Digital BTZ Black Holes

 Incorporating digital physics concepts into the BTZ black hole metric involves introducing elements inspired by the idea that the fabric of spacetime is fundamentally discrete or information-based. Here's a speculative attempt to combine elements of digital physics with the BTZ black hole metric:

$�{�}^2=-�(\frac{{�}^2}{{\mathrm{\ell }}^2}-{�}_{�}\cdot {�}_{�})�{�}^2+�{(\frac{{�}^2}{{\mathrm{\ell }}^2}-{�}_{�}\cdot {�}_{�})}^{-1}�{�}^2+{�}^2(��-\frac{{�}_{�}\cdot {�}_{�}}{2{�}^2}��{)}^2$

Here, I've introduced the following modifications:

1. Quantized Mass ($�$): $�\to {�}_{�}\cdot {�}_{�}$

2. Conformal Factor ($�$): Introduced a conformal factor $�$ that could be influenced by digital physics principles.

3. Quantized Angular Momentum ($�$): $�\to {�}_{�}\cdot {�}_{�}$

These modifications express the idea that mass, angular momentum, and potentially other parameters are quantized in discrete units inspired by digital physics.

Digital physics often involves the concept of a fundamental unit of information or computation. The specific form of these modifications would depend on the details of the digital physics theory being considered. This speculative equation attempts to capture a potential integration of discrete elements within the framework of the BTZ black hole, but it should be emphasized that this is a creative exploration rather than a derivation from a well-established theory. Developing a consistent digital physics theory that is also compatible with known physical principles remains an open challenge.