ISSN 0202-2893, Gravitation and Cosmology, 2014, Vol. 20, No. 4, pp. 264–273. c Pleiades Publishing, Ltd., 2014.

The Truth about the Energy-Momentum Tensor and Pseudotensor R. I. Khrapko* Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, Moscow, 125993 Russia Received July 19, 2013; in final form, April 9, 2014

Abstract—The operational and canonical definitions of an energy-momentum tensor (EMT) are con- sidered as well as the tensor and nontensor conservation laws. It is shown that the canonical EMT contradicts the experiments and the operational definition, the Belinfante-Rosenfeld procedure worsens the situation, and the nontensor “conservation laws” are meaningless. A definition of the 4-momentum of a system demands a translator since integration of vectors is meaningless. The mass of a fluid sphere is calculated. It is shown that, according to the standard energy-momentum pseudotensor, the mass-energy of a gravitational field is positive. This contradicts the idea of a negative gravitational energy and discredits the pseudotensor. And what is more, integral 4-pseudovectors are meaningless in general since reference frames for their components are not determined even for coordinates which are Minkowskian at infinity. DOI: 10.1134/S0202289314040082

Some Notations the system is conserved. A part of the field energy is simply converted to the particles’ kinetic energy. Indices: i, j, ··· = t, x, y, z or t, r, θ, ϕ; α, β, ··· = x, y, z or r, θ, ϕ. However, at the final stage when the atom forms, part of the kinetic energy travels away in the form of The (standard) Einstein√ –Eddington–Tolman radiation. As a result, the mass of the hydrogen atom i i − pseudotensor: H∧k = Hk g∧. is smaller than the sum of masses of the free proton ik The Landau-Lifshitz pseudotensor: h∧∧ = and electron by 13.6 eV. However, the mass of the ik h g∧∧. system “atom + radiation” is conserved. Mass-energy of a body with its gravitational field, The situation is different for gravitational attrac- according to the pseudotensor: J. tion. Consider, instead of the distant electron and Mass-energy of a body, i.e., the absolute value of proton, a dust cloud surrounded by its own grav- its 4-momentum: P . itational field. As the cloud contracts, the parti- cles, as in the previous case, acquire kinetic en- Pressure in the interior of a massive ball: p. ergy, and as a result, the mass-energy of the cloud increases (a positive mass defect). But the gravi- 1. INTRODUCTION tational field of the cloud does not disappear, it is strengthened and extends to the volume that has If particles or bodies attract each other by forces of become free from the cloud. However, the mass of a certain real field and join, the joint mass turns out the system “cloud + gravitational field” is conserved to be smaller than the summed mass of the original according to Birkhoff’s theorem and remains equal to bodies. This is called the (negative) mass defect. The the Schwarzschild parameter m. Therefore one has to simplest example is given by electrostatics. A proton ascribe a negative mass-energy to the gravitational and an electron located at distance attract each other field. by electric forces. The mass-energy of their own fields is a part of their masses. Another part is formed by To take into account this negative gravitational the electron or proton substance (if the latter exists). energy, one uses the standard pseudotensor of the As the electron and proton join to form a neutral gravitational field together with matter contained in i i i hydrogen atom, part of the field disappears due to in- it, H∧k = T∧k + t∧k [1, (89.3)]. It is equal to a sum of terference while the corresponding energy turns into the matter energy-momentum tensor (EMT) and the the kinetic energy of these particles. Therefore, as gravitational field pseudotensor. It is the latter that, the particles approach each other, the total mass of according to the existing opinion, provides a negative contribution because it is claimed that the mass- *E-mail: [email protected],[email protected] energy of the body together with its gravitational field

264 THE TRUTH ABOUT THE ENERGY-MOMENTUM TENSOR 265 √ ∧ − 2 equals m. This is expressed by the relation [1, (91), datr = datr g∧ = dθ dϕ r sin θ; (92), (97)]  the same surface element extended in time is a 3- i i ∧ volume with the components Jt = (T∧t + t∧t)dVt = m. ∧ tθϕ ∧ dVr = dV trθϕ = dt dθ dϕ, In the present paper we show that this conclusion is ∧ ∧ ∧ dV = dV = dV =0. wrong. In fact, the gravitational field pseudotensor t θ ϕ i t∧k contributes positively to this integral, and this en- The 4-momentum components in a volume at rest are tirely discredits the idea of a gravitational field pseu- i it it dP = T∧ dVt∧ = T∧ drdθdϕ dotensor. The quantity Jt is neither mass nor even √ it a temporal component of anything because it has = T −g∧dr dθ dϕ, been obtained by integration of a component dJt = i ∧ and a 4-momentum passing for dt through the area H∧tdVi of the infinitesimal 4-momentum rather than ∧ its absolute value dJ. However, we would like to datr has the components begin the article with a definition of the EMT of i ir ∧ ir dP = T∧ dVr = T∧ dtdθdϕ. matter and considering the problems connected with its integration. The metric tensor determinant is itself a scalar density of weight +2: g∧∧. It is important to note that the definition (2.1) only gives the coordinate component 2. OPERATIONAL DEFINITION dP i. The physical component is obtained by tak- OF AN ENERGY-MOMENTUM TENSOR ing into account the corresponding metric coefficient: √ ˆi i tˆ There are at present two different definitions of the dP √= dP gii. For example, the mass is dP = t matter EMT which exist in parallel. On the one hand, dP gtt. thereisalocaloperationaldefinition of matter EMT like this [2]: The operational definition of the EMT is, in par- ticular, used by Synge [6] (preserving the author’s A 3-dimensional infinitesimal element dV of a notations): “We borrow from the statistical model the medium contains or transfers through itself the in- interpretation of the energy tensor in terms of fluxes, finitesimal 4-momentum and we make the following statement: i ik ∧ dP = T∧ dVk . (2.1) (flux of 4-momentum across a polarized 3-target ij Let us comment on this definition. The EMT dSj = T dSj”. is actually a tensor density (but we will call it a Rashevsky [7] writes: “Suppose we are interested tensor for simplicity). We do not use a gothic font in the general picture of distribution and motion of while writing densities, as is often done, see, e.g., [3]. energy and momentum in space and time. To describe Instead, we mark them with the symbol “wedge” it, we must build, in the 4D space of events, an ∧. This notation was used by Kunin in his Russian appropriately selected, twice contravariant symmetric translation of the monograph [5]. However, unlike [4], tensor T ik,theEMT. ∧ we put the sign ontheleveloflowerorupperindices A local interpretation of the EMT is (sometimes) for densities of the weights +1 or −1, respectively. ∧ supported by Landau and Lifshitz: “If the tensor Tik The Levi-Civita symbol is denoted by ijkl, while a is zero at some world point, then this is the case for volume element or an elementary area with an exter- any reference frame, so that we may say that at this nal orientation, which are densities of weight −1,are point there is no matter or electromagnetic field”. ∧ ∧ denoted in space-time as dVk or daik, respectively; The local EMT definition gives an unambiguous the square√ root of the metric tensor determinant is value of the infinitesimal 4-momentum (2.1) pos- denoted as −g . For instance, it we use spherical ∧ √ ∧ sessed by the element dVk , and this quantity is ob- 2 coordinates, −g∧ = r sin θ, a 3-volume at rest has served experimentally. For example, in the case of ∧ the components an electromagnetic field, the infinitesimal area daβ , ∧ rθϕ ∧ absorbing the electromagnetic radiation, i.e., a “black dVt = dV trθϕ = dr dθ dϕ, ∧ ∧ ∧ area,” without doubt accepts the power dI, the light dV = dV = dV =0, or i i r √ θ ϕ pressure force dF and the momentum dP according ∧ − 2 to the relations dVt = dVt g∧ = dr dθ dϕr sin θ; tβ ∧ α αβ ∧ the element of a spherical surface has the components dI = T∧ daβ ,dF= T∧ daβ , α αβ ∧ ∧ θϕ ∧ dP = T∧ da dt. (2.2) datr = da trθϕ = dθdϕ, or β

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 266 KHRAPKO Here then there emerges an integral conservation law. Let αβ αγ βi β ij us transform the integral (2.7) to an integral over a 4- T∧ = g (−F F∧ + δ F F∧ /4) (2.3) γi γ ij volume Ω embraced by the closed hypersurface V = is Maxwell’s tension tensor which is the spatial part ∂Ω, according to the Gauss theorem (which certainly implies partial differentiation): of the EMT of Maxwell’s electrodynamics,   ik ij kl k im  ∧  ∧ T∧ = g (−FjlF∧ + δ FlmF∧ /4), (2.4) i ik i ik j 0= Ψi T∧ dVk = ∂k(Ψi T∧ )dΩ . (2.8) ∂Ω Ω tβ and T∧ is the Pointing vector. The resulting expression (2.8) can be simplified be- Addition of any terms to the Maxwell EMT would cause the partial derivative ∂k at point x from the violate the agreement between calculated and experi- i ik vector density Ψi T∧ at point x is equal to a covariant mental results and is therefore inadmissible. We have   derivative: ∂ (Ψi T ik)=∇ (Ψi T ik).Thuswehave already paid attention to it [8].  k i ∧ k i ∧ At a necessity to determine the 4-momentum P i  ∧  ∧ 0= ∂ (Ψi T ik)dΩ = ∇ (Ψi T ik)dΩ of a macroscopic body or its part, or, say, the elec- k i ∧ k i ∧  Ω  Ω tromagnetic field in a cavity, one has to integrate the ∇ i ik ∧ i ∇ ik ∧ elements (2.1) over a whole volume, = kΨi T∧ dΩ + Ψi kT∧ dΩ . (2.9)   Ω Ω i i ik ∧ P = dP = T∧ dVk . (2.5) It is reasonable to calculate the momentum of a i  body considering the translator Ψi (x ,x) as that of This integration does not make a problem in Carte- parallel transport. In this case, since the covariant sian coordinates. However, when using curvilinear derivative ∇k also rests on parallel transport, in flat coordinates, the components of the integral (2.5) do space we obtain not form a geometric quantity (vector) for two rea-  ∇ i sons: (1) Eq. (2.5) implies arithmetic addition of the kΨi =0, (2.10) i vector components dP belonging to different spatial and (2.9) leads to the local covariant conservation law, points where the coordinate vectorial bases can be ∇ ik not parallel. Therefore, thereisnobasisto which kT∧ =0, (2.11) the components of the integral could belong. (2) At which provides the conservation of the macroscopic i i a  coordinate transformations x = f (y ) there is no momentum P i (x) of an isolated body, calculated at transformation law for the components of P i to the  a a fixed world point. If the world point x is located components of P . For these reasons, the integral on the integration hypersurface of the integral (2.6) (2.5) in curvilinear coordinates is in general meaning-   P i (x ) less. We will call such quantities pseudovectors. and moves together with it, then the vector is transported in a parallel manner. That is what is In order that integration in curvilinear coordinates meant by a covariant conservation law of the macro- be geometrically meaningful, one should inevitably  P i x use a two-point tensor function called a transla- scopic quantity ( ). i  It is, however, important that if the body un- tor, Ψi (x ,x) [8–10]. With its aid, before inte- i der consideration is not isolated, or one considers a gration, one transfers the elementary vectors dP = medium subject to external influence, then the con- ik ∧  i  T∧ dVk to a certain common point x , dP (x )= servation law (2.11) is violated. In such a case the i  i Ψi (x ,x)dP (x), the one where integration is carried medium EMT is such that ik i out:  ∇kT∧ = f∧, (2.12)   i  i ik ∧ i P (x )= Ψi T∧ dVk . (2.6) where f∧ is the density of a 4-force acting on the medium in question from another medium or field. The integration (2.6) is integration of the elements This can be a Lorentz force, a force of light pressure, i  ik ∧ or a gravitational force (not geometrized according to Ψi (x ,x)T∧ dVk that are scalar at points x,and curvilinear coordinates do not make problems. Einstein). Thus, for Maxwell’s electromagnetic field If one carries out the integration (2.6) over a closed tensor interacting with electric charges and currents, 3D surface which twice intersects the world tube of an it will be ik ij i isolated body in space-time, and this integration gives ∇kT∧ = −g Fjlj∧. (2.13) zero,  The covariant derivative of the parallel transport i ik ∧ translator is zero (see (2.10)) in flat space where Ψi T∧ dVk =0, (2.7) this translator is path-independent. Indeed, let there

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 THE TRUTH ABOUT THE ENERGY-MOMENTUM TENSOR 267   k be a certain covector fi at the point x . Then the where the quantity T ∧i is called the canonical EMT: i c translator Ψi induces in a neighborhood of a point   ∂Λ∧ i  k − k x the covector field Ψi (x ,x+ ξ)fi , which, however, T ∧i = ∂iq δi Λ∧. (3.4) can be obtained by the same parallel transport of the c ∂(∂kq)   i  covector Ψi (x ,x)fi from the point x to the points However, the very purpose causes a bewilderment. x + ξ: Matter with a zero-divergence EMT is unobservable i  since an observation requires interaction with an ob- Ψi (x ,x+ ξ)fi server, whereas if there is an interaction, the EMT j   i  divergence is nonzero, see (2.12)! =Ψi (x, x + ξ)Ψj (x ,x)fi . (2.14) A comparison of the canonical EMT (3.4) with   i  Therefore this field, Ψi (x ,x+ ξ)fi ,turnsouttobe the operational tensor of Section 2 by using it in covariantly constant, which is proved by (2.10). Un- equations like (2.1) causes a difficulty since the La- like that, if there is space-time curvature, then the grangian of an elastic substance with mechanical   i  tensions is not clear. To the author’s knowledge, field Ψi (x ,x+ ξ)fi is not covariantly constant in a ∇ i  nobody has dealt with that. Popular is the canonical neighborhood of a point x, kΨi =0, because it will EMT of electrodynamics, be different from the r.h.s. of the equality (2.14). In this case, the local conservation law does not provide ik ∂Λ∧ i ik T ∧ = ∂ Aj − g Λ∧ conservation of the macroscopic momentum of the c ∂(∂kAj)  isolated body in time. The change of the quantity P i ij − ki k lm = g ( ∂jAiF∧ + δj FlmF∧ /4), (3.5) applied to point x gives the expression  obtained from the Lagrangian of a free electromag- ∇ i ik ∧ netic field, kΨi T∧ dΩ ij Ω Λ∧ = −FijF∧ /4. (3.6) An example of matter creation in curved space-time However, the tensor (3.5) is physically absurd as an was given in [11]. EMT: it is not symmetric and gives a negative energy density in a homogeneous electric field Ex = E [12], tt xt 2 T = FxtF /2=−E /2, (3.7) 3. CANONICAL DEFINITION c OF THE ENERGY-MOMENTUM TENSOR it is too unlike Maxwell’s experimentally justified ten- Along with the operational definition, there is sor (2.4), and its divergence, contrary to the claimed also a formal definition of matter EMT as a zero- purpose, is nonzero, divergence two-index tenor density (where a partial ik ij l ∂k T ∧ = −g ∂jAlj∧, (3.8) rather than covariant divergence is meant). To obtain c such a density, one writes down the gradient of the and different from the divergence of Maxwell’s ten- matter Lagrangian [3]: sor (2.13). This difference in the divergence means ∂Λ∧ ∂Λ∧ that even any zero-divergence additions like ∂ Ψikl, ∧ l ∂iΛ = ∂iq + ∂i∂kq ikl ilk ∂q ∂(∂kq) Ψ = −Ψ cannot convert the canonical tensor ∂Λ∧ ∂Λ∧ (3.5) to Maxwell’s tensor (2.4). And this is not to = ∂k ∂iq + ∂i∂kq mention that any additions to a true EMT are entirely ∂(∂ q) ∂(∂ q) k  k inadmissible. ∂Λ∧ Nevertheless, the Belinfante-Rosenfeld famous = ∂k ∂iq , (3.1) ∂(∂kq) procedure [13, 14] consists in precisely adding a i kl quantity like that, ∂l(A F ), to the canonical EMT using there the derivative from the Euler-Lagrange (3.5). This results in a tensor which we have call the equations, standard tensor [15]: ∂Λ∧ ∂Λ∧ ik ij − ki k lm = ∂k . (3.2) T ∧ = g ( ∂jAiF∧ + δj FlmF∧ /4) ∂q ∂(∂kq) st i kl ik i k After that, everything is put on a single side of + ∂l(A F )=T∧ + A j∧. (3.9) Eq. (3.1), and it turns out that This tensor is also nonsymmetric and is even more ∂ k =0, absurd than the canonical one since it explicitly con- k T ∧i (3.3) k c tains the current density j∧, a quantity alien to the

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 268 KHRAPKO electromagnetic field. Naturally, neither the canonical 4. A NONTENSOR CONSERVATION LAW nor the standard tensor are used in practical calcula- As was noted in Section 2, the covariant local tions. conservation law (2.11), Not having obtained Maxwell’s tensor in the ik ∇ T∧ =0, (4.1) framework of the canonical formalism but trying to k save the prestige of this formalism, the physicists, in any curvilinear coordinates in flat space-time pro- using Feynman’s words [16], preferred to put on vides conservation (in time) of the macroscopic mo-  sackcloth and ashes: mentum 4-vector (2.6), created at point x by using i “We would like to say that we have not really the translator Ψi , that is, its independence of the spacelike hypersurface V : ‘proved’ the Poynting formulas. How do we know that   by juggling the terms around some more we couldn’t i  i ik ∧ i ik ∧ find other formulas for ‘u’ and other formulas for ‘S’? P (x )= Ψi T∧ dVk = Ψi T∧ dVk . (4.2) The new S and the new u would be different, but they V1 V2 would still satisfy (the hypersurfaces V1 and V2 have a common bound- E · j = −du/dt −∇·S. ary, some closed surface). A nontensor law using the partial divergence of a It’s possible. There are, in fact, an infinite number of tensor or nontensor, no matter, different possibilities for u and S, and so far no one ik has thought of an experimental way to tell which one ∂kT =0, (4.3) is right!” provides conservation in time of the nontensor inte- gral quantity, the pseudovector (2.5), Evidently, this very strange claim completely con-  tradicts the reality (and Section 2 of the present pa- i ik , per), however, the idea that the fieldenergyisuncer- P = T dVk (4.4) tain and nonlocalizable is offered by all textbooks. For in any space and in any coordinates. Indeed, example:   “Localization of an energy flow leads to para- ik ik T dVk − T dVk doxes” [17].  V2 V1 “It is necessary to note that the [canonical] defini- ik ik tion of the EMT T ik is essentially ambiguous” [3]. = T dVk = ∂kT dΩ=0, ∂Ω “It is clear from the definition that θμν is not, in  Ω  general, symmetric. On the other hand, neither is it i ik ik λμν P = T dVk = T dVk. (4.5) unique, for we may add a term ∂λf ” [18]. V1 V2 “We will be only interested in integral dynamic quantities like the energy-momentum 4-vector P α. There is nothing good in it, however. Between αβ Eqs. (4.2) and (4.5) there is an essential difference. The structure of the tensor T , which is even am-    biguous in our presentation, is of interest by itself only The vector P i (x ) (4.2) is applied to x while the in a consistent theory taking into account gravita- four numbers P i (4.4), (4.5) are not a function of a tional effects [19]. point; this quantity has no argument; the result of integration (2.5), (4.4) is not applied to any specific (The classical books are cited more completely point in space-time and is therefore deprived of a in [8].) geometric meaning since we do not know the basis To finish this section, we recall Father Yelchani- to which the components P i refer, and there is no nov’s statement. Slightly adapted, it sounds like this: law of their transformation at a coordinate change. “In the light of love (the canonical formalism), To feel the meaningless nature of the nontensor rela- Reason adopts the apparent absurds of Faith (in the tions (4.3), (4.4) in curvilinear coordinates, consider a canonical formalism)”. plane with mechanical tensions. Let the tension ten- sor be nonzero in the right halfplane and be presented Besides, it is important to note in this paper that, in polar coordinates r, ϕ at −π/2 <ϕ<π/2 by the in the light of the above-said, the enormous con- relations structions on the basis of certain Lagrangians aimed rr rϕ ϕr T∧ = r sin ϕ, T∧ = T∧ =cosϕ, at solving the energy problem in Einstein’s gravity ϕϕ theory (see, e.g., [20]) do not look convincing. T∧ =0. (4.6)

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 THE TRUTH ABOUT THE ENERGY-MOMENTUM TENSOR 269 The nontensor conservation law (4.3) is valid for this Consider the (internal and external) Schwarzs- tensor: child space-time that describes a sphere of perfect rr rϕ − fluid. The internal space depends on two parameters, ∂rT∧ + ∂ϕT∧ =sinϕ sin ϕ =0, ≤ ≤ ≤ ϕr ϕϕ R and r1,where0 r r1 R [1]: ∂rT∧ + ∂ϕT∧ =0. (4.7)     3 r2 1 r2 2 Accordingly, the nontensor integral (4.4) obtained by ds2 = 1 − 1 − 1 − dt2 2 R2 2 R2 integration over the circle r = const does not depend 1 on the circle radius, − dr2 − r2dθ2 − r2 sin2 θdϕ2, (5.1) 1 − r2/R2  π/2 √ √ 2 r rr ∧ −g∧ = −gttgrrr sin θ. (5.2) F = T∧ dlr = r sin ϕdϕ =0, −π/2 Here R is the curvature radius of space, determined by the constant fluid density ρ =3/(8πR2), while r  π/2 1 is the coordinate of the surface where the external ϕ ϕr ∧ F = T∧ dlr = cos ϕdϕ =2. (4.8) Schwarzschild space is attached, depending on the single parameter m = r /2: −π/2  g 2m 1 These integrals pretend to give the components of the 2 − 2 − 2 ds = 1 dt − dr force F i acting on an arc of radius r in this excited r 1 2m/r halfplane and does not depend on the arc radius. − r2dθ2 − r2 sin2 θdϕ2. (5.3) These components, however, do not determine any 3 2 direction since they are not applied to any specific It is seen that at smooth matching m = r1/2R . point, while a vector with the component F ϕ =2has To calculate the fluid mass energy in the sphere, different directions in space, depending on its appli- oneshouldknowthefluid EMT, and it can be calcu- cation point. Therefore the pseudovector integrals lated from the Einstein tensor: ik ik ik ik lm (4.10) are meaningless. 8πT = G = R − g R glm/2. (5.4) Thus our comparison of the covariant condition The sphere is at rest, therefore, among T it only the (4.1) and the nontensor condition (4.3) has revealed a tt meaningless nature of the nontensor condition. How- component T is nonzero, it has been calculated t 2 ever, unfortunately, there is quite an opposite claim in [1]. By Eq. [1, (96.7)], Tt = ρ =3/(8πR ). in [3, Sec. 96], that the (covariant) equation (4.1) If one uses the nontensor formula (2.5), the inte- does not imply conservation of anything, while a non- gral pseudovector component is obtained:  tensor equation like (4.3) should be used to determine √ t − the conserved 4-momentum of the gravitational field Pt = Tt g∧drdθdϕ together with matter contained in it. Therefore an aim r1 of the theory is to introduce a nontensor construction 3 √ √ with zero partial divergence instead of an EMT with = g −g r2dr · 4π. (5.5) 8πR2 tt rr zero covariant divergence. We will follow the reason- 0 ing of [3, Sec. 96] in Sections 6 and 7. This is certainly not the mass of the sphere just be- cause a mass is the absolute value of the vector P . 5. THE MASS OF A SPHERE One could think that the mass is obtained by dividing√ OF PERFECT FLUID this quantity by the metric coefficient, P = Pt/ gtt. Recall, however, that the index t is here “pseudo- As noted in the Introduction, it is important to cor- vector,” i.e., fake since it is not defined to which rectly calculate the mass-energy of matter together point of the integration domain the pseudovector Pt with its gravitational field. Unfortunately, there is a √ is applied. It is significant because gtt changes in problem in determining the integral 4-momentum of  − 2 2 − a macroscopic body in curved space in general relativ- the integration domain from (3 1 r1/R 1)/2 to − 2 2 2 ity (GR). And, in our view, it is not quite certain that 1 r1/R . If√ one puts for clarity r1 =2, R =8, there is an adequate definition of this quantity. Not all m =1/2,then gtt changes from 0.57 to 0.71.That we want really exists! However, in some simple cases is, first, the pseudovector component Pt (5.5) does not such a definition is possible with the aid of a translator help one to find the mass, and second, it turns out to and Eq. (2.6). Thus, in particular, one can find the be much smaller than the mass P , Pt ≈ 0.64P for the mass of a sphere of perfect fluid, which is important parameter values chosen.√ It is absurd to apply Pt to a for what follows. point at infinity where gtt =1.

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 270 KHRAPKO To obtain the correct value of the fluid mass P the problem of its energy-momentum emerged from let us use the fact that in this case the infinitesimal the very beginning. Although everybody understood vectors dP i are parallel to each other, therefore one that the notion of energy is a certain luxury, not can add their absolute values which do not change at all necessary for problem solving, nevertheless when transported by the translator to a single point there was a desire to introduce the “gravitational field for summing.√ So one can readily integrate the values energy” in order to extend to gravity the total energy dP = dPt/ gtt, and instead of (5.5) we obtain conservation law.   √ √ − To materialize the idea of energy-momentum of t g∧ P = dPt/ gtt = Tt √ drdθdϕ the gravitational field and to define a conserved total gtt 4-momentum of the gravitational field with matter r 1 √ contained in it, a nontensor density with zero par- t − 2 = Tt grrr dr4π tial divergence was suggested, in accordance with a belief that a nontensor equation like (4.3) could 0 provide conservation of something. This density, r1 2 i 3 r dr which we denote H∧ ,containedfirst- and second- = √ . (5.6) k 2R R2 − r2 order derivatives of the metric tensor of the coordinate 0 system used. In [1, (89.3)] it is brought to the form of a partial divergence (but not from an antisymmetric The integration gives  quantity, as usually happens). This density is called 3R r1 the energy-momentum pseudotensor of matter to- P = (arcsin ξ − ξ 1 − ξ2),ξ= . (5.7) 4 R gether with the gravitational field: Restricting ourselves to two terms in the ξ-expansion i il im l − l H∧k = ∂lH∧k = ∂l[g∧ (Γkm δ(kΓm) in (5.7), we obtain i mn l l   − δ g∧ (Γ − δ Γ )/2]/8π, 3r2 k mn m n P = m 1+ 1 + ... . (5.8) Γ =Γn . (6.2) 10R2 m mn The partial divergence of the pseudotensor Hi is The excess of P over the Schwarzschild parameter ∧k surprisingly zero in any coordinate system: m has been called in [3, Sec. 100] the (positive) grav- i itational mass defect. The gravitational field pseudo- ∂iH∧k =0. (6.3) tensor, to be considered in the next section, should insert a negative contribution to the total mass of the Vanishing of the covariant divergence of a tensor in system “matter + gravitational field” to make this any coordinate system is not surprising. However, total mass equal to the Schwarzschild parameter m. here the zero partial divergence of the pseudotensor is preserved under coordinate changes because the It is of interest that integration of Eq. (5.5) shows geometric content of the pseudotensor itself changes that Pt is not only smaller than the mass P but also in the sense that changes the value of the contraction smaller than the Schwarzschild parameter m: i ∧ k   H∧kVi V with fixed vectors. In the case of a ten- 3r2 sor, whose geometric meaning is fixed, a zero par- P = m 1 − 1 + ... . (5.9) 10R2 tial divergence is usually violated under coordinate changes. The uniqueness of the zero-divergence pseudo- 6. STANDARD ENERGY-MOMENTUM tensor (6.2) was probably not investigated, and there PSEUDOTENSOR is apparently no data on a pseudotensor with two OF THE GRAVITATIONAL FIELD ij ij upper indices, H∧ , ∂jH∧ =0. The idea to connect matter with geometry was The expression (6.2) is called the energy-momen- realized by Einstein when he equated the EMT of tum pseudotensor of matter with the gravitational matter, having a zero covariant divergence, to the field because if the coordinate system used turns out only geometric tensor (Einstein’s) with zero covariant to be locally Minkowskian at a certain point of space- divergence: time, then the pseudotensor (6.2) coincides at this ik ik ik ik lm 8πT = G = R − g R glm/2, point with the Einstein tensor, hence with the EMT i ∼ i i ik ik of matter located at this point H∧k = G∧k/8π = T∧k. ∇kT∧ =0. ∇kG∧ =0. (6.1) ∼ (We denote by the approximate equality sign = the Thus the gravitational field was eliminated (ge- equalities valid at the center of a locally Minkowskian ometrized). All dynamic problems are being solved in coordinate system.) At other points the pseudotensor GR, but, since the gravitational field was eliminated, differs from the matter EMT. The difference is called

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 THE TRUTH ABOUT THE ENERGY-MOMENTUM TENSOR 271

i the gravitational field pseudotensor t∧k.Thatis, This result is understandable: the pseudocomponent the pseudotensor (6.2) represents a sum of matter Pt is smaller than m, as was noted in Section 5; i the gravitational field pseudotensor makes a positive EMT and the gravitational field pseudotensor: H∧k = i i contribution; the resulting sum is m. (T∧ + t∧). However, the pseudocomponents Jt and Pt do not Materialization of the idea of energy-momentum represent masses. The actual mass of matter in the of the gravitational field with the aid of the pseudo- I ball together with its gravitational field, in the context tensor H∧k is contained in its following definition: of using the gravitational field pseudotensor, is given The 4-momentum of matter together with the by an expression like (5.6):  √ gravitational field is given by the integral [1, (88.4)] −  t g∧ √ J = Ht √ dx dy dz i − ∧ gtt Jk = Hk g∧dVi  √ −g  V = (T t +3p) √ ∧ dx dy dz √ t g i i − ∧  √ tt = (Tk + tk) g∧dVi (6.4) −g∧ V = P + 3p √ dx dy dz > m. (6.9) gtt over a hypersurface V including the whole 3D space, and the components of this integral have equal values This mass is much larger than m because the mass of matter alone is P>m. A positive contribution on two hypersurfaces V1 and V2, resting on a common boundary in the form of a closed 2D surface due to from the gravitational field pseudotensor means a full (6.3). That is, break-up of the whole construction because the pseu-   dotensor should make a negative contribution corre- i i sponding to a negative gravitational mass-energy. Jk = H dVi = H dVi, (6.5) k k What is simultaneously compromised is the Ha- V1 V2 miltonian approach to solving the problem of en- as in (4.5). ergy in GR, based on certain Lagrangians and men- tioned at the end of Section 3. Indeed, Ref. [20] Naturally, the integral (6.4) causes the criticism states “a coincidence of the results of the Hamiltonian J presented in Section 4: the four numbers k (6.4) are approach and the one based on using the energy- not applied to any specific point in space-time and momentum pseudotensor in defining the total energy is therefore deprived of any geometry meaning, since in an asymptotically flat space-time”. nobody knows a basis to which the components Jk refer. But let us put aside the fatal problems related to 7. THE LANDAU-LIFSHITZ the absence of a basis supporting the components of PSEU-DO-TEN-SOR OF THE GRAVITATIONAL FIELD Jk. Let us look how the pseudotensor (6.2) is used in practice in the simplest case discussed in Section 4, Landau and Lifshitz [3, Sec. 96] found a much i.e., calculation of the mass of matter contained in a simpler construction of the pseudotensor than (6.2); ball together with the gravitational field. moreover, it is presented as a divergence of an an- In [1, (92.4), (97.2)] an explicit form is given for the tisymmetric quantity, which automatically leads to a gravitational field pseudotensor, zero partial divergence, t ik ikl 2 − i[k l]m tt =3p, (6.6) h∧∧ = ∂lh∧∧ = ∂lm( g∧∧g g )/8π, (7.1) ik where p is the pressure inside the ball. This pseudo- ∂kh∧∧ =0. (7.2) tensor is used in Eq. (6.4) in the coordinate system Unfortunately, the pseudotensor (7.1) is a density of with the isotropic metric weight +2, so that at space-time points where the ds2 = eνdt2 − eμ(dx2 + dy2 + dz2), (6.7) coordinate system used is locally Minkowskian, this pseudotensor coincides√ with Einstein’s tensor density and after a calculation, the result is presented [1, up to the factor −g∧: Sec. 97]: √  ik ∼ − ik √ h∧∧ = g∧G∧ /8π t J = (T +3p) −g dx dy dz ik ik ik t t ∧ = −g∧∧(R − g R/2)/8π = −g∧∧T . (7.3)  √ Certainly, at other space-time points the pseu- = P + 3p −g∧dx dy dz = m. (6.8) t ik ik dotensor h∧∧ differs from the tensor −g∧∧T .The

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 272 KHRAPKO difference is called the gravitational field pseudotensor as the notion of the gravitational field itself, which tik. Thus the pseudotensor (7.1) is considered as a is, according to Einstein’s GR, replaced by curved sum of matter EMT and the gravitational field pseu- space-time. It has been noticed that the mass-energy dotensor: of material systems and fields can be calculated in ik ik ik ik ∼ curved space with the aid of translators. h∧∧ = −g∧∧(T + t ),t= 0. (7.4) Rather long ago the noncovariance of the pseudo- Integration of the pseudotensor (7.1) over a hyper- tensor was discussed by M.F. Shirokov [21]: “Quan- surface V including the whole 3D space, tization of weak gravitational fields with introduction  of special particles into the theory, i.e., gravitational i ik ∧ J∧ = h∧∧dVk , (7.5) field quanta, or gravitons, is not a generally covariant formulation of the quantum theory of particles and represents, by definition, the whole mass-energy of fields since the employed gravitational field quantiza- matter with the gravitational field. Unlike (6.4), the tion procedure rests on using the Lagrange function integral (7.5) is a vector density. A calculation of and the energy-momentum pseudotensor which are the mass-energy of a fluid sphere using (7.1) was only covariant under Lorentz transformations.” apparently not carried out. In the absence of a gravitational field, in Minkow- ik i REFERENCES ski coordinates, h∧∧ =0everywhere (just as H∧k = 0 1. R. C. Tolman, Relativity, Thermodynamics and ). This means, in particular, that the integral (7.5) is Cosmology (Oxford, Clarendon 1969). zero, in agreement with that in flat space-time there 2. R. I. Khrapko, Classicalspininspacewithand is no matter and no gravitational field. However, in [3, without torsion,Grav.Cosmol.10, 91 (2004); Sec. 96] it is said about the integral (7.5),  http://khrapkori.wmsite.ru/ftpgetfile.php?id=18& module=files i − ik ik ∧ J∧ = ( g∧∧)(T + t )dVk , (7.6) 3. L. D. Landau and E. M. Lifshitz, The Classical The- ory of Fields (Pergamon, N.Y., 1975). that it is not zero in flat space but passes into 4. J. A. Schouten, Tensor Analysis for Physicists ik (Nauka, Moscow, 1965). T dVk, i.e., to the 4-momentum of matter without gravity. This contradiction has emerged due to 5. J. A. Schouten, Tensor Analysis for Physicists (Clarendon, Oxford, 1951). ambiguity of the notations. In [3, (96,7)] it is implied 6. J. L. Synge, Relativity: The General Theory (North- that Holland, Amsterdam, 1960). T ik =(Rik − gikR/2)/8π, (7.7) 7. P.K. Rashevsky, Riemannian Geometry and Tensor Analysis (Gostekhizdat, Moscow, 1967), p. 304. while in other places the tensor T ik is not related to 8. R. I. Khrapko, Energy-momentum localization and the space-time curvature. spin, Bulletin of Peoples’ Friendship University The nontensor integral (7.5) causes the same ob- of Russia, Series Physics 10 (1), 35 (2002); jections as the integral (6.4). They are connected http://khrapkori.wmsite.ru/ftpgetfile.php?module= with the absence of a basis that would support the files&id=32 The Ostrogradsky-Gauss theorem in Ji Ji 9. R. I. Khrapko, components of ∧. It√ is absurd to apply ∧ to a curved space, Abstracts of the USSR gravitational pointatinfinity where gtt =1. Sinceabasisof conference (GR-4, Minsk, 1–3 July 1976), p. 64. the components (7.5) is not defined, this “conserved” 10. R. I. Khrapko, Path-dependent functions, integral is meaningless. Teor. Mat. Fiz. 65 (3), 1196 (1985); http://khrapkori.wmsite.ru/ftpgetfile.php?id=19& module=files 8. CONCLUSION 11. R. I. Khrapko, Example of creation The energy-momentum pseudotensor does not of matter in a gravitation field, solve the problem of gravitational energy for two Sov. Phys. JETP 35 (3), 441 (1972); reasons: (1) The pseudotensor has no physical http://khrapkori.wmsite.ru/ftpgetfile.php?id=93& meaning due to its nontensor nature. Using an module=files asymptotically flat environment does not remove this 12. R. I. Khrapko, Spin flux gives rise to antisymmetric stress tensor fatal diagnosis. (2) The physical meaning ascribed to the pseudotensor is wrong because, according http://mai.ru/publications/index.php?ID=8925 (2009) to the pseudotensor expression, for example, the 13. F. J. Belinfante, Physica 6, 887 (1939). gravitational field energy of a massive ball is positive. 14. L. Rosenfeld, Sur le tenseur d’impulsion-energie, It is possible that the notion of mass-energy of Memoires de l’Academie Royale des Sciences de Bel- the gravitational field is inadequate to nature, as well giques 8 (6) (1940).

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15. R. I. Khrapko, Mechanical stresses produced by 19. N. N. Bogoliubov and D. V. Shirkov, Introduction to a light beam. J. Modern Optics 55, 1487 (2008); the Theory of Quantized Fields (GITTL, Moscow, http://khrapkori.wmsite.ru/ftpgetfile.php?id=9& 1957), p. 23. module=files 16. R. P. Feynman, R. B. Leighton, and M. Sands, The 20. L. D. Faddeyev, The problem of energy in Einstein’s Feynman Lectures on Physics (Addison–Wesley, theory of gravity,UspekhiFiz.Nauk136, 435 London, 1965), Vol. 3. 17. L. I. Mandelshtam, Lectures on Optics, Relativity, (1982). and Quantum Mechanics (Nauka, Moscow, 1972), p. 19. 21. M. F. Shirokov, On the materialist essence of rela- 18. L. H. Ryder, Quantum Field Theory (Cambridge tivity, in: Philosophic Problems of Modern Physics University Press, Cambridge, 1985). (AN SSSR, Moscow, 1959).

GRAVITATION AND COSMOLOGY Vol. 20 No. 4 2014 Русский оригинал статьи представлен по адресу http://khrapkori.wmsite.ru/ftpgetfile.php?id=112&module=files

Addition Publishing of this paper is a miracle as well as publishing of What is mass? Physics – Uspekhi, 43 1267 (2000) http://khrapkori.wmsite.ru/ftpgetfile.php?id=14&module=files Mechanical stresses produced by a light beam, J. Modern Optics, 55, 1487 (2008) http://khrapkori.wmsite.ru/ftpgetfile.php?id=9&module=files Journals reject papers, which disclose mistakes of authorities. This paper was rejected by the following journals

- ТМФ . « Ваша статья не представляет интереса для журнала ТМФ и не может быть опубликована ». Отв . Секретарь В.В.Жаринов , July 30, 2013.

- ЖЭТФ . « Бюро редколлегии ЖЭТФ рассмотрело Вашу статью . Редколлегия признала , что по своему содержанию статья носит методический характер и потому ее публикация в ЖЭТФ нецелесообразна ». May 22, 2014

- УФН . « Редколлегия рассмотрела Вашу статью и считает ее публикацию нецелесообразной в связи с тем , что мы не видим необходимости разворачивать дискуссию по данной тематике на страницах журнала УФН ». Академик О.В.Руденко . 02.08.2013

-- GRG September 01, 2013: “The paper under consideration provides an explicit example of a well-known fact, namely that the energy-momentum pseudo-tensor does not provide an invariant means for calculating the energy- mometum contribution due to the gravitational field. It is dependent on the coordinate system, or more precisely on the reference frame used. So while I believe that the paper is correct I do not think that it contributes anything new and therefore, I suggest that it be rejected.” Abhay Ashtekar. -- My reply was: Dear Abhay Ashtekar, Sorry, Your Reviewer is not correct when he writes “that the energy- momentum pseudo-tensor does not provide an invariant means for calculating the energy-mometum contribution due to the gravitational field. It is dependent on the coordinate system, or more precisely on the reference frame used”. In reality, as is well known, the energy-momentum pseudo-tensor DOES provide an invariant means for calculating the energy-mometum contribution due to the gravitational field. It is INDEPENDENT on the coordinate system, or more precisely on the reference frame used. For example, Tolman wrote: t ν “ µ is a quantity which is defined in all systems of coordinates by (87.12), and the equation is a covariant one valid in all systems of coordinates. Hence we may have no hesitation in using this very beautiful result of Einstein”. Landau & Lifshitz wrote: “The quantities P i (the four-momentum of field plus matter) have a completely define meaning and are independent of the choice of reference system to just the extent that is necessary on the basis of physical considerations”. Tolman wrote: J “It may be shown that the quantities µ are independent of any changes that we may make in the coordinate system inside the tube, provided the changed coordinate system still coincides with the original Galilean system in regions outside the tube. To see this we merely have to note that a third auxiliary coordinate system could be introduced coinciding with the common Galilean coordinate system in regions outside the tube, and coinciding inside the tube for one value of the 'time' x 4 (as given outside the tube) with the original coordinate system and at a later 'time' x 4 with the changed J coordinate system. Then, since in accordance with (88.5) the values of µ would be independent of x 4 in all three coordinate systems, we can conclude that the values would have to be identical for the three coordinate systems”. So, I think you need to use another Reviewer. -- I have no answer.

-- Classical and Quantum Gravity, September 11, 2013: “We do not publish this type of article in any of our journals and so we are unable to consider your article further”. John Fryer, Ben Sheard, Adam Day, Martin Kitts.

-- New Journal of Physics, September 17, 2013 “We are unable to consider the article for our journal as it has previously been rejected”. Kryssa Roycroft and Joanna Bewley.

-- PRD, October 11, 2013 “Your manuscript only refers to work written more than sixty years ago, and ignores the considerable relevant work since then that is related to an understanding of the issues and difficulties associated with local and global concepts of energy in gravitating systems in a (necessarily) curved spacetime”. Erick J. Weinberg. -- My reply was: Dear Erick J. Weinberg, All works written during the sixty years on this topic are founded on the first work by Einstein, Eddington, Tolman. All these works developed the Einstein’s work, interpreted it or modernized it. In contrast, my paper argues that the first work is trivially invalid owing to a simple mistake, namely, a covariant component of the energy-momentum vector, instead of mass, was calculated in the work, and this component has no sense. Thus all works, which take the first work seriously, are of no interest. -- An appeal against the decision was rejected without explanations.