The Relativity of Rotation & Gravity

[wfunction]

Imagine being on the equator (and that the Earth is a perfect sphere). Also imagine that the Earth is spinning just fast enough so that gravity is canceled out by the centripetal (or centrifugal, that's not important here) force. I have a wait of zero... no problem, right?

Now what if I say that the Earth is stationary (and so am I) and that the universe is rotating around us? This is perfectly correct, isn't it? And wouldn't that mean that my weight should in fact not be zero since there is no rotation to cancel it?

Thanks!

 

[Bob_for_short]

You are right. Doing so, you conclude that your zero weight is zero not because of your being weightless but because of the Earth rotation.

 

[Lev]

The problem is, that the spinning Earth is NOT the inertial system (you are always affected by a force because of that). This is why rotating universe and rotating Earth is not the same here.
Galileo's relativity can be applied when we talk about inertial systems.

 

[mikeph]

You can either accept the rotation, or that you are in a non-rotating system and subject to inertial forces, either is valid.

 

[Dale]

Imagine being on the equator (and that the Earth is a perfect sphere). Also imagine that the Earth is spinning just fast enough so that gravity is canceled out by the centripetal (or centrifugal, that's not important here) force. I have a wait of zero... no problem, right?

Now what if I say that the Earth is stationary (and so am I) and that the universe is rotating around us? This is perfectly correct, isn't it? And wouldn't that mean that my weight should in fact not be zero since there is no rotation to cancel it?

This would require the full math of GR, but in GR it is perfectly fine to consider a reference frame where the Earth is stationary and the universe is rotating. The underlying spacetime would still be the same, but in these new coordinates you would find that the metric tensor takes on a different form, and as a result in this new form the stationary coordinate would now be a geodesic, meaning that the force would be zero as expected. In that new coordinate system you would essentially say that frame dragging from the rotating universe was cancelling the Earth's gravity.

 

[Bob_for_short]

It does not need GR and super-string theory. Everything is clear even in the Newton mechanics.

 

[Dale]

It does not need GR ... Everything is clear even in the Newton mechanics.

How so? There is no frame dragging effect in Newtonian mechanics, a rotating homogenous universe would produce the same gravity as a non-rotating homogenous universe and would lead to the problem identified by the OP.

 

[Cleonis]

Imagine being on the equator (and that the Earth is a perfect sphere). Also imagine that the Earth is spinning just fast enough so that gravity is canceled out by the centripetal (or centrifugal, that's not important here) force. I have a wait of zero... no problem, right?

Now what if I say that the Earth is stationary (and so am I) and that the universe is rotating around us? This is perfectly correct, isn't it? And wouldn't that mean that my weight should in fact not be zero since there is no rotation to cancel it?

Gravity can't be canceled, but if a celestial body spins fast enough then at the equator the available gravity would all be spent in providing centripetal force, and effectively you would be weightless.

(As an aside: celestial bodies can't actually spin that fast; a planet with such spin would simply not form out of a proto-planetary disk.)

Now what if I say that the Earth is stationary (and so am I) and that the universe is rotating around us?

Well, for one thing, taking the idea of the Earth being 'stationary' and the universe rotating' literally is obviously not a viable physics hypothesis. No centripetal force is available to make the distant galaxies follow a trajectory around the Earth, and there is an upper limit to velocity.

On a deeper level, one can raise the following question: what if we can avoid assigning the status 'stationary' to the universe, and assigning the status 'rotating' to the Earth, replacing that with a concept of a relative rotation between Earth and Universe.

That is not the same as saying 'that the Earth is stationary and that the universe is rotating around us'; the whole point is to avoid attributing states to the Earth and the Universe separately. Instead, the idea is to specify a relative state.

The case you submitted was where the Earth spins so fast that at the equator all gravity is spent in providing centripetal force. In terms of a 'relative rotation' view: this relative rotation is thought of as eliciting a centrifugal effect. In order for everything to be consistent this centrifugal effect must exactly cancel gravity.

Cleonis

 

[D H]

How so? There is no frame dragging effect in Newtonian mechanics, a rotating homogenous universe would produce the same gravity as a non-rotating homogenous universe and would lead to the problem identified by the OP.

A rotating frame is a rotating frame. Rotation, like acceleration, is absolute in both Newtonian and relativistic mechanics. In particular, both have a concept of rotation wrt the remote stars.

 

[Andy Resnick]

Imagine being on the equator (and that the Earth is a perfect sphere). Also imagine that the Earth is spinning just fast enough so that gravity is canceled out by the centripetal (or centrifugal, that's not important here) force. I have a wait of zero... no problem, right?

Now what if I say that the Earth is stationary (and so am I) and that the universe is rotating around us? This is perfectly correct, isn't it? And wouldn't that mean that my weight should in fact not be zero since there is no rotation to cancel it?

Thanks!

We discussed "Newton's bucket" in class. The conceptual difficulty is removed if you stand, instead of at the equator, at one of the poles. Then the direction of gravity coincides with the axis of rotation, simplifying the problem. Clearly, you cannot tell if you are rotating or not.

And the reason why rotating systems are a good way to discuss gravity is because it clearly demonstrates why "ficticious" forces, like centripetal and coriolus, are simply due to comparison with a system that has a different acceleration than you. Such as is the case when discussing free-fall.

 

[Dale]

In particular, both have a concept of rotation wrt the remote stars.

Right, but in GR the rest frame of the remote stars is not assumed to have any special properties while in Newtonian gravity it is assumed to be inertial (a concept that doesn't even exist for reference frames in GR). The Newtonian theory only replicates the right result for this case with that ad hoc assumption.

 

[Bob_for_short]

I think everything about rotating and inertial RFs was already clear in physics before GR. It was not the problem resolved in GR.

 

[Dale]

Bob_for_short and D H, you are interpreting the OP's question differently than I do. Perhaps I am over-interpreting it, but here is what I believe he is asking: "what if the universe were actually rotating instead of the earth?".

My understanding is that in the Newtonian theory of gravity it is possible that we could go out into deep space, get a very sensitive gyroscope or ring interferometer, maneuver so that our rotation was zero, and then observe that the distant stars were rotating around us. Because of frame dragging the same is not possible in GR. In other words, if the universe actually were rotating Newtonian theory would predict observable consequences, but GR would not. At least, that is my understanding of the OP's question and GR.

 

[mikeph]

OP seems to be considering how valid the argument is of someone in a non-inertial frame asserting they are in an inertial frame- the answer I think is that it is completely valid, as long as they accept inertial forces. The universe may be rotating around you, but then this assertion comes with the centrifugal and Coriolis forces.

 

[Dale]

My understanding is that in the Newtonian theory of gravity it is possible that we could go out into deep space, get a very sensitive gyroscope or ring interferometer, maneuver so that our rotation was zero, and then observe that the distant stars were rotating around us. Because of frame dragging the same is not possible in GR.

Hi D H and Bob_for_short, I have looked into this a bit more and concluded that I was mistaken and you are correct. The Goedel solution is a cosmololgical solution where an inertial observer would observe the distant stars to be rotating, directly contradicting my claim above.

So in both GR and Newtonian gravity the fact that the distant stars constitute an inertial reference frame is an observed fact and not a logical consequence of the theory. There appears to be nothing gained by my introduction of GR into the discussion.