Interesting General Relativity Scenario

[OverLOAD]

I've come up with a scenario which seems to be something of a paradox to me:

Imagine, if you will, two similar rotating objects orbiting each other at a fixed distance, sharing the same rotational plane.

These two objects now experience acceleration to some fixed speed along their shared axis of rotation to the point that relativistic effects become significant:

From each object, it will appear that 1) the distance to its counterpart has increased due to 2) its partner apparently lagging behind it in the distance traveled, and 3) the effect of its counterparts gravity will be diminished by a corresponding amount as the apparent distance between these objects increases. And finally 4) as time dilation becomes a significant factor, the force of observed gravity changes

Thus the result is that without any other direct energy input, the distance between these two objects will increase, against the pull of gravity, without effecting the input of energy into their plane of rotation.

Deceleration of these paired objects would result in permanent changes to their orbit with no additional energy beyond the 'equal' amounts from both the acceleration and deceleration of the system.

There are also some more unusual questions that I have as to what should or would happen with respect to a rest frame observer:
-the observer 'should' perceive the force of gravity between the paired objects to be the same at any speed.
-If their 'relativisitic mass' increases, then the force of gravity should increase, shortening the paired objects distance, and increasing their orbital frequency.
-As the paired objects approach c, time dilation slows down their orbital frequency
-As the apparent distance from one paired object to the other (as experienced by light, and gravity) increases, the paired objects orbital frequency slows.

I know that this is a more complicated scenario then most... but the question I have is: What happens? After the paired objects return to the same seed as the original rest frame, is the distance between the two objects a) the same, b) less, or c) more?

My prevailing thought is that somehow the total energy of the system must be preserved, and the distances should be the same, but with all of the relativistic effects happening, how is it possible for them to cancel out perfectly?

Cheers,

OverLOAD

 

[yuiop]

I've come up with a scenario which seems to be something of a paradox to me:

Imagine, if you will, two similar rotating objects orbiting each other at a fixed distance, sharing the same rotational plane.

OK, let's say the two objects have similar mass and are rotating around their common centre of mass in the x,y plane.

These two objects now experience acceleration to some fixed speed along their shared axis of rotation to the point that relativistic effects become significant:[\quote]OK, let's say they are accelerated along the z axis to a constant velocity of 0.99c.

From each object, it will appear that 1) the distance to its counterpart has increased due to 2) its partner apparently lagging behind it in the distance traveled, and 3) the effect of its counterparts gravity will be diminished by a corresponding amount as the apparent distance between these objects increases. And finally 4) as time dilation becomes a significant factor, the force of observed gravity changes

There is no lagging effect while they are moving with a constant velocity of 0.99c (nertial motion). Visually and physically everything is the same as when they were stationary. Abberation effects make the opposite body look like it is at its predicted position rather than at its location when the light left it. Gravity works in the same way, working on the projected positions of the bodies rather than the lagged position. Relativity tells us that everything about the system moving at 0.99c along the x-axis is no different to when it was stationary. You can not detect the inertial motion.

Thus the result is that without any other direct energy input, the distance between these two objects will increase, against the pull of gravity, without effecting the input of energy into their plane of rotation.

It will not increase.

I know that this is a more complicated scenario then most...

Yep. Is that why you call yourself overload?

but the question I have is: What happens? After the paired objects return to the same seed as the original rest frame, is the distance between the two objects a) the same, b) less, or c) more?

a) The same.

My prevailing thought is that somehow the total energy of the system must be preserved, and the distances should be the same, but with all of the relativistic effects happening, how is it possible for them to cancel out perfectly?

Relativity is good at that.

All the above concerns inertial motion. While the bodies are accelerating, visual lagging effects will be apparent.

 

[Passionflower]

An analytical solution to this problem is clearly beyond the current state of affairs. Only numerical evaluations could give some sensible answers here.

There are some systems with two rotating bodies that give solutions where waves colliding from infinity have an impact, I do not remember if that is the case for the scenario you give but in general I would be very suspicious of any definitive statements about the dynamics and kinematics here.

 

[OverLOAD]

Well, in the scenario I described, I think that the apparent rotation rate would decrease as the speed approached c, but this would be problematic, as it would imply that the force of gravity between the two objects had decreased...

There is no lagging effect while they are moving with a constant velocity of 0.99c (nertial motion). Visually and physically everything is the same as when they were stationary. Abberation effects make the opposite body look like it is at its predicted position rather than at its location when the light left it. Gravity works in the same way, working on the projected positions of the bodies rather than the lagged position. Relativity tells us that everything about the system moving at 0.99c along the x-axis is no different to when it was stationary. You can not detect the inertial motion.

From One paired objects perspective, it would only be able to 'see' and 'experience' the effects of its paired object from the apparent positions... I would expect there to be a doppler shift, but also for the angle of the light from the paired object to be warped in such a way that as you look towards the vector of the original rest position, you would see not the original side of the paired object, but the warp would shift the view towards seeing the leading face of the object, with respect to its motion, making it appear to turn off it axis towards the paired object as it revolves...

the actual motion is a bit more difficult for me to visualize... if that the acceleration has completed, from the view of either paired object, the time it takes for light to travel from one of these entities to the other is now increased, so the field strength of the gravity between them should be weaker - as the world-line is longer.

Finally as their time-dilation causes time to slow down from the view of the paired objects, and the force of gravity is time dependent, would the force of gravity further diminish from the viewpoint of a distant observer?

Cheers,

OverLOAD

 

[starthaus]

Thus the result is that without any other direct energy input, the distance between these two objects will increase, against the pull of gravity, without effecting the input of energy into their plane of rotation.

No, actually the distance decreases.The two bodies are "in-spiralling". Google "gravitational radiation"

My prevailing thought is that somehow the total energy of the system must be preserved,

It isn't. The system is radiating energy. What you are describing is the Hulse-Taylor pulsar. Google PSR B1913+16

 

[jtbell]

These two objects now experience acceleration to some fixed speed along their shared axis of rotation to the point that relativistic effects become significant:

So from the point of view of an observer external to the two objects, they are now traveling in a helical path with constant pitch (distance between turns) along their mutual axis of rotation? The simplest way to do this is to accelerate the observer along a line parallel to the axis of rotation, without doing anything to the objects themselves. I assume that if you instead accelerate the objects themselves, you accomplish it in a way that produces the same result, because all we're interested in is the effect of the final relative motion, right?

From [the point of view of] each object,

[I added the brackets to indicate what I think you're asking] nothing changes in the scenario I described, because nothing has acted on the objects themselves, and they don't "know" that the external observer has accelerated. They don't even "know" that there is an external observer at all!

From the point of view of the external observer, the distance between the objects does not change, because that distance is perpendicular to their overall line of motion, and perpendicular distances are not length-contracted. They do revolve at a slower rate around the axis because of time dilation.

These are purely kinematic results, and must be true regardless of the nature of the forces between the two objects. They could be the balls on the two ends of a rotating dumbbell, for example, with small enough masses that gravitational force between them is negligible.

[added] Also, if the effects of gravitational radiation are negligible in the initial reference frame (i.e. the objects are not noticeably in-spiraling because of loss of energy through gravitational radiation), then they should also be negligible in the moving observer's reference frame.

 

[OverLOAD]

So from the point of view of an observer external to the two objects, they are now traveling in a helical path with constant pitch (distance between turns) along their mutual axis of rotation? The simplest way to do this is to accelerate the observer along a line parallel to the axis of rotation, without doing anything to the objects themselves. I assume that if you instead accelerate the objects themselves, you accomplish it in a way that produces the same result, because all we're interested in is the effect of the final relative motion, right?

I guess my point is that it doesn't produce the same result... increasing the speed of the rotating objects together changes their rate of time passage.

The external observer can see their rate of rotation prior to the change in speed, and determine it to be some fairly stable value (gravitational radiation does not need to be a significant factor).

From the perspective of either of the paired objects, would they still appear to be rotating about the shared axis at the same speed? My understanding is that yes they would, hence a distant stationary observer would now see them rotating at a lower speed after they increased speed (after factoring in the change in speed).

This means that from the distant observers vantage, the two objects appeared to have lost mass to maintain the equilibrium of the system...?

 

[yuiop]

I guess my point is that it doesn't produce the same result... increasing the speed of the rotating objects together changes their rate of time passage.

The external observer can see their rate of rotation prior to the change in speed, and determine it to be some fairly stable value (gravitational radiation does not need to be a significant factor).

Yes, the gravitational radiation can be very small for a system like the Earth/Sun for example where the inspiral might take hundreds or thousands of years to noticeable. For something more extreme like the Hulse-Taylor pulsar (two neutron stars in close mutual orbit?) the inspiral is more noticeable, but we need not consider such extremes. In any case, any inspiral due to gravitational radiation, will be exactly the same before and after the system is accelerated.

From the perspective of either of the paired objects, would they still appear to be rotating about the shared axis at the same speed?

Yes. Even better from the perspective of a non rotating observer that remains at the midpoint between the two co-orbiting objects.

My understanding is that yes they would, hence a distant stationary observer would now see them rotating at a lower speed after they increased speed (after factoring in the change in speed).

Agreed, the observer that remained in the original frame would see the rotation rate as slowed down.

This means that from the distant observers vantage, the two objects appeared to have lost mass to maintain the equilibrium of the system...?

Not sure about your conclusion here. They will be rotating slower and there distance apart will be the same from the point of view of the observer that remained in the original frame, but what happens to the mass depends on your definition of mass. Generally we do not divide mass into differtent types of mass, eg inertial mass and gravitational mass. Certainly the gravitational interaction between the the two objects is observer dependent and in this case the gravitational interaction appears to be reduced from the point of view of the observer in the original frame. In other threads we have determined that gravitational interaction (I hesitate to say gravitational mass) is dependent on the velocity of the masses relative to each other and and the velocity of the masses relative to the observer. I can guarantee that everything about the two body system from the point of view of an observer co-moving with the system will be the same, whatever the velocity of the system is, relative to the observer that remains in the original frame. If this does not hold, then it would be possible to determine an absolute reference frame and Special Relativity would be broken.

 

[OverLOAD]

I can guarantee that everything about the two body system from the point of view of an observer co-moving with the system will be the same, whatever the velocity of the system is, relative to the observer that remains in the original frame. If this does not hold, then it would be possible to determine an absolute reference frame and Special Relativity would be broken.

I understand how SR and GR state that any systems moving with respect to another will undergo the effects that they do, but I feel compelled to think that there must be some sort of reference frame... and the only way I can conceive of knowing it, would be by the idea that time would move fastest if you were fixed in relation to that frame.. but that doesn't work in SR or GR, and no experimental results give that any weight... whenever you move relative to any other timekeeping device, you always end up with slower time, rather then faster - no change in speed in any direction ever brings you closer to an 'absolute' reference frame.

I tried to liken the rate of passage of time to both the gravity well potential, and the relative effects of an objects velocity. They both have effects on the speed of time according to GR. Is it possible that at earthly observational points, the two conspire to hide the reference frame? According to research, and the CMB red/blueshift, the Milky way is moving somewhere between 300 KPS and 600 KPS within the local galaxy cluster. Our fastest and most distant probes move at ~15 KPS by comparison. Pioneer Anomaly anyone?