General Relativity Question

[garys_2k]

This is one I haven't gotten a handle on...

An asteroid is seen approaching earth at 0.9c and an expedition is sent to intercept and deflect it. The astronauts land on it (how doesn't matter, they just do) and set up the deflection engines to cause the asteroid to change its speed.

Earth-based techs monitor the whole operation and watch the astronauts install the asteroid-monted rockets in place, pointed directly at the earth. The astronauts fire the engines and the asteroid's approach speed drops toward 0, passes it, and later achieves 0.9c away from earth. The astronauts pack up and come home, never having to pay taxes again.

From earth, the influence of the rocket engines on the asteroids seems to follow GR's rules: during the times when the asteroid is at relativisitic speeds compared to earth its acceleration is reduced, because of its increased mass. Around the time its speed is near zero its acceleration from the known-thrust rockets is what Newton predicted.

Of course, to the astronauts sitting on the asteroid the accelerometers they installed work perfectly according to Newton the whole time. Acceleration from the known-thrust rockets is constant no matter how fast they're going relative to an arbitrary reference. The astronauts only can measure the asteroid's rest mass.

But wait! The techs on earth have been watching those asteroid-based acclerometers and could observe their readings, so they could see that the mass was not relativistically altered. In fact, they could even read the astronauts' bathroom scale when they weighed themselves every morning (they were parked so the scale was at right angles to the asteroid's accleration, they were reading only the gravitational attraction between the astronauts' bodies and the asteroid).

So, what do the earth-based techs conclude about the mass of the asteroid? That it changes with speed, per GR, or that it is constant, per the locally mounted guages and instruments they can read?

I do know that particles accelerated to near light speed deflect in magnetic fields as if they were more massive per GR, but what is "right" when you can see both the effects from your reference frame and those on the local reference at the same time?

Maybe this is no big deal, and reflects what I don't know about GR, but I've been under the impression that the velocity-based mass shift was supposed to be absolute to observers on the remote inertial frame.

 

[Ziggurat]

From earth, the influence of the rocket engines on the asteroids seems to follow GR's rules: during the times when the asteroid is at relativisitic speeds compared to earth its acceleration is reduced, because of its increased mass. Around the time its speed is near zero its acceleration from the known-thrust rockets is what Newton predicted.

First off, this is a special relativity question. Contrary to many misconceptions, SR can handle acceleration just fine. The thing it cannot handle correctly is gravity, but there's no gravity in your problem. So to clarify: we're dealing with special relativity here, not general relativity.

Second, it's really not correct to talk about mass changing with speed. That's focussing on the wrong thing. In Newtonian machanics, momentum is given by mv, and kinetic energy by mv^2. These don't hold up in special relativity, of course. But it's not because the mass changes, it's because the Newtonian relationship between mass and momentum/kinetic energy are only low-velocity approximations. You can get the "right" answer by saying the mass itself changes, but it leads to problems in understanding. The mass does not change. You cannot, for example, turn an object into a black hole by making it move really fast. Rather, momentum is not linear with velocity, and kinetic energy is not quadratic with velocity. You can handle everything correctly without ever talking about the mass changing.

As to the details, if you actually worked out everything in detail and calculated the changes in momentum (for example) using the full relativistic formula rather than the Newtonian formula, everything would work out. But you do have to be careful in comparing different reference frames, because things like the rate at which processes occur won't be the same in different frames - the astronauts might be firing their rockets for a minute at a time in their frame, and the earth-based scientists would observe (note also that what you "observe" is also different than what you see) them firing for maybe five minutes at a time in the earth frame. These things are going to need to be considered in any detailed comparison, but as long as you don't forget anything, it will all work out without contradictions.

 

[garys_2k]

DOH! Sorry for mixing up General with Special, you're right, of course, this is a SR question.

I guess it boils down to: If the earth-based observer watched the asteroid based accelerometers, and bathroom scale, he would see a constant reading on each. The gross readings would lead to a mass estimate true for an observer local to the asteroid but the earth-measured accelerations would show different conclusions. I guess if this happened it wouldn't be difficult to understand, using SR to correct and correlate the two frames of reference.

Thanks!