There is an apparent problem with the law of conservation of energy in a gravitational field: the velocity of objects moving freely in such a field changes continuously, and therefore their kinetic energy changes as well.
Physicists sought to solve this issue by introducing the concept of potential energy — not only in Newtonian physics but also in general relativity, where there is an analogous quantity known as the gravitational field energy, represented by pseudo-tensors.
However, both the Newtonian potential energy and the relativistic pseudo-tensors suffer from two major difficulties. The first is the problem of localization which is that this energy has no specific location in space — there is no clear notion of how this energy is distributed in space or where it actually resides. You can compare this with electromagnetic potential energy, whose energy density can be defined at every point in space according to the strength of the field there. The second difficulty is that this energy is not covariant — its existence depends on the choice of reference frame. For instance, if you calculate the kinetic energy of an object falling toward the Earth while you are falling alongside it, its kinetic energy will appear constant, and hence there would be no need for potential energy to maintain the conservation law.
Together, these two problems lead us to conclude that the concept of potential energy — and likewise its more advanced counterpart in general relativity, the pseudo-tensors — are merely mathematical constructs representing the portion of energy required for the conservation law to hold true.
The solution proposed here is that instead of attributing energy to the gravitational field itself, we simply redefine kinetic energy so that it remains constant. This means that the gravitational field carries no energy independent of matter; rather, it influences the energy of matter.
The proposed formula is shown in the picture:
Notice that this formula reduces to the definition of energy in special relativity when the gravitational field vanishes, and the special relativistic formula in turn reduces to Newton’s form in the limit of low velocities compared to the speed of light.
This formula is neither a hypothesis nor an experiment; it can be derived rigorously from the principles of relativity and the general foundations of physics.
It can be justified through a thought experiment as follows: imagine launching an object upward with a speed V greater than the escape velocity Uₑ. This means the object will continue moving upward until it is far from the gravitational field. In that case, we find that the energy of the object approaches, as it moves farther from the gravitational source, certain value equal to the one given by the above definition. But if the object’s total energy is conserved, then the energy with which it escapes into the gravity-free distant space must be the same energy it possessed at the start and during the entire journey.
This new expression for kinetic energy avoids both previous problems. It is necessarily covariant, because it is the sum of three covariant quantities: the rest energy (which is invariant), the relativistic kinetic energy (covariant), and the energy required for escape (also covariant). Furthermore, there is no localization issue, since this entire energy belongs to the motion of the body itself, not to the gravitational field or any ambiguous source.
The momentum can be calculated in the same way as shown in the picture
An important point arises here: how do we compute the energy of a body when its speed is less than the escape velocity?
The answer is simple — we use the same formula. In this case, we find that the body’s energy is less than its mass. Yes — less than the rest mass!
This can be inferred from the same thought experiment: if we launch a body upward with a speed less than the escape velocity, it will not escape the gravitational field. But if we add enough energy to bring it up to the escape velocity, it will leave the gravitational field and retain only its rest energy. If we subtract from that the amount of energy we added to make it reach the escape speed, we find that the energy it originally had was its rest energy minus the energy we supplied — meaning its total energy was indeed less than its rest energy.
Mueiz Gafer KamalEldeen
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