Chapter 6: Energy-Momentum Tensors

Transcription

1 49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel. We will focus in particular on the case of a scalar fiel interacting with a particle. The rate of change of energy an momentum is escribe in terms of tensor ivergence equations. 6.1 Basic Theory With fiels, we are concerne with ensities, such as charge, probability, energy an momentum ensity. The treatment of the istribution of energy an momentum within the fiel procees in the same way as for the more familiar cases of charge an probability. Conservation of these latter quantities is escribe by a continuity equation involving both a ensity ρ an a current ensity j i = ρv i (i = 1,2,3): i j i + t ρ =0 [6-1] The ensities that characterise a fiel's energy an momentum content are summarise in the form of the energy-momentum tensor T µν. The various terms in this quantity correspon to energy an momentum ensities an energy an momentum currents. In particular, the momentum ensity component in the i th irection (for example, ρv i in the case of a flui having a mass ensity ρ an no internal stresses) will have a current component in the j th irection (ρv i v j for a stressless flui). Thus we are le to a escription involving two inices i an j. In the relativistic case, the inices can separately have any of the values µ,ν = 0, 1, 2, or 3 an so the energy momentum tensor T µν consists of 16 components. In analogy to the continuity equation [6-1], energy an momentum conservation is escribe by the following set of 4 equations: j T µj + 0 T µ0 =0 [6-2]

2 50 The equation corresponing to µ = 0 contains the terms T 0j an T 00 an escribes conservation of energy. The 3 equations corresponing to µ = i = 1, 2, 3 contain the terms T ij an T i0 an escribe conservation of each component of momentum. In the relativistic case, it can be shown that conservation of angular momentum requires T µν to be symmetric in µ an ν an, as a consequence, the number of inepenent components is reuce from 16 to 10. These components have the following interpretation (ignoring any factors of c): T 00 = energy ensity [6-3a] T i0 = T 0i = three components of momentum ensity (equivalent to energy current) [6-3b] T ij = T ji = six components of momentum current [6-3c] Equations [6-2] can be written more elegantly as: ν T µν =0 [6-4] 6.2 Energy-Momentum Tensor for a Scalar Fiel It can be shown 1 that the energy-momentum tensor for a real, free scalar fiel φ escribe by a Lagrangian ensity is of the form: T µν =[ µ φ ( ν φ) gµν ] [6-5] where we are using the notation g µν 1 for µ = ν = 0 1 for µ = ν = 1,2,3 0 for µ ν µ φ φ/ x µ µ φ g µν ν φ [6-6a] [6-6b] [6-6c] [6-6] [6-6e]

3 51 Using the Lagrangian ensity [4-10] for a real, massless scalar fiel: scalar fiel =½( µ φ)( µ φ) the corresponing energy-momentum tensor is foun from [6-5] to be: T µν =( µ φ)( ν φ) ½g µν ( λ φ)( λ φ) [6-7] 6.3 Energy an Momentum for a Scalar Fiel interacting with a Particle From chapter 4, the overall Lagrangian ensity for escribing a particle in interaction with a real, massless scalar fiel is given by [4-19]: system =½( µ φ)( µ φ)+[½mv 2 qφ ] ρ [6-8] For a fiel that is exchanging energy an momentum with a particle, the basic conition expecte for conservation is that the rate of change of the particle s energy an momentum must exactly match the rate of change of the fiel s energy an momentum, thus ensuring that the total remains constant. Also, the energy an momentum changes must also occur in a local manner. This means that the net energy an momentum flux into or out of the fiel in the immeiate vicinity of the particle trajectory shoul match the particle's change of energy an momentum Energy an Momentum Conservation Equations For the system characterise by the Lagrangian ensity [6-8] above, it is possible to erive conservation equations illustrating an confirming that changes in the fiel's momentum an energy are compensate by changes in the particle's momentum an energy. The 4-ivergence of the fiel s energy momentum tensor yiels expressions corresponing to the rate of change of the fiel s energy an momentum an consequently, as a first step towars obtaining the particle-fiel system conservation 1 See, e.g., Ch. 12 in Golstein H., Classical Mechanics, 2n E. Aison-Wesley, Massachusetts (1980).

4 52 equations, we will evaluate the 4-ivergence of the fiel's energy-momentum tensor [6-7]: ν T µν fiel = ν [( µ φ)( ν φ) ½g µν ( λ φ)( λ φ)] [6-9a] =( ν µ φ)( ν φ)+( µ φ)( ν ν φ) 1 2 [( µ λ φ)( λ φ)+( λ φ)( µ λ φ)] =( λ µ φ)( λ φ)+( µ φ)( ν ν φ) 1 2 [( µ λ φ)( λ φ)+( µ λ φ)( λ φ)] =( µ λ φ)( λ φ)+( µ φ)( ν ν φ) ( µ λ φ)( λ φ) =( µ φ)( ν ν φ) [6-9b] [6-9c] [6-9] [6-9e] Now, the fiel equation which follows from the Lagrangian ensity [6-8] above is equation [4-20]: µ µ φ = qρ which is simply the free-fiel equation with a source term ae. Inserting this fiel equation into [6-9e] yiels the tensor ivergence equation: ν T µν fiel = qρ µ φ [6-10] Returning again to the Lagrangian ensity [6-8], the particle equation of motion it yiels via the integral equations [4-16] an Lagrange's equations [4-3] is the usual one for a particle in a scalar fiel: p i =q i φ [6-11] Also, from this equation for the rate of change of the particle's momentum, it is straightforwar to erive an analogous one for the particle's energy (see Appenix 3): E =q φ t [6-12] We are now in a position to write own the equations we are seeking. Inserting [6-11] into the right han sie of [6-10], we obtain: iν ν T fiel = ρ pi (i = 1,2,3) [6-13]

5 53 Similarly, inserting [6-12] into [6-10], we obtain: 0ν c ν T fiel = ρ E [6-14] These are the two esire equations. They link the local changes in the fiel's momentum an energy to those of the particle, in accorance with the requirement of conservation. Equations [6-13] an [6-14] also hol for other classical cases, such as a particle interacting with an electromagnetic fiel (see the Lagrangian ensity [4-21] earlier). In eveloping our Lagrangian approach to Bohm's moel, it will be necessary for something similar to hol in the case of a Bohmian particle interacting with a Schroinger fiel Introuction of T µν particle In the case of a particle interacting with a scalar fiel, conservation of momentum an energy can also be expresse by introucing an energy-momentum tensor for the particle an writing the following set of ivergence equations (µ,ν = 0,1,2,3): ν (T µν fiel + T µν particle) = 0 [6-15] For a relativistic particle, T µν particle has the form 2 : T µν particle = ρ 0 mu µ u ν where ρ 0, m an u µ [6-16] are rest ensity, rest mass an 4-velocity, respectively. This expression for T µν particle will be utilise in a later chapter. The set of equations [6-15] can be shown 3 to be equivalent to the relativistic versions of [6-13] an [6-14] provie expression [6-16] is chosen for T µν particle. From [6-15], the conservation of the three components of momentum (i = 1,2,3) will be escribe by the equations ν (T iν fiel + T iν particle) = 0 [6-17a] 2 See, e.g., Sec in Aler R., Bazin M. an Schiffer M., Introuction to General Relativity, 2 n E. McGraw-Hill Kogakusha, Tokyo (1975).

6 54 an conservation of energy will be escribe by ν (T 0ν fiel + T 0ν particle) = 0 [6-17b] Global Equations Equations [6-17] involve momentum an energy ensities an ensure conservation "locally" at each point in space. On the other han, the conservation of the total values of momentum an energy will be escribe by the following "global" equations (i = 1,2,3): [p i i fiel +p particle ]=0 [6-18a] [E fiel +E particle ]=0 [6-18b] Equations [6-18a] an [6-18b] can be erive from the "local" versions [6-17a] an [6-17b] by integrating over all space: iν ν T fiel 0ν ν T fiel 3 iν x + ν T particle 3 x =0 [6-19a] 3 0ν x + ν T particle 3 x =0 [6-19b] The ensities of momentum an energy will thereby be converte to total values. The equivalence of equations [6-19a] an [6-19b] to equations [6-18a] an [6-18b] will now be emonstrate. For both T µν fiel an T µν particle, the require integral over space can be written out in etail as follows: ν T µν 3 x = ( 0 T µ0 + 1 T µ1 + 2 T µ2 + 3 T µ3 ) 3 x [6-20] Uner the reasonable assumption that the energy-momentum tensor falls off to zero at plus an minus infinity (in any spatial irection), the last three terms of [6-20] will be zero an so only the term containing the time erivative survives: ν T µν 3 x = 0 T µ0 3 x = 1 c T µ0 3 x [6-21] 3 Felsager B., Geometry, Particles an Fiels, Sec. 1-6, Springer-Verlag, NY (1998).

7 55 In the last equality, the partial erivative has been replace by the total erivative because, after the spatial integration 3 x has been performe, only time epenence remains. With the ai of [6-21], equations [6-19a] an [6-19b] can be written as: i0 T fiel 3 x+ i0 T particle 3 x=0 [6-22a] 00 T fiel 3 x+ 00 T particle 3 x=0 [6-22b] Now, T i0 an T 00 can be ientifie from equations [6-3a] an [6-3b] earlier as momentum ensity an energy ensity, respectively. Therefore these equations reuce to the global equations [6-18a] an [6-18b] as require: [p i i fiel +p particle ]=0 [E fiel +E particle ]=0 6.4 Tentative Application to Bohm's Moel Having summarise the relevant theoretical formalism, we will now attempt to employ it to introuce conservation of energy an momentum into Bohm's moel. In oing so, it will be foun that some ifficulties arise. Fortunately these can all be overcome by a eeper an more careful analysis. We will look briefly here at the problems that are encountere as a pointer towars an appropriate course of action to follow in the next chapter. As iscusse in chapter 5, our propose Lagrangian ensity for Bohm's moel is: = 2m h2 [ jψ (x)] [ j ψ(x)] + ih 2 [ψ (x) t ψ(x) ψ(x) t ψ (x)] ( fiel terms) ½mρ(x x 0 )v j v j ( particle term) ρ(x x 0 )Q(x) (interaction term) [5-1]

8 56 By analogy with the classical cases of a particle in a scalar or vector fiel, we tentatively expect that equation [6-15] will continue to hol: ν (T µν fiel + T µν particle) = 0 in the case of the system escribe by [5-1]. Equation [6-15] escribes transfers of energy an momentum between the fiel an the particle, in accorance with the requirements of conservation. As a first step towars establishing whether this equation remains vali in our Bohmian case, we will erive the free-fiel energy-momentum tensor corresponing to [5-1]. Because our Lagrangian ensity is non-relativistic, it oes not possess the sort of symmetry between space an time that is characteristic of relativistic Lagrangians. It is therefore necessary to obtain separate expressions for T ij, T i0, T 0i an T 00 (i,j = 1,2,3), rather than just a single T µν expression (µ,ν = 0,1,2,3). This lengthens the erivation somewhat. The esire expressions are foun from the free-fiel part of the Lagrangian ensity [5-1] by applying the formula: T µν fiel =[ µ φ ( ν φ) + µ φ * ( ν φ * ) gµν ] fiel [6-23] which is a generalisation of equation [6-5] from the case of a real fiel to that of a complex fiel. The erivations are given in Appenix 4 an the results are: ij T fiel i0 T fiel = h2 2m {( i ψ)( j ψ )+( i ψ * )( j ψ) g ij ( k ψ )( k ψ)} ij g ih 2 (ψ t ψ ψ t ψ ) [6-24a] = ih 2 { ψ i ψ ψ i ψ * } [6-24b] 0i T fiel 00 T fiel = h2 2m {( tψ )( i ψ )+( t ψ )( i ψ)} = h2 2m ( kψ ) ( k ψ) [6-24c] [6-24]

9 57 Looking at these expressions, our first ifficulty is apparent. The energy-momentum tensor is not symmetric, since we have: T i0 T 0i [6-25] whereas a symmetric tensor ha been expecte from the relativistic iscussion earlier in this chapter. Techniques exist to symmetrise an energy-momentum tensor 4. However, as later analysis will show, the present lack of symmetry shoul not simply be remove in this way. Instea, its significance shoul an will be examine carefully. This matter will be resolve in the next chapter. Leaving this point an continuing on, we want to see whether the energy-momentum tensor above yiels conservation by satisfying equations [6-13] an [6-14]. Of these two equations, it will be sufficient to iscuss [6-13]: iν ν T fiel = ρ pi (i = 1,2,3) For our present purpose, the ivergence on the left of this equation nees to be split into separate space an time components, so that we have: ij j T fiel i0 + t T fiel = ρ pi [6-26] To check whether the energy-momentum tensor summarize in equations [6-24] is consistent with this conservation conition, expressions [6-24a] an [6-24b] will be inserte into the left han sie of [6-26]. This is one in Appenix 5. For the usual nonrelativistic situation of a single particle with no creation or annihilation, the following result is obtaine: ij j T fiel i0 + t T fiel =0 [6-27] Hence, unlike the scalar an vector fiel cases iscusse in the previous chapter, the ivergence of the fiel's energy-momentum tensor is zero here even when there is fiel-

10 58 particle interaction. This result is not consistent with equation [6-26] an forbis energy an momentum transfer between the fiel an the particle. Since we appear to nee an equation like [6-26] to hol, we are face with a secon ifficulty. Of course, this zero ivergence of the Schroinger energy-momentum tensor is well known an is one reason why people have conclue that Bohm's moel is not compatible with energy an momentum conservation 5. Nevertheless, Noether's theorem assures us that the esire conservation must exist for the Lagrangian ensity we have chosen. A closer examination of Noether's theorem will be neee to resolve this problem. However, some insight into the course to be followe can be gaine by consiering another well-known case, viz., an electromagnetic fiel an a Dirac spinor fiel in interaction. Before the interaction between these two fiels begins, the ivergences of the tensors T µν electromag an T µν Dirac are, of course, separately zero: ν (T µν electromag) = 0 [6-28] ν (T µν Dirac) = 0 [6-29] With the onset of the interaction, the expressions for T µν electromag an T µν Dirac o not change (i.e., they each still look the same), but their iniviual ivergences are no longer zero 6. Now, from our experience with the classical cases of a particle in a scalar or vector fiel, one might expect the following overall conition to hol: ν (T µν electromag + T µν Dirac) = 0 by analogy with [6-15]. However, this is not the case. The correct overall ivergence equation contains an extra term, as follows 7 : ν (T µν electromag + T µν Dirac + T µν interaction) = 0 [6-30] 4 See, e.g., Ch. 3, Sec. 4 in Barut A., Electroynamics an Classical Theory of Fiels an Particles. Macmillan, N.Y. (1964). 5 See p. 115 in Hollan P.R., The Quantum Theory of Motion. Cambrige University Press (1995).

11 59 This example emonstrates that the appearance of an aitional term T µν interaction may sometimes be neee to achieve conservation. In so oing it suggests a way in which our secon ifficulty may be tackle. Pursuing this possible approach, it seems at first sight that a suitable extra term T µν interaction coul be obtaine simply by applying the square bracket in equation [6-23] to the interaction part of [5-1] to construct the tentative expression: µν T interaction =[ µ φ ( ν φ) + µ φ * ( ν φ * ) gµν ] interaction [6-31] This oes not lea to the correct result, however, as will be seen in the next chapter. Instea the problem will be resolve more systematically by showing from first principles the necessity of an extra term T µν interaction an the precise form it must take. 6 This change occurs ue to the appearance of source terms in the two fiel equations use in evaluating the ivergences. 7 See Ch. 3, Sec. 4 in Rzewuski J., Fiel Theory: Vol. 1, Classical Theory. Iliffe Books, Lonon (1967).