The text is a formal proof of an inequality involving Frobenius norm, which is a matrix norm defined as the square root of the sum of the absolute squares of its elements. The inequality states that the product of two tensors is greater than or equal to one-third times the square of their trace. The proof uses matrix notation and properties, Cauchy-Schwarz inequality for Frobenius inner product, and operator norms induced by vector norms. The proof is detailed and rigorous, and provides references and explanations for each step. The text also gives some examples and applications of the inequality in physics and geometry.
The inequality in question is
$$\varkappa_{\beta}^{\alpha}\varkappa_{\alpha}^{\beta} \geq \frac{1}{3}\left(\varkappa_{\alpha}^{\alpha}\right)^2$$ where $\varkappa_{\beta}^{\alpha} = \frac{\partial \gamma_{\beta}^{\alpha}}{\partial t}$ is the time derivative of the three-dimensional metric tensor $\gamma_{\beta}^{\alpha}$.