Basic Einstein general relativity equations are linearized and coupled with Maxwell electromagnetic field equations to produce local gravitationally corrected Minkowsky space metrics; fundamental application on gravity action of an intense laser beam upon a weaker parallel one in dielectrics is undertaken towards evaluating of the space deviation phase shift and confirming the general equivalence principle of inertia for phonons in nanomaterials.
Top10.2. Equivalence Principle And Linear Einstein Equations Of Gravity
According with the principle of equivalence, for free fall coordinates of the system, 
, equations of motion follow “straight line” in curved space-time (Hawking & Ellis 1973; Hayashi & Shirafuji 1979):
(10.1) with proper time:
(10.2)Considering any other local coordinate system, 
, which can be Cartesian on the local flat space etc, we will have the dependence:
(10.3)With this relation (10.1) becomes:
(10.4)Multiplying the relation with 
we obtain:
(10.5) and noting:
(10.6) and the affine connection:
(10.7) we will obtain:
(10.8) which is the equation of free fall in line according with the principle of equivalence in curved space time, meaning equation geodesy.