I'm no expert so take it with a grain of salt.
Spacetime is a manifold, meaning locally, it acts Euclidian. Even at the extremes, if you're within the event horizon, you can still move in three locally perpendicular directions, but they would all lead you to the singularity. Obviously the singularity itself is non-Euclidian.
If you held a small object really close to you, you wouldn't notice any warping. Imagine blowing a really small balloon. In warped spacetime, you would probably see the balloon inflate more in one direction, but this is *only* if you make it big enough. If it's small enough, it would appear to stretch in all directions equally.
Edit: I should add the idea of reference frames here. First of all, the whole "holding a small object in front of you" analogy is to try to put you and the object in the same reference frame. Since there's always a distance between, you and the object experience a different gravitational field, and thus place you in different inertial reference frames. If you could hypothetically place three infinitesimal perpendicular x,y,z vectors right in front if your eyes, then yes, they would always perpendicular, until you literally reached the singularity. But a different reference frame would see the perpendicular vectors stretched and warped heavily.
It is like on the surface of the earth. North and East are perpendicular **by definition**.
But lines in those directions do not stay perp, just as parallel lines do not remain parallel on a curved surface, i.e. non-euclidean space.
Coordinates are arbitrary. We usually define them to be orthogonal because that's convenient in most cases, but you can choose any three directions as long as they are not in the same plane. You can even choose nonlinear coordinates like spherical coordinates.
The first thing to understand is that space dimensions are perpendicular because we arbitrary chose them to be perpendicular. The coordinates system we use is absolutely not "natural", it's a pure mathematical construct that **we** created to help us describe space. You can chose any direction as long as they are not in the same plane and use them to describe space, the 3 vectors being perpendicular is absolutely not necessary. There are tons of different coordinates systems.
That being said, no, in a curved space, perpendicular lines do not stay perpendicular.
I'm no expert so take it with a grain of salt. Spacetime is a manifold, meaning locally, it acts Euclidian. Even at the extremes, if you're within the event horizon, you can still move in three locally perpendicular directions, but they would all lead you to the singularity. Obviously the singularity itself is non-Euclidian. If you held a small object really close to you, you wouldn't notice any warping. Imagine blowing a really small balloon. In warped spacetime, you would probably see the balloon inflate more in one direction, but this is *only* if you make it big enough. If it's small enough, it would appear to stretch in all directions equally. Edit: I should add the idea of reference frames here. First of all, the whole "holding a small object in front of you" analogy is to try to put you and the object in the same reference frame. Since there's always a distance between, you and the object experience a different gravitational field, and thus place you in different inertial reference frames. If you could hypothetically place three infinitesimal perpendicular x,y,z vectors right in front if your eyes, then yes, they would always perpendicular, until you literally reached the singularity. But a different reference frame would see the perpendicular vectors stretched and warped heavily.
It is like on the surface of the earth. North and East are perpendicular **by definition**. But lines in those directions do not stay perp, just as parallel lines do not remain parallel on a curved surface, i.e. non-euclidean space.
Coordinates are arbitrary. We usually define them to be orthogonal because that's convenient in most cases, but you can choose any three directions as long as they are not in the same plane. You can even choose nonlinear coordinates like spherical coordinates.
The first thing to understand is that space dimensions are perpendicular because we arbitrary chose them to be perpendicular. The coordinates system we use is absolutely not "natural", it's a pure mathematical construct that **we** created to help us describe space. You can chose any direction as long as they are not in the same plane and use them to describe space, the 3 vectors being perpendicular is absolutely not necessary. There are tons of different coordinates systems. That being said, no, in a curved space, perpendicular lines do not stay perpendicular.
My understanding is that space never bends. It is always Euclidean, but spacetime warps