Mathematical Principles of Theoretical Physics

1.8 New Theory of Black Holes

are covariant under these symmetries. The gauge symmetry isspontaneously breaking in the
field equations (1.7.1), due to the presence of the terms in the right-hand side, derived by PID.
This term generates the mass for the vector bosons.

4.Weak charge and weak potential.By PRI, theSU( 2 )gauge coupling constantgwplays
the role of weak charge, responsible for the weak interaction.
Also, PRI induces an importantSU( 2 )constant vector{γb^1 }. The components of this
vector represent the portions distributed to the gauge potentialsWμaby the weak chargegw.
Hence the (total) weak interaction potential is given by thefollowing PRI representation
invariant

(1.7.3) Wμ=γa^1 Wμa= (W 0 ,W 1 ,W 2 ,W 3 ),

and the weak charge potential and weak force are as

(1.7.4)

Φw=W 0 the time component ofWμ,

Fw=−gw(ρ)∇Φw,

wheregw(ρ)is the weak charge of a particle with radiusρ.

5. Layered formulas for the weak interaction potential. The weak interaction is also
   layered, and we derive from the field equations the following

(1.7.5)

Φw=gw(ρ)e−kr

[

1

r

−

B

ρ

( 1 + 2 kr)e−kr

]

,

gw(ρ) =N

(

ρw

ρ

) 3

gw,

whereΦwis the weak force potential of a particle with radiusρand carryingNweak charges
gw, taken as the unit of weak chargegsfor each weakton (Ma and Wang,2015b),ρwis the
weakton radius,Bis a parameter depending on the particles, and 1/k= 10 −^16 cm represents
the force-range of weak interactions.

6. The layered weak interaction potential formula (1.7.5) shows clearly that the weak in-
   teraction is short-ranged. Also, it is clear that the weak interaction is repulsive, asymptotically
   free, and attractive when the distance of two particles increases.

1.8 New Theory of Black Holes

The concept of black holes is essentially developed following the Karl Schwarzschild’s deriva-
tion of the Schwarzschild solution for the Einstein gravitational field equations. In the exterior
of spherically symmetric ball of mass, the solution is givenby

(1.8.1) ds^2 =−

(

1 −

Rs

r

)

c^2 dt^2 +

(

1 −

Rs

r

)− 1

dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2 ,

where

(1.8.2) Rs=

2 MG

c^2