Let's consider 'up' to be a vector field, supposed to match our intuition about which way we fall. Intuitively, on a daily basis, we have some idea of which way 'up' is. Near the surface of the Earth, it would be a direction that points away from the center of the Earth. Nearer to another planet, the direction 'up' should be pointing away from its center of mass as well.
In Newtonian physics, if I were naive, I would define the direction of 'up' to oppose that of the gravitational field, however upon more inspection, this doesn't capture the local notion that one would want. If one considers a star system with planets, the gravitational field could point nearly directly towards the star everywhere throughout the planet, even if this direction is what we would like to call 'up'. The intuitive direction of up here is given by opposing the difference between the forces on the observer and their nearby planet. This is obviously relative to the choice to make the planet somehow primary over the star. I don't know a way to remove this choice and have a universal definition that matches intuition.
General relativity makes this even more complicated by introducing the geometry of spacetime. In Newtonian physics, the direction of the future in its spacetime is clear and universal. Now here I believe that there is still a perfectly valid definition of 'the future direction', but only for any particular observer, namely the direction of their 4-velocity. This in turn prescribes, even for non-globally hyperbolic spacetimes, a spatial subspace of the tangent space at any particular point along the worldline of an observer. Unlike in Newton's theory, gravity is not like forces, as falling is ultimately inertial motion in curved spacetime, so when we stand on the Earth's surface, up matches the spatial projection of the force acting on us, to accelerate us out of our inertial freefall through the surface. Is there any way to define similarly a direction 'up', local to any particular observer, given the geometry of the spacetime? Or is the direction of 'up' dependent on the overall force acting on an otherwise inertial observer? The disadvantage of this is that it becomes essentially impossible to say which way is up, the second you leave the surface.
Apologies if this seems insignificant and nitpicky, it's just a curiosity I'd like to understand. Thanks in advance.