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This is just a series of thoughts leading to a “conclusion”. I would appreciate comments/criticism, please. 1. Velocity is a vector which includes speed and direction.2. Acceleration is change in velocity.3. Change of speed with constant direction = acceleration.4. Change of direction with constant speed = acceleration.5. A body orbiting at constant speed is constantly accelerating.

6. Gravity is not a force that holds an orbiting body as though it were on a string.

8. The curve thus formed is a geodesic, and is defined as the most direct path from A to B in curved spacetime.

9. Thus, a geodesic is equivalent to a straight line in flat (non-curved) spacetime.

10. A body travelling at constant speed in a straight line in flat spacetime is not accelerating.

11. It should be reasonable to argue that a body following a geodesic at constant speed is not accelerating.

12. It should, therefore, be reasonable to conclude that an orbiting body is not accelerating.

Yes. As long as the orbit is not an ellipse.

The 4-acceleration is zero. The spatial acceleration is non-zero.

Quote from: JeffreyYes. As long as the orbit is not an ellipse.Kepler's 1st law states that planetary orbits are elliptical, and his 2nd law involves a constant change of speed, so I guess using the sort of generalised model of constant speed and circular orbit is on to a hiding to nothing. []Quote from: PeteThe 4-acceleration is zero. The spatial acceleration is non-zero.Could you explain this in simple, layman, terms, please?

Sure. 4-acceleration is an acceleration which depends on the mass of the object. Spatial acceleration is the time rate of change of a particle/bodies position with time.

The circle is just a special case of an ellipse with both focal points at the same place.

Quote from: Pete Sure. 4-acceleration is an acceleration which depends on the mass of the object. Spatial acceleration is the time rate of change of a particle/bodies position with time. Does 4-acceleration not involve change in position with time?Quote from: Jeffrey The circle is just a special case of an ellipse with both focal points at the same place. Would it not be true that Kepler’s 2nd law, applied to a circular orbit, would not involve a change of speed?

The angular velocity is constant, however objects tend to move in a straight line unless acted upon by a force. Since the direction is constantly changing due to the force of gravity you can argue that this is in fact a type of acceleration. There is an equilibrium in the kinetic energy of the orbiting body due to its initial velocity balancing with the gravitational force that is affecting its motion. Increase or decrease this critical velocity and the profile of the orbit becomes either an elliptical one or the object crashes to the surface.

When GRists say that "gravity is not a force" all they mean is that it's not a 4-force.

Quote from: Pete When GRists say that "gravity is not a force" all they mean is that it's not a 4-force.That's not quite what David Deutsch says here: “In the nineteenth century, few things would have been regarded more confidently as real than the force of gravity. Not only did it figure in Newton’s then-unrivalled system of laws, but everyone could feel it, all the time, even with their eyes shut – or so they thought. Today we understand gravity through Einstein’s theory rather than Newton’s, and we know that no such force exists. We do not feel it! What we feel is the resistance that prevents us from penetrating the solid ground beneath our feet. Nothing is pulling us downwards. The only reason why we fall downwards when unsupported is the fabric of space and time in which we exist is curved.”