# Is there zero energy cost to generate a gravity field?

It seems that all objects with defineable mass emit a gravity field, which interacts with other objects within said field.

My understanding on this may be very limited, but how is the field generated by the object without losing energy nor mass? Is there a zero energy cost to create a gravitational field?

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Energy is (by definition) accumulated work, which is, in turn, Force times displacement. $$E = W = F \Delta x$$ Energy conservation states that the energy of an isolated system cannot change, and if the system is not isolated, the energy change corresponds to the work performed by external forces on the system. Its like money - energy is what you have and work is what you pay (or are given by).

Now, let us imagine an isolated body with mass m. There is a static gravitational field (or space distortion, whatever you call it) around it, but it is not being 'paid off' by anything! For now it is only the way of saying that 'if you place another body nearby, it will undergo gravitational force and begin to accelerate towards the first body'. So, such kind of gravitational field is created, sort of, 'for free'. However, it cannot 'do' a thing for now - there are no other bodies and we cannot 'add' them because there is nothing like 'adding a body' in real physics.

We can, however, proceed as follows - let us cut this body in half - to create two bodies of mass m/2. We can now use force to separate those bodies. We have to pump energy into the system (do some work) to create such a separated situation. And now the gravitational field gains energy - precisely the one that we put inside by hand!

So, in static (or adiabatic - slowly changing) situations energy of gravitational field as a standalone entity does not exist - it is only a way of expressing the potential energy of bodies - the fact that someone/something (God? ;) ) has separated objects from each other long, long ago.

The only complication arises from finiteness of lightspeed. We can ask - what if we pull one of those m/2 bodies away extremally fast? Would the second one immediately 'feel' that we kicked the first one out? If we believe Einstein that no information can travel faster than light the answer is - no. So the 'change of force experienced by bodies' has to propagate as a 'sphere of knowledge' - which has to carry energy. These are still unproven gravitational waves, analogon of electromagnetic waves, which carry energy by themselves. This energy is a part of energy we used to wipe this first body out.

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I will try to answer within the level of your question. The true theory of gravity is not the Newtonian one, as you assume, but General Relativity. Newtonian gravity is a good approximation to the world we usually observe. In addition we will not approach the region of quantum mechanics and atoms.

In the Newtonian frame masses are constants which characterize matter always, that is what a constant means. These constants are additive, N particles of mass M in a cluster make a cluster mass N*M. Now energy is produced or needed to make clusters or destroy clusters, but this does not affect the individual particle masses. Therefore, at this level, the mass of the Earth, for example is the sum of the tiny delta(m) particles that constitute it and their combined mass create the force given by Newtons law. Do read the paragraph on the gravitational field in the first link.

In General Relativity space distortions greate what we perceive as a gravity field, and it needs a lot of physics studies to understand it. When one goes into quantum mechanics and fields then even more studies are necessary.

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