Is it permissible for an engineer to perform private work, whether related to the core work of the office or not, outside official working hours, without the knowledge of the office owner?

There is no objection to working outside the company's official hours, even if the additional work is within the company's core business. This is because you are a private employee of the company, and you are only required to be present during the period agreed upon in the contract. However, caution must be exercised against exploiting your work at the company to divert its customers to yourself, whether explicitly or implicitly. Otherwise, this would be considered a betrayal and acquiring money unlawfully.