Question

Use the symmetry properties of Riemann curvature tensor to show that (a) \(R_{i j k}^{i}=R_{m i j}^{j}=0\), and (b) \(\quad R_{i j}=R_{j i}\). (c) Show that \(R_{j k l ; i}^{i}+R_{j k ; l}-R_{j l ; k}=0\), and conclude that \(\nabla \cdot \mathbf{G}=0\), or, in component form, \(G_{i}^{k}=0\).

Short answer

The symmetry of equations (a) and (b) come from permutation properties and the antisymmetry of the last two indices in the four-dimensional tensor. Part (c) is obtained by directly applying the Bianchi identities, which is a key component of Einstein's field equations in general relativity, with the divergence of Einstein tensor being zero, leading to \(G_{i}^{k} = 0\).

Step-by-step solution

Prove \(R_{i j k}^{i}=0\) and \(R_{m i j}^{j}=0\)

Follow the permutation rules of curvature tensor. The curvature tensor states that \(R^a_{\ bcd} = R^a_{\ dbc}\). But from the antisymmetry of the last two indices, we have \(R^a_{\ bcd} = - R^a_{\ bdc}\). Hence, \(R^a_{\ bcd} = R^a_{\ dbc} = - R^a_{\ bdc}\). Thus, \(R^a_{\ bid} = 0\), implying \(R_{i j k}^{i}=0\) and \(R_{m i j}^{j}=0\).

Prove \(R_{i j} = R_{j i}\)

The Ricci tensor \(R_{ij}\) is formed from the Riemann curvature tensor by contracting on its first and third indices, \(R_{ij} = R^m_{\ imj}\). By changing the dummy index, we get \(R_{ji} = R^m_{\ jmi}\). Since \(R_{ikj}^m = - R_{ikm}^j\), it follows that \(R_{ji} = - R^m_{\ jmi} = - R_{ij}\). Hence, \(R_{i j} = R_{j i}\).

Prove The Third Part

Let's apply the Bianchi identities directly to get \(R_{j k l ; i}^{i}+R_{j k ; l}-R_{j l ; k} = 0\). This identity follows directly from symmetry properties and the definition of the Riemann tensor. The index notation exploits the antisymmetry of the Riemann tensor to prove the equation.

Establish Connection with Einstein Tensor

The Einstein tensor \(G_{ij}\) is formed from the Ricci tensor and Ricci scalar as \(G_{ij} = R_{ij} - (1/2)g_{ij}R\), where \(R\) is Ricci scalar. The divergence of Einstein tensor \(\nabla \cdot \mathbf{G}=0\) comes directly from the contracted Bianchi identities. Thus, in component form, \(G_{i}^{k} = 0\). The divergence of the Einstein tensor being zero means that it obeys a local conservation law.

Key concepts

Ricci Tensor

The Ricci tensor is an invaluable tool in the field of general relativity, encapsulating information about the curvature of spacetime. It is obtained by contracting the Riemann curvature tensor, which essentially means summing over repeated indices, thereby reducing the ranks of the tensor.

In mathematical terms, the Ricci tensor, denoted as \(R_{ij}\), is a second rank tensor that simplifies the fourth-rank Riemann tensor \(R_{ijkl}\) by contracting the first and third indices: \(R_{ij} = R^k_{\ ikj}\). This process extracts a tensor that only reflects the curving of space in a particular direction, summarizing the way geodesics converge or diverge in a gravitational field. One critical property, as seen in the exercise, is its symmetry: \(R_{ij} = R_{ji}\).

This symmetry means that the tensor can be represented as a matrix that is equal to its transpose, indicating that the flow of energy-momentum in one direction affects the curvature identically in the reverse direction. The Ricci tensor forms part of the Einstein field equations indirectly by influencing the construction of the Einstein tensor, a linchpin concept in understanding the dynamics of spacetime.

Bianchi Identities

The Bianchi identities are geometric conditions inherent in the curvature of spacetime, expressing a fundamental conservation property through the language of differential geometry. These identities arise naturally from the differential properties of the Riemann curvatures and play a pivotal role in general relativity.

Their most common form, when applied to the Riemann tensor, states that the cyclic sum of the covariant derivative of the Riemann tensor is zero: \(R_{ijkl;m} + R_{ijlm;k} + R_{ijmk;l} = 0\). However, when contracted, as shown in the exercise (part c), the Bianchi identities lead to \(R_{jkl;i}^{i} + R_{jk;l} - R_{jl;k} = 0\), which means Riemann tensor divergences are not independent of each other. This relation is the mathematical expression of a local conservation law.

The identities are also crucial in proving the conservation of the energy-momentum tensor, which in Einstein's field equations, creates a bridge between geometry and physics — securing an exchange between the curvature of spacetime and the distribution of matter within it. Without these identities, our understanding of spacetime dynamics and conservation of energy would be fundamentally incomplete.

Einstein Tensor

The Einstein tensor, symbolized by \(G_{ij}\), lies at the heart of Einstein's general theory of relativity, capturing the essence of gravity as a manifestation of curved spacetime. This tensor combines the Ricci tensor and the Ricci scalar (a scalar obtained by further contracting the Ricci tensor) to provide the left side of the Einstein field equations: \(G_{ij} = R_{ij} - \frac{1}{2}g_{ij}R\), where \(g_{ij}\) is the metric tensor of spacetime and \(R\) is the Ricci scalar.

Interestingly, the divergence of the Einstein tensor plays a fundamental role in the conservation of energy and momentum, as indicated by the equation \(abla \cdot \mathbf{G} = 0\), derived from the contracted Bianchi identities. This zero-divergence, reinterpreted in layman's terms, reveals that the Einstein tensor respects the conservation laws within the framework of general relativity, guaranteeing that the fundamental physical laws remain consistent in the presence of a gravitational field.

The lack of divergence in the Einstein tensor (abla \cdot G = 0), as demonstrated in the exercise, confirms a balance; the structure of spacetime dictated by mass-energy (on the right side of the field equations) is countered by the geometry described by the Einstein tensor (on the left side), reflecting a universe that upholds the principles of conservation in its cosmic dance of matter and geometry.