On the dissimilarity between the algorithms for computing the symmetric energy-momentum tensor in ordinary field theory and general relativity

NOTE BREVI

On the dissimilarity between the algorithms for computing

the symmetric energy-momentum tensor

in ordinary field theory and general relativity

A. ACCIOLY(1)(*) and H. MUKAI(2)

(1_{) Instituto de Física Teórica, Universidade Estadual Paulista} Rua Pamplona,145, 01405-900 Sa˜o Paulo, SP, Brazil

(2_{) Departamento de Física, Fundac¸a˜o Universidade Estadual de Maringá} Av. Colombo 5790, 87020-900, Maringá, Pr, Brazil

(ricevuto il 21 Ottobre 1996; approvato il 3 Dicembre 1996)

Summary. — The recipe used to compute the symmetric energy-momentum tensor

in the framework of ordinary field theory bears little resemblance to that used in the context of general relativity, if any. We show that if one starts from the field equations instead of the Lagrangian density, one obtains a unified algorithm for computing the symmetric energy-momentum tensor in the sense that it can be used for both usual field theory and general relativity.

PACS 03.50 – Classical field theory. PACS 04.20 – Classical general relativity.

There are at least two good reasons for wanting the canonical energy-momentum tensor to be symmetric [1, 2]. One is that the conservation of angular momentum requires a symmetric energy-momentum tensor.

The second reason for wanting a symmetric energy-momentum tensor is that, according to the theory of general relativity, it is this tensor which determines the curvature of space, according to Einstein field equations

Rmn 2 1

2 g mn

R 42 kTmn_, where k 48pG is the Einstein constant.

However, the usual prescription to compute this tensor in the framework of ordinary field theory bears little resemblance to that used in the context of general relativity. Where does such a dissimilarity between the two mentioned algorithms come from? As we shall show in the following, if one starts from the field equations instead of

(*) E-mail: acciolyHaxp.ift.unesp.br

the Lagrangian density (which, incidentally, is the way things are usually done) one obtains a simple algorithm for computing the symmetric energy-momentum tensor which can be used for both ordinary field theory and general relativity.

We will work in natural units where ˇ 4c41, and use the Heaviside-Lorentz units with c replaced by unity for the electromagnetic theory. Our conventions for relativity follow all recent field theory texts. We use the metric tensor

hmn4 hmn4

u

1 0 0 0 0 21 0 0 0 0 21 0 0 0 0 21

v

,

with Greek indices running over 0 , 1 , 2 , 3. As far as general relativity is concerned, our conventions are to use a metric with signature (1222) and define the Riemann and Ricci tensors as Rlmnr 4 2 ¯nGrlm1 ¯mGrln2 GslmGrsn1 GslnGrsm and Rmn4 Rrmnr, respectively.

To keep things simple, let us assume, without loosing any generality, that Tmn_{is the} symmetric energy-momentum tensor related to some scalar field C which obeys the field equation O× C40, where O× is a suitable operator that acts on the space of the C-field. Actually, it does not matter whether the field in hand is a tensorial or spinorial field, the procedure is the same. Now, as is well known from field theory, if we compute the derivative of Tmn _{with respect to n, we obtain}

¯nTmn4 (¯mC) O× C . Since O× C40, we find that

¯nTmn4 0 ,

which is nothing but the well-known differential conservation law which the symmetric energy-momentum tensor is known to obey. If we reverse the preceding argument, we promptly obtain an algorithm for computing this tensor. To find the symmetric energy-momentum tensor for some field multiply the field equation related to this field by a suitable (covariant) derivative of the field itself so that the resulting expression contains only one free space-time index and then rewrite it as a (covariant) four-divergence. In case the Tmnso obtained is not yet symmetric, simply symmetrize it. The ensuing examples will certainly elucidate how, in practice, the method works.

To begin with we consider the higher-order electrodynamics proposed by Podolsky in the early forties [3]. The corresponding field equations are

( 1 1a2

p) ¯nFmn4 0 , (1)

F[mn , a]4 0 ,

(2)

where a is a real parameter with dimension of length, Fmn _{is the electromagnetic field} tensor and p4¯b¯b.

To find Tmn_{for Podolsky-generalized electrodynamics we multiply (1) by F}

(2) yields ¯n

k

FamFmn1 1 4da n_F ruFru

l

1 Fama2p¯nFmn4 0 . The second term can be transformed as follows:

a2_F

amp¯nFmn4 a2[¯n(Famp Fmn) 2¯nFamp Fmn] . From (2) and the above equation we obtain

a2_F amp¯nFmn4 a2

k

¯n(Famp Fmn) 1¯a

g

1 2Fmnp F mn

h

2 1 2F mn p¯aFmn

l

. On the other hand,

1 2F mn p¯aFmn4 2 ¯n(Fmnp Fam) 1¯nFmnp Fam. Similarly, ¯nFmnp Fam4 2 ¯m(¯nFmn¯bFba) 2¯a

g

1 2 ¯nF mn ¯bFmb

h

. Hence, a2Famp¯nFmn4 4 a2¯n

k

Famp Fmn1 Fmnp Fam2 ¯bFab¯mFnm1 1 2da n_(F rup Fru1 ¯bFru¯bFru)

l

. The symmetric energy-momentum tensor is thus given by

Tan_{4 F}amFmn1 1 4h an FruFru1 a2 2 h an [Frup F ru 1 ¯uF ru ¯bFrb] 2 2a2[Fam p Fnm1 Fnmp Fam1 ¯bFab¯mFnm] . Now we apply our method to the Dirac field which is known to obey the field equations [4]

igm¯mC(x) 2mC(x) 40 , (3)

¯mC(x) igm1 m C (x) 4 0 , (4)

where C f C†_g0 _{is the adjoint spinor to C. The indices labelling spinor components}

and matrix elements were suppressed for the sake of simplicity. To find Tmn_{we multiply} (3) and (4) by ¯aC(x) and ¯aC(x), in this order:

¯aC igm¯mC 2m¯aC C 40 , (5)

¯mC igm¯aC 1mC ¯aC 40 . (6)

Equation (5) can be rewritten as

¯m(¯aC igmC) 2¯a(¯mC igmC) 1¯mC igm¯aC 2m¯aC C 40 . From (6) and the above equation we then have

¯m(¯aC igmC) 2¯a(¯bC igbC 1mC C) 40 . (7)

From (7) we promptly find

¯m(C igm¯aC) 2¯a(C igb¯bC 2mC C) 40 . (8)

To ensure the hermicity of the Dirac energy-momentum tensor we must, of course, compute the difference (8)2(7). The result is

¯m[C igm¯aC 2¯aC igmC 2dam(C igb¯bC 2¯bC igbC 22mC C) ] 40 . So, the symmetric energy-momentum tensor for the Dirac field is given by

Tmn 4 i 4(C g m ¯n_{C 1C g}n¯m_{C 2¯}nC gm C 2¯m_{C g}n_C ) 2 2 hmn

k

i 2(C g b ¯bC 2¯bC gbC) 2mC C

l

. As a last example, we compute the correct field equations for the highly nonlinear electrodynamics generated via gravitational nonminimal coupling proposed by Prasanna [5] in the early seventies. The field equations for the Fmn-field, whereupon Fmn is the electromagnetic field tensor, are

˜nFmn2 2 l˜n(RrumnFru) 40 , (9)

F[mn ; a]4 0 ,

(10)

where l is a suitable coupling constant with dimension of (length)2_{. To construct T}mn_for that theory we multiply (9) by Fam:

Fam˜nFmn2 2 lFam˜n(RrumnFru) 40 . Using (10) the first term can be expanded to

Fam˜nFmn4 ˜n

k

FamFmn1 1 4da

n_F ruFru

l

. Similarly, we obtain for the second term

2 2 lFam˜n(RrumnFru) 42 ˜n

k

2 lRrumnFamFru1 l 2R rumn_F ruFmn

l

1 l 2F ru_Fmn ˜aRrumn. But, l 2F ru_Fmn

In deriving the preceding equation we have used the identities

Rab[gd ; r]4 0 , Ra[bgd]4 0 . Now,

[˜r, ˜u] ˜n(FraFun) 42 Rrmun˜n(FruFmn) . So, the matter tensor is given by

Tan_{4 F}amFmn1 1 4 g an FruFru1 l

m

2 1 2g an RrubgFruFbg1 13 Rrum(aFmn)Fru2 2 ˜r˜u[Fr(aFn) u]

n

. The correct field equations for Prasanna’s electrodynamics are thus

˜n[Fmn2 2 lRrumnFru] 40 , F[mn ; a]4 0 , Gan 1 k

g

Fam_F mn1 1 4g an_F ruFru

h

4 4 lk

m

1 2g an_Rrubg_F ruFbg2 3 Rrum(aFmn)Fru1 2 ˜r˜u[Fr(aFn) u]

n

, where Gan_fRan

2 ( 1 O2 ) Rgan is the Einstein tensor.

R E F E R E N C E S

[1] KAKUM., Quantum Field Theory: A Modern Introduction (Oxford University Press) 1993. [2] RYDER L. H., Quantum Field Theory (Cambridge University Press) 1985.

[3] PODOLSKY B., Phys. Rev., 62 (1942) 68.

[4] MANDL F. and SHAW G., Quantum Field Theory (Wiley) 1984.