Flat Ads space evolved into ds space

Wen-Xiang Chen∗ Department of Astronomy, School of Physics and Materials Science, GuangZhou University

This article proposes an educated guess. Since the extreme value of the doubt way of thinking represents the minimum (the breakdown or decline of something into random disorder) of an open (related to tiny, weird movements of atoms) system, we saw in the previous article that the extreme value of the doubt way of thinking can be smaller, which points to that (the breakdown or decline of something into random disorder) reduction acts on the (related to tiny, weird movements of atoms) system. It will happen in a sudden and unplanned way. We assume that the macro system and the micro system are closed systems, the system (the breakdown) increases to 0, the macro open system increases in (the breakdown or decline of something into random disorder), and the micro system decreases in (the breakdown or decline of something into random disorder). We know that Ads space can make up/be equal to Ads/CFT explanation (of why something works or happens the way it does), while ds space has serious (problems, delays, etc.) (experiments prove that the universe is ds space-time). Assuming that there is an unplanned reduction process in the tiny system, the Ads space can change into a ds space. Keywords:Ads space, superradiance, evolved

I. INTRODUCTION

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally (having a left half that’s a perfect mirror image of the right half) Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872-1934), professor of (the study of outer space) at Leiden University and director of the Leiden (building where you look at the stars, etc.). Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe. Manifolds of constant curvature are most familiar in the case of two dimensions, where the surface of a world is a surface of constant positive curvature, a flat (Euclidean) plane is a surface of constant zero curvature, and an exaggerated plane is a surface of constant negative curvature. Einstein’s general explanation of relativity places space and time on equal footing, so that one thinks about/believes the geometry of a brought together (as one) spacetime instead of (thinking about) space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). So, they are exact solutions of Einstein’s field equations for an empty universe with a positive, zero, or negative (related to the stars and the universe) constant, (match up each pair of items in order).

∗Electronic address: [email protected] 2

Anti-de Sitter space generalises to any number of space dimensions. In higher dimensions, it is best known for its role in the AdS/CFT back-and-forth writing, which hints that it is possible to describe a force in (related to tiny, weird movements of atoms) mechanics (like electromagnetism, the weak force or the strong force) in a certain number of dimensions (for example four) with a string explanation (of why something works or happens the way it does) where the strings exist in an anti-de Sitter space, with one added (non-compact) dimension. De Sitter space in general relativity de Sitter space involves a difference of general relativity in which spacetime is (a) little curved without matter or energy. This is the same as the relationship between Euclidean geometry and non-Euclidean geometry. An built-in curvature of spacetime without matter or energy is modeled by the (related to the stars and the universe) constant in general relativity. This goes along with the vacuum having an energy density and pressure. This spacetime geometry results in at first parallel timelike (shaped like a soccer ball)s separating, with spacelike sections having positive curvature. Anti-de Sitter space (told apart from) de Sitter space An anti-de Sitter space in general relativity is just like a de Sitter space, except with the sign of the spacetime curvature changed. In anti-de Sitter space, without matter or energy, the curvature of spacelike sections is negative, going along with an exaggerated geometry, and at first parallel timelike (shaped like a soccer ball)s eventually intersect. This goes along with a negative (related to the stars and the universe) constant, where empty space itself has negative energy density but positive pressure, unlike the standard model of our own universe for which (instances of watching, noticing, or making statements) of distant supernovae point to a positive (related to the stars and the universe) constant going along with/matching up to (asymptotic) de Sitter space. In an anti-de Sitter space, as in a de Sitter space, the built-in spacetime curvature goes along with the (related to the stars and the universe) constant. De Sitter space and anti-de Sitter space viewed as deeply set within and part of five dimensions As noted above, the comparison used above describes curvature of a two-dimensional space caused by gravity in general relativity in a (having height, width, and depth) embedding space that is flat, like the Minkowski space of special relativity. Embedding de Sitter and anti-de Sitter spaces of five flat dimensions allows the properties of the embedded spaces to be serious and stubborn. Distances and angles within the embedded space may be directly decided from the simpler properties of the five-dimensional flat space. While anti-de Sitter space does not go along with gravity in general relativity with the watched(related to the stars and the universe) constant, an anti-de Sitter space is believed to go along with other forces in (related to tiny, weird movements of atoms) mechanics (like electromagnetism, the weak nuclear force and the strong nuclear force). This is called the AdS/CFT back-and-forth writing. Associate Professor Hasegawa Yuji of the Vienna University of Technology and Professor Masaaki Ozawa of Nagoya University and other scholars published empirical results against Heisenberg’s uncertainty principle on January 15, 2012[5]. They used two instruments to measure the rotation angle of the neutron and calculated it. The error of the measurement results obtained was smaller than the Heisenberg uncertainty principle, thus proving the measurement results advocated by the Heisenberg uncertainty principle. The restriction is wrong. However, the uncertainty principle is still correct, because this is the inherent quantum property of particles. In the article[2] follows the method I used to study superradiation and connects the uncertainty principle with the superradiation effect.This article proposes a hypothesis. Since the extreme value of the uncertainty principle 3 represents the minimum entropy of an open quantum system, we saw in the previous article that the extreme value of the uncertainty principle can be smaller, which indicates that the entropy reduction acts on the quantum system. It will happen spontaneously. We assume that the macro system and the micro system are closed systems, the system entropy increases to 0, the macro open system increases in entropy, and the micro system decreases in entropy. We know that Ads space can constitute Ads/CFT theory, while ds space has serious difficulties (experiments prove that the universe is ds space-time). Assuming that there is a spontaneous entropy reduction process in the microscopic system, the Ads space can evolve into a ds space.

II. FERMIONIC SCATTERING

1 Now[1] let us consider the Dirac equation for a spin- 2 massless fermion Ψ, minimally coupled to the same EM potential Aµ as in Eq..

µ γ Ψ;µ = 0 , (1) where γµ are the four Dirac matrices satisfying the anticommutation relation {γµ, γν } = 2gµν . The solution to takes the form Ψ = e−iωtχ(x), where χ is a two-spinor given by   f1(x) χ =   . (2) f2(x)

Using the representation     0 i 0 1 0 i γ =   , γ =   , (3) 0 −i −i 0 the functions f1 and f2 satisfy the system of equations:

df1/dx − i(ω − eA0)f2 = 0, df2/dx − i(ω − eA0)f1 = 0. (4)

One set of solutions can be once more formed by the ‘in’ modes, representing a flux of particles coming from x → −∞ being partially reflected (with reflection amplitude |R|2) and partially transmitted at the barrier

in in
iωx −iωx iωx −iωx
f1 , f2 = Ie − Re , Ie + Re as x → −∞ (5)

T eikx, T eikx
as x → +∞ (6)

On the other hand, the conserved current associated with the Dirac equation is given by jµ = −eΨ†γ0γµΨ and, by equating the latter at x → −∞ and x → +∞, we find some general relations between the reflection and the transmission coefficients, in particular,

|R|2 = |I|2 − |T |2 . (7)

Therefore, |R|2 ≤ |I|2 for any frequency, showing that there is no superradiance for fermions. The same kind of relation can be found for massive fields. 4

The reflection coefficient and transmission coefficient depend on the specific shape of the potential A0. However one can easily show that the Wronskian

df˜ df˜ W = f˜ 2 − f˜ 1 , (8) 1 dx 2 dx ˜ ˜ between two independent solutions, f1 and f2, of is conserved. From the equation on the other hand, if f is a solution ∗ 2 2 ω−eV 2 then its complex conjugate f is another linearly independent solution. We find|R| = |I| − ω |T | .Thus,for 0 < ω < eV ,it is possible to have superradiant amplification of the reflected current, i.e, |R| > |I|. There are other potentials that can be completely resolved, which can also show superradiation explicitly. The difference between fermions and bosons comes from the intrinsic properties of these two kinds of particles. Fermions have positive definite current densities and bounded transmission amplitudes 0 ≤ |T |2 ≤ |I|2, while for bosons the current density can change its sign as it is partially transmitted and the transmission amplitude can ω−eV 2 2 be negative, −∞ < ω |T | ≤ |I| . From the point of view of quantum field theory, due to the existence of strong electromagnetic fields, one can understand this process as a spontaneous pair generation phenomenon (see for example). The number of spontaneously produced iron ion pairs in a given state is limited by the Poly’s exclusion principle, while bosons do not have this limitation.

We can pre-set the boundary conditions eA0(x) = −yω(which can be µ = −yω)[3], and we see that when y is relatively large(according to the properties of the Fermions, y can be very large), |R|2 ≤ |I|2 may not hold.In the end,we can get ∆x∆p ≥ 1/2 may not hold(in natural unit system).If the boundary conditions of the incident Fermions are set in advance, the two sides of the probability flow density equation are not equal, because setting the boundary conditions implies a certain probability.

III. THE POSSIBILITY OF FLAT ADS SPACE EVOLVING INTO DS SPACE

We can pre-set the boundary conditions eA0(x) = −yω(which can be µ = yω)[3].If the boundary conditions of the incident boson are set in advance, the two sides of the probability flow density equation are not equal because of the boundary conditions. Implying a certain probability, this also explains why the no-hair theorem is invalid in quantum effects. The previous literature[1] indicates that the superradiation effect is a process of entropy subtraction. Spherical quantum solution in vacuum state. In this theory, the general relativity theory’s field equation is written completely.

1 8πG R − g R = − T (9) µν 2 µv c4 µν

The Ricci tensor is by Tµv = 0 in vacuum state.

Rµv = 0 (10)

The proper time of spherical coordinates is

1 dτ 2 = A(t, r)dt2 − B(t, r)dr2 + r2dθ2 + r2 sinθdφ2 (11) c2 5

We obtain the Ricci-tensor equations.

A00 A0B0 A0 A02 B¨ B˙ 2 A˙B˙ R = − + − + + − − = 0 (12) tt 2B 4B2 Br 4AB 2B 4B2 4AB

A00 A02 A0B0 B0 B¨ A˙B˙ B˙ 2 R = − − − − + + = 0, (13) rr 2A 4A2 4AB Br 2A 4A2 4AB

1 rB0 rA0 B˙ R = −1 + − + = 0,R = R sin2 θ = 0,R = − = 0,R = R = R = R = R = 0 (14) θθ B 2B2 2AB φφ θθ tr Br tθ tφ rθ rφ θφ

0 ∂ 1 ∂ In this time, = ∂r , · = c ∂t ,

B˙ = 0 (15)

We see that,

R R 1 A0 B0  (AB)0 tt + rr = − + = − = 0 (16) A B Br A B rAB2

Hence, we obtain this result.

1 A = (17) B

If,

1 rB0 rA0  r 0 R = −1 + − + = −1 + = 0 (18) θθ B 2B2 2AB B

If we solve the Eq,

r 1 C = r + C → = 1 + (19) B B r

When r tends to infinity, and we set C=−yey, Therefore,

1 y A = = 1 + Σ, Σ = ey (20) B r

 y X dτ 2 = 1 + dt2 (21) r

In this time, if particles’ mass are mi, the fusion energy is e,

2 2 2 2 E = Mc = m1c + m2c + ... + mnc + e (22)

The effect after the preset boundary is similar to that of Ads cosmological constant. Now we transform the coordinates to the locally non-rotating coordinate system by  √  ψ = φ − At (23)  ξ = t 6

In the new path, we see Z   Q 4 y 1 2 S[y, ϕ] = √ d x(−y)e ϕ − ∂ + ∂rf(r)∂r ϕ. (24) 2 A f(r) ξ

∂X But b = t × n and t = ∂s = Xs. By the Frenet-Serret (FS) formula, we have:[4]

ts = κn ns = −κt + τb bs = −τn (25)

Where τ is the torsion of the curve. We find:

Xt = Xs × XSS (26)

This is the differential evolution equation of the filament. Following Hasimoto, we combine the second and third FS formula to obtain:

(n + ib)s = −κt − iτ(n + ib) (27)

This suggests introducing the complex vector

N = (n + ib)eiφ (28)

Where the complex phase φ(s, t) is such that its derivative with respect to s should simply give τ(s, t). This will be R s 0 0 the case if φ(s, t) = 0 τ (s , t) ds . We find thus from

iφ NS = −κe t (29)

Let us set

iφ i R s τ s0,t ds0 ψ = κe = κ(s, t)e 0 ( ) (30)

This is a complex function replacing two scalar functions κ(s, t) and τ(s, t). It drives the evolution of the filament. Equation thus becomes

Ns = −ψt (31)

This is as a new complex FS equation where the variables n(s, t) and b(s, t) have been replaced by the complex ψ. This is called the Hasimoto transformation. Still following Hasimoto, we search for an equation to replace Xt = Xs × Xss· We first notice that

t · t = 1 N · N ∗ = 2 N · N = N ∗ · N ∗ = 0 (32)

(t, N, N ∗) is a new basis replacing the FS basis (t, n, b) Hence, we obtain the first FS equation in the new variables as 1 t = (ψ∗N + ψN ∗) = κn (33) s 2 Let us now derive the induction equation with respect to s. We find ∂ ∂X  ∂ ∂X  ∂ = = t = κ b + κb = κ b − κτn (34) ∂s ∂t ∂t ∂s ∂t s s s 7 let us express this in terms of the new variables ψ, N. We form

∗ ∗ ψsN − ψs N = −2itt (35)

Now let us express this in terms of the new variables ψ, N. We form

∗ ∗ ψsN − ψs N = −2itt (36)

For expressing Nt in the new variables, we set

∗ Nt = αN + βN + γt (37)

Using the orthogonality relations, we find

∗ ∗ ∗ N · Nt + Nt · N = 2 (α + α ) = 4 Re(α) = 0 (38)

Similarly, we have

(N · N)t = 4β = 0 (39)

And finally

t · Nt = γ = −iψs (40)

Thus, it comes

Nt = Im(α)N − iψst (41)

Let us set α = iR where R is real. We have:

Nt = i (RN − ψst) (42)

We now combine with ts and tt to compute Nst and Nts by two different ways. This will allow us to find the last unknown R and in turn to find the equation for ψ. Since Nst = Nts, we can equate the components in the basis (t, N, N ∗). This yields

i∂tψ = −ψss − Rψ i i (43) ψψ∗ = iR − ψ ψ∗ 2 s s 2 s The equation gives 1 1 R = (ψψ∗ + ψ ψ∗) = |ψ|2 (44) s 2 s s 2 s We get that 1 R(s, t) = |ψ|2 + A(t) (45) 2 If ψ becomes a function of x, y, z, t this generalizes to ∂ψ(x, y, z, t)   2 i + ~ |ψ|2 + A(t) ψ(x, y, z, t) = − ~ ∆ψ(x, y, z, t) (46) ~ ∂t 4m 2m By presupposing the fact that the boundary y and the uncertainty principle have a smaller extremum, we can get that under this condition, the effective potential can be greater than zero in a flat space-time, so fermions can produce superradiation phenomena.We see that when the Ads space undergoes the process of entropy reduction, it may evolve into a ds space. 8

IV. SUMMARY

This article puts forward a hypothesis. Since the extreme value of the uncertainty principle represents the minimum entropy of an open quantum system, we saw in the previous article that the extreme value of the uncertainty principle can be smaller, indicating that the entropy reduction acts on the quantum system. It will happen spontaneously. We assume that both the macro system and the micro system are closed systems, the system entropy increases to 0, the macro open system entropy increases, and the micro system entropy decreases. We know that Ads space can constitute Ads/CFT theory, but ds space has serious difficulties (experiments prove that the universe is ds space-time). Assuming that there is a spontaneous entropy reduction process in the microscopic system, the Ads space can evolve into the ds space.

[1] Brito, R., Cardoso, V. and Pani, P. [2015], ‘Superradiance’, Lect. Notes Phys 906(1), 1501–06570. [2] Chen, W.-X. [2020], ‘Uncertainty principle and super-radiance’, SSRN Electronic Journal . URL: https://dx.doi.org/10.2139/ssrn.3683008 [3] Chen, W.-X. and Huang, Z.-Y. [2019], ‘Superradiant stability of the kerr black hole’, International Journal of Modern Physics D . URL: https://doi.org/10.1142/S0218271820500091 [4] Driessen, P. [2020], ‘Derivation of the schr¨odingerequation from first principles’, VIXRA . URL: https://vixra.org/abs/2104.0182 [5] Erhart, Jacqueline; stephan Sponar, G. S. G. B. M. O. Y. H. [2012], ‘Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin-measurements’, Nature Physics .