dot{L}(t')}{L(t')}\right)^{2} [/latex]

 

 

Exponential guess in co-contracting coordinates and Friedman equation

 

 

If [latex] L(t') = e^{-kt} [/latex] (exponential guess)

=> [latex] a_{nullG}(t') = e^{kt} [/latex]

=>

1.st Friedman equation:

 

[latex]\left ( \frac{\dot{a}(t')}{a(t)} \right)^{2} = (\frac{8 \pi G }{3})(\frac{\rho_{matter,0}}{{a(t')^3}} + \frac{\rho_{rad,0}}{a(t')^{4}}) + k^{2} [/latex]

(Exponential L(t') seem to give similar term to 1.st Friedman equation than cosmological constant does)

 

=> Exponential guess L Fit to cosmological constant: [latex] L(t') = e^{-\sqrt{\Lambda}t'} (in co-contracting time units)[/latex]

 

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Additional components:

 

There is Third density term: compact matter

-Since particle contraction law depends on proper time, then due to time dilation differences, the density of compact

matter changes differently than density of normal matter

Length, time and energy units of black holes:

[latex] ds_{BH}'= \frac{ds_{0}'}{L(t_{birth}')} [/latex]
[latex] dt_{BH}'= \frac{dt_{0}'}{L(t_{birth}')} [/latex]
[latex] dE_{BH}= dE_0' * L(t_{birth}' [/latex]

(Units remain constant after birth of Black hole, since BH has infinite time dilation)

 

Length, time and energy units of compact matter:

[latex] ds_{comp}= \frac{ds_0}{L(t_{birth}+D(t-t_{birth}))} [/latex]
[latex] dt_{comp}= \frac{dt_0}{L(t_{birth}+D(t-t_{birth}))} [/latex]
[latex] dE_{comp}= dE_{0} * L(t_{birth}+D(t-t_{birth})) [/latex]

(Time dilation slows down contraction law of compact matter after birth of compact object, D is time dilation factor)

D = 0.7 for neutron stars

Edited February 22, 2016 by caracal

[caracal]

The unit transformation equations may cause some confusion.

 

The markings ds , dt , dE and ds' dt' dE' what i used there in unit equations are not equal differentials, they are units of measurement that observer uses, or local natural units. It may be better if i used following low indices markings:

 

Length, time and energy units in co-contracting coordinate system:

 

Length unit : [latex] s_{unit}' = L('t) s_{unit} [/latex]

Time unit : [latex] t_{unit}' = L(t') t_{unit} [/latex]

Energy unit: [latex] E_{unit}' = E_{unit}/L(t') [/latex]

 

I cannot edit my earlier post, so i make a remark here.

 

If i used the equal differentials to describe the conversion from rigid coordinate system to co-contracting coordinate system, then the L in the equations should be substituted by 1/L and vice versa:

 

Equal infinitesimal differentials between rigid and co-contracting coordinate system:

 

Length: [latex] ds' = ds/L(t') [/latex]

Time: [latex] dt' = dt /L(t') [/latex]

Energy: [latex] dE' = L(t') dE [/latex]

 

 

 

[Mordred]

While it's good your looking at the math I'm not confident your looking at the history of how these formulas came into effect.

 

some of the changes you've made I can't see working out to observational evidence but that's tricky to determine due to the incomplete sections.

 

Seems to me your modifying equations without studying the physics behind them.

 

For example earlier this thread I posted a couple of links on recent tests of the gravitational constant. Which fine tuned the value to an extremely high degree.

 

I need to ask how much do you understand on relativity which the FLRW metric derives from ?

 

The reason I ask is there is some concerns I have that the equations you posted will alter the light paths from a near critically flat universe to one that I can't even describe as being homogeneous and isotropic.

 

(Despite the fact your using equations that require a homogeneous and isotropic distribution)

Edited February 24, 2016 by Mordred

[caracal]

some of the changes you've made I can't see working out to observational evidence but that's tricky to determine due to the incomplete sections.

 

Seems to me your modifying equations without studying the physics behind them.

 

Isotropic Space transformation is new phenomenon and the theory of relativity does not describe it. In this cosmological model i assume that the matter spontaneosly undergoes very slow contraction that becomes visible only in cosmological time scales.

 

The mechanism behind this length transformation is same than mechanism behind relativistic length contraction- But since nobody knows exactly what kind of mechanism is behind relativistic length contraction this question remain open.

 

I am not sure at the moment how to derive friedman equation in rigid coordinates, since the matter undergoes transformation that changes the way how the gravity works -

 

But in the co-contracting coordinates the matter generally has all the time same relative L=1 , and the gravity works as usual. G and M does not change. (if i ignore compact matter that has time dilation)

 

Therefore the densities of matter and radiation behave similarly as in expanding universe. So these two density terms are exactly same in Friedman equation.

 

1. Matter density is inverse proportional to third power of scale factor.

 

2. Radiation has one a more in the nominator, since the photon, which stays unchanged in rigid coordinates, does not undergo transformation due to its time dilation - and therefore undergoes apparent transformation in co-contracting coordinates - loses energy and momentum in co-contracting coordinates.

 

3. The apparent expansion (= distance expansion) term may be tricky. In empty space that is static in rigid coordinates, with exponential

 

[latex] L(t')=e^{-kt'} [/latex]

 

Gives scale factor in co-contracting coordinates to be:

[latex]a(t') = 1/L = e^{kt'}[/latex]

 

and makes the Friedman equation to be just

[latex] H(t')^{2} = k^{2}[/latex]

 

You can now pretend that the energy density of this apparent expansion is constant, without actually going into the physics behind the length transformation.

 

But I can do following deduction to justify this dependency:

 

Contracting Hydrogen Atom

------------------------------------------------------------

Lets assume that hydrogen contracts isotropically, when both electron and proton contracts exactly by same factor L.

 

(k'/k) = L^2 Coulomb constant

r'/r = L distance between electron and proton

q'/q = 1/L charge of electron and proton

m'/m = 1/L mass

 

F'/F = (k'/k) (q'/q)^2 (r'/r)^-2 = 1/L^2 Coulomb force

 

E'/E = (k'/k)(q'/q)^2 (r'/r)^-1 = 1/L binding energy

 

a'/a = 1/L centripedal acceleration

 

p'/p = 1/L momentum

 

v'/v = 1 velocity of electron

 

-----------------------------------------------------

Here i assume that the electron goes into closer orbital from nucleus with radius r' = r L , after its transformation, when its mass and momentum has increased by factor 1/L, and therefore also its matter-wavelength is decreased by factor L.

 

 

You can see from the equations that binding energy of the electron - proton system is increased by factor 1/L.

 

Lets imagine that this binding energy difference is "radiated" away from the system.

 

(In the viewpoint of contracting observer, nothing has happened, but in the viewpoint of rigid coordinate system, the hydrogen has undergone isotropic length transformation with factor L)

 

In the viewpoint of contracting observer, contracting matter increases equal amount of binding energy and "radiates" equal amount of energy in every time interval, if the contraction is exponential.

 

[latex] L(t')=e^{-kt'}[/latex]

 

Therefore the energy density of the apparent expansion of the universe appears to be constant in co-contracting coordinates.

 

Now, the Friedman equation is exactly same than in Benchmark model, if

[latex] k^{2} = \Lambda [/latex]

,if i omit compact matter, which gives 4th density component to the equation.

 

And the Friedman equation in flat space is now:

 

[latex] H(t')^{2}= \frac{8 \pi G}{3}(\rho(t')) + k^{2} [/latex]

[latex] \rho(t') = \frac{\rho_{m,0}}{a^{3}} + \frac {\rho_{r,0}}{a^{4}}[/latex]

 

 

 

The reason I ask is there is some concerns I have that the equations you posted will alter the light paths from a near critically flat universe to one that I can't even describe as being homogeneous and isotropic.

 

The velocity-invariant isotropic length transformation keeps all velocities invariant or unchanged - that means that

the velocity of light is always constant c for both co-contracting coordinates and rigid coordinates (in empty space, that is static in rigid coordinates)

 

 

This phenomenon may cause only the cosmological constant - part of the expansion of universe.

Edited February 29, 2016 by caracal

[caracal]

Issue: Deriving scale factor in rigid coordinates:

 

While I am not sure of the Friedmann equation in rigid coordinates, but i know how to derive its solution a(t) in rigid coordinates.

 

The scale factor in rigid coordinates in terms of scale factor in co-contracting coordinates is just:

 

[latex] a_{rigid}(t) = a_{co-contracting}(t) * L(t) [/latex]

 

and for exponential guess [latex] L(t') = e^{-k(t'-t_{0}')} [/latex] :

 

[latex] a_{rigid}(t') = a_{co-contracting}(t') * e^{-k(t'-t_{0}')} [/latex]

 

To solve what is [latex](t'-t_0')[/latex] in terms of [latex] (t-t_0)[/latex], i need the Equal time interval equation, what is:

 

[latex] \frac{dt'}{dt} = e^{k(t'-t_{0}')} \rightarrow (t'-t_0')=\int_{t_0}^{t}e^{k(t-t_0)}dt [/latex]

 

And now the remaining problems is to solve the Friedmann equation in co-contracting coordinates:

 

[latex] \left ( \frac{\dot{a}}{a} \right )^{2} = \frac{8 \pi G}{3}\left ( \rho \right ) + k^2 [/latex]

 

,where [latex] k^{2}= \Lambda [/latex] - if cosmological constant is assumed to come solely from the distance expansion/apparent expansion that is caused by the equal contraction of matter in homogenous universe.

 

,which is, under this assumption [latex] k^2 = \Lambda [/latex], exactly same solution than Benchmark model solution is.

For example earlier this thread I posted a couple of links on recent tests of the gravitational constant. Which fine tuned the value to an extremely high degree.

 

 

Gravitation constant changes only in rigid coordinates, but not in co-contracting coordinates, if all matter in the universe contracts equally.

 

Same thing applies to all measurements.

 

In the static universe, When the contracting observer looks into distant universe, and also to distant past, due to limited velocity of light, he will however observe (assuming exponential guess)

 

1. Transformation difference in the matter in the past, [latex] L_{relative past}= L(t_{now}' -\Delta t') =e^{k \Delta t'} [/latex] and

2. Less distance expansion/apparent expansion in the past: [latex] a_{past} = L(t_{now}' -\Delta t') = e^{-k\Delta t'} [/latex]

3. Relative transformation changes in the properties of photons, such as redshift, and momentum decrease, since photons do not contract after emission due to that their proper time does not evolve.

 

But how these three observations differ from observation that the space is only expanding? I claim that there is no difference.

 

-if you extract time dilation effect from the cosmological photon due to expansion of space and scale factor increase due to expansion of space, in the expanding space model, that is similar operation than just extracting the transformation difference to L=1 in this model. You will get also result that G has been constant in the history of the universe, as if you have just cancelled the effects of cosmological time dilation and scale factor increase due to expansion of space.

 

In the static universe in rigid coordinates, the apparent expansion in co-contracting coordinates produces just Robertson-Walker metric in flat space:

 

[latex] ds^2 = -c^2dt^2 + \frac{dr^2}{L(t')} [/latex]

 

,Where the scale factor is : [latex] a(t') = \frac{1}{L(t')} = e^{k(t'-t_{now})}[/latex]

Edited March 2, 2016 by caracal

[michel123456]

 

But according to the local principle of relativity the observer O_B will measure all lenghts , time intervals and energies as all other physical observables and laws of nature in his own region B perfectly normal.

 

The observer O_A will measure exactly the inverse changes in the region B, but again according to local principle of relativity, he will measure all physical observables and laws of nature in his own region A perfectly normal and unchanged.

 

Yes.

You are talking about a generalized scale factor, aren't you?

Are you aware about the relation between a scale factor and acceleration?

If you use this, and if you can derive the equation of gravity, that would be great.

 

-----------

Any scale factor has a center, a geometric starting point.

If object A is scaled f from its own center of mass (for example), and object B is scaled the same way, the result will be a change in distance between A and B. If A expands & B expands they will eventually crash together. If this effect is not corresponding to the "generalized" scale factor, then the result should be a change of position of A and B. That would keep the (scaled) distance to appear remaining the same for A and B scaled observers. And this change of position is accelerated.

Edited March 2, 2016 by michel123456

[Mordred]

 

 

In the static universe in rigid coordinates, the apparent expansion in co-contracting coordinates produces just Robertson-Walker metric in flat space:

 

[latex] ds^2 = -c^2dt^2 + \frac{dr^2}{L(t')} [/latex]

 

,Where the scale factor is : [latex] a(t') = \frac{1}{L(t')} = e^{k(t'-t_{now})}[/latex]

 

 

a static universe wouldn't match the thermodynamic history that we can and have measured. Even if matter compresses.

 

Why would you think the FLRW metric uses rigid coordinates? you can have a conformal or commoving coordinates in the FLRW metric these aren't rigid

 

 

-----------

Any scale factor has a center, a geometric starting point.

If object A is scaled f from its own center of mass (for example), and object B is scaled the same way, the result will be a change in distance between A and B. If A expands & B expands they will eventually crash together. If this effect is not corresponding to the "generalized" scale factor, then the result should be a change of position of A and B. That would keep the (scaled) distance to appear remaining the same for A and B scaled observers. And this change of position is accelerated.

 

 

the only goemetric starting point is the fundamental observer. This isnt however a preferred location.

Edited March 2, 2016 by Mordred

[michel123456]

[

a static universe wouldn't match the thermodynamic history that we can and have measured. Even if matter compresses.

In this case there is no "compressed" matter, it is a scale factor not a compression. Edited March 2, 2016 by michel123456

[swansont]

 

Contracting Hydrogen Atom

------------------------------------------------------------

Lets assume that hydrogen contracts isotropically, when both electron and proton contracts exactly by same factor L.

 

(k'/k) = L^2 Coulomb constant

r'/r = L distance between electron and proton

q'/q = 1/L charge of electron and proton

m'/m = 1/L mass

 

F'/F = (k'/k) (q'/q)^2 (r'/r)^-2 = 1/L^2 Coulomb force

 

E'/E = (k'/k)(q'/q)^2 (r'/r)^-1 = 1/L binding energy

 

a'/a = 1/L centripedal acceleration

 

p'/p = 1/L momentum

 

v'/v = 1 velocity of electron

 

-----------------------------------------------------

Here i assume that the electron goes into closer orbital from nucleus with radius r' = r L , after its transformation, when its mass and momentum has increased by factor 1/L, and therefore also its matter-wavelength is decreased by factor L.

 

 

You can see from the equations that binding energy of the electron - proton system is increased by factor 1/L.

 

Lets imagine that this binding energy difference is "radiated" away from the system.

 

(In the viewpoint of contracting observer, nothing has happened, but in the viewpoint of rigid coordinate system, the hydrogen has undergone isotropic length transformation with factor L)

 

 

 

But this radiation is never observed.

[Mordred]

 

 

Gravitation constant changes only in rigid coordinates, but not in co-contracting coordinates, if all matter in the universe contracts equally.

 

 

read what he's trying to model.

[michel123456]

the only goemetric starting point is the fundamental observer. This isnt however a preferred location.

Correct.

That is the reason why I never could resolve this fundamental observation in relation to a scale factor. There is a geometrical obstination to any scale factor to begin from a center. It can be ignored mathematically but not geometrically. Or if it can be resolved I am not aware of it.

[Mordred]

Sounds like a topic for a seperate thread so we don't hijack this thread from the OP.

[michel123456]

Sure.

But the OP may realize that outside the cosmological model his interpretation may be able the explain entropy, the arrow of time (maybe Time itself) , and gravity. All that from a mysterious mechanism causing the scale factor and that would be great.

[Diversion closed]

Edited March 2, 2016 by michel123456

[caracal]

Michel:

 

This cosmological part -matter contraction law - hypothesis seem to give alternative mechanism for Robertson-Walker metric, and may give alternative explanation for cosmological constant if the contraction is exponential in the co-contracting time. It gives two clearly different predictions from expanding space model: existence of differently transformed matter, and additional effect to twin paradox.

 

But the local space transformation hypothesis can be more general: there may be more different kinds of local space transformations. I wont go into this issue here, and i concentrate only to one type of local space transformation: local velocity-invariant isotropic length transformation.

But this general class of local transformations may be useful in some way in explaining for example deep reason for quantum mechanics.

 

 

Yes.
You are talking about a generalized scale factor, aren't you?

 

no, i am not talking about scale factor or cosmology here.

I am talking about following situation:
--------------------------------------------------------------------
-One area of space or some physical object has undergone a space transformation such that it either contracted or expanded relative to some other area of space or physical object. The time unit, length unit and energy unit is transformed such that:

[latex] \frac{t'}{t} = \frac{s'}{s} = 1 / \frac{E'}{E} = L , 0 < L < infinity [/latex]

This is the hypothetical phenomenon what is in the topic of this threat.

 

 

Are you aware about the relation between a scale factor and acceleration?
If you use this, and if you can derive the equation of gravity, that would be great.

yes, this is the second Friedmann equation that relates pressure to acceleration of the scale factor:

[latex] \left ( \frac{\ddot{a}}{a}\right )= \frac{-4\pi G}{3}\left ( \rho + \frac{3p}{c^2} \right ) +\left ( \frac{\Lambda c^2}{3} \right ) [/latex]

i think this is again same in co-contracting coordinates. But not in rigid coordinates.

 

If you use this, and if you can derive the equation of gravity, that would be great.

yes i have to think about it. but this is only needed in rigid coordinate system. But we are contracting observers, and we use co-contracting coordinate system ,where the gravity is normal.

 

Any scale factor has a center, a geometric starting point.

 

You are ask what is the contraction center of the contraction of matter? This is actually a thing that i could clarify. I wish i could put a picture here.

 

Issue: The contraction-centers of contraction of matter

 

In the cosmological scale the center of contraction of matter is in the observer, but that is not necessarily true in macroscopic scales.

 

I dont know what is causing this velocity invariant isotropic length contraction - phenomenon. I assume that it starts in or inside fundamental particles, and because the effect is very slow, normal ordinary matter that is bound together by EM forces has also enough time to contract isotropically, when the macroscopic system contracts towards the center of the system similarly as in homogenous thermal contraction. Also the transformation equations and the principle of relativity are applicaple to macroscopic scales.

 

Planets and stars can still contract into direction of their center, and contract isotropically - but planet systems don't contract any more together towards the center. That is because they are too weakly bound together relative to strongly bound atoms in solid matter. (it open question where is the actual limit?)

 

What happens in astrophysical and cosmological scales is that matter differentiates into the distinct centers, such as planets, stars or dust particles or molecules in gas cloud. The distances between the centers remain constan. This effect is seen as distance expansion in the viewpoint of contracting observer, who is still contracting isotropically together with the matter.

 

In the cosmological scale, the center of the scale factor is always in the observer - if the space is homogenous and isotropic all the time.

In this model this is the case. I assume that:
1) all matter in universe has same relative transformation factor L=1
2) all matter is contracting at equal rate.
3) i assume similarly as in all cosmology, that space is isotropic and homogenous. and i assume for the sake of simplicity that
4) space is flat.

implication of 1) and 2) is that space stays homogenous and isotropic all the time in cosmological scale.

actually the 1) and 2) cannot be stricktly speaking true because of the slight time dilation differences of matter. Also the compact matter and matter in relativistic jets have time dilation relative to the average rest frame of the matter in universe. This is the main difference from the expanding universe and this is the main prediction of this model.

 

by the way if 4) is not true and space has some curvature radius, one implication is also that this curvature radius decreases in the viewpoint of contracting observer.

Mordred:

 

Why would you think the FLRW metric uses rigid coordinates? you can have a conformal or commoving coordinates in the FLRW metric these aren't rigid

I mean by rigid coordinate a coordinate system, that is not following the contraction of the matter , and

 

by co-contracting coordinates a coordinate system that is following the contraction of matter, where relative L(t') = 1 in the matter.

Co-moving coordinates are coordinates that follow the scale factor and in co-moving coordinates a(t)=1 all the time.
But co-moving coordinates are not same than rigid coordinates what i use here.

Maybe i could use better names such as non-transforming coordinates and co-transforming coordinates

 

It would be helpful if i could attach a picture here.

 

 

a static universe wouldn't match the thermodynamic history that we can and have measured. Even if matter compresses.

No i am not saying that universe is static. I just derive the Friedmann equations in the case of static universe, because it is simple, and the evolution of scale factor is then just:

 

Static universe with no gravity:
Rigid/non-transforming coordinates : [latex] a(t) = 1 [/latex]
co-contracting or co-transforming coordinates: [latex] a(t') = \frac{1}{L(t')} [/latex]

The real space can have both ordinary expansion and this apparent expansion or ordinary expansion and this contraction of matter. The real space has matter density, radiation density and remnant expansion from big bang, that would described by ordinary Friedmann equation, if L=1.

I just concluded that i can just add [latex] k^2 [/latex] to the Friedmann equation in co-contracting or co-transforming coordinates. to describe apparent expansion in real universe and assume it has constant energy density, if the contraction is exponential in co-contracting or co-transforming time. that is all that is different in co-contracting or co-transforming coordinates.

 

And it seems the term is similar to cosmological constant. Therefore it may substitute the lambda partly or totally.

But Note that If this apparent expansion substitutes the cosmological constant, the following situation may be possible:
--------------------------------
The space may be actually contracting due to effects of gravity, but because the matter is contracting faster than the space is contracting, it still seems to us that the space expanding.


Issue: What kind of observer uses rigid / non-transforming coordinates?

Actually only observer with infinite time dilation will see the universe in rigid or non-transforming coordinates.

Photon's path - coordinates are rigid coordinates: the photon does not change its relative properties when it travels in rigid or non-transforming coordinates if the space is static. (but ordinary expansion or contraction of space will change the properties of photon as usual.)

If you are travelling at almost light velocity in a closed circle, you would see things happening in the local space in rigid or non-transforming coordinates. (here i assume that all matter in space is close to same relative rest-frame)

You would see that matter (atoms, molecules, stars ) seems to contract isotropically into small volume, and you note that the local time of the matter seems to run faster and faster. You would see that the ordinary matter in the universe undergoes transformation.

 

But ordinary matter with no time dilation, also we as observers, follow and use co-contracting or co-transforming coordinates.

 

You can see the ordinary expansion of space in both coordinate systems. The transformation relation between the scale factors in rigid/non-transforming and co-contracting or co-transforming coordinates is

 

[latex] a_{cc}(t') = a_{rig}(t)/L(t) [/latex]

Edited March 2, 2016 by caracal

[michel123456]

That would mean that observed space expansion should happen not only between galaxy clusters but also inside galaxies, even inside the solar system. Or do I understand badly?

-----------

Also, why do you assume that the speculated phenomenon happens so smoothly? If the observer cannot understand his own shrinking, the contraction could happen a vertiginous rate. For example: each second a shrinking of 300000 km.

 

In a way, you could simply insert your speculation in mathematical terms and put a value of zero (or 1) for a non-scaling universe. Afterwards, change the value of the scale factor and see where it meets observation. Or is that what you already did?

But: still the main issue with the scale factor is that if contraction has multiple centers (for example one in each particle), the macroscopic objects would be feeling like teared apart. Maybe that is the reason why you must assume it happens smoothly. But in fact the problem is not the smooth, the problem is the center.

[Mordred]

Thanks for the terminology clarifications above.

 

Have you tried testing the formulas through proper distance calculations?

 

I still don't see how a homogeneous and isotropic condition can be maintained with the two coordinate system your adapting.

 

Though I realize your still working on the math.

 

by matter I assume your referring to fermions.?

Your going to need to clarify this term usage.

 

bosons don't count as matter particles only fermions do. This is based on the Pauli exclusion principle.

 

The next question is "How will your model work during the radiation dominant era? The % of matter is much smaller with the % of radiation being much higher leading to a higher rate of expansion per Mpc. Higher value for Hubbles constant. (not per size of universe).

[caracal]

I still don't see how a homogeneous and isotropic condition can be maintained with the two coordinate system your adapting.

 

 

There is a shift in some macroscopic or astronomic length scale from isotropic contraction of atoms and strongly bound macroscopic objects such as solid matter, planets and stars, to geometric contraction of contraction centers which happens in gas-clouds, planetary systems, galaxies and galaxy groups. I dont know what is the correct mathematical term for such contraction.

 

-Solid matter and planets can still contract isotropically due to strong binding forces relative to the very slow contraction velocity

 

-But In cosmolocigal length scales the matter appears to contract into the more or less point-like centers.

 

A. Imagine a cake that is decoriated with candies, and imagine now that all candies start to contract isotropically, but nothing happens to the cake and nothing happens to the distances between the candies.

 

B. Now imagine two coordinate grids, one that is still (=non-transforming coordinate grid) , and one that follow the length scale of the candies and start to contract such that the distances between the points in grid decrease (=co-contracting or co-transforming coordinate grid).

 

 

Lets look at the scheme pictures in static universe (static=no ordinary expansion for the sake of simplicity):

 

1. Here is a scheme picture of the isotropic contraction of solid matter in static universe in macroscopic scale:

 

X X X X

xxxx

X X X ---> xxx (strong binding forces -> isotropic contraction is possible)

xxxx

X X X X

 

relative L of matter: L=L(t) (L is decreasing function and L(t0) = 1)

 

2. Here is a scheme picture of the contraction of matter in static universe in cosmological scales:

 

X X X X X x x x x x

 

X X X X ---> x x x x (weak binding forces -> contraction towards the centers)

 

X X X X X x x x x x

 

relative L of matter: L=L(t) (L is decreasing function and L(t0) = 1)

 

In both situations, if the matter is observed in non-transforming coordinates, the ordinary matter undergo isotropic transformation in microscopic level such that:

 

time unit of matter: [latex] \frac{t '}{t} = L(t) [/latex]

length unit of matter: [latex]\frac{s'}{s} = L(t')[/latex]

energy unit of matter:[latex] \frac{E'}{E} =\frac{1}{L(t')} [/latex]

 

 

But free radiation does not change at all. (but remember that in rigid coordinates the contracted matter do start to emit blue-shifted light)

 

How does contracting observer observe both situations above?

 

1. Due to relativity principle, contracting observer observes that nothing is changed in the situation 1.

 

X X X X X X

X X --> X X apparent expansion in co-trans. coordinates due to contraction of matter

X X X X X X

 

and relative L of matter: L=1

 

2. Here is scheme picture how contracting observer observes the situation 2. above:

 

X X X

X X X

X X --> X X apparent expansion in co-trans. coordinates due to contraction of matter

X X X

X X X

 

relative L of matter: L=1

 

In both situations, contracting observer does not measure any changes in the properties of matter - no changes time rate or changes in energy in ordinary matter, due to relativity principle,and due to that all matter contracts equally, and relative transformation factor seems to be L=1 everywhere at present time.

 

But observer will measure relative changes in free radiation and in relativistic matter and compact matter, since radiation do not contract at all or contract at slower rate than ordinary matter:

 

Light in rigid/ non-transforming coordinates:

 

-------------------------------> (no changes in free radiation)

 

Light in co-transforming/co-contracting coordinates:

_________

____________----------------

------|____________ (redshift, diameter growth, energy loss, time dilation)

----------------_________

 

Relative changes in photon and light, observed by contracting observer:

wavelength: [latex] \frac{\lambda '}{\lambda} = \frac{1}{L(t')} [/latex]

light ray diameter: [latex] \frac{d'}{d} = \frac{1}{L(t')}[/latex]

energy:[latex] \frac{E'}{E} = L(t') [/latex]

velocity: [latex] \frac{c'}{c} = 1 [/latex]

 

(As the contracting matter do emit blue-shifted light in non-transforming coordinates , Co-contracting observer does not however measure any changes, no red shift in the light emission of matter in present time. The Balmer line is exactly in the same place than always for example.)

 

Have you tried testing the formulas through proper distance calculations?

 

 

I think the proper distance can be calculated in ordinary way:

 

[latex] D = c\int_{t_e}^{t_o}\frac{dt}{a(t)} [/latex]

 

,remembering the transformation equation for a between non-transforming and co-transforming coordinates

 

[latex] a_{co-transf.}(t') = \frac{a_{non-transf.}(t')}{L(t')}[/latex]

 

The velocity of light is invariant and its transformation equation between non-tr and co-tr coordinates is:

 

[latex] c_{co-tr.} = c_{non-tr.} [/latex]

 

by matter I assume your referring to fermions.?

Your going to need to clarify this term usage.

 

 

I mean ordinary matter a matter that is stable,non-relativistic and dont have significant gravitational time dilation: electrons, positrons, protons, atom nuclei and slow neutrinos, and dark matter particles.

 

The particle contraction law may apply to bosons also, but their time dilation determines how much do they contract, since i assume that the particle contraction law depends on proper time.

 

Compact matter should be treated as new density component, also because i assume that the particle contraction law depends on proper time.

 

[latex] L_{comp}(t') = L_{ord}(Dt') [/latex]

,where D is time dilation factor (which is a product of Lorentz gamma factor and gravitational time dilation factor).

 

 

The next question is "How will your model work during the radiation dominant era? The % of matter is much smaller with the % of radiation being much higher leading to a higher rate of expansion per Mpc. Higher value for Hubbles constant. (not per size of universe).

 

 

 

Then the first Friedman equation would be:

 

[latex] \left ( \frac{\dot{a}}{a} \right )^2 =\left ( \frac{8 \pi G}{3} \right )\left ( \frac{\rho_{r,0}}{a^4} \right )+k^2 +\Lambda [/latex]

 

(Here i add Lambda into equation since k^2 term is not necessarily same as Lambda)

Swansont:

 

But this radiation is never observed.

 

 

The phenomenon is very very weak.

 

The binding energy of proton - electron system is 13.6 eV.

The binding energy of aplha particle is 28.3 Mev

 

we can use linear approximation for 1/L(t') = E'/E = (1 + kt)

 

then The radiation power from contraction of matter is E* k , where E is binding energy.

 

The assuming that k = Hubble constant = 2.20 *10⁻18 1/s , which is close to right value of k,

 

possible radiation power of hydrogen atom from isotropic contraction is 13.6 eV * 2.20 * 10^-18 1/s = 3.0 * 10^-17 eV/s

 

possible radiation power of alpha particle from isotropic contraction is 28.3 * 10^6 eV * 2.20 * 10^-18 1/s = 6.23 * 10^⁻12 eV/s

Michel

 

That would mean that observed space expansion should happen not only between galaxy clusters but also inside galaxies, even inside the solar system. Or do I understand badly?

 

the validity of principle of relativity (= we are not able to measure any changes), demands that in the transforming area, most of the matter should undergo equal transformation (=transformation has to be homogenous) when the whole area contracts.

 

 

Also, why do you assume that the speculated phenomenon happens so smoothly? If the observer cannot understand his own shrinking, the contraction could happen a vertiginous rate. For example: each second a shrinking of 300000 km.

The isotropic contraction is only possible when the binding forces are greater than the contraction phenomenon.

 

If we imagine situation when the contraction would be ultra-relativistic, and if the contraction happens first in the particles, then the macroscopic system cannot follow the contraction and contract isotropically. In this situation all particles just remain into their places and contact towards their center, or even the particles may behave like cake where the candies =some things in the particles inner structure, just contracts towards their center.

 

 

In a way, you could simply insert your speculation in mathematical terms and put a value of zero (or 1) for a non-scaling universe. Afterwards, change the value of the scale factor and see where it meets observation. Or is that what you already did?

But: still the main issue with the scale factor is that if contraction has multiple centers (for example one in each particle), the macroscopic objects would be feeling like teared apart. Maybe that is the reason why you must assume it happens smoothly. But in fact the problem is not the smooth, the problem is the center.

 

Yes, this contraction of particles has to be very slow process, if macroscopic object can contract isotropically.

The linear approximation fit to Hubble's law is

 

[latex] \frac{1}{L(t')} = (1 + kt') = (1 + Ht') [/latex]

 

But you can also estimate that

[latex] L(t') = e^{kt'} [/latex]

And do fit to cosmological constant in Friedmann equation:

[latex] k = \sqrt{\Lambda}[/latex]

Edited March 3, 2016 by caracal

[caracal]

Issue: Friedmann equation in rigid/non-transforming coordinates

 

I here derive the 1. Friedmann equation in non-transforming coordinates for exponential guess, starting from scale factor equation:

 

Transformation factor functions for exponential guess:

 

Transformation factor function in co-transforming time: [latex] L(t') = e^{-k(t'-t_0')} [/latex]

 

[latex] (t-t_0)= \int_{t_0'}^{t'}e^{-kt'}dt'=\left ( \frac{1}{k}\right )[1-e^{-k(t'-t_0')}] [/latex]

 

[latex] (t'-t_0')=\left ( \frac{1}{k} \right )Ln[1-k(t-t_0)] [/latex]

 

[latex] e^{-k(t'-t_0')}=[1-k(t-t_0)] [/latex]

 

Transformation factor function in non-transforming time:[latex] L(t) = [ 1-k(t-t_0) ] [/latex]

 

(own note: it seems that even when the universe is infinite in co-transforming coordinates and the lifespan of matter is infinite in co-contracting coordinates, in principle contracting matter can have finite lifespan in non-transforming coordinates - if L(t) really manages to go to zero at some finite time)

 

Scale factor equation: [latex] a_{r}(t) = a_{c}(t) L(t) = a(t) [1-k(t-t_0)] [/latex]

 

[latex] \Leftrightarrow \dot{a_r} = \dot{a_c} L - a_{c}k [/latex] first time derivative

 

[latex] \Leftrightarrow \left ( \frac{\dot{a_r}}{a_r} \right ) = \left ( \frac{\dot{a_c}}{a_c} \right ) - \frac{k}{L} [/latex]

 

Friedmann eq. in co-contracting coordinates: [latex] \left ( \frac{\dot{a_c}}{a_c} \right )^2 = \left ( \frac{8\pi G_{c}}{3} \right )\rho_{c} + k_{c}^2 [/latex]

 

Densities in co-contracting coordinates: [latex] \rho_{mc} = \frac{\rho_{(m,0)c}}{a_{c}^3} ; \rho_{rc} = \frac{\rho_{(r,0)c}}{a_{c}^4} [/latex]

 

transf. equation for density and scale factor:[latex] \rho_{c} = \rho_{r}/L^{4} , a_c = a_{r}/L[/latex]

 

Trasf.eq for gravitation constant and k: [latex] G_c = G_r * L^2 ; k_c = k_r/ L^2 [/latex]

 

Friedmann equation in co-contracting coordinates in terms of rigid coordinate system units:

 

[latex] \Rightarrow \left ( \frac{\dot{a_c}}{a_c} \right )^2 = \left ( \frac{8\pi G_{r}}{3} \right )\left (\frac{\rho_{(m,0)r}L}{a^3} + \frac{\rho_{(r,0)r}L^2}{a^4} \right ) + \frac{k_{r}^2}{L^2} [/latex]

 

Now finally Friedmann equation in rigid/non-transforming coordinates:

 

[latex] \Rightarrow \left ( \frac{\dot{a_r}}{a_r} \right )^2 = \left ( \sqrt{ \left ( \frac{8\pi G}{3} \right )\left ( \frac{\rho_{m,0}L}{a^3} + \frac{\rho_{r,0}L^2}{a^4} \right ) + \frac{k^2}{L^{2}} } -\frac{k}{L}\right )^2 [/latex]

 

[latex] \Rightarrow \left ( \frac{\dot{a_r}}{a_r} \right )^2 = \left ( \sqrt{ \left ( \frac{8\pi G}{3} \right )\left ( \frac{\rho_{m,0}[1-k(t-t_0)]}{a^3} + \frac{\rho_{r,0}[1-k(t-t_0)]^2}{a^4} \right ) + \frac{k^2}{[1-k(t-t_0)]^{2}} } -\frac{k}{[1-k(t-t_0)]}\right )^2 [/latex]

 

(This may be wrong, there may be a mistake somewhere)

Edited March 5, 2016 by caracal

[caracal]

Problem: Gravitation constant of light and gravity constant for cross interactions

 

Here i represent a problem that is related to gravitation constant of transformed light. I am not sure if this is real problem or can it be explained.

 

When photon has apparent isotropic velocity-invariant transformation relative to the contracting observer, then according to transformation equations it would be expected that the gravitaion constant for light changes and is not constant. That seems not to be problem. But the problem is that Also the cross interaction gravitation constant may differ from ordinary gravitation constant. and that seems to cause a problem

 

frequency: [latex] \frac{f}{f_0} = L [/latex]

wavelength :[latex] \frac{\lambda}{\lambda_0} = \frac{1}{L} [/latex]

Energy: [latex] \frac{E}{E_0} = \frac{1}{L} [/latex]

 

[latex] \rightarrow [/latex]

 

Electric field: [latex] \frac{\bar{E}}{\bar{E_0}} = \frac{1}{L} [/latex]

Magnetic field:[latex] \frac{\bar{B}}{\bar{B_0}} = \frac{1}{L} [/latex]

Gravitation constant: [latex] \frac{G}{G_0} = L^{2} [/latex]

Coulomb constant:[latex] \frac{k}{k_0} = L^{2} [/latex]

Magnetic constant: [latex] \frac{\mu}{\mu_0} = L^{2} [/latex]

 

Electromagnetic stress-energy tensor: [latex] \frac{T^{\mu v}}{T^{\mu v}_0} = \frac {1}{L^{4}} [/latex]

 

The einstein field equation does not change its form for local gravitational interactions in isotropic velocity-invariant transformation:

 

[latex] R_{\mu v} - \frac{1}{2} R g_{\mu v} = \frac{8 \pi G}{T_{\mu v}} [/latex] ,

 

[latex] <R_{\mu v}> = <R> = \frac{1}{m^2} , <g_{\mu v}> = m^2 / s^2 , <T_{\mu v}> =J/m^{3} [/latex]

 

[latex] \rightarrow \frac{R_{\mu v}}{R_{\mu v0}} = 1/L^2 , \frac{g_{\mu v}}{g_{\mu v0}} = 1 , \frac{T^{\mu v}}{T^{\mu v}_0} = \frac {1}{L^{4}} [/latex]

 

[latex] \rightarrow [/latex] all terms change like [latex] 1/L^2 [/latex]

 

There is no problem for ordinary gravity of photons. This is not really a big surprise since this velocity invariant transformation is mathematically equal to math of unit conversions with new energy, time and length unit, and unit conversions does not change the form of any equation.

 

 

But in cross interactions, the gravitation constant may be [latex] G = \sqrt{G_{1} G_{2}} [/latex] which is geometric mean of the two different constants.

 

IF so, then the cross interactions of photon and matter or between differently transformed matter should depend on relative L, which makes a problem if apparently transfomed photon is identical to normal photon with another wavelength..Then photons gravitation constant for cross interaction should depend on relative frequency:

 

[latex] G_{photon cross-int} = \frac{f_0 G_0}{f_{relative}} [/latex]

 

where [latex] f_0 [/latex] is some fundamental frequency, and [latex]G[/latex] is normal gravitation constant.

 

and when the nominator goes to zero, G should go to infinity. In other words, for small relative frequency difference and long relative wavelength radiation should have very high gravitational constant.

 

how this can be?

 

One possible way out solution for this problem is that photons and matters cross-interaction gravitation constant is just always same.

 

Another way out solution is that G=0 for em radiation.

 

(note also that photon never undergo real transformation, since it has infinte time dilation, only apparent transformation, that is caused by the real transformation in the contracting observer)

 

by considering particles for the sake of wave-particle duality, if similar equation would be present also for particles, then the gravitation constant or em-constant of particles would depend on their relative velocity:

 

[latex] k_{rel} = \frac{v_{0} k_{0}}{v_{rel} \gamma_{rel}} [/latex]

Edited March 10, 2016 by caracal

[caracal]

I think this thread becomes too messy. I think i should write a short summary of everything.

[Phi for All]

I think this thread becomes too messy. I think i should write a short summary of everything.

 

!

Moderator Note

Excellent idea. I think you can place it here, as a "most updated" version, rather than starting a new thread.

[caracal]

...i first put here what kind of predictions this model gives for transformation factor L = 1 + kt where k = H = hubble constant.

some of the calculations are very simplified

Predictions:

 

 

 

Most notable predictions:

 

1. Modification to Newtons second Law of stars due to transformation difference in stars ,that caused by gravitational time dilation and orbital velocity time dilation

2. Microwave emission lines in galactic glouds that consists of mixed matter with different time dilation histories

3. Distance expansion in milky way : earth moon distance has distance expansion 2.63 cm/year

4. Atomic clock of cassini probe gains additional time dilation

5. Atomic clock of GPS satellite gains additional time contraction (clock runs faster relative to earth)

6. solar protons has gained transf. Difference due to time gravitational time dilation during 10 Billion years, therefore they should have permanent shift in Rydberg constant that has factor (1 + 0.7 * 10^-6) and cosmic protons may also have shift in Rydberg constant.

 

 

Differently transformed matter:

 

 

Matter with time dilation will gain transformation difference relative to ordinary matter:

 

[latex] L_rel = \frac{L(\Delta t')}{L(\Delta(Dt'))} [/latex]

 

[latex] L_rel \approx (1 + kDt') [/latex] linear approximation

 

where D is time dilation factor [latex] D=(1-t'/t)[/latex]

 

Whenever there is time dilation differences, the particle contraction happens at different rates and therefore the matter with different time dilations should gain slowly relative transformation difference.

 

Lets assume that k = H = 6.93 * 10^-11 1/year = 2.20 * 10^-18 1/s

 

 

Modified Newtons second law:

 

If two bodies have different transformation factors L1 and L2, then the Newtonian gravitational force remain same:

 

[latex] F'= \sqrt{G_{1}'G_{2}''}\frac{M_1'M_2''}{R^2}= G_{12}\frac{M_1M_2}{R^2} = F [/latex]

 

But since M1 has 1/L1 times more mass and M2 has 1/L2 times more mass, their accelerations changes such that:

 

[latex] a_{M1}'= F/M_1'=L_1(F/M_1) [/latex]

[latex] a_{M1}'= F/M_1'=L_1(F/M_1) [/latex]

 

You can say that the net effect in the gravitational interaction is that the newtons second law depends on the relative transformation factor:

 

[latex] a' = F/m' = L(F/m) [/latex]

 

 

Transformation differences on milky way disc

 

Lets assume that stars in milky way behind sun has velocity difference

v=100km/s, the maximum time dilation factor is then D = 5.56* 10^-8

During 1 Billion years the matter will gain transformation difference:

 

L = 1 + [6.93 * 10^-2 * 5.56 * 10^-8 ]

= 1 + 3.85 * 10^-9

 

And the Newtons 2^nd law will change such that:

 

a'=FL/m

 

And relative to the center of milky way, the matter has relative velocity 200 km/s , when D = 2.20 * 10^-7 ,when the gained transformation difference during 1BY relative to the center is:

 

L = 1 + [6.93 * 10^-2 * 2.20 * 10^-7]

= 1 + 1.52 * 10 ^-7

 

 

Trasformation difference of matter on surface of sun relative to earth:

 

The sun’s surface has gravitational time dilation (1 - 2.13 * 10^-6) relative to infinity.

 

The earth surface has gravitational time dilation (1 – 6.96 * 10^-10) and orbital velocity related time dilation (1 – 5.00 * 10^-12) ,which are about 5-6 magnitudes smaller than sun’s surface time dilation so I neglet them.

 

Lets assume that a particle has spend about 10 Billion years in the sun. Then it has gained transformation difference relative to matter on earth’s surface by

 

Lrel = (1 + 0.693 * 2.13 * 10^-6) = (1 + 1.47 * 10^-6)

 

This difference should cause equal redshift in spectral lines if the proton and electron have been equally transformed.

 

If the proton and electron are not equally transformed, then the redshift would be

 

[latex] \lambda'/\lambda = \sqrt{L_1L_2} [/latex], where L1 and L2 may vary from 1 to Lrel, and for constant distributions for L1 and L2 the mean value of this would be:

 

mean redshift = 1 + 0.74 * 10^-6

 

If a proton leaves the sun after spending 10BLY under gravitational time dilation, among solar wind and arrives to earth, and is catched by scientist who takes it into laboratory, and “feeds” it by normal electron, the scientist would observe that the proton’s Rydberg constant has permanently shifted into:

 

R’ = 13.6 eV * (1 + 0.74 * 10^-6 )

 

The protons mass to charge ratio is however exactly same (Q/m)’ = (Q/m)

 

 

Transformation difference of Cassini Probe:

 

The Saturn has orbital velocity v=9.67 km/s, while earth’s orbital velocity is v=29.78 km/s and titan has orbital velocity v = 5.57km/s relative to Saturn. The net time dilation difference between titan and earth is therefore:

 

[latex] D = (1 -1/\gamma) =1- \frac{1}{\gamma (20.1km/s)}\cdot \frac{1}{\gamma (5.57km/s)} =2.44 \cdot 10^-9[/latex]

 

Cassini probe has spent 10 years orbiting Saturn. A rude estimate for the transformation difference the cassini has gained during this time is:

 

L = (1 + 6.93*10^-11 * 2.44 * 10^-9 * 10 ) = (1 + 1.691 * 10^-19)

 

Therefore the atomic clock in the cassini probe should have additional permanent time dilation relative to earth by factor

 

(t’/t) = (1 + 1.691 * 10^-19)

 

(this time dilation is permanent property of the matter in the sense that it does not vanish when the probe is returned to earth, but it can be cancelled if the matter gains relativistic time contraction somehow, when the probe starts to gain transformation difference with factor L<1)

 

Due to transformation, Cassini probe’s mass has decreased also by factor 1/L and therefore the gravitational acceleration of Cassini probe should increase permanently by factor

 

a’/a = 1 + 1.691 * 10^-19

 

,assuming that saturnus consists of uniformly transformed matter that has same gravitational constant.

 

 

Transformed matter in satellites orbiting earth

 

GPS satellites orbiting earth have time contraction (1 - 4 * 10^-10) relative to the surface of earth. Lets assume that GPS satellite has spent 10 years in the orbit. During that time, the satellite has gained transformation difference relative to earth:

 

L = (1 – 6.93*10^-11 * 4 *10^-10 *10) = (1 – 2.77 * 10^-19)

 

note that L<1 in this case.

 

and the atomic clock in the GPS satellite runs faster by additional factor (1 + 2.77*10^-19) (together with relativistic time dilation). This effect does not vanish if the satellite is brought to the surface of the earth and the atomic clock is compared to identical clock on the surface.

 

 

Transformed matter in stars and modification to Newton’s second law

 

Lets assume that two stars with mass and size of a sun that are 10 Billion years old interacts by gravitation. The gravitational time dilation of the matter relative to infinity is (1 - 2.13 * 10^-6) and matter in both stars have gained transformation difference during 10 BYrs:

L = (1 + 0.693 * 2.13 * 10^-6) = (1 + 1.47 * 10^-6)

 

Their gravitational force remain same because the changes in G and M1,M2 cancel each other out:

 

[latex] F_{1\rightarrow 2}'= G_{12}'\frac{M_1'M_2'}{R^2}= G_{12}\frac{M_1M_2}{R^2} = F_{1\rightarrow 2} [/latex]

 

But since the mass of both stars change by 1/L , the Newtons second law changes such that:

 

[latex] a’ = F/m’ = L(F/m) [/latex]

 

,when a’/a = (1 + 1.47 * 10^-6 )

 

Generally the time dilation factor for star depends on its surface and mass such that:

 

[latex] (\frac{\tau}{t_{inf}}) = \sqrt{1-\frac{2GM_{\odot}}{R_{\odot }c^2}(\frac{M}{M_{\odot}}/\frac{R_{\odot}}{R})} [/latex]

 

and the Modified Newtons second law depends solely on the transformation difference of the most matter in the star such that:

 

a’/a = (1 + 6.93*10^-11 [1/year] * (1 - t’/t) * T[years])

 

 

Transformation difference of matter in supernova remnant

 

Lets now look at supernova remnant, a gas clouds, that have expanded 10 000 years at 10% velocity of light, and has time dilation factor

D = 0.005 .During this time the matter in the gas has gained transformation difference

 

L = (1 + 6.93*10^-11 *10^4 * 0.005) = (1 + 3.465 * 10^-9)

 

 

Transformation differences of matter in the milky way clouds and anomalous microwave emission

 

Milky way gas cloud may consist of mixed matter that has different transformation factors. Lets assume that gas consists of following components: old gas and supernova remnants and stellar remnants during last 10 Billion years. The matter may have had different time dilation histories in different stars and during supernova remnant expansions. The old gas has barely gained transformation difference, but matter from supernova remnant may have spent some million years in star and some thousand years in expanding remnant.

 

Lets crudely estimate that there are transformation differences of

 

[latex] L_{rel}= (1 + 5*10^{-9})[/latex]

 

between the electrons and protons or nuclei. The transformed proton with ordinary electron has then shift in Rydberg constant such that

 

R’ = 13.6eV * (1 -2.5*10^-9)

 

while ordinary proton has no shift in Rydberg constant.

 

Now lets assume that the electron jumps from one proton to the another. The electron should now emit a quantum of radiation with energy:

 

E = 13.6 eV * 2.5*10^-9 = 3.4 * 10^-18 eV , which corresponds frequency:

 

f = E/h = 8.2 Ghz

 

,which falls into microwave radiation range. The distribution of the emission is likely to be smooth, but may have broad spikes because the matter with similar history has a similar transformation difference, and there is average transformation differences between two clouds with different history.

 

This is interesting, since gas clouds in milky way are known to emit microwaves, which is called anomalous microwave radiation (AMR)

 

 

Transformation difference of matter in galaxies

 

the peculiar velocity of galaxy can be over 1000km/s, when the average relative time dilation factor between two galaxies can be D = (1 + 5.59 * 10^-6) If the galaxies maintain their high velocity for long period, and then two galaxies with different velocities merge, it is expected that there is now mixture of two “clouds” of matter with certain average transformation difference.

 

 

Transformation difference and Doppler redshift distribution in milky way

 

The transformation difference, when L>1 causes matter to emit light that has redshift, that may be difficult to distinguish from the redshift that is produced by relativistic Doppler effect.

 

Lets assume that a star has relative transformation difference

L = (1 + 10^-9)

then it will emit light that has additional redshift:

 

[latex] \lambda'/\lambda = (1 + 10^{-9}) [/latex]

 

which corresponds to relativistic Doppler effect of object that has relative velocity v = 15 cm/s

 

 

Transformation difference of matter in ultrarelativistic jets of Quasar

 

Matter in ultrarelativistic jets of Quasar should gain transformation difference close to cosmological L(t) as long the matter has ultrarelativistic velocity.

 

 

Schwarchild radius of black hole

 

-the Schwarchild radius of black hole should grow by factor L(t) because the matter in BH has infinite time dilation. Also the second law of newton changes such that a' = FL/m

 

 

 

B Prediction: Distance Expansion

 

 

Distance expansion in solar system and in Milky way

 

Distance expansion is weak, additional velocity component v = Hr that tend to pull celestial objects apart. It is apparent effect and is observed by only co-transforming observer, which we earthlings all are.

 

For moon-earth distance, the distance expansion is

 

dR =vt = 6.93*10^-11* 384000 km = 2.63 cm/year

 

For earth- sun distance, the distance expansion is

 

dR = vt = 6.93* 10^-11 * 150*10^6km = 10.4 m/year

 

The distance expansion can be compensated by that the planet has slightly more orbital velocity towards the sun, and can be therefore not visible.

 

The distance expansion of sun-center of milky way is:

 

dR = vt = 30 000 LYrs * 6.93*10^-11 = 2.8 * 10^-6 Ly/year

= 19.7 mrd km/year

(And here is soon article that collects the main ideas in pdf format)

Edited March 16, 2016 by caracal

[caracal]

i attach here pdf-file, it is easier to modify. This a kind of "beta" version , not official, and not checked for all errors

 

 

isotrop_length_transf_hypothesis.pdf

[caracal]

(I have made mistakes in the gravitation interactions of transformed matter in earlier posts, i think i now get it right:

-The gravitation of transformed matter changes such that - transformed matter causes equal gravitation force than ordinary matter, but its inertial mass has changed.)

 

 

A Very Short Summary:

 

 

If the universe is observed by contracting observer, the Friedmann 1.st equation would be:

 

[latex] \left ( \frac{\dot{a}}{a} \right )^2 =\left ( \frac{8 \pi G}{3} \right ) (\frac{\rho_{m,0}}{a^3} + \frac{\rho_{r,0}}{a^4} ) +k^2 +\Lambda [/latex]

 

, if the matter contracts such that transformation factor function is [latex] L(t') = e^{-kt'} [/latex]

,where t' is elapsing time of contracting observer.

 

the term k^2 is very similar to cosmological constant and may therefore be a substitute for it.

 

There is now however possible new density component, "transformed matter", which should be inserted to above equation.

 

 

 

Transformed matter.

 

Transformed matter has different transformation factor L than ordinary matter.

 

It may be possible that the universe has already contained in early ages transformed matter, that has significant transformation difference relative to ordinary matter. This may be a candidate for dark matter.

 

In principle the transformed matter may be both contracted matter L<1 or expanded matter L>1

 

Even if it is assumed that all matter has have relative transformation factors close to 1 in early universe,the time dilation differences causes compact and relativistic matte to gain so called transformation difference relative to ordinary matter (=matter that has L =1 in co-contracting coordinates ):

 

This kind of process produces "relatively expanded matter" that has L>1 relative to ordinary matter.

 

[latex] L_{relative} = \frac{L(\frac{\tau_1}{\tau_2}(t'-t'_0))}{L(t'-t'_0)} = e^{kD(t'-t'_0)} [/latex]

 

,where D describes relative time dilation difference and it is [latex] D = (1 - \tau_1 / \tau_2 ) [/latex]

 

t' is elapsing time of contracting observer.

 

According to transformation equations, the gravitational mass, gravitation constant and inertial mass change for this matter such that:

 

[latex] m(t')/m(t_0) = \frac{1}{L_{relative}} = e^{-kD(t'-t'_0)} [/latex] and [latex] G(t')/G(t_0) = L_{relative}^2 = e^{2kD(t'-t'_0)} [/latex]

 

and compact and relative matter have therefore density:

 

[latex] \rho_{comp} = \frac{\rho_{comp,0}}{a^3} \cdot e^{-kD(t'-t'_0)} [/latex]

 

In the gravitational interaction, the change in gravitation constant cancels the change in gravitational mass:

 

[latex]F(t') = \sqrt{G_1 G_2} \frac{m_1(t') m_2(t') }{R(t')^2} = L_1 L_2 G(t'_0) \frac{L_1 m_1(t'_0) L_2 m_2(t'_0)}{R(t'_0)^2 a(t')^2} = F(t'_0) a(t)^{-2} [/latex]

 

(note that R changes such that [latex] R(t') = a(t')R(t'_0) [/latex] when the universe expands)

 

And the transformed matter causes equal gravitation force than ordinary matter (=matter that has L =1 in co-contracting coordinates )

 

But the Newtons second law changes for such matter such that:

 

[latex] a(t') = (F/m(t_0)) e^{kD(t'-t'_0)} [/latex]

 

-That is - compact and relativistic matter appears to cause equal gravitational force than ordinary matter, but it accelerates more easily. ( Its inertial mass have decreased)

 

 

k is close to hubble constant H = 2.20 * 10^-18 1/s = 6.93 * 10^-11 1/year

 

 

EDIT : i made some corrections

Edited March 20, 2016 by caracal

[caracal]

again i may have made mistake but.... i dont have time to correct them now

Edited March 22, 2016 by caracal