1-3)The law of motion of negative mass and negative mass


----- r -----

- m1 --------- - m2
caption : Negative mass - m1 and negative mass - m2
(initial velocity =0, m1 > 0, m2 > 0)

- m1a1 = - G( - m1 )( - m2 ) / r^2
a1 = + Gm2 / r^2

- m2a2 = - G( - m1 )( - m2 ) / r^2
a2 = + Gm1 / r^2

Negative mass and negative mass: Both two objects are accelerated in the direction of +r which extends distance r, so as time passes, the distance between them is greater than initially given condition, and the force between them is attraction, but the effect is repulsive.

The force is attraction(-Gm1m2/r^2), thus the potential energy between them has negative value.

If negative mass and positive mass were born together at the beginning of universe, positive mass has attractive effect each other, so it forms star and galaxy structure now, but negative mass has repulsive effect each other, so they cannot make massive mass structure like star or galaxy.

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Hypothesis of Dark Matter and Dark Energy with Negative Mass :
http://vixra.org/abs/0907.0015

3. Negative mass is stable at the maximum point!
So the catastrophe to energy level of minus infinity never happens.

Nature prefers stable state, and has the tendency to go to stable state. Additionally, this can be expressed in another way that nature prefers low energy state, and has the tendency to go to low energy state.

Such an idea is frequently used as a logic which denies the existence of negative mass. That is, if there is negative mass and negative energy level, negative mass spontaneously emits energy to be stable, and goes to energy state of minus infinity, so finally it is confronted by catastrophe. Is it right?

In case of positive mass, stable state means low energy state, therefore it is not necessary to divide which one nature prefers among two states(stable state and low energy state).
By the way, does stable state mean low energy state also in case of negative mass?

We can get an answer, if we examine Harmonic oscillation.

User posted image: User posted image

fig04
caption : When there is negative mass in potential which has a point of maximum value and a point of minimum value.

We begin by considering the oscillatory motion of a particle that is constrained to move in one dimension. We assume that there exists a position of unstable equilibrium for the particle and we designate this point as the origin. Restoring force is in general some complicated function of the displacement and perhaps of the particle's velocity or even of some higher time derivative of the position coordinate.

We consider here only case in which the restoring force F is a function only of the displacement

F(x) can be expanded in a Taylor series,
F(x) = F(0) +(x/1!)F'(0) +(x^2 /2!)F''(0) + ...... +(x^n /n!)F^n(0) +...

Since the origin is defined to be the equilibrium point, F(0) must vanish, Then, if we confine our attention to displacements of the particle that are sufficiently small, we can neglect all terms involving x^2 and higher powers of x. We have, therefore, the approximate relation

F(x)=+kx

The force is alaways the opposite directed toward the unstable equilibrium position(the origin), the derivative F'(0) is positive and therefore k is a positive constant.

-m(d^2 x/dt^2) = +kx
(d^2 x / dt^2) + (w_0)^2 x =0
((w_0)^2 = k/m )

This form of differential equation is the same as that of particle which has positive mass. But we have to notice that positive mass carries out harmonic oscillation on a point of minimum value, whereas negative mass carries out harmonic oscillation on a point of maximum value. Additionally, restoring force is +kx at this time.

F = -∇U
U = - (1/2)kx^2
E_- = T+U = -(1/2)mv^2 -(1/2)kx^2
(v=dx/dt)

In phase space
x^2/(-2(E_-)/k) + p^2/(-2m(E_-)) = 1
This equation is ellipses equation, because total energy E_- < 0

As examined in the question of Harmonic oscillation,
in case of positive mass, a point of minimum value which energy is the lowest is stable. However, in case of negative mass, stable equilibrium is a point of maximum value, not a point of minimum value. So negative mass is toward a point of maximum value to be stable, not a point of minimum value which energy is low.


In the world of positive mass, ground state is a point that energy is low, but in case of negative mass, ground state is a point that energy is the highest.

Accordingly, in the world of negative mass, energy level is filled from the highest to the lowest, and stable state means the highest energy state, so the catastrophe to energy level of minus infinity never happens even if negative mass spontaneously emits energy.

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Hypothesis of Dark Matter and Dark Energy with Negative Mass :
http://vixra.org/abs/0907.0015

V-3. Negative mass has observed, but it was trashed away! -

In 1998 year, observation by HSS(The High-z Supernova Search team) team and SCP team(Supernova Cosmology Project team), I have seen a calculation that if the cosmological constant is 0,

Ω_M = -0.38( ± 0.22)

It means that they got a negative mass density from inspected field equation during 70years.
HSS team: http://arxiv.org/abs/astro-ph/9805201 :14P 26, 29 lines

However, HSS team and SCP team have asserted that cosmological constant must be revival. Because Ω_M must not be negative value in they thought.

They did not know that negative mass is stable at the maximum point!
In case of negative mass, ground state is a point that energy is the highest. Accordingly, in the world of negative mass, energy level is filled from the highest to the lowest, and stable state means the highest energy state, so the catastrophe to energy level of minus infinity never happens even if negative mass spontaneously emits energy.(Refer to II-3)

We must to know that not the equation(Einstein's Field Equation) has disposed the value, but our thought disposed that value. Einstein said that cosmological constant was my biggest mistake, because it was inserted artificially.

In hypothesis with negative mass,
E_T = positive mass(4.6) + negative mass(y) + 0Λ(72.1)
(negative mass ≥ positive mass, cosmological constant Λ = 0 )
E_T = positive mass( x ) + negative mass( y ) + U_T
We assume over 50% of the dark matter as negative mass.


- Age of universe -
If cosmological constant Λ=0 and mass density of universe is negative value Ω_M = -0.38, to get the age of the universe.
From Friedmann equation,

User posted image: <a target='_blank' href='http://4.bp.blogspot.com/_udXTGLmmdK4/Svhh3lfbI1I/AAAAAAAAADc/2yVvG7q95bE/s400/a86.jpg'>User posted image</a>

t_0 = 14.225Gyr
This value is approach to the t=13.73 Gyr calculated in a WMAP.(+3.6% large) considering the very large changes( Λ = 0 and Ω_M = -0.38 ; Λ is a coefficient of x^4), it seems to increase existence probability of negative mass.

Additionally,
if Ω_M = Ω_(visible) (4.6) + Ω_(dark) ( - 23.3) = - 18.7 ,
we get the t_0 = 14.927 Gyr

Universe' age by other method
Radioactive decay method : 15.2 ± 2.1Gyr,
HR diagram(Hertzsprung-Russel diagram):13.2 ± 2Gyr,
The arithmetic mean of two methods is 14.2 ± 2.05Gyr
So, it(14.225Gyr from negative mass density) is in the range estimated by other observations.


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Let’s arrange!
HSS team and SCP team had observed our universe, however they got a negative mass density from field equation.
They are very confused. But, they thought Einstein’s cosmological term(Λ). Wow!

But…,
Although they can be explained that universe is accelerating expansion by insert Λ, they make two new problems at least. Finally, problem has not solved but increased.

What is the origin of cosmological constant? And, what is the reason of that cosmological constant has a special value?
Physicist tried to explain with vacuum energy but it was failed.
Also, they have must explained dark matter. But they have not been observed dark matter during 30years at predicted place.

However,
if negative mass density is exist, accelerating expansion is explained rightly.
Also, I have assert that minus mass can be explained for the dark matter, decelerating expansion and accelerating expansion, fine tuning problem, inflation mechanism, the reason of that dark energy seems to has a small and constant value(like cosmological constant Λ), collision of Bullet cluster.

A certain view :
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one person(Icarus2) vs ΛCDM physists after 1998year.(about 50~60%? : relation branch) : No

(Einstein, Direc, Hubble,...Icarus2; Inspected field eq. during 70 years.) vs ( New field eq. supported by 50~60% of relation branch physists during 10 years.) : Yes
**************************************************
Cosmological constant and plus dark matter are not observed as yet.
Accelerating expansion and gravitational effect(centripetal force effect) are only observed.

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Hypothesis of Dark Matter and Dark Energy with Negative Mass :
http://vixra.org/abs/0907.0015

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