Does the rest mass of an object increase when it acquires gravitational potential energy, and if so, is this the reason why Einstein believed that the inertia of a mass increases in the presence of other masses?
I think your reply is interesting and raises more questions. I have searched the internet for an answer to this question, and surprisingly, I found very little information that gives a definitive answer. Pick any other subject in relativity and you will find tons of websites. But this particular question about gpe and rest mass seems to be unanswered or controversial. This surprises me because in view of all the recent confirming experiments of general relativity-- gravity probe b, gravity waves-- there are still questions in relativity that the answers are not clear.This is largely a matter of conventions and definitions rather than physics, as so often happens in relativity.
A free-falling object in a static gravitational field in a conventional coordinate system has constant total energy, which can be considered to be due to the rest energy being decreased or increased by the potential energy and the change being converted to kinetic energy. Note that the rest energy decreases in a lower potential, closer to other massive objects.
I think that Einstein's original idea about inertia in this context was almost opposite to this, in that he initially noted that the coordinate speed of light slows down in a potential well (by twice the fractional amount that the rest energy decreases), which means that the coordinate acceleration induced by a fixed force also decreases in a potential well, which he attributed to an increase in inertia. That was however a very early way of looking at things, and I don't think it really matches current thinking in that respect.
In more complex situations, there is no really satisfactory conventional definition of either potential energy or inertia involving multiple masses in General Relativity, although there are useful approximations in practical situations.
An interesting aspect of inertia is that if nearby masses are accelerated, GR predicts that a test particle will in theory experience a "frame-dragging" linear acceleration in the same direction (much too small to be measurable with current technology) which means that it would need a force applied to keep it at rest. If this idea is pushed to the limit, then if the whole universe were accelerated relative to the test particle, it appears that the "sum for inertia" of the effects of every mass would mean that the mass would experience an acceleration which would be of the right magnitude to keep it exactly in step with the motion of the universe. It seems neat that this apparently explains inertia as a purely relative effect (conforming to "Mach's Principle") but unfortunately the details don't work out correctly with General Relativity, as this explanation of inertia appears to require a gravitational "constant" that actually varies with location and time instead of being a universal constant as assumed in GR (and supported by experiment).
If you're not interested in what happens to the energy of the planet and you can assume the planet to be at rest, you can use that model within GR.Assume this hypothetical situation. Suppose there is a test mass on the surface of a planet far removed from any appreciable gravitational affects of other planets, stars, etc. Just the one planet and the one test mass. In this hypothetical, does special relativity or gr predict that when you lift the test mass off the surface of the planet, because its gpe has increased, relative to an obeserver on the surface of the planet, does the rest mass of the test mass increase equal to the increase in gravitational potenial energy of the test mass divided by the speed of light squared?
If you're not interested in what happens to the energy of the planet and you can assume the planet to be at rest, you can use that model within GR.
However, if you consider what happens to the energy of the planet when it is moved a little away from the test mass by the same action, you will get the same result for the change in potential energy of the planet, which doesn't add up, because the total energy imparted to the test mass plus planet system as a whole is only equal to the potential energy, so you then have to assume the loss of some other energy in the gravitational field to make up for it. But GR says there is no energy (in the sense of gravitational source energy) in the field. So at least one of these "energy" quantities is some sort of "effective" energy not real energy. And I'm puzzled about it too.[/But GR says there is not energy(in the sense of gravitational source energy) in the field]
Question about this. I thought gravitional waves do have energy in them? And does this imply that gravitational fields have energy? The reports from LIGO have indicated that the energy contained in the waves would be equivalent to about 2 or 3 solar masses.
Firstly, the relative increase due to gravitational potential energy is incredibly tiny, typically a few parts per trillion or billion, so there is no runaway regardless.Going back to my original question, I think a paradox arises. My understanding is gravitational mass is only related to rest or proper mass. But in the hypothetical with one test mass and one planet if indeed the rest mass of both the test mass and the planet increased via the increase of their gpe, then their gravitional field would increase, which would mean their gpe would even be greater, and therefore, their rest mass would even be greater, and on and on ad infintum. Is this the case?
Well, for a start the cannon would have turned into a thin film on the surface of the star...May I suggest a simple thought experiment:
A cannon firmly attached to the surface of a neutron star fires, cannonball flies one way, neutron star flies other way. The ratio of the two masses is the ratio of the two speeds.
This way we can find the mass of a cannonball on the surface of a neutron star.
Firstly, the relative increase due to gravitational potential energy is incredibly tiny, typically a few parts per trillion or billion, so there is no runaway regardless.
But as I already said, the trivial model of assuming that the rest energy of a particle is modified by gravitational potential to give it the gravitational energy simply does not work unless you also assume that there is a correcting change in energy elsewhere, for example in the field.
It is of course expected that the external gravitational field of any system should reflect the total source energy, taking any internal potential energy into account.
In General Relativity, there is an apparently consistent solution to this for any static system. The total gravitational effect of a system at rest includes an effect due to pressure. The total volume integral of the pressure over each of three perpendicular planes over the volume of a static system is equal (and opposite) to its internal potential energy in Newtonian gravity. The same effect in GR (as part of what is called the "Komar mass") appears to compensate exactly for the fact that potential energy would otherwise be applied twice, with each constituent component particle of an object both being reduced in energy by its potential energy and causing the rest of the object to be reduced by the same energy.
However, the pressure term does not provide an explanation in a dynamic system. The physicist Richard Tolman pointed out that if the pressure suddenly changed inside a star, for example as a result of some sort of collapse, then this would apparently change its gravitational field, even if no energy had been added or removed externally. This is now known as one of Tolman's paradoxes. Similarly, it does not provide an explanation for the potential energy of a system involving two bodies orbiting around one another.
As far as I know, this is still a generally unresolved area. As GR is difficult to handle analytically, especially in dynamic situations, the tiny additional effects due to potential energy modifying the source strength are not very relevant to current experimental modelling, but they are of interest on the theoretical side. If current GR is inaccurate at that level, this will not make much difference except in extremely strong gravitational situations, but it could for example affect whether black holes actually form. However, that sort of speculation is outside the scope of these forums.
Sorry, no, because the amount of energy depends only on the static configuration but the energy in a gravitational wave is proportional to some power of the rate of change.Is it possible that the correcting change in energy would be due to a tiny gravitaional wave generated when the 2 masses are momentarily accelerated and deaccelerated when they are separated?
Well, for a start the cannon would have turned into a thin film on the surface of the star...
But ignoring that, there are problems with what time and space coordinates to use, and what you mean by the "mass" and "speed" of an extended object (especially a neutron star).
What are you trying to show? If you lower something carefully, you are extracting energy from the system involving both the thing you are lowering and whatever you are lowering it onto. We would normally assume that the extracted energy is associated with the thing being lowered, and that what it is being lowered onto is at rest, and this works in most simple cases.
However, it is not a general solution. If you try to look at it from the point of view of the planet or whatever onto which things are being lowered, the energy of the planet is affected by the same amount. You can try to make up rules to resolve the ambiguity (for example specifically associating the energy with force moving through a distance, so in the simple case the planet didn't move enough to make any difference to the energy) but they don't work in all cases (for example if you move object A towards object B then move object B away by the same distance, where according to that rule the energy would apparently be removed from A but added to B).