Mathematical Principles of Theoretical Physics

62 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

Consider a free particle moving in a Riemannian space{M,gμ ν}, which satisfies the
motion equations

(2.3.5) D

(

dxk

ds

)

= 0 fork= 1 , 2 , 3 ,

whereDis the covariant derivative, and

ds=

√

gμ νdxμdxν.

According to the theory of Riemannian geometry, the motion equations (2.3.5) in the Minkowski
space are in the form
d^2 xk
ds^2
= 0 fork= 1 , 2 , 3 ,

which are the motion equations of special relativity, and (2.3.5) in the Riemannian space
become

(2.3.6)

d^2 xk

ds^2

+Γkμ ν

dxμ

ds

dxν

ds

= 0 fork= 1 , 2 , 3.

whereΓαμ νis the Levi-Civita connection, given by

(2.3.7) Γαμ ν=

1

2

gα β

(

∂gμ β

∂xν

+

∂gν β

∂xμ

−

∂gμ ν

∂xα

)

,

and(gα β)is the inverse of(gα β).
The equations (2.3.6) are the generalized Newtonian Second Law, the first term in the left-
hand side of (2.3.6) is the acceleration, and the second terms represent the force acting on the
particle. In view of (2.3.7) we see that the force is caused by the curvature of space-time, i.e.
by the non-flat metric
∂μgα β 6 = 0.
In addition, from this fact we can also think that the gravitation results in a non-flat Rie-
mann metric. However, in (2.3.3) we obviously see that the inertial force also lead to the non-
flat metrics. Thus, we encounter a difficulty that the curved Riemann space can be caused by
both gravitational and inertial forces.
Einstein proposed the principle of equivalence overcomingthis difficulty.

Principle 2.27(Principle of Equivalence).One cannot distinguish the gravitational and the
inertial forces at any space-time point by experiments. In other words, any non-inertial
system can be equivalently regarded as an inertial system located in a gravitational field.

With the principle of equivalence, the Riemann metric is regarded as the effects caused
only by the gravitation. Furthermore, by the classical gravitational theory, the acting force

(2.3.8) F=−∇φ=−

(

∂ φ

∂x^1

,

∂ φ

∂x^2

,

∂ φ

∂x^3

)

,

whereφis the gravitational potential. In comparing the force (2.3.8) with (2.3.7) which
contains first-order derivative terms∂αgμ ν, the connections (2.3.7) also play a role of acting