The symmetric difference algebroid and escape - shrink back of the Black-Hole*

This paper presents the symmetric difference algebroid theory, with which we give exact analytical solutions on escape retract mutation deformation of the Black-Hole.

The classical Courant algebroid associates with the direct sume of tangent and cotangent vector bundle : TM = TM+T^*M. The symmetric difference algebroid T^sd as a T-duality structure of the courant algebroid associates the direct difference form: T^sd = TM - T^*M. Even so the TM algebroid and the T^sd algebroid are an isomorphism, but have deep different mathematical physical means. For example, in a complex torus or circle S¹, two algebroids have completely different geometric realizations as:

TM= 2n ± √3 + 1/(2n ± √3);  T^sd= 2n ± √5 - 1/(2n ± √5). There exists an important relationship: TM(n=1) = 2±√3 + 1/(2±√3) = 4; T^sd(n=1) = 2± √5 - 1/ (2±√5) = 4, i.e. in first period: 

TM= T^sdM = 4. We assert that this is just maximal light speed v_max = 4 rather than c= 3x... km/s. This light speed c only is a transitional speed measured in the earth, which is not the speed at original head (the Sun). This important discovery on the limit light speed will rewrite the relation between mass and velocity of the universe and fundamental particles. the new formula of mass transformation is m = √(1 - v/4) including the case of Tachyon particles. It clearly is that for velocity : v= 4(1 - m²). Then Gmv = 4Gm(1-m²) = Id =1. For the freedom fundamental group, we found there exists a monodromy dimensionless gravitational constant G = 2/3 = 0.666666.....

The symmetric difference algebroid exactly describes the extremel escape retract deformations over the universes. The escape- retract radius are √5 ∼ 1/√5, extremel escape-retract  angle: θ₁=atan(√5)= 65.9^o > 60^o;  θ₂= atan(1/√5) = 24.1^o < 30^o. We asserts that if argument angle is over the stable angle 60^o or mass is over the critical mass √3/2, sub-system is possible to escape from the universe.. At a freedom escape point r_coh= √5/2 = 1.1180 > r=1, there exist two escape directions: √5/2 + 1/2 = 1.6180; √5/2 - 1/2 =0.6180, or φ = 1.6180, φ-1 =0.6180, φ(φ-1) = 1, φ²= 1 + φ, or the escape equation: φ² - φ -1 = 0, its two roots are 1/2 ± √5/2. The Black Hole is a typical escape-retract phenomena, which also is vanishing cycle mutation. We can compute the mass of a BH: m_BH = √(1- (√5)²/4) = √(-1/4) = i/2, it is complex mass as Tachyon. New stars are just retracted BHs. The BH has not earth or death.