The essence of this discussion was my claim that observer's points of view can not be simply swapped as soon as acceleration is involved. This simple swapping is only allowed when objects are moving frictionless at constant speed. As I pointed out earlier, when acceleration is involved, then the force which is responsible for the acceleration needs to be included in both reference frames. When swapping reference frames or point of observation, that force needs to applied to (and drawn on) the object which really is accelerating.

No, that force is represented by the introduction of a fictitious field, with corresponding potential energy, if the chosen frame of reference is non-inertial. This is true of both RET and FET. Under general relativity, gravity is a fictitious force which compensates for the fact that our measuring instruments experience an upward

*proper* acceleration but no

*coordinate* acceleration.

The RE explanation for this proper acceleration is that following a geodesic in spacetime would lead an object to fall towards the Earth's centre of mass. The Earth is in balance between the natural tendency of matter to follow that geodesic, and the electrostatic repulsion between atoms. The

*real* force at play is electrostatic repulsion pushing the Earth's surface up, but for most practical problems, it makes more sense to ignore general relativity, treat an observer on Earth as stationary, and consider falling objects subject to the

*fictitious* force of gravity within a gravitational field. The existence of this field, and the presence of gravitational potential energy, is entirely due to our choice of a non-inertial frame of reference.

UA works on exactly the same principle. There is no fundamental difference between gravity under general relativity and UA. If you choose a frame of reference that accelerates with the Earth, you create a fictitious gravitational field, and you have to account for gravitational potential energy.

A claims that he is accelerating towards B

If A is accelerating then a force of 2F is needed to accelerate mass 2M with an acceleration rate of a. In order for A to travel distance s towards B, an amount of energy of 2F.s is required.

Now if B claims the he is acclerating towards A

If B would be accelerating, then a force F would be required to create the same acceleration rate 'a' between both. But this requires an amount of energy equal to E = F.s, which is only half of the previous supposecly identical or reciproce situation.

This violates the law of thermodynamics.

No, it does not. At least one of A or B will be experiencing proper acceleration, which requires the use of a fictitious field if you want to consider the problem from that perspective.

So if A (mass 2M) is the body which is accerating then the only correct interpretion for both observers would be:

A claims he's accelrating towards B and B claims that A is accelerating towards B, not the other way round.

No. It is equally valid to choose a frame of reference in which A does not undergo coordinate acceleration. But you cannot change the fact that A is undergoing

*proper* acceleration, in which case B is subject to a fictitious force, and its potential energy within that fictitious field is transformed into kinetic energy.

The laws of physics (and more specifically thermodynamics), are applicable at any given point in time, which means that no matter which point in time (or space) we analyze as seen from no matter which reference frame , the observed result needs to be identical.

There are various cases where different observers see results that are not identical. Special relativity tells us that time, length and even momentum are observer-dependent.

It is not sufficient to only look at start- and end-point or to hope that differences will cancel out over time.

That depends on the context. Earlier, you seemed to be arguing that the Earth would fall out of sync with the sun and moon, in which case long-term and large-scale behaviour is very relevant. It's not clear why else you think anything would need to be "modulated".

(see also my first point) this is correct in practice because indeed the differences are not measurable, let alone sensible. But in theory - and for the sake of reasoning - the motion of these small masses are not negligible.

Indeed, we can reason about them if there is a purpose to it. Is your reasoning going anywhere?

..and that its change in velocity over that time is assumed to be constant in this model, so the kinetic energy accumulated is constant by .

The energy fed to the accelerating Earth is not accumuluted at a constant rate. The rate increases over time. in other words: the power which is required to accelerate the Earth increases over time.

Yes and no. I accept that I may not have explained myself clearly, but we seem to be in agreement that this isn't especially relevant. Let me know if you would like me to try a clearer explanation of what I meant.

But essentially this changes nothing to the principle explained.

What principle is that? Are you telling me you can't think of a single natural system that has fluctuating energy output?

No, I claimed that this modulation would be the rather unrealistic consequence.

Do you have any evidence for this claim?