Classical gravity with higher derivatives

General Relativity and Gravitation, Vol. 9, No. 4 (1978), pp. 353-371

Classical Gravity with Higher Derivatives 1

K. S. STELLE 2

Department of Physics, Brandeis University, Waltham, Massachusetts 02154

Received May 10, 1977

Abstract

Inclusion of the four-derivative terms fRl~vR#V(-g) 1/2 and fR2(-g) 1/2 into the gravitational action gives a class of effectively multimass models of gravity. In addition to the usual masstess excitations of the field, there are now, for general amounts of the two new terms, massive spin-two and massive scalar excitations, with a total of eight degrees of freedom. The massive spin-two part of the field has negative energy. Specific ratios of the two new terms give models with either the massive tensor or the massive scalar missing, with corre- spondingly fewer degrees of freedom. Tile static, linearized solutions of the field equations are combinations of Newtonian and Yukawa potentials. Owing to the Yukawa form of the corrections, observational evidence sets only very weak restrictions on the new masses. The acceptaNe static metric solutions in the full nonlinear theory are regular at the origin. The dynamical content of the lineadzed field is analyzed by reducing the fourth-order field equations to separated second-order equations, related by coupling to external sources in a fixed ratio. This analysis is carried out into the various helicity components using the transverse-traceless decomposition of the metric.

w Introduction

Upon several occasions, dating back to the early days of general relativity, it has been suggested that the Einstein equations for the gravitational field be

replaced by others involving derivatives of higher than second order. Early

suggestions [1] by Weyl and Eddington concerned an attempt to include the electromagnetic field in a unified geometrical framework, but this line of approach proved unfruitful and was eventually abandoned) Higher-derivative theories were studied in the general context of quantum field theory by Pals and

1 Research supported in part by the National Science Foundation under grant No. PHY-76- 07299.

2Present Address: University of London, King's College, Strand, London SC2R 2LS, England. 3Later suggestions were given in [ 2].

353

0001-7701/78/0900-0353505.00/0 �9 Plenum Publishing Corporation

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Uhlenbeck in 1950 [3]. Later on, quantum field theory also gave rise to suggestions that higher-derivative terms be included in the gravitational Lagrangian in order to permit renormalization of divergences in the quantum corrections to the interactions of matter fields [4] .4 This motivation was further reinforced in recent years by the demonstration that gravitation itself contains nonrenormal. izable divergences when in interaction with matter, [6].

The inclusion of terms proportional to RuvRUV and R 2 in the gravitational Langrangian produces a stabilization of the divergence structure of gravity, allow- ing it to be renormalized, along with its matter couplings. The details of this renormalization have been discussed elsewhere, [7] and will not be considered here. In the present article, the classical features of such higher-derivative models will be discussed.

The early investigators emphasized that the empty space solutions of Einstein's equations, Ruv = 0, are also solutions of the field equations derived from actions like f(-g)l/2RvvRVV or f(-g)X/2R2, and thus they thought that all the classical tests of general relativity were automatically satisfied. It was not until comparatively recently [8] that it was pointed out that although, e.g., the Schwarzschild solution is a solution to the empty space equations, it is not the one that couples to a positive definite matter distribution. In fact, those solutions of purely four-derivative models which do couple to a positive matter source are not asymptotically flat at infinity. One may see this immediately from the linearized theory, where the Green's function is generically (V 2 V 2)-1 ~ r. Similarly, the Schwarzschild solution is ruled out because V2V 2 (r -1 ) ~ V253 (x), which does not correspond to a positive definite matter distribution.

Accordingly, in this work, we shall restrict ourselves to models derived trom actions that include both the Hilbert action f(-g)X/2R and the four-derivative terms. There are only two independent additions that one can possibly make, because of the Gauss-Bonnet relation in four dimensions, (_g)1/2 (Ruvc~RU vc~ -4RuvRUV + R 2) = divs., so that we have only a two-parameter family of field equations (we do not consider a cosmological term). We write the action in the form

I = -j(-g)ll2(aR.vR ~'v - {3R 2 + 7K-2R) (1.1)

where ~2 = 321rG, and a,/3, and 7 are dimensionless numbers. It will turn out that the correct physical value for 3' is 2, as in Einstein's theory.

We shall find that the static spherically symmetric solutions to the field equations derived from (1.1) either reduce asymptotically to the sum of a Newtonian and two Yukawa potentials, or they are unbounded at infinity and must be eliminated by boundary conditions. The masses in these Yukawa potentials are only weakly constrained by the observational evidence to be/>10 -4 cm -1 . Although the magnitude of these effects is negligible at laboratory or astronomical distances, there are some interesting qualitative features. Coupling the lin-

4More recent use of higher derivatives to regularize the stress tensor is reviewed in [5].

CLASSICAL GRAVITY WITH HIGHER DERIVATIVES 355

earized theory to a pressurized fluid distribution shows that the coefficients of the Yukawa potentials depend on the pressure and the size of the distribution. This shows that Birkhoff's theorem is not valid in these models.

The limit of the linearized exterior gravitational field for an extended source tends to a large but finite value at the origin as the size of the source shrinks to zero. A similar feature is maintained in the full theory, where an analysis of the structure of the solutions near the origin shows only a one-parameter family of singular metrics, which is therefore just the Schwarzschild solution already men- tioned, and which is to be discarded because of coupling problems. There are also two other two-parameter families in which the grr and gtt components behave either like r ~ or r 2 . Only the family behaving like r ~ has a regular geom. etry at the origin.

The Yukawa potentials and the failure of Birkhoff's theorem in the static case are complemented by the existence of massive radiation in the dynamical field. Analysis of the field dynamics, both in the action and in the interaction between sources, shows that the gravitational field now has eight degrees of freedom, two corresponding to the usual massless spin-two graviton, five more corresponding to a massive spin-two excitation, and the last to a massive scalar. This decomposition is carried out into the various individual helicity components using the transverse-traceless (TT) decomposition of a symmetric tensor. The massive excitations are separated from the massless ones by introducing auxiliary oscillator variables, in terms of which the Lagrangian may be separated into a sum of Lagrangians for the Einstein, Pauli-Fierz, and massive scalar fields.

In carrying out the above separation into different masses and helicities, one feature is inescapable: The massive spin-two field comes with a minus sign relative to the other fields, both in the oscillator variable Lagrangian and in the radiation. Classically, this means that the corresponding excitation has negative energy. This presents the single major stumbling block preventing the adoption of these models as physical theories. In the quantum theory, the negative energy must be traded for quantum states of positive energy but negative metric in the state vector space [7]. ClassicaUy, the negative energy leads to a breakdown of causality in that one can make superpositions of waves with group velocities greater than the speed of light.

In our notation, we use the signature (-+++). The curvature tensor is de- k fined by R~cw -- OvI'~a + �9 �9 ", and the Ricci tensor by Ruv = R ~ x u.

w Field Equations and Bianchi Identities

The field equations following from the action (1.1), supplemented by a matter action, are

Huv = ( a - 2~)R;u;v - aRuv;n;n ( ! a - - 2 2~)guvR;n;n + 2e~RnXRunvx

_ 2 ~ R R u v _ 1 ~guv (r - ~R ~) + 7K-2Ruv - 17K-2guvR = - I ~Tu~

(2.1)

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In deriving these equations, the result has been simplified by use of the commutation relations for covariant derivatives.

As always with equations derived from an action possessing a local gauge invariance, the equations (2. I) are linked by generalized Bianchi identities,

HU";v - 0 (2.2)

It may be directly verified that the identities (2.2) are consequences of the usual uncontracted Bianchi identities and the commutation relations. The identities (2.2) enforce covariant conservation of the source, TUV;v = 0, so the equations of motion for a particle are a consequence of the gravitational field equations in these models just as in Einstein's theory.

Another familiar feature is that the Bianchi identities are necessary to ensure a consistent evolution of the field, given a certain initial set of Cauchy data. However, we shall see in Section 6 that the higher derivatives complicate somewhat the identification of the proper gauge and constraint variables, which involve various linear combinations of the guo and gij and their derivatives. At this point, we shall give only a brief sketch.

Writing out in fuller detail the equation (2.2),

ao H u ~ = -3i H u i - I'~uh HUn - Plx HUn (2.3)

we note that the right-hand side has no time derivatives higher than (30) 4 , so the equations H u~ = - �89 u~ involve at most (3o) 3 . Although some components of the metric will be seen to obey differential equations of fourth order in time derivatives, this is not true of all components, and the (~0) field equations are a mixture of constraints and lower-order dynamical equations. There is in fact only one constraint now, and, as expected, it involves the instantaneous New- tonian part of the field. If this constraint is imposed at the outset on the Cauchy data given on some initial hypersurface, then the dynamical fourth-order equations Hij = - iT.-. 2 ~1, together with the dynamical combinations of the (/~0) equations and energy-momentum conservation of the source, may be inserted into the Bianchi identities to show that the time derivative of the constraint vanishes, so it is preserved during the dynamical evolution of the field.

General covariance of the equations (2.1) implies that the evolution of some parts of the fieldis not determined by these equations. As ~usual, one must specify coordinate conditions to give a complete determination of the metric guu. For example, the standard harmonic coordinate conditions 3 u [(_g)l/2gU~,] = 0 are perfectly admissible. However, since tl~e Cauchy data now involve up to third time derivatives of some parts of the metric, it is now equally natural to use coordinate conditions that relate the gauge parts of the metric to any of this Cauchy data, including the higher time derivatives.

We shall see in the following that the various derivatives ofgu~, may be recombined into new fields which obey only second-order equations, effectively producing a model with both massive and massless excitations. Specification of


higher time derivatives of the metric as part o f the Canchy data then corresponds to specifying the values of these auxiliary fields and their first derivatives in a more familiar fashion.

w (3): Linearized Static Solutions

In order to extract the physical content of equations (2.1) for the static spherically symmetric gravitational field, we choose to work in Schwarzschfld coordinates, in which the invariant interval takes the form

ds 2 = A(r )dr 2 + r2(dO 2 + sin20d~ 2) - B(r)d t 2 (3.1)

In view of the complexity of (2.1), we begin by linearizing the field equations. In our coordinates, we let

A (r)= 1 + W(r) (3.2a)

B(r) = 1 + V(r) (3.2b)

and then keep only terms linear in W or V in the equations. The results are

HLrr = - ( a - 4/3)r-Zg ' ' ' - 2 ( a - 4 3 ) r - I V " + 2 ( a - 43)r-3V ' - 7K-2r -1 g '

+ (3a - 8/3)r -2 W" - 2 (3a - 8/3)r -4 W + 7•-2r-2 W (3.3a)

H~o = - �89 (a - 4/3)r=V iv - ~ (a - 4/3)rV'" + (a - 4/3) V" - (a - 4/3)r -1V'

= �89 �89 1 ( 3 a - 8/3)rW'"

- (3a - 8/3)r -1W' + 2(3a - 8/3)r -~ W+ �89 ' (3.3b)

HLtt=(a - 2/3)V iv + 4 ( a - 2~3)r-iV ' ' ' - ( a - 4/3)r-lW ' ' ' - ( a - 4/3)r-2W ' '

+ 2(a - 4/3)r -a W' - 2 (a - 4/3)r -4 W - 7~-2r-1 W' - 3'K =2r-2 W (3.3c)

In these equations, primes and roman numerals indicate derivatives with respect to r.

To simplify finding the general solution to H~v = 0, it is convenient to define the quantity

Y = r -2 (r W)' (3.4)

and to consider the linear combinations//~/;~ = A - 1 H ~ + 2r-2H~o - B-1HLtt and L i L t -1 L -2 L - I L H i - H ~ =A H ~ r + 2 r H o o + B Ht t :

H~" = 2(3/3 - a)V2V2V - 7K - 2 7 2 V - 4(3/3 - a )V2Y+ 27K -2 Y (3.5a)

H Li - HtLt = 2/3V2V2V - 7~-2V2V+ 2(a - 2/3)V2Y (3.5b)

Standard transform methods may now be used to solve these generally for V and Y, and then the results can be substituted back into, e.g., HrLr to determine W.

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The result is the general solution to the linearized equations H~v = 0:

c2,O era2 r e-ma r emo r e- too r V = C + + C 2 + - + C 2- ~ + C O+ - - + C o- .... (3.6a)

r r r r r

c2,o C2+em~r C2-e-m2r CO+emo r CO-e-mor W = - + +

r r r r r

1/-,2+~ ~rn~r _ 1 + ~ . . . . 2 ~ . C2-m2e - m 2 r - Co+too em~ + C~ e - m ~ (3.6b)

where m2 = ")'I/2(ot~2fl12, m o = 71/2 [2(3/3 - a ) r 2 ] -1/2, and C, C 2'~ , C 2+, C 2-, C O+ and C o- are arbitrary constants.

The rising exponentials r -1 e +mr occur in (3.6) because they are just as good solutions of the static Klein-Gordon equation (V 2 - m2)~ = 0 as are the r -1 e -mr.

Of course, as in the Klein-Gordon equation, they must be eliminated by imposing boundary conditions at infinity. Inside an empty spherical shell, however, they would be present in terms of the form e -m t(mlr)-I sinh(mr), with l the radius of the shell. When one counts the number of parameters that must be determined by source coupling and boundary conditions at infinity, the parameter C should be discarded, since it may be absorbed into a trivial rescaling of the time coordinate t. Consequently, the general solution to the linearized static spherically symmetric equations is a five-parameter family, before imposition of boundary conditions.

In the case of a point particle with T~v = ~ ~ ~ a (x), the gravitational field takes the form

-K2M r : M e -m~r K2M e -m~ v = + - - ( 3 . 7 )

87rqr 6~r7 r 24rr7 r

Comparison at infinity with the Newtonian result V = -2GMr -1 shows that the correct physical value of 7 is 2. As expected on dimensional grounds, the higher- derivative terms only produce an appreciable effect at small distances from the source of the field, at a scale set by m2 and mo. At the origin, the Newtonian 1/r is canceled, and (3.7) tends to the f'mite value K2M(247rT) -1 (rno - 4m2). We shall investigate the regularity of the metric in the full theory in the next section.

The peculiarities of the static field coupling in these models are better revealed if, instead of a point particle, we couple to an extended source with some structure. For this purpose, we consider a spherical distribution of a perfect fluid with an internal pressure that is maintained by an elastic membrane. The stress tensor for this source is

T•v = [P - �89 l e5 • - l)1 r 2 [ P - �89 (r - l )]r2sin20 (3.8)

3M(41rl 3 )- 1 t


where l is the radius of the source, M is its total energy, and P is the internal pressure. The delta functions in Too and T~r correspond to membrane tension, and are necessary to ensure equilibrium of the source, TUV;v = 0, which in spherical coordinates reads

( Trr)' + 2r -1Trr - rToo - rsin2OTr = 0 (3.9)

The exterior field for the source (3.8) is

tc2M tc2e-m2r ~ M [lcosh(m21) sinh(m2/)l - - d I .

[s (m.O lcosh(m.O 1'sinh(m.01l -P[ m~ m~ + 3m--~ JJ

ti2e -m~ ~ M [.lcosh(mol ) sinh(mol)]

2rrr [ ~ [ m2o ~oo J

[sinh(mol) lcosh(mol) 12sinh(mol)] 1 - P [ ~o 3 m~ + 3~o JJ (3.10)

In (3.10), the existence of the quantities rn2 and mo with the dimensions of mass has enabled the gravitational coupling to involve the source pressure and radius as well as the total energy M. In the limit l --->0, the pressure dependence drops out, and (3.10) tends to the field (3.7) of a point particle. The three. parameter family of static solutions (3.10) shows that Birkhoff's theorem does not hold for these models. We shall see this to be the case in the dynamical field as well, for the Yukawa potentials in (3.7) and (3.10) correspond to the virtual exchange of massive particles and have a real counterpart in massive spin-two and spin-zero radiation. An essential characteristic of this radiation is already forshadowed in the repulsive character of the potential +r -1 e-m~r: The massive spin-two field must have negative energy. This is both the cause of the most interesting features of these models and also the impediment to their physical ap- plication. We note at this point that (3.7) and (3.9) only give an acceptable Newtonian limit for m2 and mo real, so that there are only falling exponentials, and not oscillating 1/r terms at infinity. This corresponds to the absence of tachyons (both positive and negative energy) in the dynamical field. Also, we note that it is possible to remove either of the two massive fields by picking combinations of ~ and ~ that make m 2 or rn o infinite, choosing, respectively, ce = 0 or ce = 3/L

The static field (3.7) may be considered as the gravitational potential of a star for making comparisons with observation to determine bounds on the values of me and mo. However, astronomical tests are useless here, as may be seen by considering the motion of Mercury, for which the orbital precession is known to about one part in 10 9 . The corrections due to the higher-derivative terms are of order e -mr, where for r we must use the radius of Mercury's orbit ~ 5 X 10 6

360 STELLE

kan. This results in a lower bound on the masses ~ 5 • 10 -11 cm -1 . Since there is no discontinuity in the coupling to light in the limit rn2, rno ~oo, measurements of light bending by the sun do no better. Laboratory experiments on the validity of the Newtonian 1/r 2 force seem much more promising, for there one can verify agreement on the scale ~ 1 m. Unfortunately, high-precision Cavendish- type experiments appear not to have been numerous. We mention, however, the interesting measurements of Long [9], who reports a deviation from 1/r 2 on the laboratory scale, corresponding to a repulsive interaction which would require a mass ~ 1 X 10 -4 cm -~ .s This is still a long way from the quantum domain "-~ 1013 cm-1, where one might expect deviations from Newton's law.

w Static Solutions near the Origin in the Full Theory

The five-parameter family of solutions to the linearized equations found in the last section may not be expected to persist as such in the full theory, where linear superposition is no longer possible. The Schwarzsehild solution is a solution to our exact field equations, as are all empty space solutions of Einstein's equations. It is just a one-parameter family, however. The other solutions should be expected to split up into smaller families also.

To study this question, and learn about the regularity properties of the static spherically symmetric solutions at the origin, we consider power series expansions of the two metrical functions A (r) and B(r) in (3.1), with the orders of the lowest terms themselves variables to be determined. Thus, we let

A(r ) = asr s + as+lr s+l + as+2r s+2 + " " (4.1a)

B(r) = r t + bt+l r t+l + bt+2 r t+2 +" "" (4.1b)

where a s must be nonvanishing. The coefficient bt has been arbitrarily set to 1 by rescaling the time coordinate. Upon inserting these expansions into the field equations (2.1), then equating the coefficients of the lowest.order terms to zero, we obtain the following indicial equations:

[ r - ' - ~ I t " 3s2t 2 st a st 2 3s2t st 9s 2 Hrr =a / . . . . ~ + - - + - - - + 2 t 24 + 2 t

[ as 16 16 8 4 4 2 4

- 3 s + 5 ) - 4 r - 4 - a s r s - ' ]

_ _ _ ( t ~ - s t + 2 t - 4 s + 4 ) + - - + - - + - - + [31_ a s 8 4 2

- I

- 12r -4 - 2asrS-4] = 0 (4.2a)

s For observational constraints on deviation from 1 #2 at large distances, cf [ 10].


[ r - 2 - 2 s [ t 4 st 3 9s2t 2 3s3t t 3 3st 2 3s2t 9S 3 1too 1 _ _ _ + _ _ _ _ _ + - - - - + - - - + - -

| a - - - ~ 16 4 16 8 2 2 4 4

- t 2-3st+ 21s2 2 s - 5 + ~ (2s- 2 t + 4 ) + r -2 4 a s

- ~ L a---~s ( t 2 - s t + 2 t - 4 s + 4 ) -+8 8 2 + - - - 4 5s-

r-2 -s ] + - - ( - 6 t + 6 " s + 1 2 ) + 2 r -2 =0 (4.2b)

as

Htt= a L[ rt-2---S-4a2s .(--16 t4 +-st38 + ~1 ls2t 216 3s3t4 + 5st24 5s2t4 3s32

5st 7s 2 ) 4r t-s-4 + + 2 t + s + 3 + r t-

2 4 a s

- - + - - + - - _ _ _ + - - + -/3L as 2 (t 2 - s t + 2 t - 4 s + 4 ) 8 8 2 4 2

12r t - s - 4 ] + 2r t-4j = 0 (4.2c)

as

We shall not present the details of solving these algebraic equations, and shall only give the result. We note, though, that a useful check on the algebra when dealing with expressions like (4.2) is provided by the r component of the Bianchi identity, which in these coordinates i~

[Hrr~' 2Hrr B'Hrr 211oo B'Htt + Ar + 2AB r 3 +--2B 2 =-- 0 (4.3)

For general values of the parameters r and/3, the only solutions to (4.2) are

(s, t) = (1, - 1) with al undetermined (4.4a)

(s, t) = (0, 0) with ao = 1 (4.4b)

(s, 0 = (2, 2) with a2 undetermined (4.4c)

The indicial set (4.4a) corresponds to the Schwarzschild solution, which we anticipated. The other two sets correspond to families of solutions to the field equations with all components of the metric regular at the origin. Thus, one of the main features of the linearized solutions is preserved in the full theory. Regularity of the metric at the origin does not assure that the geometry is nonsingular, however, as one may see by comparing the leading nonvanishing terms of the curvature invariant R~ v~R ~v~:

(s, t) = (1,- 1) ~Ruv~R~W~ = 8a~2r -6 + ' ' " (4.5a)

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(S, t) = (0, O) ~ guv~r162 = 8a~ + 12b~ + " " (4.5b)

(s, t) = (2, 2) --'--~Ruu~RUUC~;s = 20a~2r -s + "" �9 (4.5c)

In arriving at (4.5b), we have used the fact that (s, t) = (0, 0) -+al = bl = 0, as we shall see below. This case, where the metrical functions A (r) and B(r) tend to constants at the origin, is the only one in which the geometry itself is nonsingular. The present methods do not permit us to relate the behavior of the metric at the origin with that at infinity. To determine which of our families of solutions can match onto solutions well-behaved at infinity would appear to require a com- puter solution, for it seems unlikely that the exact metric can be written in terms of known functions. In the special model with a = 3/3, where the scalar part of the field is absent and the higher-derivative terms in the action are just the square of the Weyl tensor, the exact problem might be reduced in complexity, for there the trace of (2.1) gives R = 0. With an appropriate choice of coordinates, this might be used to obtain a first integral.

To conclude this section, we present the equations that determine the first few coefficients in the expansions (4.1), for each of the three indicial cases, in order to determine the number of parameters that are left free in each case. For (s, t) = (1 , -1) , define L = b o/al + a2/a~, K = b o/2al - a2 [2al - 1 ; then

I-lrr = ar -4 (-3L + 6K) -/3r -4 (-3L + 12K) + O(r -a) (4.6a)

1 -1 -3 Hoo= ~ aal r (7L - 20K)- /3a~lr -a ( 6L- 24K)+ O(r -2) (4.6b)

Htt = aa~lr -6 (2K) -/3a~Xr -6 (-3L + 12K) + O(r -s) (4.6c)

These equations require L = K = 0, leaving a one-parameter family which consists only of the Schwarzschild solution. For (s, t) = (0, 0),

Hrr=t~[r-a(-6al + 2 b l ) + l r - 2 ( - a l + 3bl) (al +bx)] -/3r -a (-16al + 8 b l )

+ O(r -1) (4.7a)

Hoo =a[r -x (3a l - b l ) - l b l ( 3 a l - bl)] -/3[r-X(8al - 4bx) + bl

(-4al + 2bl)] + O(r) (4.7b)

Ht t= �88 - b l ) ( a l + b a ) - 2/3r -2 (2al - b l ) ( a l + b l ) + O(r -1)

(4.7c)

These equations require al = bx = 0, but leave a2 and b2 undetermined, giving a two-parameter family. For (s, t ) = (2, 2), define N = aa/a] - ba/a2 ; then

Hrr = 12(3/3 - ~)r-SN+ O(r -4) (4.8a)

Hoo = - 16(3/3 - ct)ai 1 r -SN+ O(r -4) (4.8b)

Htt = 16(3/3 - a )a i I r -SN + O(r -4) (4.8c)

Here we must have N = 0, so aa/a2 = ba, again leaving a two-parameter family.


The five-parameter family of solutions to the linearized theory splits up into a one- and two two-parameter families. The one-parameter family consists of the Schwarzschild solution itself, and we reject this because it does not couple to positive definite matter. In the other two families, boundary conditions remain to be imposed at infinity. In at least one of these, we expect that the condition of asymptotic flatness can be met. Whether this is the (s, t) = (0, 0) family with nonsingular geometry at the origin, or the (2, 2) family, remains an open question. Note that after imposition of the boundary condition at infinity, it is likely that only one parameter will be left free. This is not surprising, for in the linearized theory, (3.7) shows that for positive definite matter concentrated in a point at the origin, only the total energy M is needed to determine the field.

w Dynamics: Separation o f the Masses

We now turn to the dynamical correlate of the Yukawa potentials encoun- tered in the static field. Here again we shall work in the linearized theory, starting with the appropriate action, which is the part of (1.1) that is bilinear in the field huv = ~-1 (guy - rluv). The Abelian gauge invariance of the linearized action is made manifest by writing it in the form

I = / a 4 x { - l h"'( aK:D: - "~n~(:).,- u,p,,hP~ + l hUU[2(3~- a)K2El: - 7]

E3 e ( ~ 1 7 6 1 7 6 � 89 "v} (5.1)

where the two completely transverse projectors [11] 6p#vpo(2) . . . . o.,a p(o-S)#vpo are defined in terms of Our = rluv -couv and c%z, = 3.3v/E] 2 ,

= _ p ( O - s ) ( 5 . 2 a ) p(2).vpa 21 (O.oOv ~ + OuaOvp ) _ - . vpa

p (O- s )_ I O.vOp• (5.2b) U u p o -

Raising and lowering of indices is done with the Minkowski space metric r/up here. Note that the gauge invariance of (5.1) involves the absence of differential operators containing 6% v. An immediate consequence is the absence of the spin- one part o f h u v in the action.

The dynamical content of the linearized model is first revealed in a Lorentz covariant form by writing down the interaction &,.t, r-r(z)a7,.v(1) between

2 . . . . U v,,.~ pa J~ two conserved sources T (1) and T (2) In momentum space, this interaction is ,up Uv.

K: [(T(1)T u"(2)- 1TP(1)T~ ! a h (T(2).~T~VO) = __ 2*p *(~ 2 u.~ po: 27 L k =

T(1) T .V(2) _ 17"p (1)7"cr(2),~ UP 3~p ~o ]

Id +m~ 6The use of such projectors has become common in dealing with tensor fields. For a general

discussion of tensor Lagrangians using them, of. [ 12].

T~ 0)T~ (2) ]

+ k -5 + m--~o J (5.3)

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This expression has been written in terms of single poles in k 2 by separating multiple poles like [k2(k 2 + m2)] -1 into partial fractions. The retarded interactions with k 2 = -m22 and with k 2 = - m~ show that we do have massive radiation now, in addition to the usual interaction propagating at the speed of light. We take note once again of the minus sign associated with the pole at k 2 = -m~. This sign is an unavoidable consequence of the separation into partial fractions. There is already an indication of the spin content of the models in the numerators of (5.3). The first term is the familiar form which corresponds to the exchange of a massless spin-two field. The second term has the form that results from the exchange of a massive spin-two field, and the third from the exchange of a massive scalar.

The spin and mass content indicated in (5.3) may be made explicit in the action by introducing auxiliary variables. This is necessary in any case before one can write down a canonical first-order form, for in (5.1) there are more derivatives than fields. There are a number of ways this may be done; the classical one is due to Ostrogradski [ 13], and defines the auxiliary variables by qi = (at)i-l c l, arriving at a Hamiltonian

N - 1

H = ~ Piq i+ 1 + P N q N - L ,

where N is the order of the highest time derivative occurring in L. The result of applying this method to (5.1) is quite clumsy and does not reveal the spin and mass content as we require. Instead, we take another approach which maintains explicit Lorentz invariance and works directly with the Lagrangian without going to Hamiltonian form, reducing (5.1) to a system of independent second-order actions in which the fields are coupled to external sources in a fixed ratio. This method was used in [3] for scalar fields, where it was called decomposition into oscillator variables.

Because the spin and mass content of (5.1) is somewhat complicated, we shall explain the procedure first using a simpler gauge model, the Maxwell field with higher derivatives. Consider the action

Im = f d'x(- 1FuvFUV + �88 2 FurlS2 Fuv + AujU) (5.4)

where Fur = 3uAv - OvA u. Corresponding to (5.3), we.have the interaction be. tween conserved sources

A. (j(2))j,.0) -__ / ~ .iO)j.(2) k2 kS + mS (5.5)

so we see that there are massive degrees of freedom in this field too. To display them in covariant form in the action, we first introduce an auxiliary variable


Zu, with the action now reading

f a ( m2 m2 Im 4x 1AuD2ZU + �89 - --8- AuAU +--4- AuZU

- ~ - ZuZU + Av/v (5.6)

Varying Zu gives

Z v = 2m-2n2Au - 2rn-2Ou~vA v +A v (5..7)

and the coupled second-order equations from (5.6) are fully equivalent to the fourth-order equations from (5.4). The system (5.6) now separates cleanly into the actions for two fields, related only by the fixed ratio of their couplings to external sources, when one makes the change of variables

A u = C u + a u (5.8a)

z~ = c~ - B~ (5.8b)

In terms of C u and Bu, the action now becomes

f d ' 2 # 1 B#-~2 _ 1 m 4X [gC.[~ C +�89 2 - �89 g(O. , �89

+ (c. + B.)/"] (5.9) This is the difference of the Maxwell action for C u and the Proca action for Bu. The relative minus sign is again an essential feature. The coupling between two conserved currents that follows from (5.9) is just (5.5), as it must be. Note that consistent coupling of j u to the massless field Cu requires that/u be conserved, even though it is also coupled to a massive field Bu, which does not make such a requirement. The original Abelian gauge invariance of (5.4) has been replaced by the Abelian invariance of the C u field alone in (5.9). The consequences for coupling to external currents are the same. Self-coupling may be handled similarly to the coupling to ]u above. Gauge-invariant self-coupling terms like (FuvFUV) = in (5.4) can be replaced in (5.9) by the same structures with Cu + Bu substituted for Au.

Returning now to (5.1), the separation of massive from massless fields in the gravitational action proceeds directly along the lines described above. We skip the intermediate nondiagonal step and find, replacing huv by r + ]guy,

~/2 ~2t3 I= f d'*x "rZe(4)uv) - ')"g'E(]~gv) + ~ ~'l~v ~'#v 4a(413 - a)

q + - (~u. + 2u . )T" (5.10) 2

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where s is the linearized Einstein Lagrangian,

d~E(~/zv ) = I 2 - 1 .~.r--12.tp I eh#vT~ :a .~h I d ~ V ~ ~ ~ p "~uv IS] (~uv -~q~u ~p + ~ . vuuv~)~, - ~ . vpvvw u

(5.11)

The system (5.10) is the difference of the Einstein action for ~buv and a massive spin-two action for ~u~, but it still is not what we require, for the mass term in (5.10) is still not the Pauli-Fierz mass term of a pure spin-two theory, so ~ v also contains a spin-zero part. Normally, the scalar would have negative energy, but in this case the whole ~uu part of the action has the wrong sign, so the scalar actually has positive energy here. To complete the separation of the masses, the ~uv action must be broken up into pure spin-two and scalar parts. This involves the replacement of ~uv by ~uv and • according to

~u~' = x~m' + rluz, X + 2m22 ~u~vX (5.12)

If we were only dealing with massive fields, the last term would have to be included in the coupling to Tar. Since we also have the massless field ~buv , the gradient term may be dropped since it may be absorbed into a gauge transforma- tion of ~buv. Equivalently, it cannot contribute to the coupling because T m, is required to be conserved in order to have consistent coupling to ~uu. The final oscillator variable action is

= f a [ 7m~ (~uv~uv ' x -

- I

37 OuXO ~X - 37 + ~ J 2 --2 m~x~ 2 ((~u,, + ~u~' + rtu,'X) Tg" (5.13)

The action (5.13) leads directly to the interaction between sources (5.3). As expected, it contains a positive energy massless spin two field Cu~, a massive spin two field ~uu with mass mz and negative energy, and a positive energy scalar field X with mass too. As in the higher derivative Maxwell model, the gauge invariance of the original action is replaced by a gauge invariance for the q~u~ field alone. Although the Pauli-Fierz and scalar parts of (5.13) don't require conservation of the source, the Einstein part does. As in the lineafized Einstein theory [14], however, there is an inconsistency in the coupling to a dynamical source, since the T~v of the gravitational field must be coupled to hay as well. This is the road one follows to arrive back at the full nonlinear Einstein equations, and similarly here, where the self-coupling may be handled by substituting cuv+ ~uv + ~1~• for huu in the nonlinear structures in (1.1).

w Elimination o f Nondynamical Variables in the Radiation FieM

We now continue the analysis of the system (5.1) one step further by aban- doning explicit Lorentz covariance and exhibiting the individual helicity degrees


of freedom. The basic tool here is the transverse-traceless decomposition of a symmetric tensor which has been used in the canonical discussion of Einstein's theory [15]. We reproduce the decomposition of a purely spatial tensor hi / in to its transverse-traceless h TT, transverse h T, and longitudinal parts ~i:

h T = hii - V - 2 h i j , i j (6.1a)

h~] = 1 (6i]h T _ V-2hT , i i ) (6.1b)

h T T = ( 6 i k _ V - 2 ~ i O k ) ( g l j _ V - 2 O l O j ) h k l _ 1 _ ~ (~ i j V - 2 ~ i ~ j ) (~k l V -2OkOl )hk l

(6.1c)

(6.1d) ~; = V-~(h i j , i _ 1 V-2hkj , kji)

This decomposition is complete:

hi/= h TT + hiff + ~i, j + ~j, i (6.2)

In Einstein's theory, the decomposition of hij is all that is needed in order to obtain the dynamical variables, for the huo are purely nondynamical there. In our models, however, we know that the variables (6.1) cannot be sufficient because in addition to the helicity --- 2 excitations, the massive fields must contain helicity -+ 1 and 0 and also spin 0. For this reason, we find it necessary to extend the TT decomposition to include the h~o as well. Accordingly we define

~0 = V - 2 ( h o i , i - 1 -2 ~ V hij, ijo ) (6.3a)

h T = hio - V -2 (ho l , li + hil, tO - ~ - 2 h j k , j k i O ) (6.3b)

hoo = hoo - 2~o,o (6.3c)

The variable hi T is transverse and, together with (6.1), we now have a complete decomposition of h u v, for

hio = hiTo + ~i,o + ~o,i (6.4)

h -T-T h T h/to, and h'oo are all left un- This decomposition is also invariant, for "~t , "~ , changed by huv ~h~v + r/~,u + %,~, for arbitrary r~.

As the new variables involve time derivatives of h~ v, it is clear that they can only be relevant for dynamical analysis when the higher time derivatives of h~v are interpreted as independent dynamical quantities other than just the momenta conjugate to the undifferentiated variables. The ~ are the natural gauge variables, the hi T contain the helicity + 1, and linear combinations of h T and ho0 contain the helicity-0 and spin-0, parts of the gravitational field, as we shall see shortly.

Before proceeding to break apart the action into its individual dynamical components, we show their physical effects by reducing the interaction between conserved sources (5.3). Again, the procedure is the same as in the simpler model of Maxwell with higher derivatives, so we first reduce (5.5). The source conserva-

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tion condition in momentum space, k~/~ = 0 for k , = (p, co), enables us to eliminate the contribution from the longitudinal parts of the currents:

A/~ (](2))] tt(l) = jT(1) .jr(2) [jT(1) .jT(2) + m2/p2]o(1)]o(2)] (6.5)

(p2 _ w2) (p2 _ co2 + rn 2)

Similarly, the gravitational interaction between conserved sources (5.3) can be reduced to show only the effects of the real dynamical interactions:

g h ( T ( 2 ) 3 T # V ( 1 ) - g 2 J r, fr(1)r, rr(2)___ 2 " '"u~'~ - 2")' [ (p2 _ w2)

[Ti~rO)T7 tO) + 4m]/p2~i~)~i~ 2) + I(Tr(1)_ 2m22/p2To(~)) (Tr(2) - 2m~/p2To(~))] (p2 _ r + m ~ )

+ (TrO) + m~/p2T~ (Tr(2) + m~/p2T~ + - - . (6.6) 2(p 2 - o~ 2 + m 2) 4a(3/3- a) r2(p2) z J

where ~/o = T/o - V -2 ai~/T]o. This expression displays clearly the effects ot all eight dynamical degrees of freedom in the radiation field. The last term is instantaneous and contributes only to the static Newtonian interaction.

The dynamical parts of the field can be isolated in the action starting from the reduced covariant form (5.13), but we choose instead to return to (5.1) and discuss the spin content of the models directly in the higher derivative version. This will lead us to a better understanding of the Cauchy problem as well. Inser- tion of the decompositions (6.2) and (6.4) into the action (5.1) gives

I= f a+x { l hrrt'OC 2~2 - -~ i] ~, ~ u -"y)O2hTT+8hT[(8 ~- 3t~)K21-12 - )'] [S]2h T

- l h T [ ( 4 [ 3 - a ) ~ 2 D 2 - ~ / ] V Z h o o + ( ~ - 2 ) ~ 2 h o o V 2 V 2 h o o

1 T + ~hio (ag2l--12 _ ~[)v2hToi

+--2r hT T TT T + 4~ h r T r - rhiTo Pio + ~ hooToo (6.7)

In writing (6.7), we have used the conservation of Tuv to eliminate the ~u from the coupling. Since they have automatically dropped out of the free field part of the action, and since the other variables are gauge invariant, the ~u are the natural gauge variables of the linearized system. Note that those variables whose construction involves the huo are not constraint variables even though they are not governed by field equations with fourth time derivatives. The h/To are in fact the dynamical helicity + 1 massive components of the field, and


linear combinations of/~oo and h T make up the helicity-0 and spin-0 components.

The Cauchy data that are required on an initial hypersurface consist of the values of the hi] spatial parts of the field and up to their third time derivatives, plus the values of the huo and their first time derivatives. These Cauchy data are required to be consistent with the 61/5hoo equation from (6.7). In addition, the gauge must be fixed by choosing some arbitrary relations which determine the }u in terms of the other variables and their derivatives. Of course, in practice one would only choose gauge conditions that determine the }u in terms of the given Cauchy data on the initial hypersurface.

Finally, one can split up the four-derivative parts of ~.7) into massless and massive fields, and at the same time diagonalize the (h T, hoo) sector. The result is

_ - rn2)F~i I 4x KffT []2 Kiffr _4 *ii ,-- -~ FT ( E3=-

~/ 2 V : - Z Fo(E32 - m[)Fo +TGo(E3 - r a g ) G o - D o V2Do 4

tr tom2 F~. Toi + K + --2 (KTT + FiT~T) TiTT - (2V2) 1/2 2(6)1/= (Fo + Go) T T

K [ l { 2 m [ m~ ) + ( 8/3-3a~1/2 ~__o] } +-~ V=k6,/= Fo "- 6-7/2/2 Go \ ~ ' _ - - ~ ) ] Too (6.8)

where the new fields are related to the old by

hT.T = ~TT + p T..T (6.9a) tI *~tl - * tl

h~oi - 2m2 Fo~ (6.98) (272)1/2

2 h r = ~ (Fo + Go) (6.9c)

hoo = - ~ -6-i-]- s Fo - gyfs Go + L~(-~--7) 7 -

It can be verified that (6.8) leads directly to the interaction between sources (6.5). Since we have just a string of Klein-Gordon actions in (6.8), the ultimate reduction to canonical form is trivial. For this system, the Cauchy problem is also trivial: one must specify the values and first time derivatives of the K, F, and G fields. The D field is the constraint variable, so in the absence of sources, we take D = 0.

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w Conclusion

The reduction of the higher-derivative action in the last two sections shows the kinship of these gravitational models to the Pauli-Villars regularization scheme which has been used extensively with other field theories. The higher derivatives similarly improve the divergence structure of the quantum theory, in this case making it renormalizable instead of finite [7]. Unfortunately, they also introduce difficulties of interpretation that make the models unsuitable for a fundamental theory of gravitation. We have seen at every stage that the relative minus sign in the massive spin-two part of the field is an essential feature. The only way to avoid it is to set c~ = 0, killing the massive spin-two degrees of freedom, but also bringing back the divergence problems in the quantum theory. The other special combination that is singled out is ~ = 3/~, which kills the scalar but does not help the positivity problems.

Physically, the negative energy excitations would arise in processes such as gravitational bremsstrahlung, in which a particle would emit negative energy radiation and speed up in the course of its collision with a nucleus. Spontaneous emission and stimulated emission by positive energy incident radiation are kine- maticaUy forbidden, [16] ,7 but emission accompanying the decay of a particle into a heavier one is permitted, and the vacuum is also unstable. The existence of negative energy excitations in a model also leads to a breakdown of causality, since superpositions of positive and negative energy waves can be made for which propagation occurs Outside the light cone [17].

In quantum mechanics, there is an alternative interpretation of the minus sign associated with the massive spin.two field. Positivity of the energy spectrum can be kept if an indefinite metric is introduced into the state vector space. This alternative is actually required in order to obtain renormalizability [7], but one is then faced with the problem of negative probabilities for processes involving an odd number of the massive spin-two quanta. These massive states are presum- ably unstable, however, and one could then go round robin and trade negative probabilities for acausality once more by following Lee and Wick's prescription, [18], retaining unitarity but excluding the negative norm states from the physical sector of the vector space.S

This alternative of dilemmas makes it seem unlikely that higher derivative models of gravitation will find a place in an ultimate theory. On the other hand, the breakdown of causality might only occur on a microscopic scale if the parameters a and ~ were small enough to make the massive fields only important on distance scales near the Planck length ~ 10 -33 cm. It is therefore not unlikely that a higher-derivative model could represent an effective theory of gravitation at more familiar lengths.

7This article contains a general discussion of the alternative attitudes one may take to the minus sign in the higher-derivative Maxwell model.

8For a discussion of the causality problem in this procedure, cL [ 17].


Acknowledgments

The author gratefully acknowledges the original suggestion of this investiga- t ion by Professor Stanley Deser, and also discussions and encouragement throughout the course o f the work. Thanks are also due Professor Bryce DeWitt for stimulating conversations on this subject.

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1. Weyl, H., (1921). Raum-Zeit-Materie, 4th ed. (Springer-Vedag, Berlin) [English transla- lation (1952) Space-Time-Matter (Dover, New York], Chap. IV; Eddington, A. (1924). The Mathematical Theory o f Relativity, 2nd ed. (Cambridge University Press, London), Chap. IV. Later suggestions were:

2. Lanczos, (1938). Ann Math., 39, 842; Buchdahl, H. A. (1948). Proc. Edinburgh Math. Soc., 8, 89.

3. Pals, A., and Uhlenbeck, G. E. (1950).Phys. Rev., 79, 145. 4. Utiyama, R., and DeWitt, B. (1962). J. Math. Phys., 3, 608; DeWitt, B. S. (1965).

Dynamical Theory o f Groups and Fields (Gordon and Breach, New York), Chap. 24; Sakharov, A. D. (1967). DoM. Akad. Nauk SSSR, 177, 70 [(1968). Soy. Phys. Dokl., 12, 1040.]

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van Nieuwenhuizen, P. (1974). Phys. Rev. D, 10, 401,411; Deser, S., van Nieuwenhuizen, P., and Tsao, H. S. (1974). Phys. Rev. D 10, 3337.

7. SteUe, K. S. (1977)Phys. Rev. D. 16,953. 8. Pechlaner, E., and Sexl, R. (1966). Commun. Math. Phys., 2, 165; 3rd paper in [6] ;

Havas, P., (1977). Temple University preprint. 9. Long, D. R. (1976). Nature, 260, 417.

10. Mikkelsen~ D. R., and Newman, M. J. California Institute of Technology preprint No. OAP-475.

11. Rivers, R. J. (1964 ). Nuovo Cimento 34,387. 12. van Nieuwenhuizen, P. (1973). Nucl. Phys., B60, 478. 13. Ostrogradski, M. (1850). "'Mdmoires sur les ~quations differentielles relatives au

probldme des isop(rim~tres," Mem. Acad. St. Petersbourg, VI4, 385; see also Whittaker, E. T. (1937). Analytical Dynamics o f Particles and Rigid Bodies, 4th ed. (Cambridge University Press, London); and also [ 3 ].

14. Deser, S. (1970). Gen. Rel. Gray., 1, 9. 15. Arnowitt, R., Deser, S., and Misner, C. W. (1962), in Gravitation, an introduction to

current research, ed. Witten, L. (Wiley, New York). 16. Matthews, P. T. (1949). Proc. Cambridge Phil. Soc., 45,441. 17. Coleman, S. (1970). "Acau sality" in Su bnuclear Phenomena, ed. Zichichi, A. (Academic

Press, New York). 18. Lee, T. D., and Wick, G. C. (1969). Nucl. Phys., B9, 209; (1969). B10, 1.

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Classical gravity with higher derivatives