2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 61
- the space-time Riemannian space,
- general coordinate transformations, and
- general tensors,
under which the theory of general relativity is developed.
First, the Minkowski space is now replaced by the Riemannianspace. In Theorem2.18,
we see that the Minkowski metric
(2.3.2) ds^2 =−c^2 dt^2 + (dx^1 )^2 + (dx^2 )^2 + (dx^3 )^2
is invariant under the Lorentz transformations. However, when we consider a non-inertial
reference system
(c ̃t,x ̃^1 , ̃x^2 ,x ̃^3 ),
rotating with constant angular velocityΩaround thex^3 -axis of an inertial system(ct,x^1 ,x^2 ,x^3 ),
the coordinate transformation is given by
x^1 = ̃x^1 cosΩ ̃t− ̃x^2 sinΩt,
x^2 = ̃x^1 sinΩ ̃t+ ̃x^2 cosΩt,
x^3 = ̃x^3 ,
t= ̃t.Under this transformation, the metric (2.3.2) becomes
(2.3.3) ds^2 =−[c^2 −( ̃x^1 )^2 Ω^2 −( ̃x^2 )^2 Ω^2 ]dt ̃^2 − 2 Ωx ̃^2 dx ̃^1 d ̃t
+ 2 Ω ̃x 1 d ̃x^2 d ̃t+ (dx ̃^1 )^2 −(d ̃x^2 )^2 + (dx ̃^3 )^2.Hence in a general non-inertial system, the metricds^2 takes the form
(2.3.4) ds^2 =gμ νdxμdxν,
where{gμ ν}is a Riemannian metric different from the Minkowski metric given by (2.2.5).
In mathematics, the spaceMendowed with the metric (2.3.4), denoted by{M,gμ ν}, is
called a Riemannian space, or a Riemannian manifold, and (2.3.4) or{gμ ν}is the Riemannian
metric.
In Theorem2.12, we know that the Minkowski spaceM^4 is flat, and the Riemannian
space{M,gμ ν}is curved provided that the metricgμ νis not the same as the Minkowski
metric in any coordinate systems.
2.3.2 Principle of equivalence
In the last subsection we see that the underlying space for the general theory of relativity is
the 4-dimensional Riemannian space{M,gμ ν}, instead of the Minkowski space. Now, a
crucial step is that we have to make sure the physical significance of the Riemannian metric
{gμ ν}.