2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 63
forces in (2.3.6). Hence we need to regard the Riemann metric{gμ ν}as the gravitational
potential.
Thus, the principle of general relativity requires that theunderlying space-time manifold
be a 4-dimensional Riemann space{M,gμ ν}, and the principle of equivalence defines the
Riemann metricgμ νas the gravitational potential.
In conclusion, by Principles2.26and2.27, we derive the following crucial physical con-
clusion of the general theory of relativity, and the second crucial physical conclusion is the
Einstein field equations for the gravitational potential:
Physical Conclusion 2.28(General Theory of Relativity).The physical space of our Uni-
verse is a 4-dimensional Riemannian space{M,gμ ν}, and the Riemann metric gμ νrepre-
sents the gravitational potential. All physical laws are covariant under the general coordinate
transformations in{M,gμ ν}.
The principle of equivalence have received many supports byexperiments. In fact, the
principle is based on the fact that the inertial mass is the same as the gravitational mass. By
the Newton Second Law,
F=am,
wheremis called the inertial mass, denoted bymI, and the gravitational law provides
F=
Gm 1 m 2
r^2,
wherem 1 andm 2 are called the gravitational masses, and denoted bymg.
Theoretically both massesmIandmgare different physical quantities, and we cannota
priorclaim that they are the same. It must be verified by experiments. Newton was the first
man to check them, resulting
mg
mI
= 1 +o( 10 −^3 ),i.e.,mg=mIin the error of 10−^3. In 1890, E ̈otv ̈os obtained the precision to 10−^8 , and in
1964, Dicke reached at 10−^11.
2.3.3 General tensors and covariant derivatives
In order to obtain the second crucial conclusion of general theory of relativity, i.e., the Ein-
stein gravitational field equations for the gravitational potentialgμ ν, we introduce, in this
subsection, the concept of general tensors, general invariants, and covariant derivatives.
- General tensors. Let{M,gμ ν}be ann-dimensional Riemannian space, andx=
(x^1 ,···,xn)be a local coordinate system ofM. We call the following transformation
(2.3.9) ̃xk=φk(x) for 1≤k≤n,
a general coordinate transformation if the functionsφk(x) ( 1 ≤k≤n)in (2.3.9) satisfy that
the Jacobian
(2.3.10)
(
aij)
=
(
∂ φi
∂xj)
forx∈M