- #1

fab13

- 318

- 6

- TL;DR Summary
- Summary of My Issue with Tensor Calculus Calculations :

In my exploration of tensor calculus within a specific theoretical framework-potentially related to Horndeski theory or another complex field theory-I have encountered a significant computational issue involving the higher-order derivatives of the metric tensor, ##g_{\rho \sigma}##.

**Core Equation**:I am examining the following equation:

## \frac{\partial \mathcal{A}_{\mu \nu}}{\partial g_{\rho \sigma, \tau \pi \lambda}} \frac{\partial g_{\rho \sigma, \tau \pi \lambda}}{\partial g_{\rho^{\prime} \sigma^{\prime}, \tau^{\prime} \pi^{\prime} \lambda^{\prime}}} = \frac{\partial \mathcal{A}_{\mu \nu}}{\partial g_{\rho \sigma, \tau \pi}} \frac{\partial g_{\rho \sigma, \tau \pi \lambda}}{\partial g_{\rho^{\prime} \sigma^{\prime}, \tau^{\prime} \pi^{\prime} \lambda^{\prime}}}##**Assumptions and Context**:The assumption that third-order derivatives of ##g_{\rho \sigma}## are zero is crucial in this framework. This assumption has direct implications for how the derivatives in my calculations are evaluated.**Problem Identified**:The term:

## \frac{\partial g_{\rho \sigma, \tau \pi \lambda}}{\partial g_{\rho^{\prime} \sigma^{\prime}, \tau^{\prime} \pi^{\prime} \lambda^{\prime}}}##

acts like a delta function, being 1 when the indices match. However, the term:

## \frac{\partial \mathcal{A}_{\mu \nu}}{\partial g_{\rho \sigma, \tau \pi \lambda}}##

is zero because all third-order derivatives of ##g_{\rho \sigma}## are assumed to be zero.

**Mathematical Concern**:Typically, having a zero numerator wouldn't be problematic, but in this case, since the calculations require non-zero values for correct progression, having a zero value in these derivatives leads to a division by zero in subsequent calculations, effectively rendering the result as undefined or infinite. This represents a significant challenge to the consistency and applicability of the theoretical model.**Conclusion**:The essence of the issue lies in the presence of a zero denominator, leading to an infinite or undefined quantity. This issue may necessitate a reevaluation of the assumptions or the mathematical techniques employed in my study.

Hoping you will help me to understand why the denominator concerned equal to zero doens- 't imply the inverse to be infinite.

Regards