Geometric origin of the galactic dynamics

Many modified gravitational theories have been proposed to address the singularity as well as the rotation curve of the spiral galaxy issues in Einstein’s gravity and we find con- formal gravity a prospective option. We hereby try to search for a solution to resolve both problems in the framework of Einstein’s conformal gravity. We prove in an n dimensional Schwarzschild spacetime that firstly, there always exists a conformal factor in small scale that does not affect the physics in a great distance to make the massless and conformally coupled particles geodetically complete, hence to resolve the singularity problem and sec- ondly, by introducing a singular rescaling, instead of a regular one, of the Schwarzschild’s spacetime, which likewise does not affect the physics in small scale, the new metric im- plies an unapproachable singularity at the edge of the universe that cannot be reached in any limited affine parameter, that is, the spacetime is geodetically complete. rendering the resulting velocity profile for a conformally coupled particle on the equator approach a con- stant in far away from the event horizon. Afterwards, we explicitly demonstrate that in 4 dimension It can find a solution to the rotation curve profile of the spiral galaxy simply on the geometric level needless to introduce any dark matter that we are not familiar with and has not been detected yet as well as any modification to the gravitational theory apart from a new symmetry, conformal invariance. In the thesis, the primary focus is on Einstein’s conformal gravity but the outcome is relevant for any theories with conformally invariant symmetry which produces the Schwarzschild’s metric as a precise solution. It is important to note that any metric that is conformally equivalent to Schwarzschild’s solution in Gen- eral Relativity is a precise vacuum solution of the spacetime, and the conformal symmetry of Weyl is broken spontaneously towards the new singular vacuum which is asymptoti- cally Anti-de Sitter instead of the original Schwarzschild metric that is flat in far away. Subsequently, in the approximation of Newtonian dynamics, we evaluate the contribution from all the sources in the galaxy to a reference star to determine the physical velocity square. For each galaxy, we then match the velocity profile with observations and adjust the only unrestricted parameter from the theory as well as the mass-luminosity ratio. As a contrast, we as well apply this to the Anti-de sitter space in the Einstein’s theory where there is also only one free parameter to see whether dark matter is required. We have derived our fits by studying a group of 175 galaxies, and our findings indicate that both velocity profiles accurately fits the rotation-curve data of most spiral galaxies. The results of our fitting process suggest that the mass-to-luminosity ratio is consistently around 1, which suggests that dark matter is not present.

[1] PENROSE R. Gravitational Collapse and Space-Time Singularities[J]. Phys.rev.lett, 1965, 14 (3): 57-59.
[2] HAWKING S W. The Large Scale Structure of Space-Time[M]. Cambridge Press, 1973.
[3] HAWKING S W, PENROSE R. The Singularities of Gravitational Collapse and Cosmology[J]. Proceedings of the Royal Society of London, 1970, 314(1519): 529-548.
[4] HAYWARD S A. Formation and evaporation of non-singular black holes[J]. Physical Review Letters, 2006, 96(3): 031103.
[5] NICOLINI M P. Black holes in an ultraviolet complete quantum gravity[J]. Physics Letters B, 2011.
[6] BONANNO A, REUTER M. Renormalization group improved black hole spacetimes[J]. Phys.rev.d, 2000, 62(4): 043008.
[7] MODESTO L, PRéMONT-SCHWARZ I. Self-dual black holes in loop quantum gravity: The- ory and phenomenology[J]. Physical Review D, 2009, 80(6): 064041.
[8] SIMPSON A, VISSER M. Regular black holes with asymptotically Minkowski cores[Z]. 2019.
[9] CARBALLO-RUBIO R, DI FILIPPO F, LIBERATI S, et al. Geodesically complete black holes [Z]. 2019.
[10] CLARKE C J S. On the geodesic completeness of causal space-times[J]. Mathematical Pro- ceedings of the Cambridge Philosophical Society, 1971, 69(2): 319-323.
[11] BEEM J K. Conformal changes and geodesic completeness[J]. Communications in Mathemat- ical Physics, 1976, 49(2): 179-186.
[12] V. NAELIKAR J, K. KEMBHAVI A. Space-time singularities and conformal gravity[J/OL]. Lettere Al Nuovo Cimento Series 2, 1977, 19: 517-520. DOI: 10.1007/BF02748215.
[13] NARLIKAR J V. The vanishing likelihood of space-time singularity in quantum conformal cosmology[J]. Foundations of Physics, 1984, 14(5): 443-456.
[14] HAWKING S W, HERTOG T. Living with ghosts[J/OL]. Phys. Rev. D, 2002, 65: 103515. DOI: 10.1103/PhysRevD.65.103515.
[15] MANNHEIM P D. Ghost problems from Pauli-Villars to fourth-order quantum gravity and their resolution[J/OL]. Int. J. Mod. Phys. D, 2020, 29(14): 2043009. DOI: 10.1142/S02182718204 30099.
[16] DONOGHUE J F, MENEZES G. Arrow of Causality and Quantum Gravity[J/OL]. Phys. Rev. Lett., 2019, 123(17): 171601. DOI: 10.1103/PhysRevLett.123.171601.
[17] ASOREY M, LOPEZ J L, SHAPIRO I L. Some remarks on high derivative quantum gravity [J/OL]. Int. J. Mod. Phys. A, 1997, 12: 5711-5734. DOI: 10.1142/S0217751X97002991.
[18] MODESTO L, SHAPIRO I L. Superrenormalizable quantum gravity with complex ghosts [J/OL]. Phys. Lett. B, 2016, 755: 279-284. DOI: 10.1016/j.physletb.2016.02.021.
[19] CORNISH N J. Quantum nonlocal gravity[J/OL]. Mod. Phys. Lett. A, 1992, 7: 631-640. DOI: 10.1142/S0217732392000604.
[20] KUZMIN Y V. THE CONVERGENT NONLOCAL GRAVITATION. (IN RUSSIAN)[J]. Sov. J. Nucl. Phys., 1989, 50: 1011-1014.
[21] TOMBOULIS E T. Superrenormalizable gauge and gravitational theories[A]. 1997. arXiv: hep-th/9702146.
[22] MODESTO L. Super-renormalizable Quantum Gravity[J/OL]. Phys. Rev. D, 2012, 86: 044005. DOI: 10.1103/PhysRevD.86.044005.
[23] MODESTO L, RACHWAL L. Nonlocal quantum gravity: A review[J/OL]. Int. J. Mod. Phys. D, 2017, 26(11): 1730020. DOI: 10.1142/S0218271817300208.
[24] BISWAS T, GERWICK E, KOIVISTO T, et al. Towards singularity and ghost free theories of gravity[J/OL]. Phys. Rev. Lett., 2012, 108: 031101. DOI: 10.1103/PhysRevLett.108.031101.
[25] ANSELMI D, MARINO A. Fakeons and microcausality: light cones, gravitational waves and the Hubble constant[J/OL]. Class. Quant. Grav., 2020, 37(9): 095003. DOI: 10.1088/1361-6 382/ab78d2.
[26] KOSHELEV A S, MODESTO L, RACHWAL L, et al. Occurrence of exact 𝑅2 inflation in non-local UV-complete gravity[J/OL]. JHEP, 2016, 11: 067. DOI: 10.1007/JHEP11(2016)067.
[27] BURZILLÀ N, GIACCHINI B L, NETTO T D P, et al. Higher-order regularity in local and nonlocal quantum gravity[J/OL]. Eur. Phys. J. C, 2021, 81(5): 462. DOI: 10.1140/epjc/s1005 2-021-09238-x.
[28] Slipher V M. The detection of nebular rotation[J]. Lowell Observatory Bulletin, 1914, 2: 66-66.
[29] Pease F G. Radial Velocities of Six Nebulae[J/OL]. pasp, 1915, 27(161): 239. DOI: 10.1086/ 122444.
[30] Slipher V M. Spectrographic Observations of Nebulae[J]. Popular Astronomy, 1915, 23: 21-24.
[31] Pease F G. The Rotation and Radial Velocity of the Spiral Nebula N. G. C. 4594[J/OL]. Proceed- ings of the National Academy of Science, 1916, 2(9): 517-521. DOI: 10.1073/pnas.2.9.517.
[32] Pease F G. The Rotation and Radial Velocity of the Central Part of the Andromeda Nebula [J/OL]. Proceedings of the National Academy of Science, 1918, 4(1): 21-24. DOI: 10.1073/pn as.4.1.21.
[33] MAYALL N. Inclinations of spectrum lines in spirals.[J]. The Astronomical Journal, 1948, 54: 44.
[34] MAYALL N, LINDBLAD P. Mean Rotational Velocities of 56 Galaxies[J]. Astronomy and Astrophysics, 1970, 8: 364.
[35] ALBRECHT, BASCHEK B. The new cosmos: an introduction to astronomy and astrophysics [M]. Springer Science & Business Media, 2013.
[36] DAR A. Tests of general relativity and Newtonian gravity at large distances and the dark matter problem[J]. Nuclear Physics B-Proceedings Supplements, 1992, 28(1): 321-326.
[37] OSTRIKER J P, PEEBLES P, YAHIL A. The size and mass of galaxies, and the mass of the universe[J]. The Astrophysical Journal, 1974, 193: L1-L4.
[38] Dwornik M, Keresztes Z, Gergely L A. Rotation curves in Bose-Einstein Condensate Dark Matter Halos[A/OL]. 2013: arXiv:1312.3715. arXiv: 1312.3715.
[39] Foot R, Vagnozzi S. Dissipative hidden sector dark matter[J/OL]. prd, 2015, 91(2): 023512. DOI: 10.1103/PhysRevD.91.023512.
[40] DWORNIK M, KERESZTES Z, GERGELY L A. Bose-Einstein condensate dark matter model tested by galactic rotation curves[M]//THE THIRTEENTH MARCEL GROSSMANN MEET- ING: On Recent Developments in Theoretical and Experimental General Relativity, Astro- physics and Relativistic Field Theories. World Scientific, 2015: 1279-1281.
[41] NELSON A. On the influence of galaxy magnetic fields on the rotation curves in the outer discs of galaxies[J]. Monthly Notices of the Royal Astronomical Society, 1988, 233(1): 115-121.
[42] HEES A, FAMAEY B, ANGUS G W, et al. Combined Solar System and rotation curve con- straints on MOND[J]. Monthly Notices of the Royal Astronomical Society, 2016, 455(1): 449- 461.
[43] MILGROM M, SANDERS R H. Modified newtonian dynamics rotation curves of very low mass spiral galaxies[J]. The Astrophysical Journal, 2007, 658(1): L17.
[44] MOFFAT J, RAHVAR S. The MOG weak field approximation and observational test of galaxy rotation curves[J]. Monthly Notices of the Royal Astronomical Society, 2013, 436(2): 1439- 1451.
[45] CAPOZZIELLO S, HARKO T, KOIVISTO T S, et al. Galactic rotation curves in hybrid metric- Palatini gravity[J]. Astroparticle Physics, 2013, 50: 65-75.
[46] STABILE A, SCELZA G. Rotation curves of galaxies by fourth order gravity[J]. Physical Review D, 2011, 84(12): 124023.
[47] CAPOZZIELLO S, CARLONI S, TROISI A. Quintessence without scalar fields[A]. 2003.
[48] SCELZA G, STABILE A. Numerical analysis of galactic rotation curves[J]. Astrophysics and Space Science, 2015, 357: 1-9.
[49] FINCH A, SAID J L. Galactic rotation dynamics in f (T) gravity[J]. The European Physical Journal C, 2018, 78: 1-18.
[50] BÖHMER C G, HARKO T, LOBO F S. Dark matter as a geometric effect in f (R) gravity[J]. Astroparticle Physics, 2008, 29(6): 386-392.
[51] SHOJAI F, SHOJAI A. An f (R) f (R) model for dark matter: rotation curves and gravitational lensing[J]. General Relativity and Gravitation, 2014, 46: 1-16.
[52] Mannheim P D, O’Brien J G. Fitting galactic rotation curves with conformal gravity and a global quadratic potential[J/OL]. prd, 2012, 85(12): 124020. DOI: 10.1103/PhysRevD.85.124020.
[53] MANNHEIM P D. Solution to the ghost problem in fourth order derivative theories[J/OL]. Found. Phys., 2007, 37: 532-571. DOI: 10.1007/s10701-007-9119-7.
[54] Milgrom M. A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis.[J/OL]. apj, 1983, 270: 365-370. DOI: 10.1086/161130.
[55] Sanders R H, Verheijen M A W. Rotation Curves of Ursa Major Galaxies in the Context of Modified Newtonian Dynamics[J/OL]. apj, 1998, 503(1): 97-108. DOI: 10.1086/305986.
[56] DODELSON S. The Real Problem with MOND[J/OL]. Int. J. Mod. Phys. D, 2011, 20: 2749- 2753. DOI: 10.1142/S0218271811020561.
[57] NOJIRI S, ODINTSOV S D. Unified cosmic history in modified gravity: from F (R) theory to Lorentz non-invariant models[J]. Physics Reports, 2011, 505(2-4): 59-144.
[58] DE FELICE A, TSUJIKAWA S. f (R) theories[J]. Living Reviews in Relativity, 2010, 13: 1-161.
[59] Fich M, Blitz L, Stark A A. The Rotation Curve of the Milky Way to 2R 0[J/OL]. apj, 1989, 342: 272. DOI: 10.1086/167591.
[60] D. MANNHEIM P. Making the Case for Conformal Gravity[J/OL]. Foundations of Physics - FOUND PHYS, 2011, 42. DOI: 10.1007/s10701-011-9608-6.
[61] D. MANNHEIM P. Mass Generation, the Cosmological Constant Problem, Conformal Sym- metry, and the Higgs Boson[J/OL]. Progress in Particle and Nuclear Physics, 2016, 94. DOI: 10.1016/j.ppnp.2017.02.001.
[62] MODESTO L. Disappearance of the black hole singularity in loop quantum gravity[J]. Physical Review D Particles & Fields, 2004, 70(70): -.
[63] BAMBI C, MODESTO L, RACHWAł L. Spacetime completeness of non-singular black holes in conformal gravity[J]. Journal of Cosmology and Astro-Particle Physics, 2016, 5(5): 003-003.
[64] BAMBI C, CAO Z, MODESTO L. Testing conformal gravity with astrophysical black holes[J].Physical Review D, 2017, 95(6): 064006.
[65] Zhang Q, Modesto L, Bambi C. A general study of regular and singular black hole solutionsin Einstein’s conformal gravity[J/OL]. European Physical Journal C, 2018, 78(6): 506. DOI:10.1140/epjc/s10052-018-5987-6.
[66] MODESTO L, RACHWAL L. Finite conformal quantum gravity and spacetime singularities[J]. Journal of Physics Conference, 2017, 942(1): 012015.
[67] ’t Hooft G. A Class of Elementary Particle Models Without Any Adjustable Real Parameters[J/OL]. Foundations of Physics, 2011, 41(12): 1829-1856. DOI: 10.1007/s10701-011-9586-8.
[68] Modesto L, Zhou T, Li Q. Geometric origin of the galaxies’ dark side[A]. 2021:arXiv:2112.04116. arXiv: 2112.04116.
[69] ENGLERT F, TRUFFIN C, GASTMANS R. Conformal invariance in quantum gravity[J/OL].Nuclear Physics B, 1976, 117: 407-432. DOI: 10.1016/0550-3213(76)90406-5.
[70] ENGLERT F, GUNZIG E, TRUFFIN C, et al. Conformal invariant general relativity with dy- namical symmetry breakdown[J/OL]. Physics Letters B, 1975, 57: 73-77. DOI: 10.1016/0370-2693(75)90247-6.
[71] ZEE A. Einstein gravity emerging from quantum Weyl gravity[J/OL]. Annals of Physics, 1983,151: 431-443. DOI: 10.1016/0003-4916(83)90286-5.
[72] MATSUO N. Einstein gravity as spontaneously broken Weyl gravity[J/OL]. General Relativityand Gravitation, 1990, 22: 561-593. DOI: 10.1007/BF00756230.
[73] MALDACENA J. Einstein Gravity from Conformal Gravity[J]. Eprint Arxiv, 2011.
[74] BOLEN B, CAVAGLIA M. (Anti-) de Sitter black hole thermodynamics and the generalizeduncertainty principle[J]. General Relativity and Gravitation, 2005, 37: 1255-1262.
[75] LI Q, MODESTO L. Galactic Rotation Curves in Conformal Scalar-Tensor Gravity[J/OL]. Grav. Cosmol., 2020, 26(2): 99-117. DOI: 10.1134/S0202289320020085.
[76] LELLI F, MCGAUGH S S, SCHOMBERT J M. SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves[J/OL]. Astron. J., 2016, 152: 157. DOI: 10.3847/0004-6256/152/6/157.