General Relativity as a fully singular Lagrange system

We present some gauge conditions to eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations of general relativity; at the same time, we present the corresponding Lagrangian in which there is not any quadratic term of first time derivative that can leads to those vierbein forms of the Einstein equations without second time derivative term by the corresponding Euler-Lagrange equations. General relativity thus becomes a fully singular Lagrange system.


Introduction
As well known, if 0 x is the time coordinate of a frame of reference, then the ten Einstein On the other hand, we have presented a group of gauge conditions to eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations in Ref. [1], however, for which so far I cannot find out the corresponding Lagrangian density. In this paper, we not only present some gauge conditions that can eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations, but also, at the same time, present the corresponding Lagrangian in which there is not any quadratic term of first time derivative that can leads to those vierbein forms of the Einstein equations without second time derivative term by the corresponding Euler-Lagrange equations. General relativity thus becomes a fully singular Lagrange system.
In Sect. 1 of this paper, we present a Lagrangian density of general relativity, which is separated to kinetic energy and potential energy terms naturally; In Sect. 2, we investigate in detail a group of gauge conditions such that general relativity becomes a fully singular Lagrange system; In Sect. 3, we generate the group of gauge conditions discussed in Sect. 2 to more general cases.
As well known, the coordinate variables ( ) x x x x in general relativity are four parameters. In the following discussion, we shall assume that the decomposition of time and space has been finished by the ADM decomposition [2] and by 0 x we denote time coordinate. If 0 x is the time coordinate of a frame of reference, from (1)(2)(3)(4) we see that there is not any time derivative term in GU L , and all time derivative terms appear in GK L ; hence, GK L and GU L can be regarded as kinetic energy and potential energy terms, respectively. Especially, for the metric of an accelerated, rotating frame of reference [3,4,5] ij ij i , this is a due result, because there is not real gravitational field for an accelerated, rotating frame of reference.
2 General relativity as a fully singular Lagrange system under a group of gauge conditions 2 We therefore regard tetrad field α µê as basic variables of gravitational field, in the following discussion, µν g and µν g are as the abbreviation for  The vierbein forms of the Einstein equations have been given in Ref. [1]. For removing all quadratic terms of first time derivative of the Lagrangian (2-1), we first prove a theorem.
As well known, when we investigate a system with constraint conditions, an algebraic constraint in which there is not derivative term can be putted directly in the Lagrangian density of the system, but, generally speaking, we cannot do like so for differential constraint including derivative term. However, if a differential constraint appears in quadratic form in the Lagrangian density of a system, then we can prove the following theorem.
Theorem. If the Lagrangian density ) , ( , λ ϕ ϕ a a L of a system whose basic variables are This means that we can obtain the equation of motion of the field a ϕ by the Euler-Lagrange For example, the action of electromagnetic field in flat spacetime is where the last term is an integral of a total derivative, which does not impact on the derivation of the equations of motion. If we add the Lorenz gauge condition from which we obtain the equation of motion . (2-5)

The time gauge condition
Under local Lorentz transformation, the manner of transformation of α µê reads ) We first choose three gauge-fixing terms, so called the Schwinger time gauge condition: notice that a a Yˆ can be expressed by metric tensor.
According to (1-3), (2-9) and (2-11), we see that under the time gauge condition (2-7), we have It is obvious that there is a negative kinetic energy term in (2-9), of which the concrete form is The above forms of the negative kinetic energy term have been discussed in detail in Ref. [1].

Two coordinate conditions
We further choose the following two coordinate conditions 1 00 Notice that the second of the above three coordinate conditions asks 0 0 0 ≥ lm ij jm il g g Γ Γ , but this inequality is only a corollary of (2-12).

General relativity as a fully singular Lagrange system
Formally, an example of fully singular Lagrange system is the Dirac field in flat spacetime, h of the Dirac field we have As well-known, the system described by (2)(3)(4) and (2)(3)(4)(5) is not equivalent to the theory of electromagnetic field described by (2)(3), the equivalence asks to add the Lorenz condition  We can try to use the Dirac-Bargmann method for a singular Lagrangian system or the method of path integral to realize quantization of the fully singular Lagrange system described by ). This will be studied further.
On the other hand, all second time derivative terms in the vierbein forms of the ten Einstein equations can be eliminated, this characteristic shows that general relativity is great different from other fields, e.g., the Yang-Mills field. For non-Abelian gauge field, we can choose appropriate gauge conditions, for example, the space-axial gauge 0 3 = a A [7] , to eliminate some dynamic variables. However, no matter how to choose gauge-fixing terms, we cannot remove all second time derivative terms in the equations of motion of the non-Abelian gauge field a A µ . Hence, it is impossible to ascribe general relativity to non-Abelian gauge field.