We show how, via gravitational repulsion associated with the Schwarzschild solution to the field equations of General Relativity, the extragalactic neutrinos, IceCube-170922A and KM3-230213A, were accelerated to high energy and ultra high energy respectively. Our theory supports the presumption that a blazar is the source of IceCube-170922A. However, it does not support the presumption that a blazar is the source of KM3-230213A. On the contrary, it leads to the conclusion that the source possesses stellar mass. The muon neutrino mass has not been experimentally determined. Our theory predicts that it is in the range 0.030 to 0.092 eV, if the blazar TXS 0506+056 is the source of IceCube-170922A. If, however, the source is PKS 0502+049, then the mass of muon neutrino is: m = 0.09136 eV . We also pinpoint the fallacy in the arguments of the critics of the concept of gravitational repulsion in the Schwarzschild field.
de la Fuente Marcos, R. and de la Fuente Marcos, C. (2015) On the Angular Distribution of Icecube High-Energy Events. Astronomische Nachrichten , 336, 657-664. https://doi.org/10.1002/asna.201512210
Abbasi, R., Ackermann, M., Adams, J., Agarwalla, S.K., Aguilar, J.A., Ahlers, M., et al . (2025) Constraints on the Correlation of Icecube Neutrinos with Tracers of Large-scale Structure. https://arxiv.org/abs/2510.18119
Guevel, D. and Fang, K. (2023) Cross Correlation of Ice Cube Neutrinos with Tracers of Large Scale Structure. Proceedings of Science (38 th International Cosmic Ray Conference ), 444, 1141. https://pos.sissa.it/444/1141
Fang, K., Banerjee, A., Charles, E. and Omori, Y. (2020) A Cross-Correlation Study of High-Energy Neutrinos and Tracers of Large-Scale Structure. The Astrophysical Journal , 894, Article 112. https://doi.org/10.3847/1538-4357/ab8561
Aartsen, M.G. (2016) Search for Correlations between the Arrival Directions of Icecube Neutrino Events and Ultrahigh-Energy Cosmic Rays Detected by the Pierre Auger Observatory and the Telescope Array. Journal of Cosmology and Astroparticle Physics , 2016, Article 037. https://doi.org/10.1088/1475-7516/2016/01/037
Bellenghi, C., Glauch, T., Haack, C., Kontrimas, T., Niederhausen, H., Reimann, R. and Wolf, M. (2021) A New Search for Neutrino Point Sources with Icecube. https://arxiv.org/abs/2107.08700
Globus, N. and Blandford, R.D. (2025) Ultrahigh-Energy Cosmic Rays. Annual Review of Astronomy and Astrophysics , 63, 339-377. https://doi.org/10.1146/annurev-astro-052622-033150
Globus, N. and Blandford, R. (2023) Ultra High Energy Cosmic Ray Source Models: Successes, Challenges and General Predictions. EPJ Web of Conferences , 283, Article 04001. https://doi.org/10.1051/epjconf/202328304001
Bister, T. (2025) Probing the Sources of Ultra-High-Energy Cosmic Rays—Constraints from Cosmic-Ray Measurements. Universe , 11, Article 331. https://doi.org/10.3390/universe11100331
Alves Batista, R., Biteau, J., Bustamante, M., Dolag, K., Engel, R., Fang, K., et al . (2019) Open Questions in Cosmic-Ray Research at Ultrahigh Energies. Frontiers in Astronomy and Space Sciences , 6, Article ID: 23. https://doi.org/10.3389/fspas.2019.00023
Letessier-Selvon, A. and Stanev, T. (2011) Ultrahigh Energy Cosmic Rays. Reviews of Modern Physics , 83, 907-942. https://doi.org/10.1103/revmodphys.83.907
Uryson, A.V. (2006) Ultra High Energy Cosmic Rays: Identification of Possible Sources, Energy Spectra, and Propagation. Physics of Particles and Nuclei , 37, 347-367. https://doi.org/10.1134/s106377960603004x
Droste, J. (1915) On the Field of a Single Centre in Einstein’s Theory of Gravitation. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences , 17, 998-1011.
Droste, J. (2002) Editor’s Note: The Field of a Single Centre in Einstein’s Theory of Gravitation, and the Motion of a Particle in that Field. https://www.lorentz.leidenuniv.nl/IL-publications/dissertations/biographies/Droste.pdf
Droste, J. (1917) The Field of a Single Centre in Einstein’s Theory of Gravitation, and the Motion of a Particle in that Field. Proceedings of the Royal Netherlands Academy of Arts and Sciences , 19, 197-215.
Hilbert, D. (1916) Die Feldgleichungen der Gravitation. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse.
Hilbert, D. (1917) Die grundlagen der physik (Zweite mitteilung). Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse, 53-76.
McGruder III, C.H. and Wieb Van Der Meer, B. (2018) The 1916 PhD Thesis of Johannes Droste and the Discovery of Gravitational Repulsion. https://arxiv.org/abs/1801.07592
McGruder, C.H. (1982) Gravitational Repulsion in the Schwarzschild Field. Physical Review D , 25, 3191-3194. https://doi.org/10.1103/physrevd.25.3191
McGruder, C.H. (2017) Acceleration of Particles to High Energy via Gravitational Repulsion in the Schwarzschild Field. Astroparticle Physics , 86, 18-20. https://doi.org/10.1016/j.astropartphys.2016.10.003
Aiello, S., Albert, A., Alhebsi, A.R., Alshamsi, M., Alves Garre, S., Ambrosone, A., et al . (2025) Observation of an Ultra-High-Energy Cosmic Neutrino with Km3net. Nature , 638, 376-382. https://doi.org/10.1038/s41586-024-08543-1
Einstein, A. (1915) Die feldgleichungen der gravitation. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 844-847.
Schwarzschild, K. (1916) Über das gravitationsfeld eines massenpunktes nach der Einsteinschen theorie. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 189-196.
Hilbert, D. (1924) Die Grundlagen der Physik. Mathematische Annalen , 92, 1-32. https://doi.org/10.1007/bf01448427
Treder, H.-J. and Fritze, K. (1975) Die fallbeschleunigung in der Schwarzschild-metrik. Astronomische Nachrichten , 296, 109-110. https://doi.org/10.1002/asna.19752960303
Vincenzi, M., Brout, D., Armstrong, P., Popovic, B., Taylor, G., Acevedo, M., et al . (2024) The Dark Energy Survey Supernova Program: Cosmological Analysis and Systematic Uncertainties. The Astrophysical Journal , 975, Article 86. https://doi.org/10.3847/1538-4357/ad5e6c
Ghisellini, G., Tavecchio, F., Foschini, L., Ghirlanda, G., Maraschi, L. and Celotti, A. (2009) General Physical Properties of Bright Fermi Blazars. Monthly Notices of the Royal Astronomical Society , 402, 497-518. https://doi.org/10.1111/j.1365-2966.2009.15898.x
Shaw, M.S., Romani, R.W., Cotter, G., Healey, S.E., Michelson, P.F., Readhead, A.C.S., et al . (2012) Spectroscopy of Broad-Line Blazars From 1LAC. The Astrophysical Journal , 748, Article 49. https://doi.org/10.1088/0004-637x/748/1/49
Xiong, D.R. and Zhang, X. (2014) Intrinsic γ-Ray Luminosity, Black Hole Mass, Jet and Accretion in Fermi Blazars. Monthly Notices of the Royal Astronomical Society , 441, 3375-3395. https://doi.org/10.1093/mnras/stu755
Paliya, V.S., Domínguez, A., Ajello, M., Olmo-García, A. and Hartmann, D. (2021) The Central Engines of Fermi Blazars. The Astrophysical Journal Supplement Series , 253, Article 46. https://doi.org/10.3847/1538-4365/abe135
Gaisser, T.K., Halzen, F. and Stanev, T. (1995) Particle Astrophysics with High Energy Neutrinos. Physics Reports , 258, 173-236. https://doi.org/10.1016/0370-1573(95)00003-y
Zas, E. (1997) The Highest Energy Neutrinos. https://arxiv.org/abs/astro-ph/9704016
Aharonian, F., Akhperjanian, A.G., Bazer-Bachi, A.R., Beilicke, M., Benbow, W., Berge, D., et al . (2007) Detection of VHE Gamma-Ray Emission from the Distant Blazar 1ES 1101-232 with HESS and Broadband Characterisation. Astronomy & Astrophysics , 470, 475-489. https://doi.org/10.1051/0004-6361:20077057
Benbow, W. (2010) VERITAS Observations of Blazars. https://arxiv.org/abs/1001.0770
Costamante, L. (2012) AGNs in the VHE Gamma Ray Era: A Review. Memorie della Societa Astronomica Italiana , 83, 138.
Sol, H. and Zech, A. (2022) Blazars at Very High Energies: Emission Modelling. Galaxies , 10, Article 105. https://doi.org/10.3390/galaxies10060105
Hovatta, T. and Lindfors, E. (2019) Relativistic Jets of Blazars. New Astronomy Reviews , 87, Article 101541. https://doi.org/10.1016/j.newar.2020.101541
Eichler, D. (1979) High-Energy Neutrino Astronomy—A Probe of Galactic Nuclei. The Astrophysical Journal , 232, 106-112. https://doi.org/10.1086/157269
Berezinsky, V.S. and Ginzburg, V.L. (1981) On High Energy Neutrino Radiation of Quasars and Active Galactic Nuclei. In: International Cosmic Ray Conference. International Cosmic Ray Conference , 1, 238.
Begelman, M.C., Rudak, B. and Sikora, M. (1990) Consequences of Relativistic Proton Injection in Active Galactic Nuclei. The Astrophysical Journal , 362, Article 38. https://doi.org/10.1086/169241
Stecker, F.W., Done, C., Salamon, M.H. and Sommers, P. (1991) High-Energy Neutrinos from Active Galactic Nuclei. Physical Review Letters , 66, 2697-2700. https://doi.org/10.1103/physrevlett.66.2697
Protheroe, R.J. (1997) High Energy Neutrinos from Blazars. In: Wickramasinghe, D.T., Bicknell, G.V. and Ferrario, L., Eds., IAU Colloquium 163: Accretion Phenomena and Related Outflows. Astronomical Society of the Pacific Conference Series , Vol. 121, 585. https://doi.org/10.48550/arXiv.astro-ph/9607165
Mannheim, K. and Biermann, P.L. (1989) Photomeson Production in Active Galactic nuclei. Astronomy and Astrophysics , 221, 211-220.
Szabo, A.P. and Protheroe, R.J. (1994) Implications of Particle Acceleration in Active Galactic Nuclei for Cosmic Rays and High Energy Neutrino Astronomy. Astroparticle Physics , 2, 375-392. https://doi.org/10.1016/0927-6505(94)90027-2
Mannheim, K. (1995) High-Energy Neutrinos from Extragalactic Jets. Astroparticle Physics , 3, 295-302. https://doi.org/10.1016/0927-6505(94)00044-4
Bednarek, W. and Protheroe, R.J. (1999) Gamma-Ray and Neutrino Flares Produced by Protons Accelerated on an Accretion Disc Surface in Active Galactic Nuclei. Monthly Notices of the Royal Astronomical Society , 302, 373-380. https://doi.org/10.1046/j.1365-8711.1999.02132.x
Rachen, J.P. and Mészáros, P. (1998) Photohadronic Neutrinos from Transients in Astrophysical Sources. Physical Review D , 58, Article 123005. https://doi.org/10.1103/physrevd.58.123005
Mücke, A. and Protheroe, R.J. (2001) A Proton Synchrotron Blazar Model for Flaring in Markarian 501. Astroparticle Physics , 15, 121-136. https://doi.org/10.1016/s0927-6505(00)00141-9
Atoyan, A. and Dermer, C.D. (2001) High-Energy Neutrinos from Photomeson Processes in Blazars. Physical Review Letters , 87, Article 221102. https://doi.org/10.1103/physrevlett.87.221102
Mücke, A., Protheroe, R.J., Engel, R., Rachen, J.P. and Stanev, T. (2003) BL LAC Objects in the Synchrotron Proton Blazar Model. Astroparticle Physics , 18, 593-613. https://doi.org/10.1016/s0927-6505(02)00185-8
Aartsen, M.G., Ackermann, M., Adams, J., Aguilar, J.A., Ahlers, M., Ahrens, M., et al . (2018) Mul-Timessenger Observations of a Flaring Blazar Coincident with High-Energy Neutrino IceCube-170922A. Science , 361, Article 1378. https://arxiv.org/abs/1807.08816
Collaboration, K., Group, M., Collaboration, L., et al . (2025) Characterizing Candidate Blazar Counterparts of the Ultra-High-Energy Event KM3-230213A. https://arxiv.org/abs/2502.08484
Choudhury, S.R. and Hannestad, S. (2020) Updated Results on Neutrino Mass and Mass Hierarchy from Cosmology with Planck 2018 Likelihoods. Journal of Cosmology and Astroparticle Physics , 2020, Article 037. https://doi.org/10.1088/1475-7516/2020/07/037
Navas, S., Amsler, C., Gutsche, T., Hanhart, C., Hernández-Rey, J.J., Lourenço, C., et al . (2024) Review of Particle Physics. Physical Review D , 110, Article 030001. https://doi.org/10.1103/physrevd.110.030001
Tanaka, Y.T., Buson, S. and Kocevski, D. (2017) Fermi-LAT Detection of Increased Gamma-Ray Activity of TXS 0506+056, Located Inside the IceCube-170922A Error Region. The Astronomer’s Telegram 10791:1.
Mirzoyan, R. (2017) First-Time Detection of VHE Gamma Rays by MAGIC from a Direction Consistent with the Recent EHE Neutrino Event IceCube-170922A. The Astronomer’s Telegram 10817:1.
Paiano, S., Falomo, R., Treves, A. and Scarpa, R. (2018) The Redshift of the BL Lac Object TXS 0506+056. The Astrophysical Journal Letters , 854, L32. https://doi.org/10.3847/2041-8213/aaad5e
Drinkwater, M.J., Webster, R.L., Francis, P.J., Condon, J.J., Ellison, S.L., Jauncey, D.L., et al . (1997) The Parkes Half-Jansky Flat-Spectrum Sample. Monthly Notices of the Royal Astronomical Society , 284, 85-125. https://doi.org/10.1093/mnras/284.1.85
Padovani, P., Oikonomou, F., Petropoulou, M., Giommi, P. and Resconi, E. (2019) TXS 0506+056, the First Cosmic Neutrino Source, Is Not a BL Lac. Monthly Notices of the Royal Astronomical Society : Letters , 484, L104-L108. https://doi.org/10.1093/mnrasl/slz011
He, H.-N., Inoue, Y., Inoue, S. and Liang, Y.-F. (2018) High-Energy Neutrino Flare from Cloud-Jet Interaction in the Blazar PKS 0502+049. https://arxiv.org/abs/1808.04330
Sahakyan, N. (2019) Origin of the Multiwavelength Emission of PKS 0502+049. Astronomy & Astrophysics , 622, A144. https://doi.org/10.1051/0004-6361/201834606
Sumida, V.Y.D., Schutzer, A.d.A., Caproni, A. and Abraham, Z. (2021) The Relativistic Parsec-Scale Jets of the Blazars TXS 0506+056 and PKS 0502+049 and Their Possible Association with Gamma-Ray Flares and Neutrino Production. Monthly Notices of the Royal Astronomical Society , 509, 1646-1663. https://doi.org/10.1093/mnras/stab3022
Banik, P., Bhadra, A., Pandey, M. and Majumdar, D. (2020) Implications of a Proton Blazar Inspired Model on Correlated Observations of Neutrinos with Gamma-Ray Flaring Blazars. Physical Review D , 101, Article 063024. https://doi.org/10.1103/physrevd.101.063024
Böttcher, M., Fu, M., Govenor, T., King, Q. and Roustazadeh, P. (2022) Multiwavelength and Multimessenger Observations of Blazars and Theoretical Modeling: Blazars as Astrophysical Neutrino Sources. https://arxiv.org/abs/2204.12242
Dzhatdoev, T. (2025) The Blazar PKS 0605-085 as the Origin of the KM3-230213A Neutrino Event. https://arxiv.org/abs/2502.11434
Stickel, M. and Kuhr, H. (1993) Optical Spectroscopy of 1Jy, S4 and S5 Radio Sources. IV. Vol. 101, 521-540.
Healey, S.E., Romani, R.W., Cotter, G., Michelson, P.F., Schlafly, E.F., Readhead, A.C.S., et al . (2008) Cgrabs: An All-Sky Survey of Gamma-Ray Blazar Candidates. The Astrophysical Journal Supplement Series , 175, 97-104. https://doi.org/10.1086/523302
Bañados, E., Momjian, E., Connor, T., Belladitta, S., Decarli, R., Mazzucchelli, C., et al . (2024) A Blazar in the Epoch of Reionization. Nature Astronomy , 9, 293-301. https://doi.org/10.1038/s41550-024-02431-4
Bañados, E., Khusanova, Y., Decarli, R., Momjian, E., Walter, F., Connor, T., et al . (2024) [C II] Properties and Far-Infrared Variability of a Z = 7 Blazar. The Astrophysical Journal Letters , 977, L46. https://doi.org/10.3847/2041-8213/ad823b
Burrows, A. and Liebert, J. (1993) The Science of Brown Dwarfs. Reviews of Modern Physics , 65, 301-336. https://doi.org/10.1103/revmodphys.65.301
Chabrier, G. and Baraffe, I. (1997) Structure and Evolution of Low-Mass Stars. Astronomy and Astrophysics , 327, 1039-1053. https://arxiv.org/abs/astro-ph/9704118
Baraffe, I., Chabrier, G., Allard, F. and Hauschildt, P.H. (1998) Evolutionary Models for Solar Metallicity Low-Mass Stars: Mass-Magnitude Relationships and Color-Magnitude Diagrams. Astronomy and Astrophysics , 337, 403-412. https://arxiv.org/abs/astro-ph/9805009
Crowther, P.A., Schnurr, O., Hirschi, R., Yusof, N., Parker, R.J., Goodwin, S.P., et al . (2010) The R136 Star Cluster Hosts Several Stars Whose Individual Masses Greatly Exceed the Accepted 150 MStellar Mass Limit. Monthly Notices of the Royal Astronomical Society , 408, 731-751. https://doi.org/10.1111/j.1365-2966.2010.17167.x
Crowther, P.A., Caballero-Nieves, S.M., Bostroem, K.A., Apellániz, J.M., Schneider, F.R.N., Walborn, N.R., et al . (2016) The R136 Star Cluster Dissected with Hubble space Telescope/STIS. I. Far-Ultraviolet Spectroscopic Census and the Origin of He II Λ1640 in Young Star Clusters. Monthly Notices of the Royal Astronomical Society , 458, 624-659. https://doi.org/10.1093/mnras/stw273
Bestenlehner, J.M., Crowther, P.A., Caballero-Nieves, S.M., Schneider, F.R.N., Simón-Díaz, S., Brands, S.A., et al . (2020) The R136 Star Cluster Dissected with Hubble Space Telescope/STIS-II. Physical Properties of the Most Massive Stars in R136. Monthly Notices of the Royal Astronomical Society , 499, 1918-1936. https://doi.org/10.1093/mnras/staa2801
Spallicci, A.D.A.M. (2017) Comment on “Acceleration of Particles to High Energy via Gravitational Repulsion in the Schwarzschild Field” [Astropart. Phys. 86 (2017) 18-20]. Astroparticle Physics , 94, 42-43. https://doi.org/10.1016/j.astropartphys.2017.08.002
Boonserm, P., Ngampitipan, T. and Visser, M. (2018) Near-Horizon Geodesics for Astrophysical and Idealised Black Holes: Coordinate Velocity and Coordinate Acceleration. Universe , 4, Article 68. https://doi.org/10.3390/universe4060068
Deriglazov, A.A., Guzmán Ramírez, W. and Rojas, P. (2019) Comment on “Acceleration of Particles to High Energy via Gravitational Repulsion in the Schwarzschild Field” by C. H. Mcgruder III. Astroparticle Physics , 107, 35-37. https://doi.org/10.1016/j.astropartphys.2018.12.001
Grøn, Ø. (2018) No Gravitational Repulsion in the Schwarzschild Spacetime. Astroparticle Physics , 102, 95-97. https://doi.org/10.1016/j.astropartphys.2018.05.006
Born, M. and Einstein, A. (1947) Briefwechsel 1916-1955. Nymphenburger Verlag.
Marletto, C. and Vedral, V. (2017) Gravitationally Induced Entanglement between Two Massive Particles Is Sufficient Evidence of Quantum Effects in Gravity. Physical Review Letters , 119, Article 240402. https://doi.org/10.1103/physrevlett.119.240402
Bose, S., Mazumdar, A., Morley, G.W., Ulbricht, H., Toroš, M., Paternostro, M., et al . (2017) Spin Entanglement Witness for Quantum Gravity. Physical Review Letters , 119, Article 240401. https://doi.org/10.1103/physrevlett.119.240401
Krori, K.D., Sarmah, J.C. and Goswami, D. (1984) Gravitational Repulsion in the Einstein-Zero-Mass Scalar Theory. Canadian Journal of Physics , 62, 629-631. https://doi.org/10.1139/p84-085
P L Bragança, D. (2024) Gravitational Repulsion in an Expanding Ball of Dust. Classical and Quantum Gravity , 41, Article 075008. https://doi.org/10.1088/1361-6382/ad2d70
Polo, C.L. and Singh, H.S. (2024) Hilbert Repulsion in the Kerr-Newman Anti-De Sitter Spacetime. Astrophysics and Space Science , 369, Article No. 41. https://doi.org/10.1007/s10509-024-04304-8
Célérier, M.-N., Santos, N.O. and Satheeshkumar, V.H. (2017) Hilbert Repulsion in the Reissner-Nordström and Schwarzschild Spacetimes. https://arxiv.org/abs/1707.06994
Gorkavyi, N. and Vasilkov, A. (2016) A Repulsive Force in the Einstein Theory. Monthly Notices of the Royal Astronomical Society , 461, 2929-2933. https://doi.org/10.1093/mnras/stw1517
Kutschera, M. and Zajiczek, W. (2009) Shapiro Effect for Relativistic Particles-Testing General Relativity in a New Window. https://arxiv.org/abs/0906.5088
Herrera, L. (2005) Geodesics in a Quash-Spherical Spacetime: A Case of Gravitational Repulsion. Foundations of Physics Letters , 18, 21-36. https://doi.org/10.1007/s10702-005-2467-7
Ponce de Leon, J. (1988) Gravitational Repulsion in Sources of the Reissner-Nordström Field. Journal of Mathematical Physics , 29, 197-206. https://doi.org/10.1063/1.528172
Krori, K.D. and Barua, M. (1985) Gravitational Repulsion by Kerr and Kerr-Newman Black Holes. Physical Review D , 31, 3135-3139. https://doi.org/10.1103/physrevd.31.3135
Einstein, A. (1916) Näherungsweise integration der feldgleichungen der gravitation. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 688-696.