Birkhoff's Theorem Fails to Save MOND from Non-Local Physics

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GMc ~rak ~rca 2 2 (1 ·2 )+ vk +2va ca − c=a r − 2rca 6 X 2 3 ~va ~rak 4 ~va ~vk · (1) − · − 2 rak GMa~rak (3~va 4~vk) (~va ~vk) · 3 − − a=k − rak 6 X 2 7 G MaMc + ~rca 3 . ak 2 a=k c=a r rca X6 X6

Here ~ra = ~xa ~x and ~rab = ~xa ~xb. ~x is the spatial − − variable in the PPN coordinate frame, and ~xa is location of object a. This equation describes the free-fall of each point in the system. In order to describe the acceleration of the test particle in our system, we should remove from equation (2) the terms that are associated with the acceleration of the shell (which is much smaller than the acceleration of the point particle). The acceleration is then FIG. 1: Mass distribution: We consider a point particle inside 2 2 a large spherical shell. R denotes the radius of the shell, and d xk dvk G MaMk = ~rak (2) rk denotes the amount of displacement of the particle from 2 4 dt dt ≃− a=k rak the center of the shell. 6 2 X G MkMs R rk 2rkR = 2 ln − + 2 2 zˆ . − 4rkR R + rk (R rk) We may consider this distribution to be a very simpli- − fied version of for example a galaxy in a cluster, or a In obtaining the final result, we have replaced the sum clump of matter – star, globular cluster, clump of dark by an integral over the shell which we were able to per- matter – inside a galaxy. For definiteness we will the to- form. Notice that there is a singularity as rk R. This tal mass of the system to be typical of a galaxy cluster, → 14 singularity is not real. It appears because PPN is not Mtotal = 2.0 10 M , and take the radius of the shell ⊙ valid as rk is very close to the shell. to be fixed at× a typical cluster raius, R = 2.0Mpc. We GMs now proceed to investigate the acceleration of the point Defining d = rk/R, RSchs = 2GMs and gNs = R2 , mass in both General Relativity and MOND. equation (2) can be written as

2 d xk Mk RSchs 1 1 1 1 d 2 = gNs 2 + ln − zˆ GENERAL RELATIVITY dt − Ms R 4d 1 d 2d 1+ d − Mk RSchs 1 2 3 3 5 gNs d + d + d + z,ˆ (3) In Newtonian mechanics, a spherical shell is a well- ≃ − Ms 2R 3 5 7 ··· defined object. However, in GR, since particles change the geometry around them, the simple definition of a where the last line is an expansion for small d. sphere as a set of point equidistant from some common The gravitational acceleration clearly does not vanish point (the center) is lost in the presence of a perturbing – it points toward the center. A particle inside a spherical mass. We define a spherical shell using Parametrized shell is therefore attracted toward the center to restore Post-Newtonian (PPN) coordinates. We identify a point, spherical symmetry, where the acceleration happily does the center, such that the distance from the center in PPN vanish. However, since Mk Ms, the acceleration is coordinates is the same for each point on the shell. We very small compared to, say, the≪ Newtonian acceleration call this distance R the radius of the shell. just outside the shell, gNs. It is suppressed by both the We can calculate the acceleration of the test particle ratio of the test mass to the shell mass, and, more impor- in GR from the Einstein-Infeld-Hoffman (EIH) equation tantly, by the ration of the Schwarzschild radius RSchs of [6]: the shell to its physical radius. This suppression factor 6 2 is huge for typcial astrophysical systems, 9.7 10− for d xk dvk GMa GMb × 2 = = ~rak 3 1 4 our choice of mass and radius. This suppression factor dt dt r − rbk a=k ak b=k makes the acceleration physically insignificant in GR. X6 X6 3

MOND PERTURBATIVE CALCULATION

In 1984 Bekenstein and Milgrom introduced the La- We first consider the mass of the particle inside the grangian formulation of Modified Newtonian dynamics shell to be much larger than the mass of the shell. In this (MOND) [9]. The field equation of MOND is derived case, the density distribution of the spherical shell can from the Lagrangian be treated as an aspherical perturbation on a spherical system – the point mass – and a perturbative solution to 2 3 1 2 ( ψ) the modified Possion equation is possible. Let the density L = d r ρψ + (8πG)− a0 ∇ , (4) − F a2 distribution of the point particle and the spherical shell Z 0 be ρk and ρshell, respectively. Then the unperturbed field 2 where ψ is the gravitational potential. (y ), with y ψ0 satisfies the modified Poisson equation: F ≡ ψ /a0, is an arbitrary universal function, that together |∇ | [µ( ψ0 /a0) ψ0]=4πGρk . (9) with a0, the characteristic scale of MOND, specifies the ∇ · |∇ | ∇ theory. Varing L with respect to ψ yields a modified The exact solution for ψ0 can be found by applying Possion equation: ∇ GMk 2 Gauss’s theorem. In terms of the quantity u = a0 ~r ~rz , | − | [µ( ψ /a0) ψ]=4πGρ(x) , (5) ∇ · |∇ | ∇ ψ0 1 1 2 ∇ = u2 + u4 + u2 (10) where µ(y) ′(y ). µ(y) must approach 1 as y 1 a0 s2 4 ≡ F | | ≫ 1 r and y as y 1, in order that the field scale as r2 | | | | ≪ 1 1 near a spherical mass distribution (the usual Newtonian u 2 (1 + u + ...) . 1 ≃ 4 result) and as r far from the mass distribution to explain

flat galaxy rotation curves. GMk The expansion in small u is valid for ~r ~rk . One the widely used form of µ is | − | ≫ a0 Knowing ψ0, one can find the first-order perturbationq ∇ y equation for ψ1 using µ(y)= | | 2 ) . (6) ( 1+ y [ ψ0 /a0(( ψ1 eˆ0)ê0 + ψ1)]=4πGρshell , (11) | | ∇ · |∇ | ∇ · ∇ p (However see [7] for different form of µ function.) The wheree ˆ0 is a unit vector pointing in the direction of value of a0 is then given by phenomenological fit. We will ψ0. In the deep MOND regime where µ( ψ0 /a0) ∇ |∇ | ∼ adapt the value derived by Begeman et al [8] in the study ψ0 /a0, and in the limit R rk, one can expand ψ1 |∇ | ≫ rk of external galaxies with high quality rotation curves. just inside and outside of the shell in powers of R . The 10 2 first order aspherical contribution to the potential is a0 =1.2 10− m/s . (7) × rk Ga0 cos θ s 4 Mk M r for r R Again we consider the mass distribution in Fig. 1. The ψ1 = ≤ . (12) rk Ga0 r cos θ  q Ms 2 for r > R center of the spherical shell is chosen to be the center  4 Mk R of coordinates. Because of the axial symmetry of the q configuration, we put the point particle on the z-axis. The net force on the spherical shell due to ψ0 and ψ1 is We assume that the radius of the shell is large enough 2 2 Ms√GMka0 R rk R rk that the gravitational acceleration goes deep into the F~shell = 1+ − ln − MOND regime. For a bounded mass distribution of total 2rk 2rkR R + rk 3 mass M, we define a transition radius Ms (GMk) 2 1 1 1 (R rk) 1 2 2 + ln − − 8 2 rk R r 2rkR (R + rk) a0 k Rt = GM/a0 . (8) − 5 rk Ga0 2 p 2 Ms + zˆ . (13) Rt indicates a point at which the Newtonian field approx- 24 R Mk − r ···) imately equals a0, and this is about the point at which 2 the field switches from Newtonian 1/r to MOND’s 1/r. The net force on the shell points toward positivez ˆ, having Thus we take R Rt. For the physical parameters we the effect of restoring the spherical symmetry by moving ≫ are considering, R/Rt =4.2. the shell so that the it is centered on the point mass. The Non-linearity in the modified Possion equation negates first two terms result from ψ0 and the last term comes the superposition principle, and it makes an analytic cal- from ψ1. We note that the first term, which is dominant culation of the field a non-trivial task. We present the over other terms, is independent of the specific choice perturbative solution for the case Mk Ms, and in more of µ function. It is a consequence solely of the MOND general cases, we resort to numerical≫ calculations. condition: µ(y) y for y 1. → ≪ 4

The Lagrangian (4) from which the field equation (5) is derived is invariant under spacetime translations and spatial rotations. Energy-momentum and angular momentum conservation therefore hold for an isolated system in MOND. In particular, Newton’s third law of ac- tion and reaction is valid. Hence, the acceleration ak of the point particle in the frame of the shell can be found from equation (13). Writing Rk = GMk/a0, d = rk/R, and expanding in a Taylor series, p ak Ms Rk 1 1 1 d + d3 + d5 + ... a0 ≃ −Mk R 3 15 35 3 Ms Rk 1 1 3 + d + d3 + d5 + ... Mk R 12 10 28 2 5 Ms Rk + d zˆ . (14) 24 Mk R ) As expected, ak approaches zero as d 0. For d = 0, → 6 ak points toward the center of the spherical shell. As the point particle is displaced, the system reacts to re- FIG. 2: This figure plots the acceleration on the point par- store the spherical symmetry. We see here that when ticle as a function of d = rk/R. They all points toward the center of the spherical shell. The numerical solutions and the the system as a whole is not spherically symmetric, the corresponding perturbation solution is shown. acceleration on a test particle in a spherical shell is non- vanishing. The suppression of the acceleration compared to char- The discretization and numerical calculation on the lat- acteristic accelerations in the system is much milder in tice is done in terms of U~ . Because of the one-to-one cor- MOND than that in GR. In MOND, the scale of the respondence between ψ and U~ , one can find ψ from ∇ ∇ acceleration is a0, but suppressed by Rt/R, whereas in U~ . general relativity, it the scale is gN , but suppressed by An initial ansatz for U~i in the numerical solution is RSch/R. Since Rt RSch in general and since a0 gN given by solving for a system deep in≫ the MOND regime, the scale acceler-≫ ation is much larger in MOND than in GR. For Mtotal = Ui =4πGρ . (18) 14 ∇ · 2.0 10 M and RSch =2GMtotal, the ratio of accelera- × ⊙ 5 ~ tions is approximately (a0 Rt)/(gN RSch)=4.3 10 . The field Ui has the correct divergence, but not the × × × correct curl. The code iterates to make the curl vanish at each vertex of the lattice. U~i also serves as the NUMERICAL CALCULATION boundary condition for the numerical solution. Since (U~ U~i) = 0, U~ and U~i differ only by the curl field. It To find the acceleration of the point particle in a more ∇·can be− shown [9] that the curl field for a bounded mass 1 general case, we adapt the numerical scheme developed distribution vanishes at least as fast as r3 . Then, as- by Milgrom [10]. In this scheme, the field U is defined by suming that the physical size of the lattice∼ is large compared to the mass distribution, U Ui on the boundary. U~ = µ( ψ /a0) ψ . (15) → |∇ | ∇ We implement Milgrom’s algorithm on a spherical lat- So long as µ is monotonic, one can invert the relation tice. The point particle is placed at the center of the in equation (15) and write ψ in terms of U. For the lattice, and the center of the spherical shell is displaced ∇ specific choice of µ given by equation (6), from the center of the lattice by rk. The total number of angular grid lines is denoted by− L, and the total num- 2 ber radial grids is fixed at L/5. To meet the boundary 1 1 a0 ~ ψ = v + + 2 U. (16) condition, the radius of the outermost shell is set to be ∇ u2 s4 U u | | 100R. t The field U~ , then, satisfies the set of differential equations In the case Mk Ms, a comparison between the numerical results and≫ the perturbative solution is possible. ~ U~ =4πGρ ∇ · We make the comparison between the solutions with fixed 2 values of Ms/Mk =0.01 and Rk/R =0.25. Fig. 2shows ~ 1 1 a0 ~ that these agree to better than one percent. On the v + + 2 U =0 . (17) ∇× u2 s4 U plot shown, the percentage difference varies from 0.1% u | | t 5

FIG. 3: This figure shows the dependence of the numerical solution on the number of lattice sites. The value of d is fixed at d = 0.02. The dotted line indicates the value from the perturbation solution. FIG. 4: This figure shows the acceleration of the point particle as a function of Mk/Mtotal for particles at a variety of positions. The acceleration points toward the center of the to 0.9%. The discrepancy between the values increases shell. The numerical results are for L=300. as d = rk/R approaches 1, where the perturbative expansion becomes less reliable. Fig. 3 shows how numerical (d 1), the value for Mk/Mtotal = 0.01 exceed that for values depend on the number of lattice points. As ex- ∼ Mk/Mtotal = 0.15. This is because the peak mentioned pected, the values approach the perturbative solution as above occurs at lower value of Mk/Mtotal for larger value L increases. In the following study, we use the lattice size of d. of L = 300 and L = 600. Unlike for GR, in MOND the acceleration of the point Fig. 4 and Fig. 5 represent the results of the numerical particle is a significant fraction of gN . Especially when calculations. The mass dependence of the acceleration of the particle is close to the shell, the acceleration can be the point particle is plotted in Fig. 4, and the position de- larger than gN . pendence is plotted in Fig¿ 5. We see clearly that there is non-vanishing acceleration directed toward the center of the spherical shell. In Fig. 4, we see that the acceleration CONCLUDING REMARKS AND IMPLICATIONS vanishes in both the Mk 0 limit and as Ms 0. This → → is expected since these are two limits in which spherical In both general relativity and MOND, the accelera- symmetry is recovered. For the three curves plotted in tion of a massive test particle inside a spherical mass Fig. 4, the peak value occur around Mk/Mtotal 0.15. ∼ shell vanishes only when the particle is at the center of The position of the peak has a slight position dependence the shell. When the particle is displaced from the center, – as the particle gets closer to the shell (large d), the peak the particle experiences a force toward the center of the occurs at smaller Mk/Mtotal ratio. shell, the direction in which spherical symmetry would We note one curious feature from Fig. 4. The astro- be restored. The magnitude of the acceleraion in GR is nomically interesting region in Fig. 4 is where Mk/Ms not physically significant in most (or probably all) astro- ≪ 1. In this regime, the acceleration of the point particle physical situations of interest, since it is suppressed by is a very sharp function of its mass. This implies that the ration of the Schwarzschild radius of the shell to the galaxies with slight mass differences might experience size of the shell. quite different acceleration in theoriest that implement In MOND, on the contrary, the acceleration of the the MOND limit. point particle is a significant fraction of the surface grav- The distance dependence of the acceleration is shown ity just outside the shell. This is despite the fact that in Fig. 5. Again, the acceleration of the particle van- MOND has a Birkhoff or Gauss-like theorem which im- ishes when the particle is near the center (d 0), plies that the potential inside an isolated spherical mass and increases monotonically outward. Near the→ shell shell is constant. However, this theorem operates in 6

From Fig. 4 and Fig. 5, we can see that the acceleration inside the spherical shell can be signiﬁcant at this scale. In MOND, the applicability of Birkhoﬀ’s theorem is very limited, and in computations one needs to consider not only the local mass distribution but also the back- ground mass distribution.

ACKNOWLEDGEMENTS

We want to thank Irit Maor for helpful suggestion in development of the code. We also wish to thank Yi-Zen Chu for helpful discussion regarding EIH equation. GDS and RM are supported by a grant from the Department of Energy in support of the Particle Astrophysics theory group. DCD is supported by grants from the HEPCOS group at SUNY.

[1] Mordehai Milgrom, Astrophys. J. 270: 365-370, 1983 FIG. 5: This figure shows the acceleration on the point par- [2] Mordehai Milgrom, Astrophys. J. 270: 371-383, 1983 ticle as a function of position for fixed particle-mass to total- [3] Robert H. Sanders and Stacy S. McGaugh, Ann. Rev. mass ratio. Results are plotted for two different mass ratio. Astron. Astrophys. 40:263-317, 2002, astro-ph/0204521 The acceleration points toward the center of the shell. Simu- [4] Mordehai Milgrom, Astrophys. J. 270: 384-389, 1983 lations are performed on the lattice with L=600. [5] Jacob Bekenstein, Phys. Rev. D70, 083509, 2004, astro-ph/0403694 [6] Charles W. Misner, Kip S. Thorne, John Archibald the absence of the usual explanation for Gauss’ Law in Wheeler: exercise 39.15, ”Graviation”, ISBN 0-7167- Newtonian gravity or classical electrostatics – the bal- 0334-3 2 2 [7] Benoit Famaey and James Binney, ance between the r growth of surface areas and the r− Mon.Not.Roy.Astron.Soc.363: 603-608, 2005 force law. Thus, when the very particular conditions of [8] Begeman K.,Broeils A. and Sanders R., Birkhoff’s theorem are broken even a little, the gravi- Mon.Not.Roy.Astron.Soc. 249: 523-537, 1991 ational force re-emerges at considerable strength. The [9] Jacob Bekenstein and Mordehai Milgrom, Astrophys. J. characteristic MOND field for the values of parameters 286: 7-14, 1984 √GMtotala0 [10] Mordehai Milgrom, Astrophys. J. 302: 617-625, 1986 used in this paper is approximately R =0.24a0.