Bianchi identity

---

tags: - colorclass/differential geometry ---The Bianchi identities are fundamental relations in differential geometry and general relativity that involve the Riemann curvature tensor. These identities are crucial for the mathematical consistency of the theory of general relativity and have significant implications for the conservation laws of energy and momentum in curved spacetime.

Differential Geometry Form

In the context of differential geometry, the Bianchi identities refer to certain symmetries or constraints on the Riemann curvature tensor, (R^\rho_{\sigma\mu\nu}), and can be expressed in two forms:

1. First Bianchi Identity: It reflects the cyclic symmetry in the last three indices of the Riemann tensor and is given by: $R_{\sigma [\mu \nu ,\lambda ]}^{\rho }+R_{\lambda [\sigma \mu ,\nu ]}^{\rho }+R_{\nu [\lambda \sigma ,\mu ]}^{\rho }=0$ This identity essentially states that the cyclic sum (indicated by the square brackets implying a sum over cyclic permutations of the indices) of the Riemann tensor components equals zero.

2. Second Bianchi Identity (or Differential Bianchi Identity): It involves the covariant derivative of the Riemann tensor and is written as: ${\nabla }_{\lambda }{R}_{\sigma \mu \nu }^{\rho }+{\nabla }_{\mu }{R}_{\sigma \nu \lambda }^{\rho }+{\nabla }_{\nu }{R}_{\sigma \lambda \mu }^{\rho }=0$ This form of the Bianchi identity indicates that the covariant derivative of the Riemann tensor, taken cyclically over its last three indices, sums to zero. This identity is particularly important in the formulation of the Einstein Field Equations and the conservation laws derived from them.

Implications in General Relativity

In general relativity, the Bianchi identities are intimately related to the conservation of energy and momentum. When applied to the Einstein Field Equations, the second Bianchi identity leads to the conservation law expressed by the vanishing divergence of the stress-energy tensor, $T_{\mu \nu }$:

${\nabla }^{\mu }{G}_{\mu \nu }=0$

Given that the Einstein tensor, (G_{\mu\nu}), is constructed from the Ricci tensor, (R_{\mu\nu}), and the metric tensor, (g_{\mu\nu}), which in turn are derived from the Riemann tensor, the Bianchi identities ensure the mathematical consistency of general relativity’s description of how matter and energy influence the curvature of spacetime, and conversely, how this curvature dictates the motion of matter and energy.

The conservation of energy and momentum in curved spacetime, as dictated by the Bianchi identities, highlights one of the most profound aspects of general relativity: the deep connection between the geometry of the universe and the distribution and flow of energy and matter within it.