Relativistic Modications in Galactic Rotation Curves under a Toroidal Galactic Field

Rotational dynamics of galaxies exhibits an increase beyond the Keplerian velocity which corresponds to a missing mass up to six times the dynamic mass in the observable universe. In this paper we show that the observed increase in galactic rotation velocities is a general relativistic e⁄ect resulting from the combined e⁄ect of toroidal magnetic energy density in galaxies and spacetime dragging due to the rotating compact mass in galactic center. The e⁄ect of magnetic energy density on spacetime vorticity is derived from Maxwell equations in axially symmetric spacetime where the dragging e⁄ects are shown to extend farther in the galactic disc via the toroidal (cid:133)eld, modifying the rotational speed of the galactic matter. This is shown to lead to the diverse rotation curves of spiral galaxies, along with the Tully-Fisher relation for total galactic mass and maximum rotational velocity.

Introduction.-Rotation curves of galaxies show a marked di¤erence from the Keplerian velocity p 2GM=r, for galactic mass M and radial distance r, which becomes increasingly signi…cant over the galactic radius. This rotation curve discrepancy is now well established through a large number of astrophysical observations [1][2][3][4]. A main aspect in modeling galactic dynamics is the diversity exhibited by these rotation curves. Most of the spiral galaxies, like N GC 1566, have an almost ‡at rotation curve at large distances within the galactic disk, where as galaxies like N GC 1097 shows a marked rise in the rotational velocity.
Satellite galaxies, like M 31, even show a sharp steep in rotational speed over the galactic disk.
Despite the observed diversity of rotational patterns of di¤erent galaxies, certain features also emerge as similar in general galactic dynamics. For instance, apart from an increase over the Keplerian velocity at large distances, an almost direct proportionality exists between the galactic mass and the fourth power of the maximum rotational velocity. This Tully-Fisher relation [5][6][7][8][9] is a general scaling relation with no signi…cant dependence on other galactic properties like galaxy size or surface brightness, and has little intrinsic scatter [10][11][12][13][14]. A plausible model of galactic dynamics should explain these observed features of galactic motion with a minimum number of hypotheses.
In this paper, we give a model of galactic dynamics based on …rst principles, and deduce these features of galactic motion from directly observable galactic parameters, namely the galactic mass, radius, and the magnetic …eld. In this context the magnetic …eld generated in galaxies, mainly by accretion around galactic nuclei, provides the feedback mechanism for galactic rotation. Analysis of general relativistic Maxwell equations shows that toroidal magnetic …eld thus generated enhances spacetime vorticity. Such a coupling occurs when the central mass has non-zero angular momentum and an intrinsic magnetic …eld is present in the galactic center. Spacetime dragging is thereby extended to much larger distances under the magnetic pressure. We show that this enhanced spacetime drag corresponds directly to the velocity increase observed in the diverse behavior of galactic rotations.
The model presented here depends on the following main features of galactic structure: (1) A large proportion of galactic mass is contained in the region in and around the galactic center. In most spiral galaxies this region is fully formed as galactic bulge, and despite very di¤erent morphologies, masses, sizes, and gas fractions; matter in these galaxies can be generally divided into a disk and a central bulge or a nuclear mass. (2) Although most disk galaxies have a central bulge, pure disk galaxies contain a compact source at the galactic center. In such cases galactic dynamics is dominated by the compact source. In general, the existence of central compact gravitational sources is now well established in almost all galaxies [15][16][17][18]. (3) Galaxies have an intrinsic magnetic …eld, which we identify as toroidal …eld generated in the central region. In spiral galaxies the galactic magnetic …eld strength varies from 10 15 G within the galactic arm [19][20][21]. Observationally, the total magnetic …eld strength can be determined from the intensity of total synchrotron emission, assuming energy balance (equipartition) between magnetic …elds and cosmic rays. The typical average equipartition strength for spiral galaxies is about 10 G. In general the galactic magnetic …eld has been observed to be of order of G which extends across the galactic plane [22][23][24].
Velocity distribution.-The magnetic …eld around a rotating compact object can be derive from Maxwell equations in slowly rotating Kerr spacetime [25]. Using the standard spacetime coordinates, (r; ; '; t), where the azimuthal angle ' is measured in the galactic plane, the toroidal magnetic …eld in the galactic disk ( = =2) is given by, In the equatorial plane identi…ed as the galactic plane, the total energy density due to the magnetic …eld interior to a circular orbit of radius r around the source at origin can be calculated by …rst integrating over the variable , which gives a constant contribution absorbed in B 0 . Therefore, the magnetic energy density at any point r in the galactic disk is given by, Putting from equation (1) into equation (2), we obtain, Equation (3) gives the energy density due to the magnetic …eld coupled to the spacetime dragging, also the total energy has contribution due to the Newtonian gravitational potential energy for the net mass M contained within the orbit. Correspondingly, in the outer region r > r s , disk velocity has contribution v 0 due to the source at r = r s , the velocity …eld v in the exterior due to the magnetic energy, and the Keplerian velocity due to the gravitational potential energy. The e¤ective potential energy of gravity per unit mass is where M is the total mass contained within the orbit. De…ning the dimensionless magnetic energy density E B = B =B 2 0 , we equate the net potential energy (E B + E G ) to the kinetic energy per unit mass. This gives for the orbital velocity at any point in the galactic disk for For the zero of velocity v 0 , we  (4) for typical galaxies N GC 1097 (data points from [26]), N GC 1566 [27], M 31 [28], and M 33 [29]. The dashed plots represent the Keplerian velocity, where as the dotted plots correspond to the magnetic vorticity contribution p r + r s ln(r r s ). r = r s . Using (3) we have v 0 = p r + r s ln(r s r) j r=0 . We therefore obtain v 0 = p r s ln r s , which determines the magnitude of v 0 in equation (4). Notice that the square-root sign in equation (4) is decided by co-rotating and counter-rotating motion in the galactic disk relative to the rotation of the central compact object.
A set of typical galactic curves is shown in Figure 1 for the disk region r > r s , where the data points lie, within observational error, on the velocity curves derived from equation (4).
In plotting the rotation curves for N GC 1097, N GC 1566, and M 31 galaxies, we have used phenomena has been observed in N GC 4826 and a number of other galaxies. As shown in Figure 2, the data points below radius of one kpc lying on the inner disk correspond to velocity in opposite direction to points lying on the scale larger than a kpc.
In galaxies with well formed spiral structure, such as barred galaxies, mass distribution di¤ers from ‡at disk galaxies, and is not uniform over large radii. Thus for the Milky Way duced directly from velocity distribution formula (4). In the Tully-Fisher equation maximum luminosity corresponds to maximum rotational speed in the galactic disk. Therefore, by the extremum condition dv=dr j v=vmax = 0 , we have from equation (4), Since r 2 r s , for the galactic disk, the second term in equation (5) is negligible. Putting therefore in equation (4) By p 2GM > r s , the second term on rhs. of equation (6) is also negligible compared to the …rst, we then obtain the Tully-Fisher relation M v 4 max . Putting back the magnetic energy density we see that v 4 max 8B 2 0 GM , which shows that for the usual galactic parameters B 0 10 G and total galactic mass of approximately 10 10 M , v max is of the order of 10 2 km=s, which agrees with the observed order of magnitude of galactic rotational velocities. We notice that when the Schwarzschild radius r s = 2GM c =c 2 , where M c is the mass of the compact source in galactic center, has magnitude comparable to p 2GM , deviations from the Tully-Fisher law arise due to the second term on rhs. of equation (6). In Figure 4 we give a plot based on equation (6) for deviation from the Tully-Fisher fourth power law. We see that B 2 0 and the predicted deviations from the Tully-Fisher relation become larger for larger magnetic …eld for a given compact mass, since it increases the maximum velocity in the galactic disk. Also, the deviation increases if the central compact source is more massive. But since a larger proportion of the total galactic mass M is then contained in the central compact source than in the luminous matter in the galactic disk, maximum velocity decreases due to the logarithmic term in equation (6) which depends on the di¤erence of total mass M and the mass M c . Since the deviation corresponds to v 4 max which in turn is proportional to maximum luminosity, this implies that even if the central compact source is more massive, deviation from Tully-Fisher relation for luminosity will still be small provided the galactic magnetic …eld is su¢ciently high.
Galactic age and size.-Assuming that the primodal compact source with the toroidal …eld had moment of inertia I 10 40 kg m 2 and angular speed 10 4 =s, compatible with general relativistic equilibrium conditions [31], we have for the galactic mass to extend up to the galactic radius of 10kpc, the time t 2 =! = 2 r 3 =I = 10 17 s which agrees well with galactic age inferred from observation [32].
Since in this model spacetime vorticity coupled to the magnetic energy density determines radial spread of velocity, galactic radius can be derived, given the total angular momentum of the galaxy. To determine this we note that the e¤ects of magnetic coupling to the spacetime vorticity extend only to a …nite distance, since cos in equation (1) has principal zero at = =2. Therefore, using ! J=r 3 and for a co-moving observer the angular speed 0 = '=t, we …nd that the galactic radius is related to the total angular momentum by 0 J=r 3 = =2t. For su¢ciently long period of time, we therefore have for the galactic radius r (J= 0 ) 1=3 . For the Milky Way galaxy, total angular momentum inferred from dynamics of double galaxies is approximately 10 67 Js [33]. Given that the angular speed of the solar system is about 550km=s, this gives for the radius of the Milky Way galaxy, r 2:6 10 20 m, which is of the observed order of magnitude of 10 20 m of galactic radius [34]. Notice that this corresponds to an outer ring of maximum acceleration, where as the region close to where v = v 0 must form an inner ring of increased acceleration due to the sudden change in velocity.
Conclusions.-In equation (6) we see that the contribution to the total galactic mass v 4 max , due to the magnetic vorticity is 8GM for the dominant term, whereas the second term is negligibly small. The Keplerian velocity alone however contributes only 2GM to v 4 max for the same radius r = p 2GM . Given that the total galactic mass inferred from Keplerian velocity distribution is M , the rotational velocity (4) thus implies an additional mass of 6M . This corresponds to the missing galactic mass inferred from rotational velocity curves.
Finally, we stress that in the linear approximation to Einstein …eld equations there is the gravitomagnetic …eld which couples to the magnetic …eld in particle dynamics around rotating magnetized objects [35]. The gravitomagnetic force however is too weak to derive orbital motion at large distances. Moreover, its e¤ects decay as 1=r 3 , hence cannot support ‡at rotation curves. The e¤ect we have discussed here is distinct from Lense-Thirring frame drag, since it results from a coupling of gravitational and electromagnetic …elds. Moreover it is generally valid for any axially symmetric matter distribution endowed with a magnetic …eld structure. In astrophysical systems accretion provides an e¤ective means of magnetic …eld generation around massive objects. Spacetime around a rotating compact object, while remaining locally isomorphic to the Kerr spacetime, thus modi…es the dynamics of matter by the mechanism discussed above. The formation of central region of su¢cient mass density and magnetic …eld strength is essential for observed rotational dynamics of galaxies. In ellipticals, disk-like region formation in the galactic centers due to rotation have also been observed [36,37], indicating that the toroidal …eld coupled to the spacetime vorticity is also important in the evolution of spiral and elliptical galaxies. The above model also explains diverse rotation curves of dwarf galaxies [38], where the magnetic …eld generated by spacetime vorticity is comparable to that in disk galaxies, which cannot be otherwise accounted for by accretion and other physical mechanisms.