# 3+1 decomposition in the new action

for the Einstein Theory of Gravitation

###### Abstract.

The action of recently proposed formulation of Einstein Theory of Gravitation is written according to 3+1 decomposition of the space-time variables. The result coincides with known formulation of Dirac and Arnowitt-Deser-Misner.

Recently I proposed to use new dynamical variables to describe the gravitational field [1], [2]. In [2] the new formulation was shown to be equivalent to the classical one of Hilbert-Einstein. So the question arises why to do such an effort. The only answer I can give now is that I follow an old advice of Feynman — to generalize a theory one must work it out in many guises. And there is no doubt that we need to develop Einstein’s theory further.

In this note I continue the work in [1], [2] and develop 3+1 decomposition of the space-time variables in the action functional. I shall show how traditional formulas of Dirac [3] and Arnowitt-Deser-Misner [4] appear in my formulation.

The set of dynamical variables introduced in [1], [2] consists of 40 components — 10 covariant vector fields on four dimensional space-time with coordinates ; thus , .

In terms of these variables I define metric

and linear connection

Here are contravariant vector fields

where is, as usual, inverse to

As in [1], [2] I do not bother with the subtleties of the pseudoriemannian signature, so all scalar products are euclidean. In such situation the separation of time and space variables and seems somewhat artificial, but I continue to follow this convention to avoid minus signs. So I use a term “3+1 decomposition” instead of the space-time one.

The action is written more transparently via the contravariant components

where

and is the “vertical” projector

The expression defines the scalar curvature of the connection . The full curvature tensor

is beautiffully expressed as

and

All these formulas include only usual partial derivatives, but they are fully covariant with respect to the general coordinate transformations

where is a vector field, defining infinitesimal coordinate transformation.

In this note I shall explicitly realize the 3+1 decomposition of these formulas in coordinates and refer to as time and to as space variables. The main goal is to rewrite the action in hamiltonian-like form.

The first observation is that contains time derivatives only linearly. This allows to develop the reduction formalism following the general ideas of my paper with R. Jackiw [5]. There the dynamical variables entering the original lagrangian are divided into three classes: canonical, excludable and Lagrange multipliers. To exclude the variables of second class one is allowed to use equations of motion which do not contain the time derivatives.

Among the equations of motion, which are derived in [2], there is a set of equations which express the vanishing of the torsion of the connection

Out of these 24 equations 12 do not contain the time derivatives

and I shall use them in the reduction of action in what follows.

The formulas I plan to derive should be covariant with respect to coordinate transformation, generated by vector fields , obtained from by restriction

The covariant vector fields are compatible with this requirement

Furthermore the component defines scalars

I begin by writing

where contains all terms with time derivatives

and

We can interpret as one form using substitution

In this guise the action is an explicit example of general scheme in [5].

Now we proceed to realize the promised separation. The covariant 3-dimensional metric is given by

and the components of 4-dimensional contravariant metric , which we need, can be expressed via , , , which are 3-minesional tensor, vector and scalar, correspondingly,

The 4-dimensional determinant can be written as

where is determinant of metric . We shall also see, that terms containing will always have the form

Let us begin our rearrangement with one-form . We have

and immediately see, that terms, proportional to contain combinations

and the second terms in the RHS are annihilated by vertical projector . After this observation we see, that these terms cancel and we get the satisfactory expression

Let us do the same for and separate the contributions, corresponding to , , and . The component vanishes due to antisymmetry. The and components coincide after change of mute indeces and give

The contribution gives

where

and substituting via , , we get , where

and

Combining and and using the same trick as before we get

Thus we get

and their expressions are satisfactory also.

Now it is time to reduce the vertical projector . We have

The first two terms define the 3-dimensional vertical projector and the last can be rewritten as

Thus we have

Let us mention, that the last term has proper normalization because

Now I substitute this expression for into , and .

Begin with : we get three contributions according to the form of . The first is

The last factor can be rewritten as

where I remind the notation for and denote

Thus we have

The first term here is quite satisfactory, it is almost of Darboux form.

Now consider the second term

Here I used the orthonormality of and to rewrite

and

and introduce the 3-dimensional connection

Finally the third contribution is given by

where I used that

Let us collect all contrbutions containing

and compare it with

Using the vanishing of covarinat derivative of

we get

The second term in the RHS disappears due to mentioned above time-independent equations of motion. Indeed we have

and is symmetric to interchange together with and .

Thus the full contribution containing is a pure divergence and can be omitted.

Consider now the expression

Due to symmetry of it can be rewritten as

Using this we have

where

The second term can be dropped and we obtain the canonical expression for the one form . The normalization of canonical pairs — as contravariant density of weight 1 and as covariant density of weight — appeared first in the paper of Schwinger [6]. I used it in the survey [7].

Let us turn now to . First take

and consider three contrbutions according to form of .

In the first we use

In the second we get

Finally in the third term we have

Combining all together we get

Taking into account symmetry of and this can be written as

where we introduce Lagrange multipliers

and

Finally consider and take into account, that the first two terms in give 3-dimensional analogue of the vertical projector. Its contribution to is

where is scalar curvature of metric and connection . The last term in gives

With this can be rewritten as

where Lagrange multiplier is given by

and

This finishes calculations in this note. Let us remind that in it we used the change of 40 variables to the set . Superficially we have here 44 components, however we have 4 constraints

The main result is the formula for the action in 3+1 decomposition

which coincides with formulas of Dirac and ADM. Thus it is shown once more, that my proposal is equivalent to the classical formalism of Hilbert-Einstein.

However, as I already said in the beginning, this formulation could be a point of departure for the generalization not evident in the classical formulation.

## References

- [1] L. D. Faddeev, “New action for the Hilbert-Einstein equations,” arXiv:0906.4639 [hep-th].
- [2] L. D. Faddeev, “New variables for the Einstein theory of gravitation,” arXiv:0911.0282 [hep-th].
- [3] P. A. M. Dirac, “The Theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. Lond. A 246 (1958) 333.
- [4] R. L. Arnowitt, S. Deser and C. W. Misner, “Canonical variables for general relativity,” Phys. Rev. 117 (1960) 1595.
- [5] L. D. Faddeev and R. Jackiw, “Hamiltonian Reduction of Unconstrained and Constrained Systems,” Phys. Rev. Lett. 60 (1988) 1692.
- [6] J. S. Schwinger, “Quantized gravitational field,” Phys. Rev. 130 (1963) 1253.
- [7] L. D. Faddeev, “The energy problem in Einstein’s theory of gravitation,” Sov. Phys. Usp. 25 (1982) 130 [Usp. Fiz. Nauk 136 (1982) 435].