A Study of Einstein Field Equation

Abstract

In this paper we present our research work: ①We divide 9 connection coefficients into two groups: Group A involving (the linear acceleration) includes Γ001,Γ100, Γ111, Γ122, and Γ133; Group B not involving includes Γ212, Γ233, Γ313, and Γ323. ②Recalling the relation formula: ()R=()Nγ3, where ()R is the linear acceleration in relativistic mechanics, ()N the linear acceleration in Newtoneon mechanics, and γ Lorrentz factor. ③We have determined that (Γ001)N= -b′b-1, (Γ100)N= -c2b′b-5, (Γ111)N= b′b-1, (Γ122)N= -rb-2, (Γ133)N= -rb-2sin2θ. ④We have determined that (Γ001)R = (-b′b-1)γ3, (Γ100)R = (-c2b′b-5)γ3, (Γ111)R = (b′b-1)γ3, (Γ122)R = (-rb-2)γ3, (Γ133)R = (-rb-2sin2θ)γ3. ⑤We have proved that Γ001=A′/(2A)=-b′b-1, Γ100 = A′/(2B) = -c2b′b-5, Γ111 = B′/(2B) = b′b-1, Γ122 = -rB-1 = -rb-2,Γ133 = -rB-1sin2θ= -rb-2sin2θ. (6)We have determined that (Γ112)NR = r-1, (Γ233)NR = sinθcosθ, (Γ313)NR= r-1, (Γ323)NR = cotθ. ⑥We have analyzed Schwarzschild solution, and drawn two conclusions: (a)B=(1-2GM/(c2r))-1 = (1-r2/c2)-1 = γ2, indicating that it involves Newtoneon formula of energy conservation. (b)In the case of the weak gravitational field, GM/(c2r)1, B = (1-2GM/(c2r))-1 ≈ 1 + 2GM/(c2r) ≈ 1 + 2GM/(c2r) + (GM/(c2r))2 = (1+GM/(c2r))2 = γ2, therefore, γ= 1 + GM/(c2r) = b, it holds too, in the case of the strong gravitational field. (8)Substituting the needed expressions into equations, and applying the relation formulas, we can simplify the equations, Obtaining the relativistic solution: -c2dτ2 = c2(1+GM/(c2r))-2dt2 - (1+GM/(c2r))2dr2-r2dθ2-r2sin2θdφ2

Keywords

• general relativity
• field equation
• gravitational theory