The Relationship Between the Energy of Gravitational Field and the Representative Pseudotensor R

The Relationship between the Energy of Gravitational Field and the Representative Pseudotensor R. I. Khrapko

World Academy of Science, Engineering and Technology proton attraction. But in the case of gravitational attraction, the International Journal of Electronics and Communication Engineering gravitational field of the dust is not eliminated. Instead, the Vol:12, No:8, 2018 gravitational field strengthened and extends to the volume that http://waset.org/publications/10009409/pdf has become free from the cloud. So, if we trust that the mass- — Abstract As is known, the role of the energy-momentum energy of the system “cloud + its gravitational field” is pseudotensors of the gravitational field is to extend the conservation conserved in the process of compression, we have to ascribe a law to the gravitational interaction by taking into account the energy and momentum of the gravitational field. We calculated the negative mass-energy to the gravitational field. contribution of the Einstein pseudotensor to the total mass of a A. Guth demonstrates the negativity of the gravitational stationary material body and its gravitational field. It turned out that energy by the example of a thin sperical shell of mass in this contribution is positive, despite the fact that the mass-energy of a Apperdix A of [1]. stationary gravitational field is negative. We concluded that the The standard method to calculate the mass-energy of pseudotensor incorrectly describes the energy of the gravitational matter, M , implies the use of the energy-momentum field. Nevertheless this pseudotensor has been used in a large number ν of scientific works for 100 years. We explain this by the fact that the tensor Tµ . The standard method to calculate the mass-energy covariant component of the pseudotensor was regarded as the mass- energy. Besides, we prove the advantage of the covariant energy- of gravitational field, G , implies the use of the Einstein momentum conservation law for matter in the Minkowski space-time. pseudotensor [2]-[4] or Landau-Lifshitz pseudotensor [5] tν . µ Keywords —conservation law, covariant integration, gravitation The standard method to calculate the total mass-energy of a field, isolated system. system “matter + its gravitational field”, J = M + G , ν ν I. INTRODUCTION. ENERGY OF THE GRAVITATION FIELD IS implies the use of the sum Tµ + tµ . The contribution of the NEGATIVE pseudotensor to the mass of the system “cloud + its hen an electron is attracted to a proton from infinity by gravitational field” must be negative, G < 0 , in order to W the force of the electric field, the energy of the electric compensate for the growth of the mass of the matter under field is converted into the kinetic energy, and the compression if we trust that J = M + G = Const . corresponding part of the electric field is eliminated. As a result, in harmony with the mass-energy conservation law, we II. CALCULATION OF THE MASS -ENERGY OF MATTER have the excitation energy of the atom instead of the energy of the part of the electric field. This means that the total mass- An infinitesimal space element dV ν contains the 4- energy of the system, “electron + proton + their field”, is momentum element of the matter conserved and the mass-energy of the matter is increased due dP = T ν − gdV , ( µ, ν,... = 0,..., 3 ); (1) to the disappearance of the part of the field. Then 13.6 eV are µ µ ν

radiated, and we get the neutral atom without an external here dP µ is an infinitesimal covariant vector (this is the electric field. ν Now consider compression of a dilute dust cloud under the totality of its components), Tµ − g is the energy- action of own gravitational attraction forces. If we adhere to momentum tensor density of the weight +1, dV is the the concept of gravitational field, the cloud is surrounded by ν its gravitational field. In the compression process, the kinetic covariant vector density of the weight −1, and g is the energy appears. And this kinetic energy will be converted into determinant of the metric tensor of the coordinate system (see the thermal energy when the compression is stopped by the e.g. § 96 in [5]). For example, dV t = dxdydz , or pressure forces. As a result, the mass-energy of the matter is increased in the same way as it is increased at the electron- dV t = drd θdϕ Note that the time component dP t is not the mass of the element of the matter. The mass is the module R. I. Khrapko is with the Moscow Aviation Institute - Volokolamskoe of 4-momentum (1), dM = dP / g . shosse 4, 125993 Moscow, Russia (corresponding author; e-mail: t tt [email protected] , [email protected] ).

A calculation of the total 4-momentum of an object is not −1  r 2  possible, in general, because we cannot sum up vectors (1), Using (5) and   yields g rr = −1− 2  which components are defined by curvilinear coordinates in  R  different points of a curved space-time. However, in the case 2 r1 3 r dr 3R of a static body, the infinitesimal 4-momentums (1) are M = = (sin 1- ξ − ξ 1− ξ 2 ), ∫0 2 2 parallel to each other and have only time components dP t in a 2R R − r 4 (9) static coordinate system. So, in this case, it is possible to ξ = r / R. calculate the total mass-energy of a body M by integrating of 1 The series expansions modules of 4-momentums (1) over space using this coordinate -1 3 5 system: sin ξ = ξ + ξ 6/ + 3ξ 40/ + ... , t 2 2 4 .M = ∫Tt ( − g / g tt )dV t . (2) 1− ξ = 1− ξ 2/ − ξ 8/ + ... yields: Note, that t is the mass-energy volume density (see 3 5 2 Tt = ρ M = Rξ 2/ + 3Rξ 20/ + .. = m + 3m /( 5r1 ) + .. (10) (95.10) in [3] or § 100 in [5]). So, M > m. This excess of the matter mass M over the Let us use (2) to calculate the mass-energy of a sphere of Schwarzschild parameter m was named the (positive) perfect fluid. The space-time of such a body is discribed by gravitational mass defect (§ 100 in [5]). Thus the parameter the Schwarzschild's internal and external solutions of the m equals the initial mass of the fluid when the gravitation Einstein equation. The internal solution [6] (see also § 96 in field was negligible. [3]) depends on two parameters, R and r : 1 Because of J = M + G , the energy of the gravitation 2  3 r 2 1 r 2  field G must be negative if we wish the total mass, ds 2 =  1− 1 − 1−  dt 2  2 2  m = J = M + G , to be conserved. [In fact, it turned out that  2 R 2 R  G > 0 and m < J , see (17)]. −1  r 2  The calculation of the gravitational energy G must use the   2 2 2 2 2 2 , (3) − 1− 2  dr − r dθ − r sin θ dϕ formula of the type (2):  R  G t t g g dV , (11) 2 . = ∫ t ( − / tt ) t − g = − gtt grr r sin θ . (4) where t t − g is the component of the Einstein [2]-[4], or Here 0 ≤ r ≤ r ≤ R , r is the coordinate of the surface t 1 1 Landau-Lifshitz [5] pseudotensor density. And it must be where the external Schwarzschild spaceitime is attached, and t 2 R is the curvature radius of the inner space determined by the .G = ∫ tt ( − g / gtt )dV t = −3m /( 5r1) + ... < 0 ,..(12) fluid volume density But the pseudotensor density is very complicated. So, 3 t . (5) physicists calculated neither (11) nor .Tt = ρ = 2 8πR J = M + G = (T t + t t )( − g / g )dV The external solution, ∫ t t tt t , 2 2 −1 2 ds = (1− 2m/r)dt − (1− 2m/r) dr Instead of the total mass J , an integral without /1 gtt was (6) − r 2 dθ 2 − r 2 sin 2 θ dϕ 2 calculated U = (T t + t t ) − gdV . depends on the single (Schwarzschild) parameter m = rg 2/ , ∫ t t t which describes external gravitational field. It is seen that at This integral was chosen because the pseudotensor property smooth sewing the inner soluition together with the external provided a nontensor equality soluition ν ν ∂ν [( Tµ + t µ ) − g ] = 0 (13) 3 2 m = r1 2/ R . (7) and, consequently, the conservation of the nontensor integral In view of the Birkhoff's theorem, the parameter m U . remains constant when compressing the sphere. Now the mass-energy of the sphere, M , can be calculated III. THE STANDARD CALCULATION OF THE “TOTAL MASS - by using of (2): ENERGY ” t M = ∫Tt ( − g / g tt )drd θdϕ The Einstein pseudotensor density is [2]-[4] ν r (8) 1 t 2 tµ − g = ∫ Tt − g rr r dr 4π 0 1  ∂L  = − ∂ (gαβ − g ) + δ ν L  ,  µ αβ µ  16 π  ∂(∂ν (g − g )) 

L αβ ν µ µ ν IV. THE CONSERVATION LAW FOR SENSELESS QUANTITIES = g − g (Γαµ Γβν − Γαβ Γµν ) . (14) The idea about the energy-momentum pseudotensor of the The expression (14) is awkward. So, nobody calculated the gravitation field arose from the belief that the conservation of mass of the gravitational field (11). But luckily, we can use a an integral quantity in time is ensured by the vanishing of the result of a calculation that was performed by Tolman. Tolman partial divergence. Landau & Lifshitz wrote: “The integral integrated the sum of the covariant components ν t t ∫Tµ − gdV ν is conserved only if the condition dJ t = (Tt + tt ) − gdV t (15) for a quasi-static isolated system with respect to quasi- ν ∂ ν ( − gTµ ) = 0 (18) Galilean coordinates. The integration gives (see [7], or (91.1), is fulfilled” [5 § 96]. The covariant local conservation law (91.3), (92.4), (97.5) in [3]) ν t t ∇ν ( − gTµ ) = 0 (19) U = ∫ dJ t =∫ (Tt + tt ) − gdV t was discredited: “In this form, however, this equation does not t 1 2 3 (16) generally express any conservation law whatever”. = ∫ (Tt − T1 − T2 − T3 ) − gdV t To understand the situation, let us consider two = (ρ + 3p) − gdV = m ∫ t hypersurfaces V1 and V2 which have a closed surface as their where p is pressure. common boundary. In any space and in any coordinate system, This integral quantity (16) is named U . It cannot be named two quadruples of numbers P = − gT ν dV , P = − gT ν dV , (20) J t because it is not a component of a vector: it is obtained by 1µ ∫ µ ν 2µ ∫ µ ν V1 V2 adding up the components dJ t belonging to different points will be equal to each other, P1µ = P2µ , if (18) is fulfilled, where the coordinate bases differ from each other. And there because of the Gauss’ theorem: is no coordinate basis to which the quantity (16) could belong ν ν as a component. The quantity U is deprived of a geometrical ∫ ( − gTµ )dV ν − ∫ ( − gTµ )dV ν V2 V1 meaning and is senseless. It is not the mass of the system. ν ν Nevertheless, the result (16) is very useful! It shows that the = ∫ ( − gTµ )dV ν = ∫ ∂ ν ( − gTµ )dΩ = 0 (21) integral ∂Ω Ω t [here ∂Ω = V2 U (−V1 ) is the boundary of 4-volume Ω ]. ∫ tt − gdV t = ∫ 3p − gdV t > 0 But this constant quadruple P (20) has no geometric is positive because pressure p is positive. And this means a µ meaning if a curvilinear coordinate system is in use. We very important fact: the contribution G (11) of the Einstein repete, the integrals (20) are not applied to any specific point pseudotensor to the mass of the system, is also positive [8] in space and is therefore deprived of a geometric meaning because g > 0 : tt since we do not know the basis to which the components Pµ .G = t t ( − g / g )dV > 0 . (17) refer, and there is no law of their transformation at a ∫ t tt t coordinate change. The Landau-Lifshitz pseudotensor gives the same result; it is To feel the wantonness of the nontensor equations (18), (20) stated by formula (105.23) in [5]. in curvilinear coordinates, consider a plane with mechanical So, the pseudotensors give positive value for the mass- tensions. Let the tension tensor density be nonzero in the right energy of the gravitational field. The pseudotensor does not halfplane and be presented in polar coordinates r, ϕ at provide the conservation law. M > m, G > 0 , so, J = M + G > m . − π 2/ < ϕ < π 2/ by the equations ( − g is included): This means that, in reality, the pseudotensors do not T rr = r sin ϕ, T rϕ = T ϕr = cos ϕ, T ϕϕ = 0 . (22) represent mass-energy of the gravitational field, which is The condition (18) is fulfilled: negative. (If we adhere to the concept of gravitational field rr rϕ and trust that the mass-energy of the system is conserved). ∂ rT + ∂ϕT = sin ϕ − sin ϕ = ,0 Tolman wrote: “This satisfactory result (16) can serve to (23) T ϕr T ϕϕ increase our confidence in the practical advantages of ∂ r + ∂ϕ = .0 Einstein's procedure in introducing the pseudo-tensor densities So, integrals of the type (20) obtained by integration over the of potential gravitational energy and momentum” [3, p. 250]. circle r = Const , Unfortunately, this is an illusion. The conserved quantity (16) π 2/ r rr t t F = ∫T dl r =∫ r sin ϕdϕ = ,0 (24) U = ∫ (Tt + tt ) − gdV t −π 2/ π 2/ has no geometrical and physical sense. ϕ ϕr F = T dl r = cos ϕdϕ = 2 (25) ∫ ∫−π 2/ do not depend on the circle radius r .

These integrals pretend to be the components of the µ ν + {Ψ (x′, x)∇ [T (x) − g(x ]}) dΩ . (33) i ∫ µ′ ν µ (constant) force F acting on an arc of radius r in this the Ω µ strained halfplane. The component (25), however, does not ∇ν [Ψµ′ (x′, x)] = 0 in the Minkowski space in any determine any direction since it is not applied to any specific coordinate system [9]. Thus the covariant equation (19), ϕ point, while a vector with the component F = 2 has ν T g , different directions in space, depending on its application ∇ν ( µ − ) = 0 point. Therefore the pseudovector integrals (24), (25) are ensures the conservation of the energy-momentum Pµ in meaningless. Minkowski space. In a space-time of General Relativity, the energy-

V. TRUE CALCULATION OF INTEGRAL QUANTITIES momentum Pµ of matter does not conserve, in general, Counting the mass-energy of matter when using curvilinear because of the term coordinates requires parallel transfer of the 4-momentum µ ∇ν[Ψµ′ (x′, x)] elements of the matter (1) in (33). An example of creation of matter in a gravitation field ν dP µ = Tµ − gdV ν was presented in [10]. to the point x′ of counting using a special two-point tensor But, even worse, the energy-momentum of matter Pµ (28) function called a translator [9], does not have a single-valued meaning in a space-time of Ψ µ (x′, x) . (26) µ µ′ General Relativity because the translator Ψµ′ (x′, x) (26) Before the integration, one must transfer the elementary depends on the path of the transfer. vectors dP µ by the firmula: VI. CONCLUSION dP (x′) = Ψ µ (x′, x)dP (x) . (27) µ′ µ′ µ We have to admit that we do not know how to describe the Then the integral 4-momentum can be obtain energy of a stationary gravitational field in the frame of the P (x′) = Ψ µ (x′, x)T ν (x) − g(x)dV . (28) General Relativity. The pseudotensors are not suitable for this. µ′ ∫ µ′ µ ν The nontensor equation (13) [11 (35)] The integration (28) is integration of the elements ν ν ∂ν [( Tµ + t µ ) − g ] = 0 Ψµ (x′, x)T ν (x) − g(x)dV that are scalar at points x , µ′ µ ν means the conservation of the quantity (16) and curvilinear coordinates do not make problems. t t U = ∫ (Tt + tt ) − gdV t = m = Const , Let us consider two hypersurfaces V1 and V2 which have a closed surface as their common boundary as in (21). Then the which has no geometrical and physical sense. And here we do not touch the problem of the energy of two 4-momentums occur at the point ′ : x gravitational waves. µ ν P ′ (x′) = Ψ ′ (x′, x)T (x) − g(x)dV . (29) 1µ ∫V µ µ ν 1 REFERENCES µ ν P2µ′ (x′) = ∫ Ψµ′ (x′, x)Tµ (x) − g(x)dV ν . (30) [1] A. Guth, The Inflationary Universe: The Quest for a New Theory of V2 Cosmic Origins , Random House, 1997. Their difference equals the integral over the closed [2] A. Einstein, “Das hamiltonisch.es Prinzip und allgemeine Relativitatstheorie,” Sitzungsber. preuss. Akad. Wiss ., 1916, 2, pp. hypersurface ∂Ω = V2 U (−V1) : 1111—1116. µ ν [3] R. C. Tolman, Relativity Thermodynamics and Cosmology, Dover Books P ′ − P ′ = Ψ ′ (x′, x)T (x) − g(x)dV . (31) on Physics, 2011. 2µ 1µ ∫∂Ω µ µ ν [4] A. S. Eddington, The mathematical theory of relativity, Cambridge Now the Gauss theorem (which certainly implies partial University Press, 2010. differentiation) gives: [5] L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Pergamon, µ ν N. Y., 1975. Ψµ′ (x′, x)Tµ (x) − g(x)dV ν [6] K. Schwarzschild, Berl. Ber . 1916, p. 424. For an English translation see ∫∂Ω “On the Gravitational Field of a Sphere of Incompressible Fluid µ ν according to Einstein’s Theory,” arXiv:physics/9912033. = ∂ν [Ψµ′ (x′, x)Tµ (x) − g(x ]) dΩ . (32) ∫Ω [7] R. C. Tolman, “On the Use of the Energy-Momentum Principle in µ ν General Relativity,” Phys. Rev . 35, p. 875, 1930. But Ψµ′ (x′, x)Tµ (x) − g(x) is a vector density at x . So [8] R. I. Khrapko, “Goodbye, the Pseudotensor,” Abstracts of ICGAC-12, Moscow, 2015. we can change: ∂ ν → ∇ν , and as a result we have: http://khrapkori.wmsite.ru/ftpgetfile.php?id=141&module=files [9] R. I. Khrapko “Path-dependent functions” Theoretical and Mathematical P2µ′ − P1µ′ Physics . Volume 65, Issue 3, (1985), p. 1196 [10] R. I. Khrapko, “Example of creation of matter in a gravitation field”, µ ν 35 = {∇ν [Ψµ′ (x′, x)] Tµ (x) − g(x }) dΩ . Sov. Phys. JETP (3), 441 (1972) ∫Ω [11] A. Einstein, “Über Gravitationswellen”. Sitzungsberichte. preuss. Akad. Wiss ., 1918, 1, 154—167.