Does Velocity or Acceleration Determine the Energy of a Moving Object?

[BruceW]

Experimentally, an inertial reference frame is one where all accelerometers read the same as the second time derivative of their position. Neither condition requires reference to any other reference frame.

Going back to this, an accelerometer works because its internal mechanism is not subject to any forces from outside (apart from the force transmitted by contact by whoever might be swinging it around). So by assuming we can make reliable accelerometers, we are automatically assuming that we can tell that the internal mechanism has no external forces acting on it. This seems to me to be the same as assuming that we can tell if a particle has no forces acting on it. Well, the internal mechanism could be a beam of light. But in this case, we must be able to say whether the internal mechanism is being acted on by an external field or not.

This is a bit similar to the question "can we make reliable clocks?". And we do assume this in relativity, so maybe I am just being pedantic when it is really not necessary to be. I am trying to think of a counter-example, but I can't think of any. And judging from other posts, it seems that it is common to assume that we can make reliable accelerometers?

These are two different questions that are independent of each other.

Ah yeah, sorry about that. I didn't really explain very well. I meant that if we answer the question "are we in a Minkowski spacetime, and using metric diag(-1,1,1,1)?" then this answers the question "how do we know a particle has no forces acting on it", because we can just look and see if the particle moves in a straight line or not.

There's a distinction in GR between a global inertial reference frame and a local inertial reference frame. Global inertial frames exist in Minkowski space-time and in these frames you can set up coordinates in which the metric tensor takes the form diag(-1,1,1,1) everywhere...

Yeah, I was just thinking of the global inertial frame. I have no problem with the local inertial reference frame, because it is clear to me that you can define this at wherever you are, so It is not necessary to define local inertial frame as being 'relative' to another reference frame.

 

[PeterDonis]

if we answer the question "are we in a Minkowski spacetime, and using metric diag(-1,1,1,1)?" then this answers the question "how do we know a particle has no forces acting on it", because we can just look and see if the particle moves in a straight line or not.

True, but it's worth noting that this only works if the answer to the first question is yes; it doesn't if the answer is no. The method I gave, computing the path curvature of the particle (or, physically, attaching an accelerometer to the particle), works regardless of whether spacetime is Minkowski or not and regardless of what coordinates we use to express the metric.

 

[Dale]

This is a bit similar to the question "can we make reliable clocks?". And we do assume this in relativity, so maybe I am just being pedantic when it is really not necessary to be. I am trying to think of a counter-example, but I can't think of any. And judging from other posts, it seems that it is common to assume that we can make reliable accelerometers?

Yes. Recall that this is for the experimental definition of an inertial frame. The problem you mention here is not specific to accelerometers but rather it is a general problem with ANY experiment using any type of measurement device to measure any quantity. You always need to evaluate the reliability of your measurement devices, so this is just "standard fare" for experimental work.

 

[BruceW]

The definition I was using (when this thread began) was that two reference frames need to be moving at constant relative velocity to be inertial 'with respect to each other'. So my experimental definition required that we could make reliable devices that could measure this.

I was a bit apprehensive at the 'reliable accelerometers' definition, which people seem to be more familiar with. But really I guess it is not much more radical than my old definition was. Thanks for helping explain to me, everyone :) Also, am I right in thinking that in a Minkowski spacetime there is only one unique set of global inertial reference frames? (and therefore, it is obvious that we don't need to specify what they are inertial 'relative to', because there is only one set which they can be inertial 'relative to', so we might as well drop the words 'relative to' in this case).

Also, sorry for wandering away from the topic of the thread a bit.