Power calculation for gravitational radiation: oversimplification and the importance of time scale
aa r X i v : . [ a s t r o - ph . C O ] S e p Astron. Nachr. / AN , No. 88, 789 – 791 (2011) /
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Power calculation for gravitational radiation: oversimplification and theimportance of time scale
Alan B. Whiting ,⋆ Astrophysics and Space Research Group, School of Physics and Astronomy, University of Birmingham, Edgbaston, Birm-ingham B15 2TT, UKReceived 2010 June, accepted 2010 SeptemberPublished online 2010 September 17
Key words gravitational waves—relativityA simplified formula for gravitational-radiation power is examined. It is shown to give completely erroneous answers inthree situations, making it useless even for rough estimates. It is emphasised that short timescales, as well as fast speeds,make classical approximations to relativistic calculations untenable. c (cid:13) Gravitational radiation, the vibration of the space-time met-ric produced by masses in motion, forms one of the acceptedpredictions of General Relativity. Though not yet directlydetected, current projects are producing astrophysically in-teresting upper bounds and planned instruments may wellmake the first real observations .The calculation of gravitational radiation in all but thesimplest cases can be difficult and tedious, however. Baker(2006) presented a simplified formula for the gravitationalpower radiated by an object, or collection of objects, ‘inorder to render astrophysical applications more apparent.’This paper examines that formula to determine its limits ofapplication. The starting point for Baker’s (2006) derivation is the for-mula for total averaged power radiated by a body, under var-ious simplifying assumptions (the gravitational waves are ofsmall amplitude, the stress-energy tensor can be reduced tothe mass density). From Landau & Lifshitz (1975), p. 355,eq. 110.16, this is − d E d t = P = G c (cid:12)(cid:12) ... D αβ (cid:12)(cid:12) (1)(I have used G for the gravitational constant instead of Lan-dau & Lifshitz’ k ), where the moment of inertia tensor isgiven by ⋆ Corresponding author: e-mail: [email protected] The literature on gravitational radiation is vast and even a brief surveyis beyond the scope of this paper. Basic derivations are found in most Gen-eral Relativity texts, and specific references to Landau & Lifshitz (1975)will appear below. Dietz (2010), to take a recent example, shows the usefulapplication of non-observations to astronomical objects. D αβ = Z ρ (cid:0) x α x β − δ αβ r (cid:1) d V, (2)the integral to be taken over the volume of the body in ques-tion, and transformed into a sum when considering a collec-tion of point masses. (This is their equation 110.10, p. 355; Ihave substituted ρ for the mass density instead of Landau &Lifshitz’ µ , since the latter symbol appears with a differentmeaning below.)Citing considerations of ‘symmetry’ (which he does notspecify), Baker equates D αβ with a scalar moment of inertia I , taken to be a mass δm times the square of the radius ofgyration r . In taking the third time derivative he obtains d I d t = 2 rδm (cid:18) d r d t (cid:19) + . . . (3)(Baker does not say what the missing terms on the right are,nor why he chose to ignore them). He then identifies δm times the third time derivative of r with the time derivativeof force; and finally averages the change in force over sometime interval, giving as his formula for power P = 1 . × − (cid:18) r ∆ f t ∆ t (cid:19) (4)the numerical coefficient chosen to give watts . He then ap-plies his formula to two-body motion and rigid-rod rotation,finding numerical agreement, and considers his formula tohave general application. I have not been able to duplicate Baker’s numbers, which he gives intwo incompatible equations. In all numerical calculations I will use . × − s kg − m − as the coefficient to the moment-of-inertia term inEq. (1); the difference, large as it is, does not affect my conclusions. c (cid:13)
90 A. B. Whiting: oversimplification
A thorough investigation of Eq. (4) would set out the condi-tions under which the simplifications and assumptions of itsderivation hold good. For present purposes it is sufficient tolook at three examples.First, we follow on from Problem 1, page 356 of Landau& Lifshitz (1975). Two masses, m and m are in a circularorbit a distance r apart. They move with angular frequency ω and at time zero lie along the x -axis, rotating in the x, y plane. The components of the moment of inertia tensor are D xx = µr (cid:0) ( ωt ) − (cid:1) (5) = 32 µr cos(2 ωt ) + 12 µr D yy = µr (cid:0) ( ωt ) − (cid:1) = 12 µr − µr cos(2 ωt )D xy = µr (3 cos( ωt ) sin( ωt ))= 32 µr sin(2 ωt )D zz = − µr where µ = m m / ( m + m ) is the reduced mass. Next,we add an identical pair of masses, the same distance apartrotating in the same orbit with the same speed, but placethem one-quarter of the way around the orbit with respect tothe first pair. They add the following terms to the momentof inertia tensor: D ′ xx = 32 µr cos(2 h ωt + π i ) + 12 µr (6) D ′ yy = 12 µr − µr cos(2 h ωt + π i )D ′ xy = 32 µr sin(2 h ωt + π i )D ′ zz = − µr . It is easy to see that ˙D ′ xx = − ˙D xx , ˙D ′ yy = − ˙D yy , ˙D ′ xy = − ˙D xy and ˙D ′ zz = ˙D zz = 0 . That is, the time derivative(to all orders) of the magnitude of the moment of inertiatensor is identically zero; there is no gravitational radiation.Baker’s formula, Eq. (4), however, predicts double the radi-ation of the two-body situation.Next, consider two bodies of mass m and m con-strained to move along a straight line, which we will identifywith the z -axis. We fix the centre of our coordinate systemat their centre of mass. Their distance apart is r . The com-ponents of the moment of inertia tensor are D xx = − µr (7) D yy = − µr D zz = 2 µr with µ the reduced mass, as before. (By allowing one of themasses to be much larger than the other we can use these formulae to analyse the motion of a single body.) From thesewe calculate (cid:12)(cid:12) ... D αβ (cid:12)(cid:12) = 24 µ ( r ... r + 2¨ r ˙ r ) (8)or, to compare with Baker’s formula, (cid:12)(cid:12) ... D αβ (cid:12)(cid:12) = 24 (cid:16) r ˙ f + 2 ˙ rf (cid:17) . (9)If the force (a sort of reduced force, acting on the reducedmass) is constant, Baker’s formula gives no gravitational ra-diation; but Eq. (9) shows that there is still some given off.Baker’s formula has thus been shown to be completelyin error both ways, in predicting gravitational radiation whenthere is none, and predicting none when there is some. Thusit cannot be used even as a rough guide. The examples givenare only slightly different from those in Baker (2006) andwould be a reasonable first approximation to some astro-nomical objects (orbits of more than two objects are fairlycommon, as are linear jets), so his assertion in that paper ofthe usefulness of his formula in astrophysics is unfounded.Such a conclusion would not appear to have a big im-pact on the gravitational-wave community, since Baker’sformula has not been used in astrophysical circles (wheremore accurate techniques are customary). It has been em-ployed in another context, however, and that forms our thirdexample. Baker, Li & Li (2006) describe an apparatus intended togenerate and detect high-frequency gravitational waves ina laboratory. Only the generation side concerns us here.Two targets, 20m apart, are hit with high-intensity laserpulses, directed such that they are momentarily acceleratedin opposite directions; the authors consider them to emu-late, for the duration of the pulse, a two-body orbiting sys-tem. The 23TW pulses last for 33.9fs and are repeated tentimes per second. Using Eq. (4) the authors calculate thatthey generate . × − watts of gravitational radiation .There is one major problem with this analysis. Baker etal. apply their formula for the duration of one laser pulse, . × − s. Light travels only 10.6 µ m in this time (some-thing they mention, but without any apparent considerationof its implications). To analyse bodies not in causal contactas if they were part of a Newtonian object, moving in New-tonian ways, is a very questionable procedure. Indeed, it isnot clear that the essentially classical definition of a momentof inertia tensor and its time derivatives can be made rela-tivistically meaningful, nor that it would retain its role as The authors tacitly assume that the pulse changes intensity linearlyover its duration, inserting the calculated radiation-pressure force for23TW and the duration of 33.9 fs in Baker’s formula. It might be more ac-curate to take the pulse as a square wave, staying at something like its peakintensity for the duration. In principle, the difference is important since thefactors enter as squares, and the average of a square is not the square of theaverage. When I calculate some numbers I will look at both possibilities. c (cid:13) stron. Nachr. / AN (2011) 791 a source of gravitational radiation if it were. And it is cer-tainly not justified to apply Eq. (1), which is averaged overa complete period of the gravitational waves, to a tiny frac-tion of an orbit. By looking at such a short pulse Baker etal. (2006) have removed themselves from the assumptionsunderlying the starting point of Baker’s (2006) derivation,Eq. (1) and so have no grounds for believing their resultingnumbers.As far as one can apply Eq. (1) to the apparatus of Bakeret al. one must be restricted to a region in causal contact,a single laser target. Indeed, not all of that: ordinary matteris not rigid on this time scale. The impact of the laser pulseon the rear of a target cannot be known at the front until asound wave can traverse the intervening distance. If this pe-riod is much shorter than the pulse, the whole object cannotaccelerate (as Baker et al. tacitly assume); instead a soundwave is set ringing through the target.To clarify the picture it is useful to have some numbers.Since Baker et al. (2006) give no details about their lasertargets, I will use some nominal values (exact figures arenot important here, as will be clear). For a nominal soundspeed in steel of 5790 m s − , the pulse of . × − s haspenetrated a distance of . × − m by the time it ends:a layer of atomic dimensions . Assuming a laser spot sizesimilar to their quoted detector laser, . × − m (notall of which is in causal contact sideways!), and a density ofsteel of . × kg m − , we have an accelerated mass ofsomething like . × − kg.We now turn to Eq. (9) to calculate the gravitational ra-diation of this accelerated mass. First using the Baker et al.assumption of a constant change in force, (cid:12)(cid:12) ... D αβ (cid:12)(cid:12) = 983 ˙ f t µ (10)which gives a radiated power of . × − W. If, on theother hand, we assume a constant force, (cid:12)(cid:12) ... D αβ (cid:12)(cid:12) = 96 f t µ (11)resulting in . × − W. The difference between thesenumbers and the calculation in Baker et al. (2006) amountsto twenty-seven orders of magnitude. This difference mayin principle arise either from the simplifications of Baker(2006) or from the possibility that Eq. (1) is simply inappro-priate for very short periods of time; in either case, Eq. (4)must be discarded.
A simplified formula for a relativistic effect has been shownto be completely unreliable, giving infinitely wrong answersin two instances, and an error of something like twenty-seven orders of magnitude in a third. The latter number is A true analysis of this interaction of radiation with matter should, ofcourse, be done in a quantum context. not firm, since there is some question as to whether its ownbasis is justified; but it is certain that the formula of Baker(2006) cannot be used. Gravitational waves are not to begenerated in the laboratory in the forseeable future.The lesson of this episode is that simplified formulaemust be justified and carefully handled, since it is quitepossible to push them beyond their applicability. In addi-tion, Newtonian approximations of relativistic effects mustbe carefully examined for tacit assumptions that make theirresults untenable. In particular, short time scales (as well asspeeds comparable to light) make Newtonian expressionsunreliable.
References
Baker Jr., R.M.L.: 2006, AN 327, 710Baker Jr., R.M.L., Li, Fangyu, Li, Ruxin: 2006, in M.S. El-Genk ed.,
Space Technology and Applications InternationalForum—STAIF 2006 , AIP Conf. Proc. , pp. 1352-1361Dietz, A. (on behalf of the LIGO Scientific Collaboration and theVirgo Collaboration): 2010, Proceedings of the 14th Gravita-tional Wave Data Analysis Workshop, Rome, January 2010;astro-ph/1006.3393Landau, L.D., Lifshitz, E.M.: 1975,
The Classical Theory ofFields , Fourth Revised English Edition, Pergamon Press c (cid:13)(cid:13)