Can anyone explain how does an object move upward from the surface of the earth defying the spacetime curvature?

Question

I am trying to understand the while concept of spacetime curvature. Space is a fabric which is bent by heavy masses. But I don't understand that why objects follow the geodesics and get attracted to the earth's surface. The equator-moving-to-pole analogy works for some extent but fails to explain why don't the objects follow longitude rather than latitudes. I am a beginner.

Answer 1

The question of why do objects follow geodesics in spacetime is discussed on this question, so I'll focus on the comparison with Earth.

In spacetime, particles move following geodesics. However, to determine the motion, one always needs two initial condition: a starting point and a starting velocity. Once you give this data, you can figure out what is the motion of the particle. More specifically, you can solve the geodesic equation.

In the earth analogy, this would also mean you must choose a starting point on the surface of the Earth and a starting velocity. Say you want to start, for example, at São Paulo. There are many geodesics passing through São Paulo (more specifically, all circles on the surface of the Earth which have the same radius as the Earth). To know which geodesic you'll follow, you still need to choose an initial velocity. For example, you might want to move north, or west, or south east, and so on, at some chosen speed. Once all initial data is fixed, you can figure out the correct geodesic. Notice that nothing forbids you to "move along latitudes", but the geodesic motion is always in a great circle (those circles with radius coinciding with the Earth radius).

Notice, though, that motion in geodesics on the surface of the Earth is not a consequence of Relativity, which states point particles move on geodesics on spacetime. The Earth case is a mere consequence of the classical dynamics of a particle constrained to move on a sphere, but otherwise free. This is not a relativistic effect and will still hold on non-relativistic Physics, it is merely an analogy between both cases. Furthermore, spacetime is a Lorentzian manifold, while the Earth surface is Riemannian, meaning distances on the Earth surface are always positive, while on spacetime they can be negative. This changes some technical details of what I just mentioned. For example, we always fix that massive particles move through spacetime with constant "spacetime speed" (more specifically, four-velocity norm) $-1$.

Now how can curvature produce gravity on the surface of the Earth?

The curvature that produces gravity is not the Earth's spherical curvature, but rather spacetime curvature. More specifically, the time component of the spacetime metric is the main responsible for gravity, the Earth surface has nothing to do with it. See, for example, this beautiful video by PBS Spacetime. As a consequence, the "gravitational geodesics" are not the great circles I mentioned. They are curves through spacetime, not only at the surface of the Earth. In fact, geodesic motion with no angular momentum would have you falling into the Earth, but the surface of the Earth keeps you from falling. In other words, someone standing on the surface of the Earth is not following a geodesic, but instead is being accelerated upwards by the soil they are standing on. Hence, the way curvature produces gravity on the surface of the Earth is just as it happens anywhere else: spacetime is curved and the curvature leads to gravity, the fact that the surface of the Earth also happens to be curved (in a different sense) makes no difference at all. The relevant curvature for gravity is off spacetime, not of space alone.