My discovery about mathematics

Substitute the radius of the particle for the constant which means the radius
of the universe in Einstein equation. Robertson-Walker metric is a solution.
For example,the proton and the neutron in the deuteron are kept in a ball.
You can transform this ball into arbitrary forms by transformations in the space.
T^0,0 and the density don't change then. the mass and volume
don't change. General relativity(?) explains the invariance of the volumes of
the neucleons in the nuclear. The other theories cannot do it.
Maxwell equations in general relativity(?) explain the quadrupole moment
and the spin-orbit interaction. In general relativity(?), the gravity and the
nuclear force are the same force like the case of Newtonian mechanics.

You can explain results of general relativity and the quantum physics in
Newtonian mechanics.
Let (r,θ) be the circular coordinate system. Put u=r^-1.
The orbit of the particle by the central force is decided by
θ=∫(E-U(u))^2-1du+C.
You may assume that U(u) is an analytic function ∑_n∈Nc_nuⁿ.
When r is large enough,you can negrect the terms c_nuⁿ (n≥4).
u(θ) becomes an elliptic function whose period isn't 2π. When u is large,
you must treat U(u) as it is. This may be the nuclear force.
(u_θ)²=E-U(u)≥0
In the case of the electric force,E decreases by the radiation of the light.
(u_θ)²=E-U(u)=0, The orbit becomes a circul.
u_θθ=2^-1(E-U)^-2-1U_uu_θ=2^-1U_u.
When the orbit is a circle, 0=u_θθ=U_u
E=U(u) is a local maximum point of U(u). The E's are discrete values.

(r,θ)を極座標とし、u=1/r, p=du/dθ とおく。中心力による質点の軌道は、
p²+u²=V(u) で定まる。　V(u) を　uの解析関数と仮定しよう。V(u)=∑_i∈Nc_iuⁱ.
rが十分大きいと、3次以上の項は無視して、　p²+u²=c₀+c₁u+c₂.
Newtonの万有引力では、c₂=0. 軌道は三角関数で表される。
一般相対論は、3次までの近似で、軌道はWeierstrassの楕円関数で表される。楕円関数の周期は2πではない
から、軌道は楕円から少しずつ、ずれていく。さらに緻密な理論では、4次以上の多項式によって近似し
軌道は超楕円関数で表される。これは多価関数
rが小さいと　V(u)を、そのまま扱うことになり、Newtonの万有引力は、今の物理学が、そうしてるように、
無視できる小さな力ではなくなる。これが核力なのでは？　つまり重力と核力は同じ万有引力なのでは？
uが有界な軌道は、p²+u²=V(u) のグラフの中の閉曲線の一つを一意化する
周期関数によって表される。
水素原子の中の電子の軌道も同様だが、光の放出によって、エネルギー定数Eが減少してしまう。　c₀は
E　に比例するとしていいから、c₀も減少し、y=V(u)-u² のグラフは下方へ
平行移動し、0≤p²=V(u)-u² から　極大値がu軸に一点で接するところで止まる。軌道を
表す閉曲線は一点に収縮し、軌道は円になる。光を吸収するとグラフは上方へ平行移動し、一点は
閉曲線なり他の閉曲線と接触し合体し、また光を放出すると、一点に収縮する。　電子の安定した軌道は
V(u)-u² の極大値を与える様なrを半径とする円で、そのエネルギー準位はDiscrete。
Bohrの理論もNewton力学で説明できるのだ。ただし、Maxwellの方程式は原子のなかでは成立しない。
エネルギー準位は無限個だから、V(u)-u² の極大値も無限個で　V(u)　は超越関数。

Galois　が長生きしてたら、数学だけでなく物理学も今とは、大分違ったものになっていただろう。

                             1

I indicate defect of special relativity and reconstitute the theory to explain phenomena
explained by relativity.

                             2

Assume that light is radiated from an observer and is reflected by a mirror and is observed
by the observer. For the light, the observer moves. The speed is the light velocity. The time
for the observer stops. This means that the light is radiated and come back simultaneously
for the observer. The light velocity is infinite. Remark that the light and the observer
don't change the speed.

                             3

Put F_{i k}=∇_iA_k-∇_kA_i=∂_iA_k-∂_kA_i.
∇_iF_{k h}+∇_hF_{i k}+∇_kF_{h i}=∂_iF_{k h}+∂_hF_{i k}+∂_kF_{h i}=0 then.
Put A¹=e^{-∫ Γk _{k 1}dx1}, A^k=0 (k≠1). ∇_kA^k=0 then.
Put j_k=∇_iF_kⁱ=∇_i(∇_kAⁱ-∇ⁱA_k).
j^k and A_k satisfy Maxwell equation and Lorentz condition.
∇_k∇^kA_i=-j_i ⇒ ∇_k∇_iA^k=0 ⇒
R_{i 1}A¹=R_{i k}A^k=(∇_i∇_k-∇_k∇_i)A^k=0 ⇒ R_{i 1}=0
Similarly, R_{i 2}=R_{i 3}=R_{i 0}=0.
Let's assume the equation R_{i k}=0 for the spacetime.
This equation is common to any observers and doesn't include quantities decided by
observation.
Let g_{i k}dxⁱdx^k be a solution.
Allow only transformations for which g_{i k}dyⁱdy^k=g_{i k}dxⁱdx^k.
If the metric is Minkowsky metric,you get results of special relativity. Remark that
there is no transformation for which y⁰=f(t,x¹),y¹=g(x¹-ct)　(g(0)=0), yⁱ=xⁱ (i≥2),
g_{i k}dyⁱdy^k=g_{i k}dxⁱdx^k.
You cannot set up a moving coordinate system whose speed is the light velocity.
The principle of relativity is denied.
If the metric is Schwarzshild metric,you get the results of general relativity except
cosmology.

                             4

Invariant one isn't the light velocity but the metric. If the metric(gravity) changes,
the light velocity may change too.

光（電波）が地上から発射されて、静止衛星か何かに反射して地上にもどってきたとする。
光から見ると地球が光速で等速運動して往復したように見えるから、地上では時間が静止してる
ことになる。　これを地上から見ると、光は発射されたと同時にもどってくることになり、
光速は無限ということになる。光は速度を変えず方向だけ変えるから加速系にいる時間は　０．
相対運動する地球も同じ。

実際の光速が有限なのは地球の重力場のためだろう。光の軌道は　g_ikdxⁱdx^k=0
で決まり、許される変換　xⁱ→yⁱ は g_ikdxⁱdx^k=g_ikdyⁱdy^k　となるものだけ。
Minkowsky計量で近似すれば、特殊相対論を得る。　もっとも、これは、もはや相対論ではなく、
地球を静止系の原点とする天動説です。不変なのは計量であって光速ではない。計量（重力）が変われば光速も変わるはず。
太陽系とシリウス系とでは光速や万有引力定数がことなるのでは？
重力場の一部だけ空間を平坦にすれば相対論により、そこだけ光速が無限になり未来が見えるのでは？

Let (r,θ) be the polar coordinates.
The orbit by central force is
(du/dθ)²=V(u)+C-u² (u=1/r,V(u)=U(u)/h²,C=E/h²,E:the energy. h:the angular momentum.)
Assume that V(u) is an analytic function about u.
V(u)=c₀+c₁u+c₂u²+…
Let's think periodic solutions. If c₂=0 and r is large enough,you may put Newton equation
(du/dθ)=C+c₀+c₁u-u².
The period is 2π. If r isn't so large,the period isn't 2π. You can explain the orbit of
Mercury in classical physics.
When you approximate V(u) by polynomials,hyperelliptic function which mathematicians
gave up to think appear.

Let's think the orbit of the electron in the hydrogen atom.
When it's a periodic solution, a closed curve in the (u,(du/dθ)) plane corresponds to it.
When E and C decrease by radiation of light,the graph v=V(u)+C-u² goes down (parallel translation).
The closed curve becomes small and becomes a point (u₁,0) at a maximum of V(u).
If the atom absorbs the light,E and C increase. The graph v=V(u)+C-u² goes up and
the point (u₁,0) becomes a closed curve again. When E and C decrease by radiation of
light,the closed curve becomes small and becomes a point (u₂,0) at another maximum of V(u).
E takes discrete values at the maxmums of V(u). (Forget Maxwell equation.)
(1990)
Physicists needn't to think so. But the above gives a problem to mathematicians.
Problem. Make an analytic function V(u) whose maximums are the elementsof a given set A.
For Bohr's model A={ Rhc/n² | n∈N}.

Though I omit explanation,the next formula is expected.
H=c h /(r√(3+(Mcr)²/ h ²))
α_s=(3+(Mcr)²/ h ²)²α
H: the binding energy. r: the radius of the particle. M: a mass of separated particles.
c: the light velocity.  h : Planck's constant. α:the fine structure constant.
α_s:the coupling constant of a strong interaction of separated particles. (1996)
For combined two nucleons in the nuclear,
M=1.67×10^-27 (kg),  ³√2×1.1×10^-15≤ r ≤ ³√2×1.4×10^-15 (m).
Then, 6.52≤ H/2 ≤10.4 (MeV).
For the reaction p + π^- → n + γ + Q ,
H=E_γ=131 (MeV).
For π^- , M=2.49×10^-16 (kg).
Then, r=8.24×10^-16 (m). This corresponds to the radius of p. And α_s=0.08 .
For p , M=1.67×10^-27 (kg). If r=8.24×10^-16 (m) like the above,
H=55.9 (MeV). This may corresponds to the next reaction.

 p + π- →   n + π0 
                 ↓
                 2γ   (Eγ≥55 (MeV).)

As a by-product,it is expected that the time goes backward in the virtual bozon field .
(I omit the explanation ,though.)
If you can combine isobars like Ar and Ca²⁺ by a virtual bozon,you may be able to make
a macroscopic virtual bozon field.
U.S.A , Russia and China seem to be able to make such a field F and may have spectacles
by which you can see the future.

Let think the disk D(m) in which the index of reflaction is n=arm. r is the 
distance from the center of the disk.
For the orbit of the light in the disk,
 δ∫nds=δ∫n√(r2+(rθ)2)dθ=0  ( rθ=  dr   )
                                     dθ
(1-rθ ∂   )  n√(r2+(rθ)2) =b .   (b is a constant.)
      ∂rθ

Then, r=(b/(a•sin((m+1)(θ+d))))^1/(m+1). (d is a constant.)

When m>-1, the center of the disk attracts the light .
(Try to write the orbit.)
When m<-1, the center repulses the light.
Put D(m) (m>-1) and D(k) (k<-1) in a plane in the vacuum.
D(m) continues to absorb　light and heat. (It resembles green house efect.)

D(k) continues to radiate light and heat.
(They resemble the earth and the sun.)
It continues even if D(m) becomes hotter than D(k) for only radiation of light
conveys the heat. Clausius' principle is doubtfull.
You may be able to make a perpetuum mobile of the 2nd kind by using ferrites.
(Try to change permeability instead of the index of refraction and radiate
electromagnetic wave（nuclear waste's γ rays etc.).) (1990)