Gravitation and Energy Localization

The Failure of Pseudotensors and the Alternative Solution of the Energy Problem in General Relativity Mueiz Gafer KamalEldeen Mueizphysics@gmail.com October 2025 Abstract In this article, I show that the introduction of pseudotensors as a solution to the energy problem in general relativity stems from neglecting the proper method for addressing the violation of the law of energy conservation — a method that had already been successfully applied in special relativity. Failure of Pseudo-Tensor Approach. When special relativity first appeared, it was initially a correction and completion of the concepts that preceded it, concerning space, time, reference systems, and the transformations of lengths and time intervals between them. Then these new concepts were introduced into problems of dynamics, and they influenced them. Here, we want specifically to draw attention to how this influence occurred in the problem of energy. The energy E=12mv2 which had been conserved in pre-relativistic physics, became non-conserved after the changes introduced by special relativity—changes that become apparent at high speeds. The optimal solution to this problem was to search for another conserved quantity that could play, in the new context, the same role the previous quantity played in the old one. Naturally, it inherited its name as well. The new quantity is: E =m c2 1-v2c2 This does not mean that all the old concepts were completely abandoned; they are now simply regarded as special cases. We can derive from this a very valuable conclusion—one that is valid for any situation in which we develop our concepts of space-time—namely that energy is fundamentally defined through the law of conservation. Every quantity we once chose and called “energy” was so chosen and named because it was conserved; when changes in our understanding of space-time occur that disrupt this conservation law, we must abandon that old definition and seek a new conserved quantity that becomes the true heir to the concept that failed to keep pace with the new transformations. Unfortunately, the method that succeeded in special relativity was not applied when we faced the same problem in general relativity. When we found that the quantity defined in special relativity as energy—which, together with momentum, forms the energy-momentum tensor—was no longer conserved, we did not think of modifying the definition. Instead, we insisted on retaining the same definition that failed to fit with the concepts of general relativity. Then, rather than adjust the definition, we began to distort general relativity itself to forcibly make it compatible with these outdated definitions—by partially sacrificing one of its most fundamental principles, the principle of general covariance—through the introduction of what are called pseudo-tensors to represent one part of the energy after dividing it between matter and the gravitational field. We ended up with an energy conservation law that can best be described as a law for the sake of having a law—constructed merely so we could say that general relativity “has” an energy conservation law! The very first step in deriving this law is based on an unjustified assumption—that energy must be divided into two parts: a material part and a gravitational part, and that the material part is defined exactly as it was before we were aware of the new space-time concepts and reference systems introduced by general relativity, and the changes they may bring to dynamics. All these solutions begin a mathematical step like this: Tij + tij;i = 0 Then they lead to multiple results that only confirm their weakness. This distortion caused some physicists [1] to consider—even to begin—abandoning general relativity and searching for alternatives. To properly understand the type and extent of the error found in the pseudo-tensor method, we can return to the problem of energy in special relativity and ignore Einstein’s correct solution, attempting instead to solve it using a method analogous to the pseudo-tensor approach. Let us study a simple experiment involving collisions at low speeds. We find that the law of conservation of energy, as defined by Newtonian mechanics, holds perfectly well. Then, if we observe the same experiment from another reference frame moving at high speed relative to the first, we find that the law of energy conservation breaks down. We try to keep the Newtonian definition of energy, which requires us to search for another “container” of energy, beyond matter, to account for the imbalance in the material side of the conservation law. In this experiment, there is nothing that can represent this container except the reference frame itself, so we say that the reference frame has energy. We then reach results full of problems and complications, the most important of these are the issues of covariance and localization [2], but in any case, they help to calm the scientific community’s insistence on maintaining the principle of energy conservation with its strong influence—while preserving special relativity with its the superb logical, mathematical, and experimental consistency. Before presenting an application of the correct method that solves the problem of energy conservation in general relativity, I must respond to an objection that might be raised by those who adopt the division of energy into material and gravitational parts. They may ask: “Isn’t the existence of energy in the gravitational field necessary to explain the observed transfer of energy through gravitational fields between distant bodies, which is also confirmed by modern experiments that detected gravitational waves?” We can reply simply: if it has been established that the gravitational field affects the mathematical form that defines energy, then the question of whether the gravitational field possesses energy without matter becomes meaningless. The answer, whether yes or no, makes no difference—because the only way to test that is through the interaction of the field with matter, and in both cases, the gravitational field will cause changes in the energy of the matter within it. Therefore, the effect of the gravitational field on the energy of material bodies does not necessarily mean that the gravitational field itself possesses energy, independently of matter. The Proposed Solution After responding to this objection, we are now prepared to make the necessary modification to the definition of energy that will satisfy the conservation law. Fortunately, the modification we propose will not affect Einstein’s equation or any formula derived from it, because the steps used in deriving Einstein’s equation did not involve the general energy conservation law that applies in all reference systems within a gravitational field. Only the form of the law valid in flat space-time was used, and that remains correct regardless of later changes concerning the general case. Thus, the energy-momentum tensor appearing in Einstein’s equation will remain unchanged. It resembles, in this respect, the concept of rest mass in special relativity—a concept still used in many equations even though we know perfectly well that it is only a special case of the broader concept of energy. Determining the mathematical form of the concept of energy that fulfills the conservation law requires a precise description of the behavior of particles in a gravitational field, which in turn requires exact solutions to the equations of general relativity. This is provided by the Schwarzschild solution: ⅆs2=( 1-2GMc2r )C2ⅆt2-ⅆr21-2GMc2r-r2ⅆ2-r2θⅆ2 We can summarize the result of this solution and its effect on a particle in a gravitational field as follows: We will limit our analysis to radial (vertical) motions for simplicity.as localization problems appear in theses motions, general motions can be addressed by analyzing the motion into radial and peripheral (horizontal) directions. Suppose we place a body stationary relative to a point O at another point P, at a distance r from O, in the absence of a gravitational field. Let the measurements of distance and time changes along this line by this observer be ⅆr, ⅆt. Now, if we place a mass M at point O, the Schwarzschild solution tells us that this will lead to changes in the measured distance and time for the same stationary observer at P, as follows: ⅆr'=1-2GMc2rⅆr , and ⅆt'=ⅆt1-2GMc2r . Compare this with : ⅆr'=1-v2C2ⅆr and ⅆt'=ⅆt1-v2c2 of Lorentz transformation. Because velocity is the small change of distance divided by the small change of time, any velocity whose direction lies along the radial direction whether it is directed toward the center or outward in the opposite direction, will decrease by an amount determined by the factor vg0=2GMr as follow: vg0 ≡ 2GMr vg=v -vg0 According to the rules of relativistic subtraction of velocities vg= vg0-v1-vg0vc2 = constant We take the difference between the magnitudes of the two velocities, and the direction of the equivalent velocity is the same as that of the greater velocity. This is the effective or equivalent velocity which replaces the (ordinary) velocity in all applications. In all cases of free fall, this velocity remains constant. In short, we apply all laws and definitions of physics that apply in the absence of gravitational field, but after adjusting the value of the velocity to this equivalent value. From this equation one can see that motion at velocity vg0 in gravitational field radially toward or outward the center of the source is equivalent to rest in the absence of gravitational field while rest in gravitational field is equivalent to moving radially at this velocity. Now, it becomes clear that the effects of a gravitational field on physical quantities are similar to those of velocity transformation. However, it differs from velocity in its effect on velocities themselves, because the gravitational field has the same effect on velocities in both opposite directions, it decreases them both, whereas velocity transformation affects opposite velocities differently; it increases the velocity in one direction and decreases the velocity in the opposite direction according to the rules of relativistic addition and subtraction of velocities [3]. It is known from special relativity that we can apply the laws of physics in given space and reference frame, and then apply the same laws in the same space but in another reference frame moving uniformly with respect to the first one. However, in this case all the physical quantities must be transformed into their corresponding values in the new frame. The same concept applies to the gravitational field; because the presence of gravitational field itself constitutes a kind of transformation, it is a transformation between two references which are static with respect to each other but with gravitational field in one of them and without gravitational field in the other, Therefore, the physical laws and definitions in the absence of gravitational field retain their same form in the presence of a gravitational field, provided that we replace the values of physical quantities with their corresponding values within the gravitational field. It would be a very appropriate name for it if we called the gravitational field “the Static Transformation”. We notice that moving with a speed v greater than vg0 away from the center in the presence of a gravitational field is equivalent to moving with a speed vg=v -vg0 way from the center in the absence of a gravitational field. Similarly, moving with a speed vg greater than vg0 toward the center in the presence of gravitational field is equivalent to moving with a speed to vg=v -vg0 toward the center in the absence of gravitational field. This will not change whether the velocity vg0 is taken toward the center or away from the center, because the greater velocity is v, and it is the one that determines the direction of the equavilent velocity. From this, we observe that any motion in the absence of a gravitational field has one corresponding equivalent motion in the presence of a gravitational field, but the opposite is not true. In a gravitational field, there exist motions that have no specific conceivable equivalent in the absence of gravity. If a body moves in a gravitational field with a speed less than vg0 whether toward the center or away from it, this motion is equivalent to motion at velocity vg in two opposite directions: away from the center and toward the center, which is equivalent to half of the body moving in one direction while the other half moves in the opposite direction. This is a mathematical result arising from the existence of two zero states of motion: motion at velocity vg0 vaway from the center and motion at speed to vg0 toward the center. Calculating the equivalent velocity on each of these two zero states gives opposite results, because the greater velocity which determines the direction of the equivalent velocity in this case vg0 consists of two velocities in opposite direction.. Therefore, a body projected with a speed less than vg0 (the escape speed) moves upwards and then reverses its motion. However, its total momentum is not affected by this reversal, because motion at a speed less than vg0 in a gravitational field is equivalent, in the absence of gravity, to the body being at rest with increased energy. This is analogous to the body’s components splitting into two halves moving in opposite directions at speed vg —meaning that the total kinetic energy is affected by this velocity and equals mc21-vg2c2 , while the total momentum equals zero. Therefore, a body moving with velocity v in a gravitational field will have energy equal to ; Eg= mc21-vg2c2 This is the definition of the energy of a body moving in a gravitational field which satisfies energy conservation law. We can also calculate its momentum: pg= mvg1-vg2c2 When the body moves at velocity v less than vg0 in magnitude, pgequals zero because vgwill have two opposite directions. The reader should note that it is quite natural for a gravitational field to be richer than empty space devoid of any field, and that within it exist phenomena that have no non-gravitational equivalent that can be easily conceived or imagined. It is sufficient for us to obtain an equivalent that can be computed. Our earlier discussion about the division of a body into two halves moving in opposite directions was meant only to illustrate that it is possible to have motion that affects energy without affecting momentum. It was not intended to suggest that this is exactly what happens to a body in a gravitational field. Let’s apply this to a body moving at velocity v radially (vertical) in a gravitational field, it has an equivalent effective velocity vg given by: vg =v -vg0 = v-2GMr1-v 2GMrc2 =constant If a body is projected upward, its velocity v will continuously decrease, but this will not change the effective velocityvg, because the quantity vg0 also decrease in a such a way that the effective velocity is saved. The process continues until the velocity v becomes zero. When the body begins to fall and the effective velocity remains the same since the both quantities v and vg0 increase. Thus, the total energy of the body remains constant. This may seem to contradict simple observations, since we see that the impact becomes more violent — with more energy converted into heat, sound, etc. — as the vertical distance between the point of release and the point of impact increases. However, this apparent contradiction disappears if we note that the body which the falling object collides with also influences the intensity of alignment. The closer it is to the center, the greater the velocity that corresponds to the gravitational field — which, in this case, is directed upward — and this velocity increases as one approaches the center. As for momentum in this case, it always equal to zero. After modifying the definition of kinetic energy and momentum for a particle in a gravitational field, we can also modify the energy-momentum tensor, since it expresses the energy and momentum density of a collection of bodies in a region of space. Conclusions and Outlook From the Schwarzschild solution, we learn that the effect of the gravitational field in physical quantities can be compared to the effect of the transformation between frames that are in uniform motion relative to each other. This allows us to introduce a definition of effective velocity of a body when situated in a gravitational field, which in turn leads to a redefinition of energy and momentum as well in such a way that the conservation laws are satisfied. References [1] See for example: The Relativistic Theory of Gravitation. A. Logunov and M. Mestvirishvili. The works of Logunov are considered among the best sources for explaining general relativity and correcting misconceptions about it, even though his main purpose was to criticize, reject, and propose alternatives to it. [2] An excellent presentation of the energy problem in general relativity and its solutions, can be found in: The Energy-Momentum Problems in General Relativity, Sibusiso S. Xulu. [3] The equivalence principle is usually formulated in terms of acceleration rather than velocity. However, I argue that formulating it in terms of velocities is more complete, as it aligns with the Schwarzschild solution. In this framework, the effect of the gravitational field becomes equivalent to that of velocities. It can also be observed that the formulation based on acceleration can still be derived, since velocity at each point changes along the radial direction. However, we must understand that neither velocity alone nor acceleration alone can represent all the properties of the gravitational field.