Poblems with Mannheim s confomal gavity pogam June 4, 18 Youngsub Yoon axiv:135.163v6 [g-qc] 7 Jul 13 Depatment of Physics and Astonomy Seoul National Univesity, Seoul 151-747, Koea Abstact We show that Mannheim s confomal gavity pogam, whose potential has a tem popotional to 1/ and anothe tem popotional to, does not educe to Newtonian gavity at shot distances, unless one assumes undesiable singulaities of the mass density of the poton. Theefoe, despite the claim that it successfully explains galaxy otation cuves, unless one assumes the singulaities, it seems to be falsified by numeous Cavendish-type expeiments pefomed at laboatoies on Eath whose wok have not found any deviations fom Newton s theoy. Moeove, it can be shown that as long as the total mass of the poton is positive, Mannheim s confomal gavity pogam leads to negative linea potential, which is poblematic fom the point of view of fitting galaxy otation cuves, which necessaily equies positive linea potential. 1 Intoduction Recently, Mannheim s confomal gavity pogam has attacted much attention as an altenative to dak matte and dak enegy [1,, 3]. Howeve, so fa, the only way its validity could be tested was though cosmological consideations. In this pape, we suggest that Mannheim s confomal gavity pogam seems poblematic accoding to numeous Cavendish-type expeiments on Eath. One of ou ideas is that Mannheim s confomal gavity pogam pedicts that the gavitational foce due to an object depends on its mass distibution even in the case that in which it has a spheical symmetic mass distibution. (i.e., the mass density only depends on the distance fom the cente of the body.) Fo example, Newtonian gavity pedicts that we can calculate the gavitational foce due to the Eath as if all the mass of Eath wee at its cente. This is not tue 1
in the case of confomal gavity; the gavitational foce heavily depends on the mass distibution. In Secs. and 3, we intoduce and eview confomal gavity. In othe sections, we give a couple of aguments why confomal gavity is poblematic. Mannheim s confomal gavity Instead of Einstein-Hilbet action, in confomal gavity, we have the following action. S = α g d 4 x gc λµνκ C λµνκ = α g d 4 x g[r µν R µν 1 3 R ] (1) whee C λµνκ is the confomal Weyl tenso, and α g is a puely dimensionless coefficient. By adding this to the action of matte and vaying it with espect to the metic, one can obtain the confomal gavity vesion of the Einstein equation. 3 The metic solution in confomal gavity The following deivation closely follows Mannheim and Kazanas s in Ref. [3]. [See in paticula Eqs. (9), (13), (14) and (16) in thei pape.] In case thee is a spheical symmety in the distibution of the mass, one can wite the metic as follows: ds = B()dt + d B() + dω () Plugging this into the confomal gavity vesion of the Einstein equation, and assuming that all the matte is inside the adius, Mannheim and Kazanas obtain B( > ) = 1 β +γ (3) 4 B() = f() (4) whee β = 1 6 γ = 1 f() without any appoximation whatsoeve. d 4 f( ) (5) d f( ) (6) 3 4α g B() (T T ) (7)
4 Non-Newtonian potential If we ignoe T in the above equation, as it is small, set T get: = ρ, and use B() 1, we β = 1 ( 1 d 4 ρ) (8) 8α g Compae this with the Newtonian case, which is the following: β = G c d 4π ρ = GM total c (9) Thus, unlike in the Newtonian case, we see that in Mannheim s confomal gavity, the gavitational attaction depends not only on the total mass, but also on the mass distibution. Theefoe, if two spheically symmetic objects with the same mass but diffeent density distibutions yield the same stength of gavitational foces, then confomal gavity is toublesome. On the othe hand, if they yield diffeent stengths of gavitational foce, in pecisely the manne that confomal gavity pedicts, then confomal gavity will be veified. Notice that the diffeence of the gavitational foce would be big; it would be in leading ode, not in next-to-leading ode. Fo example, if the mass of two objects is the same, but the fist one s size is double that of the second one, the fome will exet quaduple the amount of gavitational foce. Confomal gavity seems toublesome, as many Cavendish-type expeiments have been pefomed, and none of them has detected that gavity depends on the density distibution [4]. We intoduce Mannheim and Kazanas s cicumvention of this dilemma in the next section. 5 The wong sign of the linea potential tem Mannheim compaes the gavitational potential in Newtonian gavity and confomal gavity in Ref. [1]. He consides the case in which all the matte is inside the egion ( < R), and the mass distibution only depends on (i.e., spheically symmetic). In the case of Newtonian gavity, the potential is given by φ( < R) = 1 φ( ) = g( ) (1) φ( ) = 1 4π φ( > R) = 1 d 3 g( ) d g( ) 3 (11) d g( ) (1) d g( ) (13)
In the case of Mannheim s confomal gavity, we have 4 φ( ) = h( ) (14) φ( ) = 1 8π d 3 h( ) (15) φ( < R) = 1 6 φ( > R) = 1 6 d 4 h( ) 1 d 4 h( ) d 3 h( ) Then, Mannheim notes that the following h(): h( < R) = γc N i=1 δ( i ) yields the following gavitational potential: 3βc φ( > R) = Nβc i=1 d h( ) (16) d h( ) 6 N ][ ] [ δ( i ) 1 4 + Nγc d h( ) Thus, by assuming the singulaities of the mass density of the poton, Mannheim ties to cicumvent the poblem we aised in Sec. 4: we can abitaily make 1/ potential fom a single poton, and if we add them up, they would epoduce Newton s law. Howeve, a close look at the last tem of Eq. (16) shows that this cicumvention will not wok. Notice that h is the mass density up to a cetain positive coefficient. Theefoe, (17) (18) (19) h d () should be equal to the total mass of the poton divided by the positive coefficient. This is obvious fom the following elementay fomula: M = 4π ρd (1) Theefoe, the last tem of Eq. (16) yields the negative linea gavitational potential. Howeve, we know that we want the positive linea gavitational potential to fit the galaxy otation cuve; what we need thee is not exta epulsion but exta attaction. Theefoe, Mannheim s confomal gavity pogam seems poblematic, unless someone can come up with an agument that α g in Eq. (1) can take a negative value. Finally, we want to note that Mannheim s confomal gavity pogam, with which we have dealt in this pape, should not be confused with Andeson-Babou-Foste- Muchadha confomal gavity [5]. 4
Refeences [1] P. D. Mannheim, Altenatives to dak matte and dak enegy, Pog. Pat. Nucl. Phys. 56, 34 (6) [asto-ph/5566]. [] P. D. Mannheim, Making the Case fo Confomal Gavity, Found. Phys. 4, 388 (1) [axiv:111.186 [hep-th]]. [3] P. D. Mannheim and D. Kazanas, Newtonian limit of confomal gavity and the lack of necessity of the second ode Poisson equation, Gen. Rel. Gav. 6, 337 (1994). [4] G. T. Gillies, The Newtonian gavitational constant: ecent measuements and elated studies, Rept. Pog. Phys. 6, 151 (1997). [5] E. Andeson, J. Babou, B. Foste and N. O Muchadha, Scale invaiant gavity: Geometodynamics, Class. Quant. Gav., 1571 (3) [g-qc/11]. 5