What's the Flaw in the Relativity Paradox of Gravity and Speed?

[Shaw]

Consider a small particle traversing the Sun's gravitational field near the speed of light. Without reference to the rest of the universe, we are unable to say anything about which object has undergone the acceleration. The particle could be at rest and the Sun could be moving near the speed of light, in which case General Relativity requires no change in the Sun's force of gravity, while time has slowed to a crawl within its frame of reference.

This appears to mean that the particle will experience normal gravitation for a much longer period of time than if the Sun had not received all the acceleration, so the particle could be made to curve through almost any angle, depending on the Sun's velocity.This is unlikely to be true, but where is the flaw.

 

[Hurkyl]

According to GR, neither has been accelerated.

 

[Shaw]

Paradox

I understand that should be the case, but don't you run into trouble with the twin paradox. Some argue that either twin could be younger, but this is a misreading of GR. Only the twin who's undergone acceleration is affected.

 

[Hurkyl]

Not quite. The proper time experienced along a path is computed with the metric. (analogous to simply computing the length of a curve)

The fact you quote is a theorem about flat space -- the longest proper time between two points is experienced along a straight line. Among multiple such paths, the one that deviates the least from a geodesic (i.e.a straight line) experiences less proper time.


In a general space-time, this theorem is only true locally. If two paths are almost the same, the one that deviates less from the geodesic still undergoes the greater proper time. However, when two paths are not almost the same, all bets are off -- you need to find some other way to compare the two proper times. (Such as actually computing them)

 

[pervect]

Consider a small particle traversing the Sun's gravitational field near the speed of light. Without reference to the rest of the universe, we are unable to say anything about which object has undergone the acceleration. The particle could be at rest and the Sun could be moving near the speed of light, in which case General Relativity requires no change in the Sun's force of gravity, while time has slowed to a crawl within its frame of reference.

This appears to mean that the particle will experience normal gravitation for a much longer period of time than if the Sun had not received all the acceleration, so the particle could be made to curve through almost any angle, depending on the Sun's velocity.This is unlikely to be true, but where is the flaw.

One of the flaws is assuming that the gravitational field of a moving object is the same as that of a stationary object. It's not the same, and GR does not say that it is the same.

The space components of the space-time curvature will prevent any model of the situation "as a force" from giving quantitatively correct answers.
[clarification] when the velocity of the moving object is a large fraction of the speed of light.

Qualitiatively speaking, though, the situation will be very similar to that of the electric field of an object moving at a relativistic velocity. The field lines will be considerably concentrated in the direction perpendicular to the direciton of motion, meaning that the transverse field will be very much stronger than the "normal" field of a statioanry body.

[add] Because the transformation of the electric field is a very much simpler mathematically, I generally urge people to become familiar with how the electric field transforms with velocity first, before worrying about how gravity transforms.