GR & Coriolis Forces: Inertial Trajectories & Resources

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In summary: My $.02In summary, the frame of reference in which a body is not rotating experiences a Coriolis pseudo-force. If you adopt a non-rotating coordinate system, the body will appear to move in a straight line.
  • #1
dand5
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How does GR account for coriolis forces on a solid spherically symmetric body in a frame of reference where the solidy body rotation has been eliminated?
Are these trajectories inertial, as in if a free falling object has an initial velocity along the angle between the plane of rotation and the axis of rotation, are the coriolis trajectories geodesic paths?
Also please let me know of any good references that might help me
understand this.
Thanks in advance for any replies.
 
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  • #2
dand5 said:
How does GR account for coriolis forces on a solid spherically symmetric body in a frame of reference where the solidy body rotation has been eliminated?

It appears that you meant that solid body is in fact rotating then you attempt to define a frame of reference for which it is not rotating. First of all it is not possible to define such a frame of reference when each point in the body not only has a different frame of reference but the frame of reference for each point is constantly changing. By constantly changing frames of reference means that it is accelerating, i.e. constantly changing either direction or magnitude of motion. Note that acceleration is not relative. All observers regardless of the frame of reference agree on what is and isn't accelerating. Observers can only disagree on how much acceleration. The only frame of reference in which the body is not rotating is when the body is not rotating in any frame of reference meaning no coriolis forces.
 
  • #3
my_wan said:
Note that acceleration is not relative. All observers regardless of the frame of reference agree on what is and isn't accelerating.
This seems to fly in the face of the equivalence principle.
my_wan said:
Observers can only disagree on how much acceleration.
This is better. There is a frame in which an observer would say that zero is how much. The observer attributes observations to the fact that there is a gravitational field.

There is a frame in which the rotation of the body is eliminated, but it is not an inertial frame.
 
  • #4
My $.02

Relativity explains rotating frames with Christoffel symbols, but I don't really think that's the type of explanation that's being asked for here.

So let's go with the Newtonian explanation.

If you are on a rotating body, you experience a Coriolis pseudo-force if you adopt a rotating coordinate system.

If you are on a rotating body and you use a non-rotating coordinate system, there are no pseudo-forces, and the body that was experienceing the coriolis psuedo-force in the rotating coordinate system is seen to be moving in a straight line in the non-rotating coordiante system.
 
  • #5
my_wan said:
Note that acceleration is not relative. All observers regardless of the frame of reference agree on what is and isn't accelerating.
jimmysnyder said:
This seems to fly in the face of the equivalence principle.

Why? I do not see how my statement implies that gravitational and inertial mass are not equivalent.

my_wan said:
Observers can only disagree on how much acceleration.
jimmysnyder said:
There is a frame in which an observer would say that zero is how much. The observer attributes observations to the fact that there is a gravitational field.

Zero how? No wait, first you defined a frame with zero acceleration. Then you say the observer attributes the observations (observations presumably meaning g forces) to a gravitational field. So you have g forces with zero acceleration? If that's not a violation of the Principle of Equivalence I don't know what is. The only frame for which you could say acceleration is zero is one moving -at- the speed of light relative to the sphere. Even that is debatable.
jimmysnyder said:
There is a frame in which the rotation of the body is eliminated, but it is not an inertial frame.

Here you have conceded that the frame is not an inertial frame, which -means- it is accelerating.

To be fair it is possible to define a frame for which locally (meaning for a given infinitesimal point within the sphere) there is no rotation, only acceleration. This is what is meant when it's said that the Principle of Equivalence only applies locally. Globally it is always possible to to prove a spin even without referring to an outside frame of reference. Just try to operate a gyroscope inside such a spinning sphere.
 

1. What is the concept of inertial trajectories in GR and Coriolis forces?

Inertial trajectories refer to the paths that objects follow in the absence of external forces. In the context of General Relativity (GR) and Coriolis forces, these trajectories are affected by the curvature of spacetime and the fictitious forces that arise due to rotating frames of reference.

2. How do GR and Coriolis forces affect the motion of objects in space?

GR and Coriolis forces can cause objects to deviate from their expected trajectories in space. In GR, the curvature of spacetime can cause objects to follow paths that differ from the straight lines predicted by Newton's laws. Coriolis forces, on the other hand, arise due to the rotation of a frame of reference and can cause objects to appear to curve in a rotating system.

3. What are some real-life applications of GR and Coriolis forces?

GR and Coriolis forces have many practical applications, such as in navigation systems, weather forecasting, and space exploration. In navigation, these forces are taken into account when calculating the trajectory of a moving object, such as a plane or a ship. In weather forecasting, Coriolis forces play a significant role in the formation and movement of storms. In space exploration, GR is used to accurately predict the motion of planets and other celestial objects.

4. Can GR and Coriolis forces be observed on Earth?

Yes, both GR and Coriolis forces can be observed on Earth. For example, the curvature of spacetime can be observed through the bending of light around massive objects, such as stars. Coriolis forces can also be observed in the rotation of hurricanes and the direction of water draining in a sink or toilet bowl.

5. Are there any limitations to the concept of inertial trajectories in GR and Coriolis forces?

Yes, there are some limitations to these concepts. In GR, the concept of inertial trajectories only applies in the absence of external forces. In the presence of strong gravitational fields, such as near a black hole, the effects of gravity can be so strong that even inertial trajectories can be significantly altered. Additionally, Coriolis forces are only significant in systems that are rotating with high velocities, and their effects can be negligible in everyday situations.

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