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THE THEORY of DYNAMICAL FRICTION Andrey Katz

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THE THEORY of DYNAMICAL FRICTION Andrey Katz THE THEORY OF DYNAMICAL FRICTION Andrey Katz with A. Kurkela and A. Soloviev, to appear soon OUTLINE Motivation: why is this important? What do we already know? Description of the formalism Calculation of the drag force: ideal gas: recover the known cases, generic expression for arbitrary distribution and arbitrary “bullet” velocity recovery of the ideal liquid results viscous liquids generic results of interacting gases in the relaxation time approximation, interpolation between the gas and the liquid Comments on the boundary conditions effects Outlook !2 Motion of black holes leaves a “wake” of stars and gas in their trail… the gravitational tug of this wake on the black hole then removes energy and angular momentum from blackhole orbit (dynamical friction) WHAT IS DYNAMICAL FRICTION? The problem is not new: A massive particles Dynamical friction propagates through a medium with a constant velocity. What is the gravitational drag force, induced by the wake? First analysis: Chandrasekhar, 1942 !3 3 DYNAMICAL FRICTION AND MODERN HEP Traditionally: used to calculate motion of bodies in the galactic dynamics — with classical NR results being perfectly adequate More modern questions: effects of self-interacting DM (component) on the galactic dynamics — more complicated, self- interactions effects should be taken into account Recently: calculate the the energy released by (primordial) black holes going through neutron stars and white dwarf — effects of matter in extreme conditions must be properly accounted for Exotic objects in the accretion discs of the black holes !4 the spatial derivatives. These two approaches are fully equivalent. Note that for δfeq we p0 assume a locally thermal distribution δfeq = f0 T (~v ~u)+δT/T . Of course we expect this ⌧ · P 1 e´ (144) assumption to break down for ⌧ , namely“ ´ in the free streaming case. However since the !1 1 importance of this term scales as ⌧ − , we are not sensitive to this assumption in this limit. ∆φ !∆t 1 (145) When we know the perturbation δf„we can„ O finallyp q calculate the correlator δT µ⌫ R R D d3p pµp⌫ (146) Gµ⌫↵β = withE „ δTS µ⌫ = δf (23) kin δh (2⇡)3 p0 ↵ a Z ⌧ c P 1 e´ (144) where the dependence on h comes fromσy the Christo↵el symbols. Interestingly, the(147) object “ ´ ↵ À RS! Gkin is not fully equivalent to the object that we have defined in the Eq. (7). Nonetheless, ∆φ !∆t 1 !∆(145)t 1 (148) „ one„ canOp showq that they merely di↵er by contact„ terms, i.e. terms that do not contain new poles in the kz complex plane and therefore theδ contribution of these terms will vanish in the RE RSD ⇢(146)r´ (149) „final answer. (Aleksi, make sure you agree„ with this explanation – ak) Finally, once a 3 δ δ´1 wec have the object Gkin, we can go ahead andRE construct´ the dressed graviton propagator. σy Reff R(147)E (150) À RS! 9 r ˆ 0 ˙ !∆t A.1 Free streaming scenario tˆ (148)a c (151) „ var „ s{ Letδ us first try to reproduce the results⇢ for⇢¯ the non-interacting gas in the non-relativistic ⇢ r´ δ ´(149) 1 (152) „ ” ⇢ „ Op q and ultra-relativistic cases. The force exerted on a body by a gas of non-interacting rela- 3 δ δ´1 tivisticR particles´ using the results of [1] and [4] is: R R E γ (150)1 v2 c2 (153) eff 9 E r “ ´ { ˆ 0 ˙ 2 2 4⇡aG mb ⇢v WHAT DO WE KNOWF = 1ABOUTlog ⇤ . (24) W vE~2 D~ B~ H~ d3x (154) tˆ a c “ 8⇡ (151)p ¨ ` ¨ q var „ s{ ª THE DYNAMICAL FRICTION?e2 dr (Translate to the relativistic language – ak) In this expression ⇢ is the energy(155) density 2 r2 ⇢ ⇢¯ ª δ ´of a gas1 of non-relativistic particles, namely(152)⇢v = mn, where n is the number density of ” ⇢ „ Op q 2 2 v particles whose velocity does not4⇡ exceedG mb mp the bullet2 velocity,. In the opposite regime of the 1 1 1 Ideal non-relativistic gas F 2 dv v f v log ⇤ (156) “ v 0 p q γ ultrarelativistic1 v2 c2 gas the result is derived in(153)ª [10] was “ ´ { a 2 2 2 2 2 4⇡m G γ 164⇡v 2 ⇢02 P02 1 F F =p ` Gq mp ⇢`vγ qloglog ⇤ ✓. v cs (157) (25) Photons gas,W relativisticE~ D~ bulletB~ H~ d3x “ 3v2 (154)b p ´ q “ 8⇡ p ¨ ` ¨ q ª e2 dr –10– How do we interpolateNote that between the result in two the intermediate these regimes? case(155) and the interpolation between the two limiting 2 r2 ª cases to the best of our knowledge does not appearOstriker; in the 1998 literature and we present it here Ideal 2liquids:2 v 4⇡G mb mforp the first2 time. As we will shortly see, our techniques allows us to get to an analytic F dv1v1 f v1 (156) “ v2 p q Supersonic: expressionª0 that smoothly interpolates between10 two these limits. 4⇡m2G2γ2 1 v2 2 ⇢ P F p ` q p 0 ` 0q log ⇤ ✓ v c (157) “ v2 p ´ sq Vanishes subsonically, though certain 11 boundary effects can change this conclusion !5 Fig. 3.— Solid curves: Dynamical friction force in a gaseous medium as a function of Mach 10 number = V/c .Curvescorrespondtoln(c t/r )=4, 6, 8,...,16. Dashed curves: M s s min Corresponding dynamical friction force in a collisionless medium with particle velocity dispersion σ = c and r Vt= c t. s max ≡ M s CALCULATION OF DYNAMICAL FRICTION — HISTORICAL APPROACH Chandrasekhar’s approach: calculate the probability function of occurrence of force F acting on a star and an average time during which the force acts In the case of the ideal fluid (Rephaeli & Salpeter, Ostriker) one calculates the drag due to the asymmetry in the flow associated with the front of the shock wave. !6 and uµ = γ(1, 0, 0,v) is the four-velocity vector of the bullet. Note that hereafter we consider the simplest possible case, namely the bullet moving in a infinite medium with constant We also notice, that, maybe surprisingly, the dynamical friction in the interacting gas has velocity for sufficiently long time. With this approach we will not be able to capture the never been properly calculated. In our paper we close this gap presenting a full calculation boundary sensitive e↵ects, like for example discussed in [7]. However, this will be mostly relaxation time approximation. The core observation of this paper is that the force exerted suffioncient a bullet, for our propagating needs. We through will also gas, briefly can easily consider be calculated boundary from e↵ aects metric and perturbation. see how, for example,In order one to reproduces calculate the the latter, results it is of su [7]fficient in our to formalism calculate the in Sec graviton XXX. propagator, dressed Theby the linear correlators perturbation of collective in the gravitationalmodes of matter. field The caused correlators by the bullet of the will collective cause modesa linear perturbationhave been in calculated the background in various medium, limits. For which example, will evolve in the in hydrodynamical time in a way that limit depends they have on thebeen material recently properties computed of the in [12]. medium. The For case now of the we interacting will not specify gas of further the ultra-relativistic the properties of theparticle medium. has been However, analyzed in general, in [13]. the In this medium work responsewe go further, of the calculating energy momentum the momentum tensor δT µ⌫transport, which should correlations be thought functions about in the as a relaxation perturbation time to approximation the (3) to the of gravitational the interacting field in genericgas of particles medium withcan be arbitrary characterized mass. This by the further response allows function us to computeGmedium dressed graviton propagators in momentum space and eventually the drag forced exerted on to a bullet, µ⌫ 4 µ⌫,↵ bullet δTwake(x)= d x0Gmedium(x0,x)h↵ (x0). (6) propagating through the interacting gas. Z We willOur later paper slightly is structured abuse the as follows. notations In Sec and II drop we presentδ, while our the formalism perturbation in great and detail the and un- perturbedexplain background the logic behind will the be calculation. distinguishable The fromcalculation the context. itself is performed This response in Sec function III. As a is nothingwarm but up, the we start retarded from propagator the free streaming of the gas energy regime, momentum reproducing tensor, the well or explicitly known literature results of both the non-relativisticµ⌫,↵ and the ultra-relativisticµ⌫ ↵ gas approximation. We then Gmedium(x, x0)= i✓(t t0) [T (x),T (x0)] (7) switch to the the hydrodynamical case,− analyzing− both the well known inviscid case and the The response function depends in detail on the⌦ microscopic dynamical↵ properties of the slightly more complicated scenario with viscosity. Finally we perform the full calculation for mediumthe interacting and eventually gas that needs interpolates to be computed. between all While these cases. some In particular the last section cases, e.g. we drawthe our free- streamingconclusions gas and discuss the ideal the liquid, application are ofvery our well results known, for real-words the most problems. generic Thecases technical must be calculated,details are as relegatedwe show in to the appendices. next section and in the appendices. Finally, the wake itself will have its own gravitational field, which can again be computed usingAII. the DESCRIPTIONNEW Green function APPROACH of OF the THE linearized FORMALISM Einstein equation wake 4 grav ↵ h (x)= d x0G (x, x0)T (x0). (8) Our main objective isµ to⌫ calculate a force,µ⌫, exerted↵ onwake a bullet that propagates through Z Thatagasofparticles.Onecannoticethatinthelinearizedregime,theforceisexertedonthe is, hwake represents the gravitational field that was created by a wake that was created (Thermal) field theory approach + Gravity as an EFT by abullet gravitational by the gravitational field that was field created of its own by thewake bullet.
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