Modified Newtonian Dynamics

J. Astrophys. Astr. (December 2017) 38:59 © Indian Academy of Sciences https://doi.org/10.1007/s12036-017-9479-0

Modiﬁed Newtonian Dynamics (MOND) as a Modiﬁcation of Newtonian Inertia

MOHAMMED ALZAIN

Department of Physics, Omdurman Islamic University, Omdurman, Sudan. E-mail: [email protected]

MS received 2 February 2017; accepted 14 July 2017; published online 31 October 2017

Abstract. We present a modified inertia formulation of Modified Newtonian dynamics (MOND) without retaining Galilean invariance. Assuming that the existence of a universal upper bound, predicted by MOND, to the acceleration produced by a dark halo is equivalent to a violation of the hypothesis of locality (which states that an accelerated observer is pointwise inertial), we demonstrate that Milgrom’s law is invariant under a new spaceÐtime coordinate transformation. In light of the new coordinate symmetry, we address the deficiency of MOND in resolving the mass discrepancy problem in clusters of galaxies.

Keywords. Dark matter—modiﬁed dynamics—Lorentz invariance

1. Introduction the upper bound is inferred by writing the excess (halo) acceleration as a function of the MOND acceleration, The modified Newtonian dynamics (MOND) paradigm g (g) = g − g = g − gμ(g/a ). (2) posits that the observations attributed to the presence D N 0 of dark matter can be explained and empirically uni- It seems from the behavior of the interpolating function fied as a modification of Newtonian dynamics when the as dictated by Milgrom’s formula (Brada & Milgrom gravitational acceleration falls below a constant value 1999) that the acceleration equation (2) is universally −10 −2 of a0 10 ms . Milgrom (1983) noticed that the bound from the above by a value of the order a0, rotation curves of disk galaxies can be specified, given = η , only the distribution of visible (baryonic) matter, using a† a0 (3) the formula where η is a dimensionless constant which is of order unity. This prediction was confirmed from the rota- gμ(g/a ) = g (1) 0 N tion curves for a sample of disk galaxies (Milgrom and which relates the observed gravitational acceleration Sanders 2005). g to the Newtonian gravitational acceleration gN as On the other hand, in Einstein’s special theory of calculated from the baryonic mass distribution. The relativity, when Lorentz invariance is extended to accel- interpolating function μ(g/a0) satisfies μ(g/a0) = erated observers, it is assumed that the behavior of g/a0 when g a0,andμ(g/a0) = 1wheng a0. measuring rods and clocks is independent of acceler- The direct observational evidence for Milgrom’s for- ation (Mashhoon 1990). This in fact, is a statement of mula is the fact that the mass discrepancy in galaxies of the hypothesis of locality which asserts that an accel- all sizes always appears below the acceleration scale erated observer makes the same measurements as a a0 (Famaey & McGaugh 2012). It follows from the hypothetical momentarily co-moving inertial observer. appearance of an acceleration scale where dark matter For instance, the rate of an accelerated clock is assumed halos are needed, that there is a universal upper bound to be independent of its acceleration and identical to to the acceleration that a dark halo can produce (Brada that of the instantaneously co-moving inertial clock ‘the & Milgrom 1999). The difference between the MOND clock hypothesis’ (Moller 1972; Rindler 1982). acceleration g and the Newtonian acceleration gN can If, however, we assume that the characteristic maxi- be explained by the presence of a fictitious dark halo and mum acceleration that appears in the behavior of dark 59 Page 2 of 9 J. Astrophys. Astr. (December 2017) 38:59 halos predicted by MOND equation (3)isinvariant The problems of modified inertia formulations of under transformations from inertial to accelerated ref- MOND can be alleviated by interpreting Milgrom’s for- erence frames, then our assumption apparently con- mula as a modification of Newtonian gravity. tradicts the kinematic rule that the acceleration a† as Bekenstein and Milgrom (1984) proposed a non- measured in an inertial frame S is given by French relativistic theory of MOND as a modification of = + (1971): a† a† A,wherea† is the acceleration Newtonian gravity (called AQUAL); the theory contains as measured in an accelerated frame S and A is the a modified gravitational action while the kinetic action acceleration of the frame S with respect to the inertial takes its standard form, and thus conservation laws are = frame S. Thus, if we assume that a† a†, then the preserved. The attempts to formulate a covariant gen- accelerated measuring rods and clocks must behave in eralization of AQUAL culminated with the emergence such a way that the relative acceleration between the of the Tensor-Vector-Scalar theory (TeVeS)(Bekenstein two reference frames S and S becomes undetectable 2004), the first consistent relativistic gravitational field and therefore it becomes unreasonable to assume that theory for MOND. But even in this theory there is still the hypothesis of locality is still valid upon making a need for a predefined interpolating function that inter- such assumptions about the maximum halo accelera- polates between the Newtonian and MONDian regime. tion. So, probably the most suitable way to derive the Instead of focusing our attention on constructing maximum halo acceleration equation (3) from physi- modified actions for MOND, let us reconsider the non- cal assumptions (not from the near coincidence of a0 conservation of momentum problem from a mathemat- with cosmological parameters (Milgrom 1983)orintro- ical point of view; the non-conservation of momentum ducing any new assumptions about the nature of dark exhibited in equation (5) can be attributed to the fact matter) is to assume that the hypothesis of locality is that each body in the isolated system is subject to a false in the low acceleration limit g a . non-Newtonian force of magnitude ma0, thus 0 √ Our derivation of the maximum halo acceleration is √ √ p˙ = F( m a − m a ), (6) based on interpreting Milgrom’s formula as a modifica- 1 0 2 0 tion of Newton’s second law of motion (Milgrom 1994), which means that it is not possible to isolate the two interacting bodies in the low acceleration limit from F = mgμ(g/a0), (4) the influence of all sorts of external forces. So, why should we expect Newton’s first law to be valid in where F is the total force exerted on the particle, and m the MOND regime? Perhaps, the isolated body in the is the inertial mass (response of the particle to all forces). MOND regime behaves differently from the isolated However, this law is not enough by itself to represent a body in the Newtonian regime. consistent modification of Newtonian inertia, because, Newton’s first law states that for an isolated body, if we consider an isolated system consisting of two bod- far removed from all other matter, the vector sum of ies interacting gravitationally with small masses m1 and all forces vanishes F = 0, hence, the isolated body m2 such that equation (4) applies, and differentiate the moves with uniform velocity. Let us instead assume that total momentum p = p1 + p2 using equation (4)we in the case of an isolated body in the MOND regime; obtain (Felten 1984) the sum of the magnitudes of all forces is constant and = η η √ √ is proportional to ma0, thus, F ma0 where is p˙ = a0 F( m1 − m2). (5) the proportionality factor, a0 is the MOND acceleration constant, and m is the inertial mass. The isolated body The total momentum of this isolated system is not then moves with uniform acceleration gμ(g/a0) = ηa0, conserved, p˙ = 0 unless m1 = m2. This problem we will refer to this assertion as the modified Newton’s can be avoided if there is a nonstandard kinetic action first law. Note that this is an assertion that cannot be from which the equation of motion equation (4)is confirmed experimentally, like Newton’s first law. derived. Milgrom (1994) constructed such modified Suppose an observer is placed in a reference frame kinetic actions and showed that they must be time- S in which the modified Newton’s first law holds, nonlocal to be Galilean invariant, but it is unclear how to then the observer in this frame will measure a force construct a relativistic generalization of such a scheme. F = mgμ(g/a0) = mηa0. If there is another observer It should be mentioned here that there are other possible in a frame S which is moving with respect to S with approaches to MOND inertia, for example, inertia could acceleration, then the second observer will also measure be the product of the interaction of an accelerating parti- the same force F = mg μ(g /a0) = mηa0 (assuming cle with the vacuum (Milgrom 1999; McCulloch 2012). that the mass and the acceleration constant ηa0 are the J. Astrophys. Astr. (December 2017) 38:59 Page 3 of 9 59 same in S as in S). The relative acceleration between We can demonstrate that the inertial mass of the test the two frames S and S is not dynamically detectable particle governed by the equation of motion equation due to the invariance of ηa0. (7) is equivalent to its gravitational mass by performing Therefore, the modified Newton’s first law, coupled the spaceÐtime coordinate transformations, with the invariance of ηa0, defines an infinite class of 1 2 equivalent reference frames in accelerated motion rel- x = x − gDt , (8) ative to one another, and suppresses the appearance of 2 = , inertial forces. Hence, the uniformly accelerated frames t t (9) S and S are equivalent and Milgrom’s law is the same where the spatial origins of the two coordinate systems = μ( / ) = μ( / ) = in both frames F mg g a0 mg g a0 F . S and S coincide at t = t = 0, the unprimed sys- When a MOND theory is fully compatible with this tem is the freely falling frame and the primed system coordinate symmetry (i.e. the impossibility of detect- is an inertial frame. Since the gravitational field gD is ing a coordinate change) it must satisfy conservation uniform (it does not depend on t or x), the equation of laws such as the conservation of momentum, because motion becomes the coordinate symmetry implies the homogeneity of 2 space. The modified Newton’s first law is a key feature d x = , m 0 (10) of our derivation of the upper bound equation (3) from dt 2 physical assumptions. the gravitational force mgD is canceled by an inertial force. Hence, at any spaceÐtime point in a uniform gravitational field we can specify a locally inertial reference 2. The maximum halo acceleration and its frame in accordance with the principle of equivalence. consequences But, motivated by the existence of the acceleration scale equation (3), let us assume that at some spaceÐtime Any physical process that involves the dynamics of par- points in the uniform gravitational field gD, we can- ticles and fields plays out on a background of space and not specify a locally inertial frame, in particular, let us time. Consequently, the physical laws must be adapted postulate that there exists a universal constant of the to any changes that might occur in the background (such order of the MOND acceleration constant a† = ηa0, as replacing the Galilean transformation by the Lorentz which is invariant under transformations from inertial transformation); this scientific way of thinking about to accelerated frames. Thus, by performing spaceÐtime space and time, initiated by Einstein, led to modifica- coordinate transformations analogous to equations (8) tions of the existing physical laws that are not Lorentz and (9) when the magnitude of the gravitational field invariant (Rindler 1982). Therefore, we can establish gD is equal to a†, the equation of motion equation (7) an elegant physical basis for MOND, if Milgrom’s law must become equation (4) is invariant under a new spaceÐtime coor- d2x dinate transformation. m = ma , 2 † (11) Let us consider a test particle of mass m freely falling dt in a uniform gravitational field of a dark matter distri- this result is what we referred to earlier as the modified bution, where the density distribution of dark matter is Newton’s first law, an observer placed in the reference derived from the rotation curves of disk galaxies; the frame S is assumed to be isolated from the influence discrepancy between the rotation curve expected from of any other matter; and yet the observer experiences a v2 = / the distribution of baryonic matter, B GMB r,and force of magnitude ma†. This is due to the fact that a† the rotation curve measured by utilizing the Doppler is an invariant of coordinate transformations, 2 effect, v = GM/r, yields the distribution of dark mat- d2x d2x ter, v2 = v2 − v2 = GM /r. Then, the force acting = = a . D B D 2 2 † (12) on the test particle is given by Newton’s second law dt dt 2 Then, according to our postulate, the spaceÐtime coor- d x GmMD m = mgD = . (7) dinate transformations equations (8)and(9)must dt2 r 2 be accommodated to the condition equation (12). As a consequence of the equality of inertial and gravita- Although this postulate has not been confirmed by tional mass, a freely falling reference frame constitutes any experiment and cannot be demonstrated from an inertial reference frame; the uniform gravitational first principles, we will demonstrate that this pos- field cannot be detected in the freely falling frame. tulate is the main reason for the emergence of the 59 Page 4 of 9 J. Astrophys. Astr. (December 2017) 38:59 upper limit equation (3) that has been confirmed from in S, who wants to measure the length of this rod, must observations. measure the coordinates of the ends of the rod at the Consider a uniformly accelerated reference frame S same time t. Using the coordinate transformation equa- moving with an acceleration gD relative to an inertial tion (14), we have reference frame S, if the origins of both reference frames = = = α + 1 2 , = α + 1 2 . coincide at t t 0 the origin of the reference frame x1 x1 gDt x2 x2 gDt S which has the coordinate x = 0 will be at a dis- 2 2 1 2 (17) tance x = gDt from the reference frame S. Thus, 2 it is reasonable to assume that x is proportional to the Then, the length of the rod as measured in the reference ( − 1 2) same x 2 gDt factor as in the familiar coordinate frame S is transformation equation (8): − l = x2 x1 = − , x2 x1 (18) 1 2 α α x = α x − gDt , (13) 2 if both frames S and S are equivalent and the length where α is the proportionality factor. The same argu- of the same rod moving in these frames at the same ment applies if we take the coordinate system S to be acceleration gD must be the same, we must have l/α = the inertial frame, in this case, the origin of the refer- l/α . Consequently, ence frame S has the coordinate x = 0 and moves with α = α. (19) acceleration −gD relative to the reference frame S ,so =−1 2 In accordance with the postulate equation (12), if that x 2 gDt . Hence, the space transformation now takes the form there is an object moving at acceleration a† in an accelerating reference frame S, then the trajectory of this 1 2 x = α x + gDt , (14) object as measured by an observer in an inertial refer- 2 ence frame S is where α is the proportionality factor. In order to deter- 1 2 mine the relation between the two factors α and α , x = a†t , (20) 2 let us consider two observers placed in the uniformly while the trajectory of the same object as measured by accelerated frames S and S . Since the observers in both an observer in the accelerating frame S is frames experience the inertial force caused by the acceleration gD (only the sign of gD is different in the two 1 2 x = a†t . (21) frames), the uniformly accelerated frames S and S must 2 be equivalent for the description of physical events, for Substituting these trajectories into equations (13)and example, the length of the same measuring rod moving (14), we obtain in these frames at the same acceleration gD must be the 2 2 a†t = αt (a† − gD), (22) same. 2 = α 2( + ), = − a†t t a† gD (23) Suppose a rod of length l x2 x1 is at rest in the reference frame S which is moving with an accelera- from which we obtain a Lorentz-type factor tion gD relative to the reference frame S. The observer a2 in S, who wants to measure the length of this rod, must α2 = † , ( + )( − ) (24) measure the coordinates of the ends of the rod at the a† gD a† gD 1 same time t. Using the coordinate transformation equa- α = . (25) tion equation (13), we have 1 − g2 /a2 D † = α − 1 2 , = α − 1 2 . The appearance of the Lorentz-type factor concludes x1 x1 gDt x2 x2 gDt 2 2 our derivation of the maximum halo acceleration from (15) physical assumptions, since it implies that it is a physical impossibility for a spherical dark matter halo to attain Therefore, the length of the rod as measured in the ref- −1 erence frame S is a surface density greater than a†G . − To get the time transformation, we can substitute l x2 x1 = = x2 − x1. (16) equation (13) into equation (14) to obtain α α Let us now interchange S and S . Suppose the same rod 1 2 1 2 x = α α x − gDt + gDt (26) is at rest in S, where length l = x2 − x1, the observer 2 2 J. Astrophys. Astr. (December 2017) 38:59 Page 5 of 9 59 and then we can rewrite the Lorentz-type transformation equations (30)and(31) in the four-vector notation: 1 2 = x −α − 1 2 = α 1 2+ 1 −α gDt x gDt gDt x g 2 α 2 2 α x 1 = α x1 − D x0 , x 2 = x2, x 3 = x3, (35) (27) a† g x 0 = α x0 − D x1 (36) whereas, from equation (24), we have a † g2 which we can write in the matrix form as equation (32), − D − α2 2 a2 2 where the components of the transformation matrix are: 1 g † g = 1 − D =−α D . (28) ⎛ ⎞ α 2 g2 2 α −α / a† − D a† gD a† 00 1 2 ⎜ ⎟ a μ −αg /a α 00 † = ⎜ D † ⎟. (37) ν ⎝ 0010⎠ which leads to 0001 g2 1 2 = α 1 2 − α D . Note that the components of the transformation matrix gDt gDt 2 x (29) 2 2 a† do not depend on the coordinates, as long as the gravitational field gD is uniform. Thus, the spaceÐtime coordinate transformations equa- A four-vector can now be defined as any set of tions (8)and(9) are the low acceleration limit gD a† four components that transform under the Lorentz-type of the Lorentz-type transformations transformation equation (32), the same way (x0, x1, x2, x3) does; for example, the time difference between any 1 2 x = α x − gDt , (30) two events and their spatial separation can be repre- 2 sented by the displacement four-vector ⎛ ⎞ 2 2 2gD t = α t − x . (31) a†tdt 2 ⎜ ⎟ a† μ dx dx = ⎜ ⎟. (38) ⎝ dy ⎠ Unlike the Galilean and Lorentz transformations, the dz transformation equations (8)and(9), and the Lorentz- type transformation equations (30)and(31) are nonlin- the components of this vector as specified relative to the ear in the time coordinate. Hence, if we represent the coordinate system S are related to the components of the μ nonlinear transformation for the space and time coordi- same vector dx as specified relative to the coordinate nates by a 4 × 4matrix system S by the Lorentz-type transformation equation (32) 3 μ μ ν 3 x = ν x , (32) μ μ ν dx = ν dx . (39) ν=0 ν=0 then the elements of the transformation matrix must Let the inverse transformation to equation (39) read as ( ) be coordinate dependent x . However, the new trans- follows: formation equations (30)and(31) have an advantage 3 over the classical equations (8)and(9): it has a free ν ν β dx = βdx , (40) parameter with the dimensions of acceleration a†. Since, ν=0 a is the same in all coordinate systems, it can be recog- † μ ν nized (in analogy to the speed of light) as a conversion where the matrices ν and β are inverse to each other; factor that converts time measurements in seconds to ⎛ ⎞ 1 2 2 0 ααgD/a† 00 meters t(m) = a†×t (s ). Thus, we can define x with ⎜ ⎟ 2 ν αg /a α 00 0 = ⎜ D † ⎟. (41) the factor a† so that x has the dimensions of length, β ⎝ 0010⎠ 0001 0 1 2 x = a†t (33) 2 By substituting equation (40) into equation (39), we can and if we number the x, y, z coordinates, so that verify that the transformation matrix of the Lorentz-type transformation equation (32) satisfies the orthogonality x1 = x, x2 = y, x3 = z (34) condition 59 Page 6 of 9 J. Astrophys. Astr. (December 2017) 38:59

3 Fμ μ ν μ where is a four-vector force (the force due to the ν β = δβ , (42) excess acceleration four-vector). We can find an appro- ν=0 priate interpretation of the time component of the force μ α where δβ is the Kronecker delta. four-vector by Taylor, expanding the factor, μ Therefore, the scalar product of dx with itself is an 2 0 ma† 1 gD invariant quantity F = = ma† + m +···. (52) 2 2 2 a μ 2 2 2 2 1 − g /a † dx dxμ = (a†tdt) − (dx) − (dy) − (dz) , (43) D † μ 2 2 2 2 dx dxμ = (a†t dt ) − (dx ) − (dy ) − (dz ) . Let us now return to Milgrom’s formula equation (1)and (44) choose the following form of the interpolating function as chosen by (Bekenstein 2004) If an observer is at rest in the frame S , then the spatial √ components of the displacement vector in this frame is 1 + 4x − 1 μ(x) = √ , (53) zero 1 + 4x + 1 μ 2 2 2 2 dx dxμ = (a†tdt) − (dx) − (dy) − (dz) where x = g/a0, hence, in the low acceleration limit 2 the total acceleration due to Newtonian gravity can be = (a†t dt ) . (45) μ expressed as follows Hence, the scalar product dx dxμ is proportional to the √ time interval measured by an observer in its rest frame. g = gN + a0gN . (54) We can employ this fact to define a transformation- = − τ Thus, the excess acceleration gD g gN can take the invariant coordinate time (the proper time ) which will following form allow us to obtain a four-vector when differentiating a √ four-vector, gD = a0gN . (55) μ 2 2 2 2 2 dx dxμ =(a†tdt) −(dx) −(dy) −(dz) =(a†τdτ) . Since, in equation equation (52) we can neglect the 2 (46) terms divided by a† and higher in the limit of small accelerations g a , we can assume that F0 is made Thus D † up of two parts: the first part gives identical results to 1/2 1 dx2 + dy2 + dz2 Milgrom’s law equation (4) in the low acceleration limit τdτ = tdt 1 − (when the numerical factor takes the value η = 1 )and a2 t2dt2 2 † in the presence of Newtonian gravitational forces 1/2 = tdt 1 − g2 /a2 , (47) D † F0 = ma† − = 1 ma† mgN (56) the invariant time unit can be obtained by integrating − 2 / 2 1 gD a† both sides / τ = − 2 / 2 1 4. and the second part is a constant force t 1 gD a† (48) F0 = The trajectory of the observer in the frame S can be 2 ma† (57) parameterized by the proper time τ; which is the force experienced by an observer at rest, 1 μ a t2(τ) and it can be interpreted as a location independent x (τ) = 2 † , (49) x(τ) weight. In contrast, the weight of an object in New- tonian physics is defined as the product of the object’s where x is the three-dimensional position, therefore the mass and the magnitude of the gravitational accelera- excess acceleration four-vector can be obtained by dif- tion which depends on the location. We deduce from ferentiating the position four-vector equation (49) with this that F0 = F0 + F0 = F is the total force acting respect to the proper time equation (48) 1 2 on the particle mass m. 2 μ Aμ = d x = α a† = α a† . In the Newtonian limit g a0 or equivalently 2 2 (50) dτ 2 d x/dt gD a† →∞,i.e.α = 1, the spatial components of the Hence, we can generalize Newton’s second law equa- force four-vector equation (51) reduce to Newton’s sec- tion (7) to the covariant form ond law equation (7). Note that there is no need for a predefined interpolation function, but instead, it is the Fμ = Aμ = α a† = α a† , Lorentz-type factor α that allows the transition between m m m 2 (51) gD GMD/r the Newtonian and MONDian regime. J. Astrophys. Astr. (December 2017) 38:59 Page 7 of 9 59

In order to write the equation of motion equation Let us first write the transformations equations (30) (51) using the Lagrangian formalism, the classical and (31) in the differential form 1 2 Lagrangian L = mu − V (x) must be invariant under 2 dx = α (dx − g tdt) , (67) the Lorentz-type transformations. The Lagrangian must D be a function of the coordinates equation (49)andtheir gD t dt = α tdt − dx . (68) derivatives with respect to the invariant parameter Ð the 2 a† proper time τ. Suppose the force equation (51) is a con- Consider a rod of length dx placed at rest in a frame servative force derivable from a potential, ∂ ( μ) of reference S whichismovingrelativetoaframeof Fμ =− V x . reference S with an acceleration of g . To measure the μ (58) D ∂x rod’s length in the frame S, the end points of the rod Using the velocity four-vector must be observed at the same time t. Since the observer μ in the frame S must measure the distance between the μ dx 1 a†t u = = α 2 , (59) two end points simultaneously dt = 0, we have from dτ u equations (67), we can suggest the covariant Lagrangian 1/2 dx = 1 − g2 /a2 dx. (69) 1 μ μ D † L = mu uμ − V (x ). (60) 2 If, for example, D is the distance between two stars in It follows from Hamilton’s variational principle a binary pair whose positions are observed simultaneously, then b δ = δ τ = 1/2 S Ld 0 (61) D = 1 − g2 /a2 D , (70) a D † 0 that the Lagrangian equation (60) must satisfy the where D0 is the distance between the two stars as mea- Lagrange equation sured in a frame of reference in which gD is equal to ∂ ∂ zero. Therefore, the distance between two uniformly d L − L = . μ μ 0 (62) accelerating stars with an acceleration of gD is reduced dτ ∂u ∂x ( − 2 / 2)1/2 by a factor 1 gD a† . Since the MOND effects can be attributed to the pres- Consider a clock placed at rest in a frame of reference ence of a fictitious dark halo, the time component of S and it measures a time interval dt, suppose that S Lagrange’s equations determines the distribution of the is moving relative to a frame of reference S with an dark halo from the distribution of baryonic mass acceleration of gD. Since the clock is stationary (there is no spatial displacement dx = 0) in the frame S ,we d ∂ L ma† = , (63) have from equation (67), dτ ∂u0 − 2 / 2 1 gD a† dx = α(dx − gDtdt) = 0 (71) ∂ L ∂V =− = mgN + ma†, (64) Substituting equation (71) into equation (68) we obtain ∂x0 ∂x0 0 t where F = mgN + ma† is the total force exerted on = , t 1/4 (72) the particle. While the spatial components of Lagrange’s 1 − g2 /a2 equations determine the motion of test particles in the D † gravitational field of the dark halo we conclude from the above formula that a uniformly d ∂ L du accelerating clock with an acceleration of gD runs slow = mα , (65) by a factor (1 − g2 /a2)1/4 relative to the clocks in the dτ ∂ui dt D † frame S. ∂ ∂ L V Radiation emitted from a source while moving =− = mαgD. (66) ∂xi ∂xi directly toward a receiver will be shifted in frequency The formulation of MOND, illustrated above, not by the Doppler effect. Suppose a light source is at rest only reproduces the predictions of Milgrom’s law but in a frame of reference S which is moving towards a it also leads to a number of physical consequences that receiver in a frame of reference S with speed u, thus arise from replacing the spaceÐtime coordinate trans- if the light source sends a signal at a time interval dt formation equations (8)and(9) by the Lorentz-type as measured by a co-moving observer at the source, transformation equations (30)and(31). then during that time the signal is sent from position 59 Page 8 of 9 J. Astrophys. Astr. (December 2017) 38:59 = 1 x udt, so this signal arrives at the receiver time Then, in high acceleration systems g 2 a0 the term 1 a appears as an anomalous acceleration dT = dt − dtu/c (73) 2 0 1 apart. But the effect of time dilation equation (72) mod- g = gN + a0, (77) 2 ifies equation (73)to 1 while in the low acceleration limit g 2 a0, we obtain dt − dt u/c = , dT 1/4 (74) 2 2 1 2 1 − g /a g = a gN + a (78) D † 0 2 0 where dt and dT are inversely proportional to the fre- which might becomes relevant within large clusters of quency ν0 of the source in S and the frequency ν of the galaxies, particularly within their central regions, since source as seen by the observer in S, respectively, MOND fails to completely resolve the mass discrepancy problem in these systems (Sanders 2003; Aguirre ν 1 − u/c 0 = . (75) et al. 2001). We can, in principle, demonstrate this by ν 1/4 1 − g2 /a2 considering a cluster in hydrostatic equilibrium, using D † equation (78) the dynamical mass can be determined Therefore, if we consider a spectroscopic binary star from the density and temperature distribution of the X- system placed in a frame of reference in which gD is ray emitting gas not equal to zero, then equation (75) predicts that there 1 kT d ln ρ d ln T should be a detectable Doppler shift at any given time + 2 2 =− + . a0GM a0r even when the inclination of the star’s orbit relative to 2 μm p d ln r d ln r the line of sight is zero, i.e. the radial component of the (79) star’s velocity is zero u = 0. In contrast, the classical This relation is apparently more convenient for clusters Doppler relation for non-relativistic speeds ν = ν(1 − 0 than the mass-temperature relation M ∝ T 2 predicted u/c) predicts that there will be no detectable Doppler by MOND (Aguirre et al. 2001), because clusters are shift at instants of time when u = 0. mostly isothermal, and isothermality (in the case of the mass-temperature relation) corresponds to a point mass not to an extended object. 3. MOND and clusters of galaxies Even though, the formula equation (76) seems to be helpful in removing the remaining mass discrepancy in We shall illustrate how the weak equivalence principle MOND, it is not obvious how this formula is consistent or the universality of free fall which allows us to equate with the fact that rotation curves are asymptotically flat. the Newtonian gravitational field with an accelerated The resolution of this apparent contradiction lies in the reference frame can be incorporated into our formu- principle of equivalence; since the ratio of inertial to lation of MOND. In Newtonian mechanics, an object gravitational mass is the same for all bodies mi = mg freely falling in a uniform gravitational field is con- then it should be possible, in the case of uniform gravita- sidered to be weightless, however, it seems from the tional field, to transform to spaceÐtime coordinates such = η modified Newton’s first law F m a0 that the state that the effect of a gravitational force will not appear. of weightlessness cannot actually be achieved even if the object is in a state of free fall in a uniform gravita- mi g μ(g /a0) = (mg −mi )gN +mi ηa0 = mi ηa0. (80) tional field. So combining the universality of free fall Hence, the analogue of Einstein’s principle of equiva- with the modified Newton’s first law yields the follow- lence in MOND states that: it is possible, in a sufficiently ing empirical formula that replaces Milgrom’s formula, small regions of spaceÐtime such that the Newtonian gravitational field changes very little throughout it, to specify a coordinate system in which matter satisfies gμ(g/a0) = gN + ηa0, (76) the law of motion equation (51), and hence it is possi- where the interpolating function can be chosen to resem- ble in these regions to observe the asymptotic flatness ble the μ-function of Milgrom’s formula, such that it of rotation curves. Therefore, any consistent general- satisfies μ(x) = 1whenx η,andμ(x) = x when ization of the afore mentioned Lorentz-type invariance x η. The dimensionless constant η is equal to 1/2 to non-uniformly accelerated coordinate systems must according to the argument above equation (56). reproduce the results of the formula equation (76). J. Astrophys. Astr. (December 2017) 38:59 Page 9 of 9 59

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