ENERGY LOCALIZATION IN GENERAL RELATIVITY

ABSRACT 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. 1 FAILURE OF PSEUD-TENSOR APPROCH 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: TµV + tµV ;V = 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. 2 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: 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 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 =2GMr Our analysis will be limited to vertical motion only, and generalizing this analysis to arbitrary motion will require nothing more than mathematical tools and the rules of vector addition. We can obtain the required transformations by following a simple and clear method: If a body moving freely in a gravitational field keeps the same energy and momentum, then we can calculate the corresponding values of energy and momentum in the absence of gravity by looking at their values when the body continues to move far away from the source. If the body moves with a speed greater than the escape velocity U=2GMr, this means as it gets farther from the source its speed approaches vg= V-U1-VUc2. And if the body moves with the escape velocity itself, this means its speed approaches zero as it gets farther from the source. This means that the escape velocity in a gravitational field corresponds to rest in space where gravity is absent. This speed vg is the speed in the absence of gravity that corresponds to the speed V > U in the gravitational field. This means that the energy in the gravitational field, in this case, is calculated by the relation: Eg= mc21-vg2c2 for (V > U) And the momentum is calculated by the relation: . pg= mvg1-vg2c2 for (V > U) As for motion with speeds less than the escape velocity, to know what corresponds to it in the absence of gravity we need a simple treatment by which we can calculate the energy and momentum in each case. If the body moves with a speed V less than the escape velocity U=2GMr , this means it needs an amount of energy [ mc21-U2c2-mc21-V2c2 ] to reach the escape velocity. This means that this body’s energy is less than the rest energy mc2 by this amount, and therefore the energy of the body in a gravitational field moving with a speed V less than the escape velocity is given by: Eg= mc2 - [ mc21-U2c2-mc21-V2c2 ] for (V < U) As for the momentum, we can also calculate it simply through a hypothetical experiment: suppose a body is moving in a gravitational field with a speed V less than the escape velocity U. We can show that its equivalent momentum in the absence of gravity equals zero by imagining that the body splits into two equal parts and we add to both parts the same amount of momentum but in opposite directions. This will not affect the total momentum of the two parts together, regardless of the amount added to each one. Let us suppose that the momentum added to one of the parts in the same direction as its motion makes it move with the escape velocity U, which means its equivalent momentum in the absence of gravity equals zero. The other part will still have a speed less than the escape velocity because we added to it a velocity in the opposite direction to its original one. Now, we repeat the same process with this part and split it into two parts again. This repeated process, which does not affect the total momentum, makes all its parts move at the scape velocity and leave the gravitational field with zero velocity. Therefore, the total momentum equals zero for any body moving in a gravitational field with a speed less than the escape velocity. From these equations one can see that motion at velocityU 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 in some ways 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 U away from the center in the presence of a gravitational field is equivalent to moving with a speed vg= V-U1-VUc2. way from the center in the absence of a gravitational field. Similarly, moving with a speed v greater than U toward the center in the presence of gravitational field is equivalent to moving with a speed vg= V-U1-VUc2. toward the center in the absence of gravitational field. This will not change whether the velocity U is taken toward the center or away from the center. 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. A body projected with a speed V less than U moves upwards and then reverses its motion. However, its total momentum is not affected by this reversal, because motion at a speed less than U 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, The total kinetic energy is affected by this velocity and equals Eg= mc2 - [ mc21-U2c2-mc21-V2c2 ] , while the total momentum equals zero. 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 less than U radially (vertical) in a gravitational field, it has energy given by: Eg=Constant= mc2 - [ mc21-U2c2-mc21-V2c2 ] = Constant Differentiate the equation with respect to r: (1-U2c2 )-3/2U dUdr = (1-V2c2 )-3/2 Vdvdr In the case of small U and V compared to the speed of light, this leads to the known result; Acceleration (g) = GMr2. If a body is projected upward, its velocity v will continuously decrease, but this will not change the energy, because U and V change in a such a way that the energy is saved. The process continues until the velocity v becomes zero then the body begins to fall but the total momentum of the body remains constant and equals 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. 3 CONCLUSION 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. The gravitational field does not have energy and momentum separate from matter; rather, it enters into the formula that defines energy and momentum of matter. This formula can be obtained by studying the case in which a body moves away from the gravitational field after its motion within it, where its energy and momentum are conserved, which 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.