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The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity

Year 2022, Volume: 12 Issue: 2, 134 - 145, 24.12.2022

Halil Kuyrukcu

Abstract

We study the case in which the five dimensional theory is the transformed Weyl–Yang–Kaluza–Klein gravity. The dimensionally reduced equations of motion are derived by considering an alternative form of the main equation of the theory in the coordinate basis. The conformal transformation rules are applied to the invariants. We also discuss the possible specific cases and the new Lorentz force density term, in detail.

Keywords

Weyl–Yang–Kaluza–Klein theory Dimensional reduction Conformal transformation Field equations

References

  • Baekler, P., Yasskin, PB. 1984. All torsion-free spherical vacuum solutions of the quadratic Poincaré gauge theory of gravity. General Relativity and Gravitation, 16: 1135-1155. doi: 10.1007/BF00760237
  • L., Bielecki, J., Cederwall, M., Nilsson, BEW., Persson, D. 2008. U-Duality and the compactified Gauss–Bonnet term. Journal of High Energy Physics, 07: 048. doi: 10.1088/1126- 6708/2008/07/048
  • Başkal, S. 1999. Radiation in Yang–Mills formulation of gravity and a generalized pp-wave metric. Progress of Theoretical Physics, 102 (4): 803-807. doi: 10.1143/PTP.102.803
  • Başkal, S., Kuyrukcu, H. 2013. Kaluza–Klein reduction of a quadratic curvature model. General Relativity and Gravitation, 45: 359-371. doi: 10.1007/s10714-012-1476-7
  • Camenzind, M. 1975. On the curvature dynamics for metric gravitational theories. Journal of Mathematical Physics, 16: 1023-1031. doi: 10.1063/1.522654
  • Camenzind, M. 1975a. Theories of gravity with structuredependent ’s. Physical Review Letters, 35: 1188-1189. doi: 10.1103/PhysRevLett.35.1188
  • Camenzind, M. 1977. On the Yang–Mills structure of gravitation: A new issue of the final state. General Relativity and Gravitation, 8: 103-108. doi: 10.1007/BF00770729
  • Camenzind, M. 1978. Weak and strong sources of gravity: An SO (1,3)-gauge theory of gravity. Physical Review D, 18: 1068. doi: 10.1103/PhysRevD.18.1068
  • Camenzind, M. 1978a. Homogeneous and isotropic world models in the Yang–Mills dynamics of gravity: The structure of the adiabats. Journal of Mathematical Physics, 19: 624-634. doi: 10.1063/1.523693
  • Carneiro, DF., Freiras, EA., Goncalves, B., de Lima AG., Shapiro IL. 2004. On useful conformal transformations in general relativity. Gravitation and Cosmology, 10 (4): 305-312.
  • Chen, P., Izumi, K., Tung, N-E. 2013. Natural emergence of cosmological constant and dark radiation from the Stephenson– Kilmister–Yang–Camenzind theory of gravity. Physical Review D, 88 (12): 123006. doi: 10.1103/PhysRevD.88.123006
  • Cook, RJ. 2008. The gravitational-electromagnetic analogy: A possible solution to the vacuum-energy and dark-energy problems. arXiv: gr-qc/0810.4495 (preprint).
  • Cvetic, M., Gibbons, GW., Lu, H., Pope, CN. 2003. Consistent group and coset reductions of the bosonic string. Classical and Quantum Gravity, 20: 5161-5194. doi: 10.1088/0264- 9381/20/23/013
  • Çelik, S. 2021. Compatibility of the dimensional reduction and variation procedures for a quadratic curvature model with a Kaluza–Klein ansatz. Master of Science, Middle East Technical University, 92 p.
  • Dabrowski, MP., Garecki, J., Blaschke, DB. 2009. Conformal transformations and conformal invariance in gravitation. Annalen der Physik, 18: 13-32. doi: 10.1002/andp.200810331
  • Dean, BH. 1999. Variational results and solutions in gauge gravity and a bifurcation analysis of black hole orbital dynamics. Graduate Theses, West Virginia University, Source DAI-B 61/03.
  • Dereli, T., Üçoluk, G. 1990. Kaluza–Klein reduction of generalized theories of gravity and nonminimal gauge couplings. Classical and Quantum Gravity, 7: 1109-1115. doi: 10.1088/0264-9381/7/7/009
  • Eddington, AS. 1924. The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, pp. 141-144.
  • Fairchild, JrEE. 1976. Gauge theory of gravitation. Physical Review D, 14: 384-391. [erratum: 1976. Physical Review D, 14: 2833]. doi: 10.1103/PhysRevD.14.384
  • Fairchild, JrEE. 1977. Yang–Mills formulation of gravitational dynamics. Physical Review D, 16: 2438-2447. doi: 10.1103/ PhysRevD.16.2438
  • Gibbons, GW., Wiltshire, DL. 1986. Black holes in Kaluza– Klein Theory. Annals of Physics, 167: 201-223. doi: 10.1016/ S0003-4916(86)80012-4
  • Hawking, SW., Ellis, GFR. 1999. The large scale structure of space-time. Cambridge, UK, Cambridge University Press.
  • Kaluza, Th. 1921. Zum unitätsproblem der physik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, K1: 966-972.
  • Kilmister, CW., Newman, DJ., Bondi, H. 1961. The use of algebraic structures in physics. Proceedings of the Cambridge Philosophical Society, 57: 851. doi: 10.1017/ S0305004100036008
  • Klein, O. 1926. Quantentheorie und fünfdimensionale relativitätstheorie. Zeitschrift für Physik, 37 (2-3): 895-906. doi: 10.1007/BF01397481
  • Klein, O. 1926a. The atomicity of electricity as a quantum theory law. Nature, 118: 516. doi: 10.1038/118516a0
  • Kretschmann, E. 1915. Uber die prinzipielle bestimmbarkeit der berechtigten bezugssysteme beliebiger relativitätstheorien (I). Annalen der Physik, 353 (23): 907-942. doi: 10.1002/ andp.19153532303
  • Kuyrukcu, H. 2013. PP-wave solutions of the Weyl–Yang theory of gravity with a five-dimensional Kaluza–Klein ansatz. Classical and Quantum Gravity, 30 (15): 155013. doi: 10.1088/0264-9381/30/15/155013
  • Kuyrukcu, H. 2014. The non-Abelian Weyl–Yang–Kaluza–Klein gravity model. General Relativity and Gravitation, 46: 1751. doi: 10.1007/s10714-014-1751-x
  • Kuyrukcu, H. 2016. A black hole solution of higher-dimensional Weyl–Yang–Kaluza–Klein theory by the Wu–Yang ansatz. arXiv: gr-qc/1602.02418v1 (preprint).
  • Kuyrukcu, H. 2021. A black hole solution of higher-dimensional Weyl–Yang–Kaluza–Klein theory. Classical and Quantum Gravity, 38 (17): 175009. doi: 10.1088/1361-6382/ac161a
  • Lanczos, C. 1938. A remarkable property of the Riemann– Christoffel tensor in four dimensions. Annals of Mathematics, 39: 842-850. doi: 10.2307/1968467
  • Lanczos, C. 1949. Lagrangian multiplier and Riemannian spaces. Reviews of Modern Physics, 21 (3): 497-502. doi: 10.1103/ RevModPhys.21.497
  • Lanczos, C. 1957. Electricity and general relativity. Reviews of Modern Physics, 29 (3): 337-350. doi: 10.1103/ RevModPhys.29.337
  • Lee, HC. 1983. An introduction to Kaluza–Klein theories: Proceedings of the workshop on KaluzaKlein theories. Chalk River, Canada, p. 187.
  • Lichnerowicz, A. 1958. Sur un procedé de quantification du champ de gravitation. C. R. Acad. Sci., 247: 433-436.
  • Liu, H., Wesson, PS. 1997. The physical properties of charged five-dimensional black holes. Classical and Quantum Gravity, 14 (7): 1651-1663. doi: 10.1088/0264-9381/14/7/006
  • Loos, HG. 1963. Spin connection in general relativity. Annals of Physics, 25: 91-108. doi: 10.1016/0003-4916(63)90335-X
  • Loos, HG., Treat, RP. 1967. Conditional dynamic equivalence of free Yang–Mills fields and free gravitational fields. Physics Letters A, 26 (2): 91-92. doi: 10.1016/0375-9601(67)90117-X
  • Mandel, H. 1926. Zur herleitung der feldgleichungen in der allgemeinen relativitätstheorie. Zeitschrift für Physik, 39 (12): 136-145. doi: 10.1007/BF01321980
  • Misner, CW., Thorne, KS., Wheeler, JA. 1973. Gravitation. Freeman WH and company, San Francisco.
  • Ni, W-T. 1975. Yang’s gravitational field equations. Physical Review Letters, 35 (5): 319-320. [erratum: 1975. Physical Review Letters, 35 (25): 1748. doi: 10.1103/PhysRevLett.35.1748]. doi: 10.1103/PhysRevLett.35.319
  • Öktem, F. 1985. Doğa Bilim Dergisi A1(9): 3.
  • Palatini, A. 1919. Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rendiconti del Circolo Matematico di Palermo, 43: 203-212. doi: 10.1007/ BF03014670
  • Pauli, W. 1919. Zur theorie der gravitation und der elektrizitaät von H Weyl. Physikalische Zeitschrift, 20: 457-467.
  • Pavelle, R. 1974. Yang’s gravitational field equations. Physical Review Letters, 33 (24): 1461-1463. doi: 10.1103/ PhysRevLett.33.1461
  • Pavelle, R. 1975. Unphysical solutions of Yang’s gravitational field equations. Physical Review Letters, 34: 1114. doi: 10.1103/ PhysRevLett.34.1114
  • Pavelle, R. 1976. Unphysical characteristics of Yang’s pure-space equations. Physical Review Letters, 37 (15): 961-964. doi: 10.1103/PhysRevLett.37.961
  • Pavelle, R. 1978. Mansouri–Chang gravitation theory. Physical Review Letters, 40 (5): 267-270. doi: 10.1103/ PhysRevLett.40.267
  • Perry, M. 2009. Applications of Differential Geometry to Physics. p. 36. https://sgielen.files.wordpress.com/2018/01/diffgeo.pdf Pope, CN. Kaluza–Klein Theory. p. 4. http://people.physics.tamu. edu/pope/ihplec.pdf
  • Stephenson, G. 1958. Quadratic Lagrangians and general relativity. ll Nuovo Cimento, 9 (2): 263-269. doi: 10.1007/ BF02724929
  • Thompson, AH. 1975. Yang’s gravitational field equations. Physical Review Letters, 34 (8): 507-508. doi: 10.1103/ PhysRevLett.34.507
  • Thompson, AH. 1975a. Geometrically degenerate solutions of the Kilmister–Yang equations. Physical Review Letters, 35 (5): 320-322. doi: 10.1103/PhysRevLett.35.320
  • Weyl, H. 1918. Gravitation und elektrizität. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 1918: 465-478. doi: 10.1007/978-3-663-19510_811
  • Weyl, H. 1919. A new extension of relativity theory. Annalen der Physik, 59: 101-133. doi: 10.1002/andp.19193641002
  • Weyl, H. 1921. Über die physikalischen grundlagen der erweiterten relativitätstheorie. Physikalische Zeitschrift, 22: 473-480.
  • Yang, CN., Mills, RL. 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical Review, 96 (1): 191-195. doi: 10.1103/PhysRev.96.191
  • Yang, CN. 1974. Integral formalism for gauge fields. Physical Review Letters, 33 (7): 445-447. doi: 10.1103/ PhysRevLett.33.445

Beş Boyutlu Dönüştürülmüş Weyl–Yang–Kaluza–Klein Kütleçekim Teorisi

Year 2022, Volume: 12 Issue: 2, 134 - 145, 24.12.2022

Halil Kuyrukcu

Abstract

Beş boyutlu teorinin, dönüştürülmüş Weyl–Yang–Kaluza–Klein kütleçekim olduğu durum incelendi. Boyutsal olarak indirgenmiş hareket denklemleri, teorinin ana denkleminin alternatif formu düşünülerek koordinat bazında türetildi. Değişmezler için konformal dönüşüm kuralları uygulandı. Ayrıca olası özel durumlar ve yeni Lorentz kuvvet yoğunluğu terimi ayrıntılı olarak tartışıldı.

Keywords

Kaluza-Klein teorisi Weyl-Yang Teorisi Boyutsal İndirgeme Konformal Dönüşüm Alan denklemleri Weyl–Yang–Kaluza–Klein theory Dimensional reduction Conformal transformation Field equations

References

  • Baekler, P., Yasskin, PB. 1984. All torsion-free spherical vacuum solutions of the quadratic Poincaré gauge theory of gravity. General Relativity and Gravitation, 16: 1135-1155. doi: 10.1007/BF00760237
  • L., Bielecki, J., Cederwall, M., Nilsson, BEW., Persson, D. 2008. U-Duality and the compactified Gauss–Bonnet term. Journal of High Energy Physics, 07: 048. doi: 10.1088/1126- 6708/2008/07/048
  • Başkal, S. 1999. Radiation in Yang–Mills formulation of gravity and a generalized pp-wave metric. Progress of Theoretical Physics, 102 (4): 803-807. doi: 10.1143/PTP.102.803
  • Başkal, S., Kuyrukcu, H. 2013. Kaluza–Klein reduction of a quadratic curvature model. General Relativity and Gravitation, 45: 359-371. doi: 10.1007/s10714-012-1476-7
  • Camenzind, M. 1975. On the curvature dynamics for metric gravitational theories. Journal of Mathematical Physics, 16: 1023-1031. doi: 10.1063/1.522654
  • Camenzind, M. 1975a. Theories of gravity with structuredependent ’s. Physical Review Letters, 35: 1188-1189. doi: 10.1103/PhysRevLett.35.1188
  • Camenzind, M. 1977. On the Yang–Mills structure of gravitation: A new issue of the final state. General Relativity and Gravitation, 8: 103-108. doi: 10.1007/BF00770729
  • Camenzind, M. 1978. Weak and strong sources of gravity: An SO (1,3)-gauge theory of gravity. Physical Review D, 18: 1068. doi: 10.1103/PhysRevD.18.1068
  • Camenzind, M. 1978a. Homogeneous and isotropic world models in the Yang–Mills dynamics of gravity: The structure of the adiabats. Journal of Mathematical Physics, 19: 624-634. doi: 10.1063/1.523693
  • Carneiro, DF., Freiras, EA., Goncalves, B., de Lima AG., Shapiro IL. 2004. On useful conformal transformations in general relativity. Gravitation and Cosmology, 10 (4): 305-312.
  • Chen, P., Izumi, K., Tung, N-E. 2013. Natural emergence of cosmological constant and dark radiation from the Stephenson– Kilmister–Yang–Camenzind theory of gravity. Physical Review D, 88 (12): 123006. doi: 10.1103/PhysRevD.88.123006
  • Cook, RJ. 2008. The gravitational-electromagnetic analogy: A possible solution to the vacuum-energy and dark-energy problems. arXiv: gr-qc/0810.4495 (preprint).
  • Cvetic, M., Gibbons, GW., Lu, H., Pope, CN. 2003. Consistent group and coset reductions of the bosonic string. Classical and Quantum Gravity, 20: 5161-5194. doi: 10.1088/0264- 9381/20/23/013
  • Çelik, S. 2021. Compatibility of the dimensional reduction and variation procedures for a quadratic curvature model with a Kaluza–Klein ansatz. Master of Science, Middle East Technical University, 92 p.
  • Dabrowski, MP., Garecki, J., Blaschke, DB. 2009. Conformal transformations and conformal invariance in gravitation. Annalen der Physik, 18: 13-32. doi: 10.1002/andp.200810331
  • Dean, BH. 1999. Variational results and solutions in gauge gravity and a bifurcation analysis of black hole orbital dynamics. Graduate Theses, West Virginia University, Source DAI-B 61/03.
  • Dereli, T., Üçoluk, G. 1990. Kaluza–Klein reduction of generalized theories of gravity and nonminimal gauge couplings. Classical and Quantum Gravity, 7: 1109-1115. doi: 10.1088/0264-9381/7/7/009
  • Eddington, AS. 1924. The Mathematical Theory of Relativity, Cambridge University Press, Cambridge, pp. 141-144.
  • Fairchild, JrEE. 1976. Gauge theory of gravitation. Physical Review D, 14: 384-391. [erratum: 1976. Physical Review D, 14: 2833]. doi: 10.1103/PhysRevD.14.384
  • Fairchild, JrEE. 1977. Yang–Mills formulation of gravitational dynamics. Physical Review D, 16: 2438-2447. doi: 10.1103/ PhysRevD.16.2438
  • Gibbons, GW., Wiltshire, DL. 1986. Black holes in Kaluza– Klein Theory. Annals of Physics, 167: 201-223. doi: 10.1016/ S0003-4916(86)80012-4
  • Hawking, SW., Ellis, GFR. 1999. The large scale structure of space-time. Cambridge, UK, Cambridge University Press.
  • Kaluza, Th. 1921. Zum unitätsproblem der physik. Sitzungsberichte der Preussischen Akademie der Wissenschaften, K1: 966-972.
  • Kilmister, CW., Newman, DJ., Bondi, H. 1961. The use of algebraic structures in physics. Proceedings of the Cambridge Philosophical Society, 57: 851. doi: 10.1017/ S0305004100036008
  • Klein, O. 1926. Quantentheorie und fünfdimensionale relativitätstheorie. Zeitschrift für Physik, 37 (2-3): 895-906. doi: 10.1007/BF01397481
  • Klein, O. 1926a. The atomicity of electricity as a quantum theory law. Nature, 118: 516. doi: 10.1038/118516a0
  • Kretschmann, E. 1915. Uber die prinzipielle bestimmbarkeit der berechtigten bezugssysteme beliebiger relativitätstheorien (I). Annalen der Physik, 353 (23): 907-942. doi: 10.1002/ andp.19153532303
  • Kuyrukcu, H. 2013. PP-wave solutions of the Weyl–Yang theory of gravity with a five-dimensional Kaluza–Klein ansatz. Classical and Quantum Gravity, 30 (15): 155013. doi: 10.1088/0264-9381/30/15/155013
  • Kuyrukcu, H. 2014. The non-Abelian Weyl–Yang–Kaluza–Klein gravity model. General Relativity and Gravitation, 46: 1751. doi: 10.1007/s10714-014-1751-x
  • Kuyrukcu, H. 2016. A black hole solution of higher-dimensional Weyl–Yang–Kaluza–Klein theory by the Wu–Yang ansatz. arXiv: gr-qc/1602.02418v1 (preprint).
  • Kuyrukcu, H. 2021. A black hole solution of higher-dimensional Weyl–Yang–Kaluza–Klein theory. Classical and Quantum Gravity, 38 (17): 175009. doi: 10.1088/1361-6382/ac161a
  • Lanczos, C. 1938. A remarkable property of the Riemann– Christoffel tensor in four dimensions. Annals of Mathematics, 39: 842-850. doi: 10.2307/1968467
  • Lanczos, C. 1949. Lagrangian multiplier and Riemannian spaces. Reviews of Modern Physics, 21 (3): 497-502. doi: 10.1103/ RevModPhys.21.497
  • Lanczos, C. 1957. Electricity and general relativity. Reviews of Modern Physics, 29 (3): 337-350. doi: 10.1103/ RevModPhys.29.337
  • Lee, HC. 1983. An introduction to Kaluza–Klein theories: Proceedings of the workshop on KaluzaKlein theories. Chalk River, Canada, p. 187.
  • Lichnerowicz, A. 1958. Sur un procedé de quantification du champ de gravitation. C. R. Acad. Sci., 247: 433-436.
  • Liu, H., Wesson, PS. 1997. The physical properties of charged five-dimensional black holes. Classical and Quantum Gravity, 14 (7): 1651-1663. doi: 10.1088/0264-9381/14/7/006
  • Loos, HG. 1963. Spin connection in general relativity. Annals of Physics, 25: 91-108. doi: 10.1016/0003-4916(63)90335-X
  • Loos, HG., Treat, RP. 1967. Conditional dynamic equivalence of free Yang–Mills fields and free gravitational fields. Physics Letters A, 26 (2): 91-92. doi: 10.1016/0375-9601(67)90117-X
  • Mandel, H. 1926. Zur herleitung der feldgleichungen in der allgemeinen relativitätstheorie. Zeitschrift für Physik, 39 (12): 136-145. doi: 10.1007/BF01321980
  • Misner, CW., Thorne, KS., Wheeler, JA. 1973. Gravitation. Freeman WH and company, San Francisco.
  • Ni, W-T. 1975. Yang’s gravitational field equations. Physical Review Letters, 35 (5): 319-320. [erratum: 1975. Physical Review Letters, 35 (25): 1748. doi: 10.1103/PhysRevLett.35.1748]. doi: 10.1103/PhysRevLett.35.319
  • Öktem, F. 1985. Doğa Bilim Dergisi A1(9): 3.
  • Palatini, A. 1919. Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rendiconti del Circolo Matematico di Palermo, 43: 203-212. doi: 10.1007/ BF03014670
  • Pauli, W. 1919. Zur theorie der gravitation und der elektrizitaät von H Weyl. Physikalische Zeitschrift, 20: 457-467.
  • Pavelle, R. 1974. Yang’s gravitational field equations. Physical Review Letters, 33 (24): 1461-1463. doi: 10.1103/ PhysRevLett.33.1461
  • Pavelle, R. 1975. Unphysical solutions of Yang’s gravitational field equations. Physical Review Letters, 34: 1114. doi: 10.1103/ PhysRevLett.34.1114
  • Pavelle, R. 1976. Unphysical characteristics of Yang’s pure-space equations. Physical Review Letters, 37 (15): 961-964. doi: 10.1103/PhysRevLett.37.961
  • Pavelle, R. 1978. Mansouri–Chang gravitation theory. Physical Review Letters, 40 (5): 267-270. doi: 10.1103/ PhysRevLett.40.267
  • Perry, M. 2009. Applications of Differential Geometry to Physics. p. 36. https://sgielen.files.wordpress.com/2018/01/diffgeo.pdf Pope, CN. Kaluza–Klein Theory. p. 4. http://people.physics.tamu. edu/pope/ihplec.pdf
  • Stephenson, G. 1958. Quadratic Lagrangians and general relativity. ll Nuovo Cimento, 9 (2): 263-269. doi: 10.1007/ BF02724929
  • Thompson, AH. 1975. Yang’s gravitational field equations. Physical Review Letters, 34 (8): 507-508. doi: 10.1103/ PhysRevLett.34.507
  • Thompson, AH. 1975a. Geometrically degenerate solutions of the Kilmister–Yang equations. Physical Review Letters, 35 (5): 320-322. doi: 10.1103/PhysRevLett.35.320
  • Weyl, H. 1918. Gravitation und elektrizität. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), 1918: 465-478. doi: 10.1007/978-3-663-19510_811
  • Weyl, H. 1919. A new extension of relativity theory. Annalen der Physik, 59: 101-133. doi: 10.1002/andp.19193641002
  • Weyl, H. 1921. Über die physikalischen grundlagen der erweiterten relativitätstheorie. Physikalische Zeitschrift, 22: 473-480.
  • Yang, CN., Mills, RL. 1954. Conservation of isotopic spin and isotopic gauge invariance. Physical Review, 96 (1): 191-195. doi: 10.1103/PhysRev.96.191
  • Yang, CN. 1974. Integral formalism for gauge fields. Physical Review Letters, 33 (7): 445-447. doi: 10.1103/ PhysRevLett.33.445
There are 58 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Research Articles
Authors

Halil Kuyrukcu 0000-0002-5585-9838

Publication Date December 24, 2022
Published in Issue Year 2022 Volume: 12 Issue: 2

Cite

APA Kuyrukcu, H. (2022). The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity. Karaelmas Fen Ve Mühendislik Dergisi, 12(2), 134-145. https://doi.org/10.7212/karaelmasfen.1009039
AMA Kuyrukcu H. The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity. Karaelmas Fen ve Mühendislik Dergisi. December 2022;12(2):134-145. doi:10.7212/karaelmasfen.1009039
Chicago Kuyrukcu, Halil. “The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity”. Karaelmas Fen Ve Mühendislik Dergisi 12, no. 2 (December 2022): 134-45. https://doi.org/10.7212/karaelmasfen.1009039.
EndNote Kuyrukcu H (December 1, 2022) The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity. Karaelmas Fen ve Mühendislik Dergisi 12 2 134–145.
IEEE H. Kuyrukcu, “The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity”, Karaelmas Fen ve Mühendislik Dergisi, vol. 12, no. 2, pp. 134–145, 2022, doi: 10.7212/karaelmasfen.1009039.
ISNAD Kuyrukcu, Halil. “The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity”. Karaelmas Fen ve Mühendislik Dergisi 12/2 (December 2022), 134-145. https://doi.org/10.7212/karaelmasfen.1009039.
JAMA Kuyrukcu H. The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity. Karaelmas Fen ve Mühendislik Dergisi. 2022;12:134–145.
MLA Kuyrukcu, Halil. “The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity”. Karaelmas Fen Ve Mühendislik Dergisi, vol. 12, no. 2, 2022, pp. 134-45, doi:10.7212/karaelmasfen.1009039.
Vancouver Kuyrukcu H. The Five Dimensional Transformed Weyl–Yang–Kaluza–Klein Theory of Gravity. Karaelmas Fen ve Mühendislik Dergisi. 2022;12(2):134-45.
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