Mathematical Principles of Theoretical Physics

60 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

By the expressions ofγμ, we infer from (2.2.66) that

R=cosh

θ

2

I−sinh

θ

2

γ^0 γ^1 =









coshθ 2 0 0 −sinhθ 2

0 coshθ 2 −sinhθ 2 0

0 −sinhθ 2 coshθ 2 0

−sinhθ 2 0 0 coshθ 2







(2.2.67) .

The four-component functionψsatisfying (2.2.63) under the Lorentz transformation
(2.2.61) is called the Dirac spinor, which ensures the Lorentz covariance for the Dirac equa-
tions.
Noting that

coshθ=

1

2

(eiθ+e−iθ), sinhθ=

1

2

(eiθ−e−iθ),

and cosh^2 θ−sinh^2 θ= 1 ,we infer from (2.2.67) that

(2.2.68) R−^1 =R†.

2.3 Einstein’s Theory of General Relativity

2.3.1 Principle of general relativity

First Principle of Physics2.1amounts to saying that

(2.3.1)

Laws of Physics = Differential Equations

Universality of Laws = Covariance of Equations.

In retrospect, Albert Einstein must have followed the spirit of this principle for his dis-
covery of the general theory of relativity. As mentioned earlier, coordinate systems, also
called reference systems, are just an indispensable tool toexpress the laws of physics in the
form of differential equations. Consequently the validityof laws of physics is independent of
coordinate systems. Hence Einstein proposed the followingprinciple of general relativity.

Principle 2.26(General Relativity).Laws of physics are the same under all coordinate sys-
tems, both inertial and non-inertial. In other words, the models describing the laws of physics
are invariant under general coordinate transformations.

In Section2.1.5, we mentioned that each symmetry in physics is characterized by three
ingredients:

space, transformation group, tensors.

The special theory of relativity or the Lorentz invariance is dictated by

2.2.2 Minkowski space and Lorentz tensors.

The three ingredients of the Theory of General Relativity are