# Why do we fall towards earth and not hover during free fall as per General relativity? [duplicate]

So this is what I understand from General Theory of Relativity:

A body freely falling towards earth's surface would be in an inertial frame of reference (air removed) with zero net force acting on it. This would cause weightlessness and would be equivalent to a body in spacetime (under no acceleration) with no gravitational masses around it (and hence no gravity). Any physical experiment in these two frames would thus give equal results making them indistinguishable. But the question is, why is the freely falling body near Earth's surface in motion in inertial frame and not at rest. Why does a ball dropped of the cliff sets in motion and not stay there suspended ?

Does the geodesic formed due to spacetime curvature near earth set the ball in motion or is every body on earth having an initial velocity which is maintained once the free fall (and hence inertial frame) begins? But I also know that the body will speed up as it approaches the earth's center of gravity and hence won't remain in uniform motion. Suppose a tunnel is dug diametrically through the surface, the body would perform Simple Harmonic Motion across this tunnel speeding up and down between the tunnel ends. How is the geodesic playing a role in imparting an acceleration which we so call acceleration due to gravity in Newtonian mechanics?

Could it be, since the motion of an accelerating body follows a curvature in spacetime, a curvature in spacetime is automatically imparting an acceleration to the body?

I'm so confused.

• Abishek: "[...] Why does a ball dropped of the cliff sets in motion and not stay there suspended ?" -- From a "Why?"-question there may arise more specific and constructive "How?"-questions. I'd suggest asking: "How do we determine whether some given constituent of Earth's surface, such as a cliff, is held (from dropping); or to which extent it is not ?", and: "How do we determine whether some object or participant, such as a ball, is moving freely; or to which extent it is not ?" and "How do we determine the (most probable) distribution of masses, charges, fields, in a given trial ?". – user12262 Jun 30 '20 at 5:21
• @user12262 The "How" of it can be worked out using the equations laid out by all these theories. Einstein Field equations, Maxwell equations etc. would perfectly describe the questions you mentioned. But my mind is concerned with the physical reality of it. What causes these phenomenon to happen that can be understood and accepted by our brains. This is where the "Why" comes in. – Abhishek Jun 30 '20 at 5:52
• Abishek: "The "How" of it can be worked out using the equations laid out by all these theories." -- Indeed; provided that to each individual term there is a definite operational "How" laid out to begin with. (The ostensible equations are thus induced as theorems of the respective theory.) "But [...] What causes these phenomena to happen that can be understood and accepted by our brains." FWIW, I'm quite content with asking "Given coincidence data which quantities could we define (and thus measure)?". – user12262 Jun 30 '20 at 10:37

You are forgetting the gist of relativity: the man falling is in motion with respect to you. According to him, he experiences no forces so he would call himself at rest, as the man floating in space would. It has nothing to do with geodesics. For him, he would just hover around. So, what makes him touch he ground then? He can argue that the ground moves up towards him, while he hovers at his place. That may sound absurd, but relativity says that it is possible. The falling man sees the Earth moving relative to him, not himself moving relative to the Earth.

In the other scenario, the man would just see the earth move in simple harmonic motion, accelerating as the center gets near him, and decelerating as the center gets away from him. Keep in mind that this is in the falling man's frame of reference, though, where he hovers and the Earth moves around him. In yours, or any other person on Earth's reference frame, you would be at rest and he would be oscillating.

Sure, all this can be explained using geodesics, but understanding the Equivalence Principle will give you the charm for General Relativity.

• I agree with you (and Einstein) in arguing that it's the Earth accelerating towards the body and not the other way round because that is the Equivalence principle. Consider this: two bodies are in free fall diametrically opposite to earth's COG. Now if the earth is acc. towards them, is the earth's surface expanding at both ends? By this argument if observers were in free fall along the Earth's perimeter, would Earth be inflating like a balloon in trying to acc. towards each of them? – Abhishek Jun 30 '20 at 3:21
• It would be inflating if spacetime were flat. The geometry is not flat, so every point in the earth’s surface can accelerate away from the center without the surface getting larger or further from the center. – Dale Jun 30 '20 at 4:30
• Yeah, this is the point which led Einstein to think about incorporating curvature in GR. Even the above situation can be explained by geodesics. In fact the falling observer does indeed track a geodesic in your frame of reference. But, in the terms of your question, with just one observer, it is easier to be accounted by the Equivalence Principle in itself. – PNS Jun 30 '20 at 4:33
• @Dale you are correct. – PNS Jun 30 '20 at 4:34
• @Dale so it's started to make some sense to me. But two more questions come up to me: Q1) Suppose a body is slowly descending in a downward firing rocket in a tunnel dug through Earth's surface and switches off the rocket just as it reaches earth's COG. My brain says that it would remain suspended there with the COG and body locally becoming inertial (I'm assuming the body is point mass else the tidal force would rip it apart). Is this what actually would happen? – Abhishek Jun 30 '20 at 4:52

An inertial frame near the Earth's surface is heading towards the center of the earth with an acceleration 9.81 ms$$^{-1}$$. A frame stationary wrt to the surface is not inertial in GR. Instead in this non-inertial surface frame we experience the "fictitious" force that we call gravity. The curvature of spacetime is responsible for the tidal effects i.e the various inertial frames are not moving parallel because they are aiming to meet at the center of the earth. We can (temporarily -- until we hit the ground) get rid of the fictitious force of gravity by jumping off a cliff, but we cannot eliminate the tidal force that makes the different part of your body want to go in slightly different directions. You are being (very) slighly compressed by the tidal force and this compression is what is being generated by the mass of the earth.

The Theory of General Relativity is a theory of geometry not of forces. The major issue here is that the distance between things in free fall changes accelerated or in other words the geodesics of such things are accelerating relative to each other.

Why does a ball dropped of the cliff sets in motion and not stay there suspended ?

The ball can't stay suspended because "dropped" means free fall and thus the ball's and the earth's geodesic accelerate towards each other.

Could it be, since the motion of an accelerating body follows a curvature in spacetime, a curvature in spacetime is automatically imparting an acceleration to the body?

Well "acceleration" in this sense needs the information relativ to what, e.g. relative to the earth. Then yes, you can say so. Spacetime curvature which according to Einstein is due to the existence of energy density e.g. the earth, means deviating geodesics, see above. In our accelerated expanding universe things (galaxies) are falling away from each other accelerated or in the vicinity of the earth are falling towards it accelerated.

Spacetime curvature can be visualized quite nicely with the rubber sheet analogy, see here: