The Uniqueness Of Minimal Coupling Of Gravity Theory

The theory of general relativity is unique in the case of minimal coupling,that is,the minimal coupling form of the equation of motion and the minimal coupling form of the action are consistant,and the minimal coupling of the action which is equivalent in the flat spacetime is also equivalent.But when more degrees of freedom are considered,the form of gravity theory becomes non-unique,that is,the minimal coupling form of the equation of motion and the minimal coupling form of the action are no longer consistant,and the minimal coupling of the action which is equivalent in the flat space-time is not equivalent.This article examines the uniqueness of the theory of gravity from the view of gauge principle,and discusses the limitations of Poincaré symmetry to the theory of gravity.It is found that the form of gravity theory is not unique under the restriction of gauge symmetry.The choice of covariant derivative in the theory is non-unique.It is allowed that the affine connection plus a proper third-order tensor field.Because the Einstein-Cartan theory introduces more degrees of freedom,the equation of motion and the action cannot take the minimal coupling form at the same time.To solve this problem we discuss the modification of Einstein-Cartan theory from the three aspects:modifying the integral measure,modifying the form of covariant derivative,and modifying the space-time background.On the other hand,unlike the gauge transformation in field theory,Poincaré gauge transformation is a coordinate transformation.In order to ensure that the action is invariant under the gauge transformation,it is necessary to find a conserved integral measure under the coordinate transformation.This leads to the fact that the minimal coupling form of the matter field is non-unique.There are two choices: the first-order derivative form and the second-order derivative form.This paper discusses different gravitational theories based on the uniqueness of the minimal coupling form.It is found that the minimal coupling form of Poincaré gauge theory and Einstein-Cartan theory is non-unique,while the general relativity,parallel gauge theory and modified Einstein-Cartan theory is unique.The main contribution of this paper is to propose two new forms of gravity theory from the perspective of modifying the covariant derivative and modifying the space-time background.They introduce more degrees of freedom than the general theory of relativity,while also ensure that the action and equation of the motion can be taken the minimal coupling form at the same time.Moreover,we systematically compared the difference between the firstorder and second-order form’s action of different gravity theories,and judged the uniqueness of the minimal coupling form of different gravity theories.