Astion. N d i r . 307 (1986) 5, 277-279 277

On the concept of gravitons in General Relativity Theory

H.-H. VON BORZESZKOWSKI, Potsdam-Babelsberg

lI:iii~teiii-I,rLIJ[)r=itoriiiiii lur ’1 licorctisclie Pliysik clcr hkddeiiiic der Wisscnschafteri der DDK

(Received 1986 February I)

I )iscussing llirce argiiisicn ts wliicli verify t h e quantum-field hypothcsis in electrodynamics, it is shown that in GRT analogous investiga- xtc: tlie opposite, namcly a liinitation of this conccption to the low-frequency region.

JSs u i~rcleii tlrei Arpiinciitc tlisliaticrt , dic Ikleg Iur die Gultigkcit dcr 1;eldquanten-Hypothese in der Elektrodynaniik sind. Dabei wird gczcigt, d a U dcr \’ersuch, dicse Argumuntc auf die ART zu ubertragen, scheitert : Die Giiltiglreit dcs GravitoneIi-Konzepts ist in der ,1HT i d das Gebiet niedriger Frcquenzen beschraiikt.

1. Introduction

Hecausc ol t l ic matliematical and experimental successes of quantum electrodynamics the conception of pliotons is certain to reiliain a part of future theory encompassing electromagnetism. This conception is singled out as the correct description of electromagnetic interaction on the whole frequency scale, from low to high Irequencies of electromagnetic fields. The photon conception is appropriate to unify the principles of quantum mechanics and classical electromagnetic field theory.

Considering gravitation instead of electromagnetism one is, of course, led to the question whether tlie conipatibi- lity between quantum physics and (now gravitational) field theory is again established by a field-quantum (now graviton) conception. This question cannot, however, be answered automatically in an affirmative sense. The need to be cautios in answering is already signaled by the fact that GlCT is not a field theory of the same type as tlie Maxwell electrodynamics (remember the energy-momentum problems). Therefore, the problem of the relation between the quantum principle and classical GRT could require another solution as that one which was given for electromagnetism.

The thesis, in favour of which there will be given some arguments in the following, reads as follows: To harmonize the quantum principle and classical GRT the graviton conception is useful or even ~iecessary in the weak-field and low-frequency region, where gravitation is governed by linearized GRI’. In difference to electromagnetism the field-quantum conception has, however, no physical iiieaning f o r tlie Lull theory; in particular, i t lias no sense for high frequencies, In other words, in cjuantized G X I tlrcrr. arch no nonlincar but o d y linear gravitons. (l‘or details, cf. BORZESZKOWSKI and THEDER (1986)).

Before turning to the announced arguments sonic words on tlit: notion “quantizcd GRT”. Tt must be stressed that our thesis refers to strong GIIT, i.e., to tliat tlieory whicli is based on the strong principle of equivalence and not to any modification of GKT destroying this basis. This means especially that one has to evade approximation procedures which linearize GRT too rigorously. ’Phis caution is required becausc thcre can be a great difference between using approximation procedures in GIiT and in usual linear field theories. A procedure as, for instance, the weak-field approximation means in usual linear ficld theories only that one considers a special physical case. But it does not mean (and must not mean) that one destroys any physical features of the theory under consideration. In nonlinear GRT linearization can mean, however, that one considers a modified theory, namely GRT without non-linearities, and this is no longer a typical GRT situation. Incleed, nonlinearity is just one expression of the strong principle of equivalence, and a too figorous linearization o f GRT means, accordingly, to consider a theory which does not satisfy this principle, i.e., to fail typical GRT cases. (This is true for most procedures using a fixed back- ground.) To consider quantized GRT implies, therefore, to look for appropriatc methods evading this danger. Only then one can answer the question concerning tlic scopc of the field-quantum conception, i.e., to say if there are also gravitons, which one can attribute to the full nonlinear GKT (“nonlinear gravitons”).

2.

Let us turn now to some arguments in favour of our thesis. For this purpose, it is useful t o remind of three proofs for the existence of photons which played a great role in the history of the discovery of photons (for details cf. PAIS (1982)). In section 3 we shall show then that all these arguments break down in the case of gravitation.

(i) As is well known, EINSTEIN was led to his light-quantum postulate by an analogy between electromagnetic radiation in the Wien regime, where the spectral energy density e(v, 2’) of radiation of temperature T and frequency v

Arguments in favour of the photon hypothesis

20 Astroii. Naclir., Ud. 307, t l . 5

278 Astron. Nadir. 307 (1986) 5

is given by Wien’s ansatz

e(v, T ) = av3 e-bvlT, (1)

and a classical ideal gas of 18 particles, where the entropy S obeys the volume-dependence law:

R N (:) S(V , T) - S(7jO, T ) = --In -

( R is the gas constant and N Avogaclro’s number). Comparing the> entropy change

A S = S(v, ‘0, T ) - S ( Y , 110, T ) , ( 3 ) following from (I), with eq. ( 2 ) EINSTEIN deduced his light-quantum hypothesis, according to wliicli monochromatic electromagnetic radiation behaves as if it consists of mutually independent energy quanta

E === Izv . (4) The point we want to stress liere is that EIX’STEIN was led to his hypothesis by considering vacuum liigli-frequency

radiation. Indeed, Wien’s law is a high-frequency approximation of Planck’s law because Planck’s density function

contains IVien’s law (I) for

(ii) Discussing now not only the light-quantum hypothesis (4), attributing energy parcels to tlie field, but also the full photon postulate one is led to the Compton effect as a crucial experiment of the plioton hypothesis. Tlie photon hypothesis ascribes the electromagnetic field particles carrying energy E = hv and momentum f i = hvlc. From this result the following relations governing the kinematics for the scattering of a photon off an electron at rest

I& = 3 + h c ,

Izclhl 1 - m c 2 = Izclkl 1- (p -1- r n Z c 4 ) ” 2

( 7 )

(8) -+

-+ + ( k is t he momentum of the plioton before scattering, lz’ after scattering, the momenturn of the electron after scat- tering). These equations imply that the wavelength difference A;! between tlic final and the initial photon is given by

(0 is the photon scattering ariglc). Because this relation was found to be satisfictl the plioton postulate was finally accepted.

Because of this nieaniiig of tlic Compton effect for tlie acceptance of tlic piloton hypothesis tlicre ariscs the qut~stion on the gravitational Compton effect.

(iii) After COMPTON’S discovcry ~ H K , KIUMM~S, and SLATEH (i()zq) niack theoretical proposals on tlic interaction of radiation and matter wliicli were to avoid the need to draw tlie plioton conclusion froiii (:OMITON’S nieasureiiiviits. BOHR et al. wanted to protect the free electromagnetic field from quantization. All peculiarities of the radiation tkieory should not be due to the particle nature of free fields but to peculiarities from the interaction between the virtual field of radiation and the illuminated atonis. According to tlie BKS proposal tlic encrgy ol tlie fieltl slioultl change continuously a i d the energy of the atonis discontinuously. l‘liis prol)osal contradicts oI course a gciic.r;il law of energy conservation. Coiisecluently, the BKS answcr was to abandon tlie conservation ol energy and nionieiituiii for radiation transitions. The conservation law sliould not hold for individual elenientary processes but only statisti- cally, i.c., as an average over niany processes. Accordingly, COMIJTON’S iiieasurenients on AA sliould only rcler to the average change of the wavelength such that the conservation laws (7) and (8) are not tested for individual processes. A s is known, the BKS approach was refuted b y the experiments of BOTHx and GEIGER, aiitl C C ~ ~ Y T O N ; ~ i i d Smoti.

3. On the limitations restricting the graviton conception

Discussing now these three arguments in the gravitational case, one sees that they break down. (i’) To analyse the vacuuni high-frequency case of GRT it would, o f course, be best to evade any approximation

procedure. Only then one could be bure to avoid the “linerization-danger” mentioned introductorily. Otlierwisc, the mathematical complexity of Einstein’s equations renders this impossible. It was used, therefore, the high-fre- quency approximation developed by BRILL, HARTLE, WHEELER, and ISAACSON to discuss high-frequency fields in GRT. Due to the ansatz

g p == y,cu t (14

UORZESZKOWSKI, H.-H. v. : Gravitons in GRT 279

where

dy N yIL , ah - hlh , E 5 AIL < I , (11)

this method assumes again a weak-field situation ; but now the high-frequency assumption leads to a backreaction such that the nonlinearity of Einstein’s equations is taken more seriously into consideration than in the low-fre- quency case. As there was shown (BORZESZKOWSKI 1985), then one obtains tlie result that thc high-frequency assump- tion (IO) , (11) together with the field-quantum hypothesis (4) presupposed now for tlie h@,, field are not generally compatible. More precisely, only for wavelengths

?, 2 (IPL)l/Z ?E A. (12)

Einstein’s relation (4) can be satisfied. ( I , = (hG/c3)lI2 is Planck’s length, and L is the characteristic lciigtli over which the classical abckground yBY, in front of which the quantunn perturbations hp,, move, changes significantly.)

Accordingly, in contract to electromagnetic theory the field-quantum hypothesis is not generally compatible with the high-frequency field described by Einstein’s equations. The graviton concept is thus only a low-frequency conception in GRT.

(ii‘) Turning to the Compton effect, due t‘o (IZ), one finds that Compton effect measurcincnts are limited to the region, where

2‘4 > A; . (13)

I. > I, . (14)

(A is tlie wave length of the field scattered by a particle, whose Compton wave length is given by A = h/mc. For I, A relation (13) yields (cf., BORZESZKOWSKI 1985)

,. 1 his means that such a “graviton-matter” Compton effect occurs for a matter-dominated background, where the self-interaction of the the gravitational field can be neglected. All high-frequency effects are cut off. This becomes even more evident if one considers cases where A = Lo < L because one then obtains from relation (13) the estimate:

I n accordance to tlie rciiiarks made under (i’), tlie gravitational Compton effect thus cannot be used as a high- frccluency (i.e., nonlinear) graviton test.

(iii’) Finally, we state that the BKS approach, wliicli was reiuted in electrodynamics first experimentally by Compton-type experiments and second by showing that it violates the laws of energy-momentum conservation, cannot be refuted for gravity. Indeed, as i t was mentioned under (ii’) Compton-effect measurements do not produce counter-arguments, and the theoretical arguments concerning energy-momentum conservation do not hold true because in GRT one has no general laws of energy-momentum conservation.

To summarize, one can say that in GRT gravitons are a low-frequency conception because, a t high-frequencies, GICT is a ,,BKS-type-theory“.

References

v. ~ ~ O R Z ~ S Z K O W S K I , H.-H., TREDBK, H.- J , : 1986, On Quantuni ‘L’lieory of Gravitation, 1). Keidel Publishing Company, Dordrecht Boston, Lancaster.

v. UOKLBSZKOWSKI, H.-H. : 1985, “’L‘lic Concept oI Gravitons anti tlic hfcr~sureniciit of Effects of Quantum Gravity”, in: M. A. MARIZOV, V. A. BERBZIS, V. 1’. FKOLOV (cds.), Proceedings of the 3rd Seminar on Quantum Gravity, Singapore. (Cf. also the references cited therein.)

I’AIS, A . : 1982, Subtle is the Lord . . ., ’I’hc Science and the Life of Albert Einstein, Oxlord UP.

Xdtlresr; ot the autlior:

1 l , - t i , VUN UOHZKSZKOWSKI ~insteiii-l,aboratoriu~~i fur Theoretische Physik der AdW der 1)DK IZosa-Luscmburg-Str. 17 a DDK- 1502 1’otsdani-Babelsberg Geriiian I.)eniocratic liepublic