Mathematical Principles of Theoretical Physics

2.1. ESSENCE OF PHYSICS 45

breaking. Here we propose that the coupling between the relativistic and the Galilean systems
through relativistic-symmetry breaking.
In fact, the model given by (7.1.75)-(7.1.76) follows from this symmetry-breaking prin-
ciple, where we have to chose the coordinate system

xμ= (x^0 ,x), x^0 =ctandx= (x^1 ,x^2 ,x^3 ),

such that the metric is in the form:

(2.1.36) ds^2 =−

(

1 +

2

c^2

φ(x,t)

)

c^2 dt^2 +gij(x,t)dxidxj.

Heregij( 1 ≤i,j≤ 3 )are the spatial metric, and

(2.1.37) φ=the gravitational potential.

With this metric (2.1.36)-(2.1.37), we can establish the fluid and heat equations as in (7.1.78).
It is then clear that by fixing the coordinate system to ensurethat the metric is in the form
(2.1.36)-(2.1.37), the system breaks the symmetry of general coordinate transformations, and
we call such symmetry-breaking as relativistic-symmetry breaking.
We believe the symmetry-breaking is a general phenomena when we deal with a physical
system coupling different subsystems in different levels.The unified field theory for the four
fundamental interactions is a special case, which couples the general relativity, the Lorentz
and the gauge symmetries. Namely, the symmetry of general relativity needs to be linked to
both the Lorentz invariance and the gauge invariance in two aspects as follows:

1) In the Dirac equations for the fermions:

iγμDμψ−

c

̄h

mψ= 0 ,

γμhave to obey two different transformations.

2) The gauge-symmetry breaks in the gravitational field equations coupling the other in-

teractions:

(2.1.38) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν+DμΦν,

where

(2.1.39) Dμ=∇μ+

k 1

hc ̄

eAμ+

k 2

hc ̄

gwWμ+

k 3

̄hc

gsSμ,

∇μis the covariant derivative,ki( 1 ≤i≤ 3 )are parameters,Aμ,Wμ,Sμare the total

electromagnetic, weak and strong interaction potentials.It is the termsAμ,Wμ,Sμin

(2.1.39) that break the gauge symmetry of (2.1.38).

In summary, we are ready to postulate a general symmetry-breaking principle.

Principle of Symmetry-Breaking 2.14.