 axiom

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Introduction

In this article, I shall fulfil two tasks. Firstly, I shall deduce total negative action in universe from particle's negative action, d(-S)=-mcuidxi-qAidxi. Total negative action in universe has the following form:

(1) d(-S)=-mcuidxi-qAidxi+aFik2dVdt-bRdVdt

In physics, ui, dxi/ds, is explained as four-dimensional velocity, Fik is explained as electromagnetic field tensor, R is explained as space-time curvature, a and b are positive constants. (1) can be explained differently in other pursuit systems.

Secondly, I shall prove that universe, the best pursuit system, will expand, because particles will contintuously absorb energy from electromagnetic field.

I would like to choose negative action as the basic quantity and largest negative action principle as the basic principle, but, in order to be in accordance with physicists' custom, I use action and least action principle in this article. I will deduce all the other quantities, including space-time and field, from the principle. So "action" is a quantity unnecessary to explain, and other quantities are explained by "action". It is better to explain all concepts from one, not to explain one by all(although it is permitted and possible).

This might be a little abstract, because it throws away all simple concepts for human beings' feelings to deduce from a less familiar concept.

The abstraction of "action" comes from that human beings do not have the same feelings as the particles, so you can not understand their pursuit.

We need not know what the mathematical expression for negative action means. We can construct other forms of expression, but that will be different universes. In science of pursuit, those universes will be worse(in Truth Evolutionism, it means smaller observation probability) than the present universe.

1 Establishing Classical Physics on Seven axioms

1.1 Axioms

In this part, I will give seven axioms about action.

Axiom 1: Particle's apparent momentum along a direction is proportional to its velocity along the direction, the ratio is a constant.

In order to have the same form as physics, the constant is marked as mc, and define velocity along a direction as

(2) ui=dxi/ds

So, axiom 1 is

(3) pi=mcui

Axiom 2: Real momentum along a direction is apparent momentum plus qAi.

From axiom 1 and 2, particle¡¯s real momentum along a direction is:

(4) Pi=mcui+qAi

Axiom 3: For motion along a direction, its action is product of real momentum along the direction and displacement along the direction.

So action along the direction is Pidxi.

Axiom 4: When a particle moving along several perpendicular directions simultaneously, the total action is sum of action along each direction.

From axiom 3 and 4, particle¡¯s action Sm is

(5) dSm=sumi(Pidxi)

Axiom 5: Action of field is expected total change of action for all possible particle states in the field.

Since every state is possible to be occupied, the natural way to evaluate field is sum of all possible influences to action. If action pursuit in universe is rational, the expected total change of action will be equal to the actually realized total change in a large scale.

Axiom 6: Sum of all realized action and expected change of action in whole universe is total action in whole universe. Total action is pursued to be the least.

Action of particles is realized action; action of field is expected change of action, Sf. Total action in universe is:

(6) S=Sm+Sf

Axiom 6 corresponds to principle of least action.

Axiom 7: For a particle, there are four perpendicular directions, among them there is only one direction along which action is negative.

So there is only one dimension with imaginary momentum and displacement; there are three with real. The direction with negative action is defined to be time.

These axioms form a set of rules to calculate action. If universe is uniquely decided by axioms about particles, which means knowledge about space-time and field can be deduced from mathematical expression for particle's action, dS=mcuidxi+qAidxi, physics is greatly simplified. Space-time metric, Fik2 and R are not knowledge foundation about universe.

1.2 Theorems

Theorem 1: In local unit, space-time metric satisfies

(7) ds2=dxt2-dx2-dy2-dz2

The three directions with positive action are x, y and z.

Proof: From (3) and (5), for a particle moving across dxi, its action is mc(dxi/ds)dxi. From axiom 6, action is pursued to be the least. For ds direction, velocity is 1, so action is mcds.

(8) mc(dxi/ds)dxi=mcds

So

(9) ds2=sumi(dxi2)

From axiom 6 and 7, action along dt and ds is negative. In order that dt and ds for a particle¡¯s orbit be real number, replace dxt2 and ds2 in (9) with -dxt2 and -ds2, then (7) is proved. So geometry is result of action calculation. Sum of all possible qAidxi is zero because opposite charges have opposite value. So ds is expected action along a path. Area and volume of a region is total expected action in the region.

Currently, calculation of action in physics and Euclidean geometry are independent knowledge. Notice that there is no new hypothesis in the proof. Logic simplicity, even without new experimental results, also means knowledge progress.

For electromagnetic field, expected total change of action is proportional to -Flk2.

Proof: From (5), by variation, a classical particle's best pursuit behavior satisfies:

(10) mc(dui/ds)=qFikuk

In (10),

(11) Fik=Ai,k-Ak,i

If the field area is small enough, Fik can be treated as constants. Sum of all possible qAidVi is zero. So, for all possible states on a closed three-dimensional hypersurface C, after moving unit ds, expected total change of action is

(12) dSf=¨° CPidVi=Fik¨° CqukdVi=-Fik2 ¨°C(q2/mc)dVdt

It is symmetrical for q and ¨Cq, so it is not zero.

Statistically, charges always absorb energy from electromagnetic field and increase particles¡¯ static energy. The expected absorption rate is proportional to (E2+H2)/2.

Proof and discussion: From theorem 2, we have action of electromagnetic field, from which we can deduce that momentum-energy of electromagnetic field is proportional to EXH and (E2+H2)/2. If action of field represents expected total change of action, momentum-energy of field represents expected total change of momentum-energy for all possible particle states.

Suppose charges enter an enough small region uniformly from all directions. After unit time, there are three net movements. Firstly, there is net current, which is proportional to E, toward E, because positive charges are accelerated toward E and negative charges toward -E. Secondly, there is net current, which is proportional to H, around H, because positive charges are accelerated clockwise and negative charges anti-clockwise (viewed along H). Their contribution to real energy is proportional to (E2+H2)/2. Thirdly, there is net momentum, which is proportional to ExH, toward ExH, because the current toward E will interact with H. (E2+H2) is always positive, so charges absorb energy from electromagnetic field statistically.

Kinetic energy of particles, including photons, is called unstable energy. Particle¡¯s static energy is called stable energy. From theorem 3, unstable energy is gradually absorbed by particles, so stable energy increases monotonously in electromagnetic interaction. Unstable energy will become part of static energy. If there were no new unstable energy, unstable energy would be exhausted.

Ultimately speaking, all unstable energy is generated from stable energy through strong and weak interaction, and turns into stable energy through electromagnetic interaction. From theorem 3, this circulation is self-magnified. If particles have larger static energy, they have stronger ability to ¡°invest¡±¡ªcreate more unstable energy in one fusion, so the universe can expand forever. Since electron¡¯s static energy increases monotonously, atom radius reduces monotonously.

Theorem 4:

Past unstable energy depreciates with time relative to particle¡¯s growing stable energy.

Proof:

When measured with a fixed energy-momentum unit, static energy increases with time. Suppose the speed is uniform,

(13) d(mc2)/dt=Hmc2

So, in fixed unit,

(14) mt=m0exp(Ht)

Space-time geometry is decided by atom radius of rulers and clocks. When atom radius reduces, clock is slower and ruler is shorter,

(15) dxi(t)=dxi(0)exp(-Ht)

So ruler is shorter and clock is slower with time. In order to measure correctly, observer needs to appoint his development. From time component of (15), we can deduce

(16) dt/dt0=1/(1+Ht0)

From (16), for a pursuit system with uniform developing velocity, no matter when you measure, its origin time is -1/H. Similarly, reciprocal of developing velocity is nominal origin time. For example, if developing velocity is 1%/y, economy is 100 years old in present unit. But if we use local time unit to measure, there is no origin time. This is a property for exponential development.

Atom's energy level is proportional to mass, so energy level in the past is smaller than that at present, and light will have a red-shift,

(17) df=f0[exp(-Ht)-1]

When Hdt is very small, the red-shift is f0Hdt. This is Hubble red-shift.

From (14) and (15), mcui is invariant with uniform development, so for observer at t0, his measured action is

(18) Pidxiexp(-Ht)=m0cds

Define

(19) dxi=dxiexp(-Ht)

Then, from (18),

(20) ds2=hiidxidxi

(21) hii=exp(-Ht)

Define

(22) Pi=m0c(dxi/ds)=hiiPi

Pi is momentum-energy along displacement measured with local unit. Pi is momentum-energy along displacement measured with fixed unit.

Space-time curvature comes from development difference.

Proof:

If development not only changes with time, but also with spatial coordinates, general form for (19) is

(23) dxi=hijdxj

So, similar to theorem 1,

(24) ds=(hijdxj/ds)dxi

Then

(25) ds2=hijdxidxj

So development distribution makes space-time curved.

For gravitational field, its expected total change of action is proportional to space-time curvature.

Proof:

By variation, particle¡¯s best acceleration is

(26) dui/ds=Gkij (dxk/ds)(dxj/ds)

For all possible uj, after moving ds, their total contribution to action is:

(27) dSg=mc¨° CG kij(dxk/ds)dxj=Rkijlpkdfjl

C is a closed path. If k is not equal to l or m, -pk and pk will have opposite contribution to action. So index k and l, or j, ought to contract. dSg is sum of all components of Ril,

(28) R=hilRil

Since distribution of development changes action, by variation, the best distribution of development is not uniform development and existence of gravitational field is unavoidable. From above, Sg is result of absorbing unstable energy unevenly, so gravity is not an independent interaction.

Electromagnetic interaction decides how to distribute and absorb unstable energy the best. How to produce unstable energy the best ought to be decided by strong and weak interaction. All three interactions have the same aim¡ªaction pursuit. Matter and space-time are links in a development circulation, in which every link is developing. If one link broke, the other links could not develop.

From theorem 2 and theorem 6, (1) is proved.

So the non-quantum universe is formalized. Any pursuit system satisfying these axioms will have the same results. For example, imagine universe is a society and each particle is a person. Treat a particle's negative action as happiness. Thinking about every complex interaction between particles as happiness exchange between persons, then a particle choosing state to have least action cooresponds to a person choosing state to have largest happiness.

So the non-quantum universe can be logically reproduced in any pursuit system.

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