Energy Reference Frames: Consensus on Object Energy Across Different Observers?

[pmb_phy]

I think it is you that is confused!

You can think it but it won't make it true. What I've told you is basic GR. Any GRist will tell you the exact same thing that I've told you.

Free falling means just that in the EEP.

That is incorrect. You're confused on this point and what Einstein's equivalence principle is. It is now clear to me that this is the exact source of your confusion and why your argument is flawed. You stated that

Over a period of time you notice that its kinetic energy is rapidly increasing as it accelerates towards you, ...

You have specifically chosen a region of spacetime which is too large and now you cannot ignore tidal forces. You can ignore tidal forces if an only if you can ignore acceleration. Your entire argument is based on the acceleration of a particle in free-fall as observered in a free-fall frame. However since you said the kinetic energy is increasing then that statement implies the spacetime is curved and it implies you're not ignoring tidal forces.

In a curved spacetime Einstein's equivlence principle only says that the gravitational field can be transformed away at only one point in spacetime. That's the EEP as stated by Einstein himself.

I.e. when you transform to a free-fall frame as you have in your example, then the gravitational field has vanished only at one point in your frame of reference - the origin of your coordinate system. At points off the origin there is still a gravitational force. Since you have indicated that the particle you're observering is accelerating then you have not chosen to ignore tidal forces.

I assume that you have Weinberg's text. In chapter 3 please read section section 1 - Statement of the Principle then read section 2 - Gravitational Forces. This time don't skip by when Weinberg states what I've explained to you on page 68 in the third paragraph.

Although inertial forces don not exactly cancel gravitational forces for freely falling systems in an inhomogeneous or time-dependant gravitational field we can still expect an approximate cancelation if we restrict out attention to such a small region of space and time that the field varies very little over the region.

You've chosen not to make your region of space and time small enough to ignroe the gravitational forces and hence you can't claim that the gravitational forces is zero since it hasn't vanished.

Global energy is conserved, ..

Nobody knows if energy is conserved globaly. See
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html

 

[Garth]

Pete - I am not disputing the existence of tidal forces, and indeed their existence can question the validity of the EEP as you seem to indicate, and which is my intention in the first place, however these forces are second-order compared to the large Newtonian attraction that the curvature of space-time has replaced.

According to the theory this large gravitational force, as understood by Newton, has been replaced, and the work it did may no longer be used either to create or use potential energy. You will not find potential energy mentioned in Weinberg or MTW except in Weinberg's popular explanation of gravitational red shift, and then he says that it is "otherwise without foundation". It is the absence of gravitational potential energy that is at the core of the local non-conservation of energy.

As I said above the observer, freely falling towards the Earth, experiences a metric with time dependent components, and therefore, by your lecture notes, in that observer's frame of reference energy is not conserved.

Earlier you said you also have to take time dilation into account; this actually makes the situation worse.

As far as the global conservation of energy is concerned I was willing to allow that even though energy is a very slippery concept - for example how do you measure it at infinity? I am happy to accept Weinberg et al's calculations of the total energy of a static spherically symmetric Schwarzschild solution which turns out to be M - a constant. (G&C Eq. 8.2.16) [By the way I have now found Weinberg's explicit law of conservation of mass derived from the EEP - in the PNA - Eq. 9.3.2]

 

[pmb_phy]

Pete - I am not disputing the existence of tidal forces, and indeed their existence can question the validity of the EEP as you seem to indicate, and which is my intention in the first place, however these forces are second-order compared to the large Newtonian attraction that the curvature of space-time has replaced.

If you measure an increase in kinetic energy then you've measured a gravitational force.

Yes. I know that tidal forces is a second-order effect. One of the reasons for that is the magnitude. All that means is that the gravitational force in your free-fall frame is small. As such the kinetic energy change is small. That does not mean that the gravitational force is zero and it does not mean that the small gravitational force is not the source of the small changes in kinetic energy.

Mind you, there are various statements of the equivalence principle. It is rigorously stated as follows (as Einstein often spoke of it)

A uniform gravitational field is equivalent to a uniformly accelerating frame of reference.

Its the uniform gravitational field that you were speaking of, i.e. in a uniform gravitational field the entire field can be transformed away. That is not true for a non-uniform gravitational field. You have been speaking about a non-uniform gravitational field and the effects of the lingering gravitational forces and the changes in kinetic energy that it produces and therefore there is no problem with energy or the work that the field does. Although in the problem you spoke of energy isn't conserved in Newtonian physics anyway.

Pete

 

[Haelfix]

Garth, global energy is not in general conserved in GR.. Only in a few situations can it be made rigorous.

When you take the Schwarzschild metric and measure its 'global' energy, you are explicitly doing something known as the far field approximation, you are sort of measuring the energy relative to what all faraway stars would feel. Fortunately, this is an example where that procedure works, b/c the asymptotic limit is flat and the metric admits nice isometries.

If you want an easy example where global energy is meaningless, I invite you to write down the energy for a flat Friedmann Robertson Walker universe.

 

[Garth]

Garth, global energy is not in general conserved in GR.. Only in a few situations can it be made rigorous.

Thank you, that's what I meant when I said it was a slippery concept.

All I am doing is upholding Noether's insight that GR is not an example of Hilbert's "proper energy theorems"; it is an improper energy theorem. I was conceeding that globally energy may be thought of as conserved - in the Schwarzschild case - but that in general, in particular in the local case, it is not.

My work seeks to address this problem, basically by modifying GR and the EEP.

 

[pervect]

You can always choose a convention in which the allocation of changes in mass and energy are hidden. The question is, "Is this is a useful convention?". In the fission of a uranium atom the reallocation of potential energies releases a hell of a lot of energy and some would say the total mass of the fragments (consisting of more tightly bound components) is less than that of the original atom; but if your convention includes this energy in with those fragments as a total system then it tells you that no energy has been released. However, whether this is a useful way to describe the exposion of an atomic bomb or not is debatable to say the least.

You might try the sci.physics faq on "Does GR conserve energy"

link

One can, in GR, define the mass of an isolated system if certain conditions are met (the space-time is asymptotically flat).

But systems that are interacting gravitationally are tricky. You've noted that two bodies that are bound by nuclear forces have a different mass as a system because of the binding. The same applies for objects that are bound by gravitation, the system mass includes the binding energy. Unfortunately, one can't take a gravitationally bound system, and express its total energy as the sum of energies of mass1, mass2, and the energy of some sort of field. At least not in a way that's observer independent. There's some schemes for doing this in an observer dependent way (called pseudo-tensors, if I recall correctly) - but because different observers don't assign energy in the same way, these approaches are somewhat "ad-hoc".

So when one does something like separate two charged particles, which requires work, one can point to the electric field and say "the energy went into the electric field". When one separates two masses, one still changes the energy of the system, but one can't point to any specific location and say "the energy went there".