aa r X i v : . [ a s t r o - ph ] M a y Energy Loss by Gravitational Viscosity
Ernst Fischer ∗ e.fi[email protected] Abstract
Due to Lorentz invariance of General Relativity gravitational interac-tion is limited to the speed of light. Thus for particles, moving withina matter field, retardation leads to loss of energy by emission of grav-itational radiation. This ’gravitomagnetic’ effect, applied to motion inhomogeneous mass filled space, acts like a viscous force, slowing downevery motion in the universe on the Hubble time scale. The energy lossrate exactly equals the red shift of photons in an expanding universe, thusshowing the equivalence of wavelength stretching in the wave picture andenergy loss in the photon picture. The loss mechanism is not restrictedto an expanding universe, however, but would also be present in a staticEinstein universe.
Today the theory of general relativity (GRT) is accepted as the correct de-scription of gravitation, but due to its non-linear character and the complicatedmathematical formalism practical applications have been rather limited and,wherever it appears acceptable, the much simpler formalism of Newtonian the-ory is used, which appears as a good approximation in the limit of weak fieldsand low velocities. One of the most important differences from Newtoniangravity is the Lorentz invariance of GRT, following from the requirement thatgravitational interaction should be independent from any preferred referencesystem other than the complete universe.One consequence of this requirement is the fact that gravitational interactionis not instantaneous but limited to the speed of light, leading to similar radiationeffects as we know them from electromagnetic interaction. Changes of the mat-ter or energy distribution lead to changes of the metric, expanding into spaceby the speed of light. Though this effect has been already discussed by Einsteinin his famous quadrupole formula, it remained of little practical interest up tothe last two decades. The first observation, which could be attributed to thisgravitational radiation, was the detection that the frequency of double pulsarswas slowing down exactly according to GRT (Weisberg and Tailor (1984) [6]). ∗ Auf der H¨ohe 82, D-52223 Stolberg, Germany t is justat equal distance r from both masses, the gravitational force at this moment isdetermined by the distance at the time t − r/c . That means that the distanceto the mass in direction of the motion is increased and the distance to the othermass is reduced. Thus the particle feels a force, which is directed opposite tothe direction of motion, thus reducing its momentum and energy of motion.Of course, if we try to sum up the forces exerted on a test particle bythe total matter in a Euclidean universe, we are confronted with the infinityproblems, well known from the Olbers paradox. But also here general relativitysupplies us with a remedy, or better to say, with two possible solutions. Oneis the assumption, favoured by main stream physics of today, that the universeis expanding and thus, even if the size of the universe is infinite, interactionis limited to that fraction of matter, which can be causally connected to thetest particle. But possible is also the other explanation, originally proposed byEinstein, the assumption that space is curved and thus of finite size and mattercontent. In the sequel we will consider both possibilities and show that theylead to similar results. To demonstrate, how the finite speed of gravitational interaction affects theenergy balance of moving particles, as a toy model we consider the motion of atest mass in a universe, expanding at a constant rate, but keeping the density2onstant by some creation process similar to that proposed by Hoyle et al.(1993) [2]. This does not mean that this model appears more attractive thanthe presently favoured ’concordance model’. But it allows a mathematicallyvery simple description of the effects, as the relevant quantities, matter densityand expansion velocity, do not depend on time.Starting with Newtonian physics, the gravitational potential at some pointgenerated by masses m i at distances r i is given by U = X i Gm i r i (1)( G is the gravitational constant). But if gravitational interaction is limited bythe speed of light, similar to electromagnetic interaction, if the position of themasses changes with time, we have to use retarded potentials analog to theLi´enard-Wiechert potentials. Instead of the distance at the local time t , wehave to insert the distance at time t − τ , where τ is the running time of thesignal τ = r i /c . U ∗ ( t ) = X i Gm i r i ( t − τ ) . (2)In the static case this does not change the result, but for a moving particle theretardation parameter t − τ changes with time. As a result there occurs anadditional gradient of the potential, leading to a retarding force on the particle.We should stress here that analogous to electromagnetic interaction retar-dation does not lead to aberration effects, as Lorentz invariance requires thatthere exists no preferred reference frame. Thus interaction can depend only onthe scalar retarded potential, not on the direction of forces as determined fromEuclidean geometry. A detailed discussion of this fact has been given by Carlip(2000) [1]. While a test particle, linearly moving in the field of homogeneouslydistributed masses, feels a retarding force, as the distance to the individualmasses changes with time, in circular motion of an isolated two-body systemthere is no retarding force. As the distance of the masses does not change,angular momentum is conserved. Thus in the solar system deviations fromNewtonian physics occur only as tiny corrections of higher order in v/c due torotation, tidal effects and eccentricity of the moving bodies. In linear motionthere exists an effect of first order in v/c , however.Let us assume a particle moving at velocity v in positive x-direction of Eu-clidean space. Using a comoving coordinate system, we can regard all the othermasses as moving with respect to it with speed − v . Thus their distance changeswith time by dr i dt = dr i dx · dxdt = x i r i · dxdt = − v x i r i (3)In the comoving system there is a gradient of the retarded potential, which isfelt by the particle as a retarding force. It can be expressed by a power series dU ∗ dx = X i Gm i ddx ∞ X n =0 d n dτ n (cid:18) r i (cid:19) τ n n ! . (4)3estricting to the linear term of the series expansion with τ = r i /c we get dU ∗ dx = X i Gm i ddx (cid:18) r i + vx i r i r i c (cid:19) = X i Gm i (cid:20) − x i r i + vc (cid:18) r i − x i r i (cid:19)(cid:21) (5)We can now apply this formula to the motion of particles in an homogeneousmatter filled space of density ̺ . In this case the mass in a toroidal volumeelement at distance r i is m i = ̺dV i = 2 π̺ r i cos ϑ dϑ dr , where the projecteddistance in the direction of motion is expressed by x = r sin ϑ . Then integrationof the potential gradient induced by the matter is dU ∗ dx = 2 πG̺ Z r max Z π/ − π/ Y ( r, ϑ ) r cos ϑ dϑ dr (6)with Y ( r, ϑ ) = (cid:20) sinϑr + vc (cid:18) r − ϑr (cid:19)(cid:21) (7)The quantity r max has been introduced to express that integration has tobe limited to the causal sphere, that means, to the volume range, which caninteract, when this interaction is limited to the speed of light. Thus, if space isexpanding at a rate given by the Hubble constant H , the limit is r max = c/H .From symmetry considerations it is immediately clear that the first termof the integral is zero, but the second term gives a non-vanishing contributionproportional to v/c : dU ∗ dx = 2 πG̺ r max · vc (cid:18) sin ϑ −
23 sin ϑ (cid:19) π/ − π/ (8)corresponding to a force on a particle of mass mF = − m dU ∗ dx = − π G̺ r max · mvc (9)Introducing r max = c/H there is a loss of momentum p proportional to theactual momentum dpdt = − π G̺H · p, (10)and correspondingly the loss of kinetic energy is dEdt = ddt (cid:18) p m (cid:19) = − π G̺H · E. (11)As in an expanding universe the case of zero curvature corresponds to the crit-ical density ̺ cr = 3 H / (8 πG ), this energy loss corresponds exactly to that ofphotons dE/dt = − HE . That means that every moving particle in a mass filled4niverse of critical density experiences the same energy loss as we ascribe it tophotons and explain it by stretching of wavelength by the expansion of space.Or looking at it the other way round: What we explain by stretching of wave-lengths in a pure wavelike picture of electromagnetic radiation, may as well beconsidered as part of a general energy loss mechanism of all matter, if we regardphotons as individual entities, which have inertia hν/c and feel gravity in thesame way as massive particles.The situation is much the same as in the experiments by Pound and Rebka(1960) [5] to measure gravitational red shift. The results can be explained ina geometrical way by time dilatation, caused by spacetime deformation by thesurrounding matter, but as well by the difference of the gravitational poten-tial between the points of emission and detection of radiation. Equivalence ofboth pictures results from conservation of energy, which is valid, when potentialenergy is included.Of course, that extending the quasi-Newtonian approximation to photonsis allowed, has not been proved. But the fact that the numerical values areexactly equal and that energy conservation should be valid, independent of thenature of the moving mass or quantum, is an indication that the energy losseffect is a common phenomenon to massive and massless particles. Otherwisethe principle of equivalence, which is a corner stone of GRT, would be violated.Mass filled space appears as a kind of viscous medium, in which energy of allmoving particles is dissipated to the gravitational potential. In the last section we have shown that the cosmic energy loss mechanism mustnot necessarily be attributed to an expansion of space. The basic propositionis the finite speed of gravitational interaction, which is a consequence of theLorentz invariance of the basic GRT equations. It is this Lorentz invariance,which requires that any homogeneous space solution of the basic equations mustbe either expanding or spatially curved. Energy loss by gravitational viscosity,as we have called it, must be present, too, if space is not expanding. We onlyhave to find another explanation, why gravitational interaction is limited to afinite amount of matter, if this amount is not limited by causal connection.There is of course such a possibility, when the total amount of matter in theuniverse is limited, as it was proposed in Einstein’s first introduction of GRT.He proposed that the universe is static, but has a positive curvature, so that thetotal matter content is limited. We can easily adjust the derivation of cosmicenergy loss to this case, we only have to change the size of the volume elementsand the limits of integration.We consider a homogeneous spherical universe with a positive radius of cur-vature R . As in Euclidean space we assume that gravity acts along the geodesiclines and that the strength of interaction decreases with the square of the lengthmeasured along these lines. Denoting this distance by r = R · ϕ , the main dif-ference compared to Euclidean space is the reduction of the volume element at5istance r by the factor (sin ϕ/ϕ ) . There exists, of course, no limitation to thelength of the geodesic lines, so that the value of ϕ may extend to infinity. Butdue to the limited size of the volume elements the value of the integral remainsfinite. Thus in eq.(6) we only have to replace the integral Z r max dr = r max by R Z ∞ sin ϕϕ dϕ = Rπ , (12)leading to an energy loss dEdt = − π G̺Rc · E. (13)The relation between the radius of curvature and the density of matter in thestatic Einstein universe is given by R = p c / (4 πG̺ ). Thus, if we again identifythe energy reduction factor in eq.(13) with the Hubble constant H , the densityof matter is related to H by ̺ = 9 H π G , (14)which differs not much from the critical density in an expanding universe. Thefactor is ̺/̺ cr = 6 /π . From the viewpoint of red shift or global energy lossthere is no reason, to prefer one model against the other. As has been shown in the last sections, the gravitomagnetic effects inherent toGRT or, as we call it in the context of global effects, ’gravitational viscosity’can explain observed red shift with and without any expansion of space as well.To prove, if the general energy loss mechanism on cosmic scale really exists, ourpossibilities are rather limited. To prove the existence, we would have to followthe path of moving particles over very long periods of time and to know theinitial energy of motion exactly. But unfortunately the only particles, for whichthe initial energy is known from the emission process, are photons - at least ifwe assume that the laws of quantum electrodynamics have hot changed withtime.Observations within the solar system have unambiguously shown, that thegravitomagnetic effects exist. The planetary and lunar ranging experiments (see[3]), by which the distance from earth to moon could be measured down to anaccuracy limit of cm, have demonstrated the validity of GRT in an impressiveway. The effect of global energy loss is just at the accuracy limit of these mea-surements. It would change the distance to the moon by about 3 cm per year.But there are so many perturbating effects of the same or higher magnitudewithin the solar system that the observed changes cannot be unambiguouslyattributed to the global loss mechanism. And even if we succeed to isolate theglobal change from all the other perturbations, it remains the question, if it isdue to expansion or curvature. 6hat we need to decide this question, are measurements, which do not re-lay on red shift or similar energy loss, but only on geometric observations likeangular size or mean number density of distant galaxies. But also the interpre-tation of these measurements is hampered by the possibility that the propertiesof galaxies may have changed with time. Thus a conclusive decision is stillmissing. But an increasing number of observations shows that the most distantgalaxies look quite similar as those nearby, casting doubts on the assumptionthat a strong development in size or properties has taken place since the lighthas been emitted, which is reaching us now.There is one indirect hint, however, that gravitational viscosity, at least bypart, is responsible for the observed energy loss of moving matter and conse-quently also for red shift of photons. Without assuming some dissipation ofkinetic energy, which matter acquires while contracting into galaxies or clus-ters, formation of these structures cannot be explained. Gravitational viscositysupplies this dissipation effect. The fact that we observe galaxies as spirals andnot as circular disks, can easily be attributed to the loss of angular momentumduring formation of structures on the Hubble time scale.Thus, if we take GRT serious as the correct theory of gravitation, we shoulddo it with all its consequences. We should not regard Doppler effect in an ex-panding universe as the only possible explanation of observed red shifts. Takinginto account energy loss by gravitational viscosity, may help us to a better un-derstanding of the formation phenomena of large scale cosmic structures and itmay even question present days assumptions on the age of the universe, derivedfrom observed red shift.
References [1] Carlip S., 2000, Physics Letters A, 267, 81-87[2] Hoyle F. et al., 1993, AJ 410, 437[3] Nordtvedt K., 2003, arXiv:gr-qc/0301024[4] Poisson E., 2004,