No, an object cannot be a black hole to one observer but not another. The nature of a black hole is intrinsic to the object itself and is not dependent on the observer. A black hole is a region in space where gravity is so strong that nothing, including light, can escape its gravitational pull. This phenomenon arises due to the extreme curvature of spacetime caused by a highly compact mass.
The defining characteristic of a black hole is its event horizon, which is the boundary beyond which nothing can escape. Once an object crosses the event horizon, it is considered to be inside the black hole and is inaccessible to external observers.
From an outside observer's perspective, an object that has collapsed into a black hole will appear to have a gravitational field that is so strong that nothing can escape it. This observation is consistent across different observers in different reference frames. While different observers may perceive the black hole from different angles or positions, they would all agree on its fundamental properties, such as its event horizon and the fact that nothing can escape from it.
It's important to note that while different observers may have different perspectives on a black hole's appearance due to relativistic effects like gravitational lensing, these effects do not alter the basic nature of the black hole itself.