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https://www.physicsforums.com/showthread.php?t=391796

where is was intended to argue that in the weak field limit where Newtonian gravity is a good approximation gravitational mass is gamma*m_{0}=E/c^{2}, as opposed to the rest mass argued for by several in that thread.

The System

Consider an object moving in a circle with constant angular velocity w, radius R and rest mass m_{0}from the frame of the center of rotation. If wR << c the classical results can be used.

dw/dt = w^{2}

V = wR

a = w^{2}R

P = m_{0}wR

F = dP/dt = m_{0}dw/dtR = m_{0}w^{2}R

The Observer

The physics of this system should be the same for all observers. Let us consider this system from an observer moving at V parallel to the axis of rotation. The motion is now a helical spiral along a cylinder with constant acceleration towards the axis.

gamma = 1/sqrt(1-(V/c)^{2}) = constant

dt/dt' = 1/ gamma

R' = R //Distance perpendicular to the motion independent of motion.

w' = w / gamma // Angular rotation acts like hands on a clock. (corrected by starthaus)

m_{0}= m_{0}// Rest mass is invariant.

dw'/dt' = d w/gamma /dt' = dw/dt dt/dt' / gamma = w^{2}/gamma^{2}= dw/dt/gamma^{2}= w'^{2}

a' = Rdw'/dt' = Rdw/dt/gamma^{2}= a/gamma^{2}= Rw'^{2}

Because the motion is perpendicular to the original motion the magnitude of the momentum is:

|P'| = gamma * |P|

The axial momentum is constant for a set V and the radial momentum is invariant.

P'_{r}= P_{r}= m_{0}wR = gamma * m_{0}w'R

F' = dP'/dt' = gamma * m_{0}dw'/dt'R = 1 / gamma m_{0}dw/dtR = F / gamma

The Force

So far we have said nothing about the nature of the force other than how it must transform in SR. In the limit of a weak gravitational field the correspondence limit states that GR can be approximated by Newtonian gravity.

Let M_{g}by the barycenter mass needed to cause the motion above. Since wR << c we can ignore the time delay of the light. The subscript g indicates gravitational mass, as the point of this exercise is to determine if gravitational mass is the rest mass or gamma * rest mass.

|F| = GM_{g}m_{g}/r^{2}

For the observer the time delay for an arbitrary V might be significant. By examining a triangle with hypotenuse C and leg along the cylinder of V we can determine the effects of this delay.

One effect is that that the observed distance between m_{0}and where M was when the force originated should be stretched by gamma. Another is that the force should have components in the axial and radial directions.

r' = gamma R

|F'| = GM_{g}'m_{g}'/r^{2}= GM_{g}'m_{g}'/R^{2}/ gamma^{2}

F_{r}' = |F'| / gamma = GM_{g}'m_{g}'/R^{2}/ gamma^{3}

but from above we find that

F_{r}' = F / gamma

GM_{g}'m_{g}'/R^{2}/ gamma^{3}= GM_{g}m_{g}/R^{2}/ gamma

M_{g}'m_{g}' / gamma^{2}= M_{g}m_{g}

M_{g}'m_{g}' = M_{g}m_{g}* gamma^{2}

This suggests m_{g}= gamma * m_{0}, not the rest mass (but consistent with rest energy + kinetic energy / c^{2})

SR + weak field -> Frame Dragging?

F' above has a component in the radial direction that is not balanced by another force, requiring a pseudo force to restore the original dynamics.

F_{a}' = |F'| V/c / gamma = GM_{g}'m_{g}'/R^{2}V/c / gamma^{3}

This force is in the direction of motion, similar to frame dragging.

The Force II

What if this force is from electromagnetism, where the Q is invariant? Parallel to above we find that:

F_{r}' = |F'| / gamma = kQq/R^{2}/ gamma^{3}= kQq/R^{2}/ gamma / gamma^{2}= kQq/R^{2}(1-(V/c)^{2}) / gamma

This force can be restored by adding a pseudo force

PF_{r}' = kQq/R^{2}(V/c)^{2}/ gamma = k/c^{2}QVqV/R^{2}/ gamma

which clearly is magnetism, as expected from the original derivation of SR from Maxwell's laws.

Both the electric force and magnetic force have a frame dragging component as above.

Hmmmmmm

Originally I had thought that the difference in the optical eclipse and tidal maximum was evidence for the different formulations above. Now I am not so sure they are in fact different.

Is there a mathematical difference between these ontologically different statements:

1) an inverse square law of some property of an object causes that property to scale by gamma in SR, with no pseudo forces.

2) an inverse square law of some property of an object in SR keeps the property invariant but produces a magnetism-like pseudo force.

If these statements are equivalent, by convention the second statement prevails.

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# Circular Motion in SR/weak gravitational field limit

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