Why don’t we feel Earth’s orbital acceleration if it’s constantly changing direction?

Question

If the earth has acceleration (that is, its direction of velocity is constantly changing, which means that the earth has acceleration) so why don't we feel the acceleration? When we drive the car and it has the constant acceleration rate, we feel it right? In a car we feel push, but the earth is not us pushing as well?

Answer 1

This is a typical example where the notions of "acceleration" in Newtonian mechanics and General Relativity differ wildly. In Newtonian mechanics, it is assumed that a body with a certain momentum will keep travelling in a straight line at the same speed unless an external force acts on it, and the corresponding change in momentum is then called "acceleration" according to Newton's law $$ \vec F = m\vec a = \frac{d\vec p}{dt} $$ Now, this definition works well enough even at cosmic scales if one is only interested in computing the equations of motion without giving too much thought to their physical interpretation apart from the spacetime trajectory of an object.

However, in General Relativity, the picture is different. We again assume that in the absence of external forces, particles travel on "straight lines", but the underlying space is no longer considered to be flat. This means that we need to replace the concept of a "straight line" with the concept of a geodesic on a manifold, which is, roughly speaking, the shortest curve connecting two points on a manifold, like a great circle on a sphere. Gravity now no longer is considered as a force changing the momentum of classical particles away from travelling on straight lines in flat space, but instead an effect of mass-energy bending the spacetime manifold so that "straight lines" point towards more massive objects. In the absence of external forces, the law of "bodies follow straight lines" is then replaced by "bodies follow geodesics", which are usually curved. However, there is no net acceleration being felt while moving along these curves, because there is no force involved. This is also the reason why there seems to be no gravity on the ISS, even though a simple classical calculation shows that Earth's gravity is indeed only diminished by 11% at its average orbital distance. The ISS along with its crew is in constant free fall with (almost) no external forces acting on it, and as such, there seems to be no measurable acceleration.