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We usually treat the earth frame as approximately inertial. When we want to apply corrections, we use the centrifugal, coriolis and azimuthal forces. Are they all the corrections we would ever need?

That would imply the frame I am talking about (attached to the centre of the earth but not rotating with it) would be inertial. Is it? Or do we need to take the earth's motion around the sun into account? What more corrections will be there?

Will there ever be enough corrections? Is there a truly inertial frame out there? I know GR says the universe expands etc etc. (I am at a freshman level, so I only have a vague pop sci idea of SR and GR), but what did people think about inertial reference frames during the classical period - say 17 or 1800s? And what do people think now?

Say newton definitely would have been aware of the fictitious forces like coriolis to account for the earth's rotation. But did he ever think of taking the earth's motion around the sun into account? Did he stop there as he thought the solar system was a fixed thing?

My question is mostly about the classical view on this, though how we look at it currently using GR would also be good but please note I am at a very beginning level in college physics.

Qmechanic
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Neeladri Reddy
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If you are in a frame which is fixed to the Earth's center, but not rotating with the Earth, is it inertial?

A non-rotating frame with its origin at the center of the Earth is approximately inertial from a classical perspective. However, as the Earth does accelerate toward the Moon and the Sun (and other bodies) due to their gravitational influence on the Earth, this makes a non-rotating Earth-centered not quite inertial. The image below shows how various perturbative effects on the motion of a satellite orbiting the Earth varies with orbital distance.

Perturbations on Newtonian spherical gravity as a function of orbital distance. The line labeled GM is the Newtonian spherical gravity. The other lines and curves denote perturbations. Note that the graph is a log-graph.

Those perturbations from the Moon and Sun (and from other bodies) are oftentimes called tidal forces by physicists. However, in aerospace we use tidal forces to denote something different. The Moon and Sun distort the shape of the solid Earth, and these distortions in turn subtly impact a satellite's orbit. This is the line denoted "dynamic solid tide" in the above diagram. In aerospace, we use "third body effects" as the generic term for perturbations caused by using a non-rotating but gravitationally accelerating frame of reference. In the diagram, these third body effects are the lines labeled "Moon", "Sun", "Venus", and "Jupiter".

David Hammen
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As you say, the Earth orbits the sun. But in GR, this is because the Earth follows a geodesic around the sun. So this is inertial.

There are further corrections. The Sun orbits the Milky Way. Again a geodesic.

mmesser314
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An inertial frame can be defined as the frame in which laws of physics take their simplest form. A frame of reference where the earth is non-rotating is a good candidate for that (as certain pseudo forces do not exist). However, as you pointed out, earth still goes around the Sun, which itself goes around the center of the Galaxy, and so on. In that sense, any frame is only locally inertial.

A good candidate for an approximately global inertial frame is the one w.r.t. which the heavens do not move. These observers are the ones that see the Universe as homogeneous and isotropic on large scales. A non-rotating earth is also very roughly in this category.

Also, geodesic motion in GR is again only locally inertial (see equivalence principle). For a large enough frame, one will start feeling tidal forces.

S.G
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  • What I understand from your answer is that we don't know a true inertial frame, even in theory since everything is moving, even space itself from expansion of the universe(I have no earthly idea what I'm talking about). By experiment - all frames we tested are only approximate, locally inertial. But a frame in which heavens don't move(is it the fixed stars frame?) might work since everything is isotropic in that so it's sort of the center of the universe and thus at rest? But if space itself is moving, then locally in that area if you are at rest wrt that space, it might be inertial? – Neeladri Reddy Oct 30 '23 at 11:11
  • Yes it's the fixed stars frame. There is no one center to the universe though. Every observer that perceives the universe to be homogeneous and isotropic can consider itself as at the center. – S.G Oct 30 '23 at 11:14
  • I thought the expansion of the universe as like a sphere expanding, is it not the right then? And Ok, it's sort of a centre.. but why isn't the normal ground frame ok for that? In a normal ground frame the stars will still appears to be homogenosly distributed.. Are there fictitious forces associated with the expansion of the universe? – Neeladri Reddy Oct 30 '23 at 11:18
  • And why is the the non-rotating earth such a good inertial reference frame, for when all we know - the galaxy could be moving too fast? What does locally inertial mean? – Neeladri Reddy Oct 30 '23 at 11:21
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    Thanks for taking the time trying to explain - but as of now perhaps I would do best to leave GR alone haha and get the answer to this - why can the non rotating earth frame treated as approximately inertial? What does it mean, locally inertial? Is it possible to explain that by any classical ideas or do we just say - by experiment? – Neeladri Reddy Oct 30 '23 at 11:32
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    how would you define "simplest" in the context of laws taking "their simplest form"? – hyportnex Oct 30 '23 at 11:34
  • @hyportnex Newton's laws holding for example, or maxwell's laws holding - I think. – Neeladri Reddy Oct 30 '23 at 11:40
  • @NeeladriReddy Adding a plug for my answer here which spells out the idea that an inertial frame is one "in which laws of physics take their simplest form": https://physics.stackexchange.com/a/681138/153305. Essentially, the collection of frames identified as inertial is model-dependent, depending on how one models forces between particles. Under a given framework, in particular under the standard picture of classical forces (Newtonian gravity + Maxwell E&M), there are some inertial frames out there, but they won't be tied to a physical body's trajectory without very delicate engineering. – jawheele Oct 30 '23 at 15:41
  • @jawheele What do you mean tied to a trajectory? – Neeladri Reddy Oct 30 '23 at 18:27
  • @NeeladriReddy I mean that it would be an extremely exceptional scenario if there exists an inertial frame (under anything resembling the standard framework of classical forces) in which any particular physical body (e.g. Earth) is exactly stationary. – jawheele Oct 30 '23 at 18:35
  • @hyportnex a frame where newtons 1st law holds is the frame where laws of nature are simplest. – S.G Oct 30 '23 at 19:45
  • @S.G that is a rather unnatural anthropocentric view; simple is in the eye of the beholder. A dish antenna on the B-1 bomber tracking a communications satellite while the plane is maneuvering would never be able to point to its target location while avoiding gimbal lock if it attempted that control in its "simplest" form. – hyportnex Oct 30 '23 at 19:59
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    @hyportnex the definition I quoted is not just my view. It's how an inertial frame is defined as far as I know https://en.m.wikipedia.org/wiki/Inertial_frame_of_reference#:~:text=An%20inertial%20frame%20of%20reference%20is%20one%20in%20which%20the,straight%20line%20at%20constant%20speed. – S.G Oct 30 '23 at 21:16
  • @jawheele Oh yes. – Neeladri Reddy Oct 30 '23 at 22:53