Mathematical Principles of Theoretical Physics

46 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

1) The three sets of symmetries,

(2.1.40)

the general relativistic invariance,

the Lorentz and gauge invariances, and

the Galileo invariance,

are mutually independent and dictate in part the physical laws in different levels of

Nature; and

2) for a system coupling different levels of physical laws, part of these symmetries must

be broken.

This principle of symmetry-breaking holds the key component for us to establish the
PID unified field theory for four fundamental interactions inChapter 4 , the field equations
for multi-particle systems in Chapter 6 , and the astrophysical fluid dynamical equations in
Chapter 7 , resolving a number of important physical problems.

2.2 Lorentz Invariance

2.2.1 Lorentz transformation

In 1903, H. A. Lorentz discovered the Lorentz coordinate transformation, under which the
laws of electromagnetism are invariant. Consequently, physicists discovered two different
invariances:

(2.2.1) the Galilean invariance in classical mechanics,

2.2.5 Lorentz invariance of electromagnetism.

In 1905, Albert Einstein introduced the special theory of relativity, based on the following
two first principles: the principle of special relativity and the principle of invariance of the
speed of light.

Principle 2.15(Special Relativity).Physical laws are covariant in all inertial systems.

Principle 2.16(Invariance of speed of light).The vacuum speed of light is a universal con-
stant.

A few remarks are now in order. First, a system of reference isalways needed to describe
the nature, and an inertial system is a system on which a freely moving object moves with
constant velocity (Landau and Lifshitz, 1975 ). Second, it is clear that under the Galilean
invariance, the speed of light changes in different inertial systems. Consequently, the Galilean
invariance and the Lorentz invariance are incompatible. Third, the invariance of the vacuum
speed of light was verified by the Michelson and Morley experiment. Einstein discovered
his special theory of relativity by postulating, based on Principles2.15-2.16, the following
principle of Lorentz invariance.

Principle 2.17(Lorentz Invariance).Physical laws are invariant under Lorentz transforma-
tions.