In what sense do we say that the Earth surface is almost inertial in Newtonian mechanics?

Question

From what I understand, inertial frames are the ones in which the momentum of every particle in the universe gets well accounted for. Like if there's any particle losing momentum, another particle somewhere must gain the same momentum, and this exchange of momentum can always be attributed to one of the four fundamental forces.

Non-inertial frames are the ones in which particles gain momentum out of nowhere, with no account of what fundamental force caused it (as no fundamental force causes it).

If we talk about the Earth surface frame, particles here gain momentum out of nowhere all the time. Just look at the falling particles. In reality (seen from an actual inertial frame), this momentum is accounted for by an equal momentum lost/ gained by Earth, but as Earth is at rest in the Earth frame, Earth gains no momentum in the Earth frame, and hence the extra momentum gained by literally every falling object gets unaccounted for in the Earth frame!

But then why do we say things like 'Earth is an almost inertial frame as centrifugal and Coriolis forces are negligible'. I know they're negligible, but why don't we talk about the unaccounted momentum that falling objects gain out of nowhere?

Answer 1

If we talk about the Earth surface frame, particles here gain momentum out of nowhere all the time. Just look at the falling particles. In reality (seen from an actual inertial frame), this momentum is accounted for by an equal momentum lost/ gained by Earth, but as Earth is at rest in the Earth frame, Earth gains no momentum in the Earth frame, and hence the extra momentum gained by literally every falling object gets unaccounted for in the Earth frame!

You have a misunderstanding here. The concept of "particles gaining momentum out of nowhere" is from the fact that in non-inertial reference frames objects can accelerate without any forces being applied to them. If you are watching an object fall on the surface of the Earth, it's gain in momentum is coming from the force of gravity, so the momentum isn't "coming from nowhere".

A correct example of momentum coming from nowhere would be if you are on a train with a ball on the floor of the train, and then the train begins to slow down. In the non-inertial frame of the train you will suddenly see the ball start to roll forward without any forces acting on it. This is a momentum change that "came from nowhere".$^*$

Therefore, I suppose one could say that in the Earth frame there is a small amount of additional unaccounted momentum gained by the object due to the upward acceleration of the Earth, but this is very negligible. The momentum change we actually can detect however isn't due to non-inertial effects. A real force of gravity is acting on the object, and it is accelerating, just how it should in an inertial frame of reference.

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$^*$Of course, many people don't like this, so instead of allowing Newton's second law to be violated we instead bring in "pseudo-forces" to explain these momentum changes. The price to pay is that now Newton's third law is no longer valid; these pseudo-forces don't come from any sort of interaction.

Answer 2

In Newtonian mechanics there are two ways to account for the momentum gained by a falling object:

1. Work in a reference frame in which the earth is at rest, and introduce gravitational potential energy $mgh$. The gain in momentum of a falling object is then accounted for by the potential energy that it loses.
2. Work in a reference frame in which the centre of mass of the object and the earth is at rest. In this reference frame the net momentum of the falling object and the earth is always zero, since gravity is now an internal force. Because the earth is so much more massive than the object, the earth is almost at rest in this reference frame too (a one kg mass falling by one metre moves the earth by about $10^{-10}$ of the diameter of an atomic nucleus).

Answer 3

Because the particle is small, the momentum change of the Earth produces insignificant change on the acceleration of the Earth. Thus the coordinate transformation between appropriate inertial frame and Earth's frame is almost an identity. The difference between the two frames is unmeasurable. In the limit, we are basically considering Earth to be of infinite mass. Then it can gain momentum without accelerating and the rest frame of the Earth becomes inertial frame.

The lost momentum is simply understood to be the momentum gained by the Earth. However, in the calculations we do not care about this momentum. We are doing "calculations in the box", that is, we know our particle is interacting with the Earth, but we do not care about the Earth and all we care about is our particle. Thus we do not demand conservation of momentum, since we know it is "leaking out toward the Earth". The measure of this "leaking out" is then retained in computations as external force acting on our particle by some source outside.