Simultaneity, Rotation & Gravity: Agree?

In summary, using Einstein synchronisation, the orbits of satellites will have the same speed in both directions according to stationary observers, but they will have different speeds according to ZAMO observers.
  • #1
yuiop
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We have had a number of threads on how to synchronise clocks around a rotating ring. One method of doing this is to start all the clocks on the ring via a signal from the centre of the ring. This method has the advantage of being transitive, but has the disadvantage that the local one-way speed of light depends on which way it is going around the ring. Einstein synchronisation has the advantage that the local one way speed of light is independent of direction. I think it was Pervect that pointed out that Einstein synchronisation also has the advantage that locally things still work according to our Newtonian expectations. For example two equal masses sent simultaneously with equal speed in opposite directions around the ring, would collide and come to rest with the ring at the mid point, if the collision is inelastic. This would not happen with the transitive synchronisation.

I was wondering what would happen if we use gravity to define simultaneity. Consider a balance with equal length horizontal arms. I would expect that if I place equal weights at the ends of the arms simultaneously, the balance would not rotate, but which definition of simultaneity applies to gravity if the balance is attached to a rotating ring? My initial thoughts are that for the balance to 'balance', the weights would have to attached simultaneously according to the Einstein synchronisation convention. If the weights were attached simultaneously according to the transitive clocks, the leading weight would appear to be attached to the balance before the trailing weight (according to the Einstein clocks) and the balance would rotate about its fulcrum in the same direction as the rotation of the ring in the time interval between the first and second weights being attached. It seems as far as gravity is concerned, Einstein synchronisation is the 'natural' method. Agree?
 
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  • #2
The arm of the balance can't be perfectly rigid. Transverse waves will travel along it at some speed v<c. So I think this scheme amounts to doing something like Einstein synchronization, but carrying it out using mechanical waves rather than light waves.
 
  • #3
On a related subject, consider geodesic orbits in the Kerr metric. The coordinate geodesic orbital velocity is given by:

##\frac{r}{a\pm \sqrt{r^3/m}}##

The coordinate speed of light is given by:

##\frac{2 m a \pm r \sqrt{r^2+a^2-2mr}}{(r^2+a^2+2ma^2/r)}##

where in both cases, the positive sign is for prograde.

Taking the ratio, the geodesic orbital velocity relative to the speed of light is:

##\frac{r(r^2+a^2+2ma^2/r)}{(a\pm \sqrt{r^3/m})(2 m a \pm r \sqrt{r^2+a^2-2mr})}##

The implication is that even a local stationary observer will see satellites at the same altitude orbiting at different velocities, with the retrograde satellites orbiting faster. Things get worse from the point of view of a ZAMO observer. However, all the above is assuming that observers on a ring of radius (r) synchronise there clocks in a transitive manner. The question is, if we use Einstein synchronisation, would the orbits have equal speeds in both directions according to the stationary or ZAMO observers? If there a unique rotation speed for a ring that gives equal orbital speeds? Is there anything else special about the frame in which the orbital velocities are equal?
 

Related to Simultaneity, Rotation & Gravity: Agree?

What is simultaneity?

Simultaneity refers to the concept that two events happening at different locations can occur at the same time in one frame of reference, but may appear to occur at different times in another frame of reference.

How does rotation affect simultaneity?

Rotation can affect simultaneity by changing the relative motion between two different frames of reference. This can cause events that were simultaneous in one frame to appear non-simultaneous in another frame.

What is the relationship between rotation and gravity?

Rotation and gravity are both caused by the curvature of spacetime. The presence of mass causes spacetime to curve, resulting in the force of gravity. Objects in motion also cause spacetime to curve, resulting in the effects of rotation.

Do all observers agree on the simultaneity of events?

No, all observers do not necessarily agree on the simultaneity of events. This is because simultaneity is relative and can vary depending on an observer's frame of reference and their relative motion to the events.

How does Einstein's theory of relativity explain simultaneity, rotation, and gravity?

Einstein's theory of relativity explains simultaneity, rotation, and gravity by describing how these concepts are all related to the curvature of spacetime. In this theory, gravity is not seen as a force, but rather as a result of the curvature of spacetime caused by the presence of mass. Rotation is also a result of this curvature, and the concept of simultaneity is relative to an observer's frame of reference and the curvature of spacetime in that frame.

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