I Distinguishing gravity from fictitious forces

[Hiero]

TL;DR Summary

‘Locally gravity is indistinguishable from fictitious forces but on large scales gravity is distinguishable by the tidal forces produced.’

Is the above statement accurate?

I am just wondering, aren’t tidal forces also produced by the centrifugal force of a rotating frame?

So then what really distinguishes gravity from fictitious forces?

(I don’t know any general relativity so if you reference things like the Riemann curvature tensor then please explain well.)

 

[Dale]

Summary:: ‘Locally gravity is indistinguishable from fictitious forces but on large scales gravity is distinguishable by the tidal forces produced.’

I am just wondering, aren’t tidal forces also produced by the centrifugal force of a rotating frame?

No, although I can see how you would think that they are. A good way to determine if there are tidal forces present is to consider a bunch of small particles. The traditional example is a bunch of coffee grounds. If there is tidal gravity then the cloud of grounds will get stretched in the radial direction and squished in the transverse directions. So if it started out spherical then it will become ellipsoidal. This does not happen in a rotating frame.

 

[Hiero]

No, although I can see how you would think that they are. A good way to determine if there are tidal forces present is to consider a bunch of small particles. The traditional example is a bunch of coffee grounds. If there is tidal gravity then the cloud of grounds will get stretched in the radial direction and squished in the transverse directions. So if it started out spherical then it will become ellipsoidal. This does not happen in a rotating frame.

Thanks for the reply. Sorry I am still not understanding why.

Just looking at the form of the centrifugal force field, it grows radially and varies angularly. Placing coffee grounds in that field would surely cause it to move apart?

I know it’s of course not the same tidal forces as gravity. For instance since its outward it will be stretched laterally, and of course it will be unaffected along the axis of rotation. But isn’t this still considered tidal forces?

 

[Dale]

Placing coffee grounds in that field would surely cause it to move apart?

No. You are forgetting the Coriolis force. When you include both then a spherical cloud of coffee grounds stays spherical while in free fall.

I know it’s of course not the same tidal forces as gravity. For instance since its outward it will be stretched laterally, and of course it will be unaffected along the axis of rotation. But isn’t this still considered tidal forces?

This would change the volume of the coffee grounds cloud. What you are describing here would be the greatest discovery of the last century. The only way that the free falling cloud of coffee grounds can increase in size is if it contains a gravitationally significant amount of exotic matter which has negative mass.

 

[Hiero]

No. You are forgetting the Coriolis force. When you include both then a spherical cloud of coffee grounds stays spherical while in free fall.

Interesting statement. I neglected the Coriolis force because it is proportional to velocity and I imagined the cloud as initially stationary.

If we consider the motion of a radial line of particles though it seems reasonable; the further away particles pick up a bit more radial speed, then the Coriolis due to that radial speed gives them a bit more lateral speed, then the Coriolis force due to the larger lateral speed tends to pull it back together.

Thanks for the insight! I’ll have to think it through in more quantitative detail to see that it really does balance, but that’s certainly what I was overlooking.

Edit:
Of course it should work; a “free falling” cloud would just be motionless in an inertial frame.

I still am curious to see how it works out mathematically though.

 

[jbriggs444]

Of course it should work; a “free falling” cloud would just be motionless in an inertial frame.

Good insight.

Note that a cloud that is initially at rest in the rotating frame will disperse slowly. The individual bits are, after all, not at relative rest in the inertial frame.

 

[Dale]

I neglected the Coriolis force because it is proportional to velocity and I imagined the cloud as initially stationary.

I thought about this some more, because this argument is actually pretty persuasive. I read the following links: http://www.zenfox42.com/GR1c-Relativity.pdf
http://math.ucr.edu/home/baez/gr/ricci.weyl.html

I realized that I forgot some important descriptors. The individual coffee grounds are always inertial and collectively they are initially comoving. I have not proven it but I believe that those two conditions are not possible to satisfy if all of the grounds are initially at rest in the rotating frame. So there must be some Coriolis force at the beginning.

 

[jbriggs444]

initially at rest in the rotating frame. So there must be some Coriolis force at the beginning.

If they are initially at rest in the rotating frame, then naturally there can be no Coriolis force initially.

I see no contradiction with a cloud of inertial grounds that all just happen to be momentarily and simultaneously stationary against a set of rotating coordinates.

 

[vanhees71]

An example is a rotating heavenly body like a star: It's an ellipsoid rather than a sphere (in non-relativistic approximation and due to its "self-gravity" + "centrifugal force" as seen in the corotating frame).

 

[Dale]

If they are initially at rest in the rotating frame, then naturally there can be no Coriolis force initially.

Right. Which is why they cannot be initially at rest in the rotating frame. There must be a Coriolis force so that they can all accelerate the same thereafter.

Start from the inertial frame and choose one coffee ground. Let this one coffee ground be momentarily at rest in the chosen rotating frame. Since the grounds are comoving then every ground's velocity is the same in the inertial frame. As we transform each ground's velocity into the rotating frame we find that the chosen ground is stationary as are any grounds in a line with it parallel to the axis. For all other grounds, the velocity of the ground in the inertial frame will not match the velocity of a momentarily co-located rotating particle. Therefore, in the rotating frame all the other grounds will have a non-zero velocity. If they do not then it will violate the comoving condition.

Alternatively, start in the rotating frame with a bunch of grounds that are all at rest in the rotating frame. Transform to the inertial frame and note that in the inertial frame their velocities are not the same. Therefore they are not comoving.

 

[etotheipi]

Right. Which is why they cannot be initially at rest in the rotating frame. There must be a Coriolis force so that they can all accelerate the same.

What if we position cannons in a circle about the axis, each pointing just slightly to the right of the axis. Let us also assume that there is no gravitational field. When we fire the cannons at the same time, the projectiles will all travel in straight lines, each just missing the axis. At the instant the velocity vector of each projectile is orthogonal to the vector from the axis to the particle, we will instantaneously have a circular velocity field around the axis. Then, we transform into a rotating frame of reference, where all the projectiles are initially at rest with no Coriolis force acting on them, and they will begin to disperse.

 

[Dale]

What if we position cannons in a circle about the axis, each pointing just slightly to the right of the axis. Let us also assume that there is no gravitational field. When we fire the cannons at the same time, the projectiles will all travel in straight lines, each just missing the axis. At the instant the velocity vector of each projectile is orthogonal to the vector from the axis to the particle, we will instantaneously have a circular velocity field around the axis. Then, we transform into a rotating frame of reference, where all the projectiles are initially at rest with no Coriolis force acting on them, and they will begin to disperse.

You certainly could do that, but it would violate the "initially comoving" condition.

 

[jbriggs444]

Right. Which is why they cannot be initially at rest in the rotating frame. There must be a Coriolis force so that they can all accelerate the same.

Ahh, so by "co-moving" you have in mind that they must not only momentarily move together but must do so for a finite interval. You've stipulated that the particles are moving inertially. So the only available forces are centrifugal and Coriolis. Centrifugal cannot do the job. Coriolis is zero and cannot do the job.

But of course, that was already obvious. A rotating cloud whose particles move inertially would not stay confined in the inertial frame. That is an invariant fact of the matter. Changing to rotating coordinates does alter the fact.

 

[Dale]

Ahh, so by "co-moving" you have in mind that they must not only momentarily move together but must do so for a finite interval.

No, sorry, I was unclear.

The co-moving stipulation is only at the beginning. The first link that I cited above describes in technical detail what is meant by co-moving.

Additionally, in flat spacetime if the grounds are both initially co-moving and always inertial then they will remain co-moving.

So they do remain co-moving for a finite interval but that is because they were momentarily co-moving and then they subsequently moved inertially in flat spacetime. To do that they have to have the same acceleration in the non-inertial frame so all of the fictitious forces must add to the same.

I was putting too many unstated things together in the same sentence. Sorry about the confusion

 

[jbriggs444]

To do that they have to have the same acceleration in the non-inertial frame so all of the fictitious forces must add to the same.

Be careful here. If by "non-inertial frame" you mean non-inertial coordinate system then the coordinate accelerations of the individual bits will vary over the extent of the cloud even while the actual physical separations remain fixed in a Born-rigid manner. [Hopefully no heated discussion will drift over here about this usage of "frame"].

Consider by way of example a stationary U.S. quarter placed so that one rim is at the origin of a rotating cartesian coordinate system. Clearly the point at the origin has zero coordinate acceleration. Meanwhile the point on the opposite rim has a non-zero coordinate acceleration that varies over time.

 

[Dale]

@jbriggs444 yes, I believe that you are right.

 

[Hiero]

Great discussions!

I completely overlooked the fact that a lack of relative motion in an inertial frame requires relative motion in the rotating frame (and vice versa).

I am still left confused though. It seems what is being said is that fictitious forces produce no tidal forces on co-moving clouds... but then “co-moving” seems to be defined with respect to an inertial frame? (Sorry, the parallel transport definition is beyond me.) This reasoning seems circular, because in my mind the whole idea is to treat the rotating frame as if it’s inertial with some ‘gravity-like forces’ and then to see that the ‘gravity-like forces’ produce no tidal effects.
(This thought process is in analogy with a linearly accelerating frame which produces a ‘gravity-like’ force minus the tidal effects.)

Sorry if I’m oversimplifying, but to be fair, the question came to mind when discussing tidal forces with a non-technical friend, so I’m trying to understand in a somewhat simple way.

I know there are a lot of subtleties involved with gravity. Feel free to use my thread for discussions that don’t involve me; I enjoy reading them!

 

[Dale]

but then “co-moving” seems to be defined with respect to an inertial frame? (Sorry, the parallel transport definition is beyond me.)

It is defined by parallel transport and geodesics, but if you don’t understand the parallel transport then you can think of it as though it were defined as co-moving in a local inertial frame.

This reasoning seems circular

Well, it isn’t circular, but to avoid the circularity requires the parallel transport stuff. All of it is based in coordinate-independent concepts, so there is no basic distinction between inertial and non-inertial frames at all. The goal isn’t to treat non-inertial frames as though they were inertial but to look at the things that don’t depend on the arbitrary and non-physical choice of reference frame at all.