On The Formulation of Bianchi Identity from Action Principle

In this letter, we investigate the basic property of the Hilbert-Einstein action principle and its infinitesimal variation under suitable transformation of the metric tensor. We find that for the variation in action to be invariant, it must be a scalar so as to obey the principle of general covariance. From this invariant action principle, we eventually derive the Bianchi identity (where, both the 1 st and 2 nd forms are been dissolved) by using the Lie derivative and Palatini identity. Finally, from our derived Bianchi identity, splitting it into its components and performing cyclic summation over all the indices, we eventually can derive the covariant derivative of the Riemann curvature tensor. This very formulation was first introduced by S Weinberg in case of a collision less plasma and gravitating system. We derive the Bianchi identity from the action principle via this approach; and hence the name ‘Weinberg formulation of Bianchi identity’ .


INTRODUCTION
The principle of general covariance (GC) along with the principle of equivalence (PE) plays a very crucial role in defining the parameters of a purely gravitational field. Following the principles of equivalence and general covariance, the affine connection (Γ) and the metric tensor ( ) are sufficient to describe all the intrinsic properties of a local inertial frame within any gravitational field, relatively respective.
In this letter, we are concerned to derive the Bianchi identity (dissolved for 1 st and 2 nd identities) from the Hilbert-Einstein ( ) or the gravitational action defined by the active lagrangian of the field. We perform an analogous gauge transformation for the metric tensor under consideration to construct an invariant action which is essentially be scalar. Our primary goal of this letter is to derive and formulate the Bianchi identity (in contracted and dissolved from) from the action principle due to purely gravitational lagrangian as proposed by Prof. Steven Weinberg first in Gravitation and Cosmology, 1977; and hence the name to be used as "Weinberg formulation of Bianchi identity".
All in this letter, we use the hyperbolic metric signature: (+, −, −, −). The constituents of this letter are as: in first section we define and its variation properties, then we discuss the lie derivative for an infinitesimal vitiations for the metric tensor and its invariant transformation, next we derive the Bianchi identity. Finally, we discuss some mathematical properties of the contracted Bianchi identities.

The Action Principle
We consider a purely gravitational field where the intrinsic properties of locally inertial frame by the virtue of transformation ( → ) within, is sufficiently defined by and Γ . For such a field the action is given by Where, ( ) is the curvature scalar defined as ( ) = ≡ or .Where is the Ricci tensor defined as Next, an infinitesimal change (√ ( ) ) is given as Now, using the Palatini identity, we have the infinitesimal change or as Using equation (4) for in (3), we have the last term √ as Using the Divergence theorem for tensors in (5), we have

Derivation of Bianchi Identity
Now, if we apply the principle of general covariance to equation (1) we may eventually conclude that for to be invariant, it must be a scalar quantity. Thus, from equation (5) it is evident that for any invariant transformation the change in action vanishes or → 0. However, this invariance is not altered by any variance in metric tensor as This analogous gauge transformation is accompanied by Thus, the variation of in (8) is given by the Lie derivative for ( ) as So, in equation (7) Which is the contracted from of the Bianchi Identity. The validity of this equation lies in the fact that, we can eventually derive the 1 st and 2 nd Bianchi Identities from equation (12) with the contracted form of Ricci tensor (2) and the cyclic summation of the Riemann-(Christoffel) curvature tensor.

RESULT and DISCUSSION
Thus, from the derivation of the Bianchi identity from the Hilbert-Einstein action can be eventually derived as suggested by Weinberg, if we consider an invariant transformation in the action principle. However, if we consider the curvature tensor as = , whose cyclic summation gives the complete Bianchi identities as Which eventually simplifies into Now, by symmetry of the Ricci tensor by (2) and (14c), we have = Then, we consider: ; = ; = ; .
This above equation is most easily conceivable at a given point x adopting all locally inertial coordinate system in which the affine connection (but not necessary its derivatives) tends to vanish at x.

ACKNOWLEDGEMENT:
The author is highly grateful to Prof. B Majumder, for many helpful discussion.