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Study of gyratonic pp-waves by using the Noether symmetry approach

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Abstract

This paper is devoted to study the gyratonic pp-waves using the Noether symmetry approach. We consider gyratonic pp-waves that travel in the z direction with unknown profile function \(H(t,z,\rho , \phi )\) and the gyratonic source function J(t, z). We evaluate velocities, kinetic energy and angular momentum of free particles for different cases of unknown metric coefficients of the spacetime. We show that the kinetic energy may decrease or increase for arbitrary choices of unknown metric coefficients of gyratonic pp-wave spacetime. The change of the kinetic energy of the free particle depends on the choices of the unknown functions under certain condition that the profile function \(H(t,z,\rho , \phi )\) should be harmonic in the Cartesian coordinates and J(t, z) should be function of t and z coordinates only. We also investigate the components of angular momentum in each case and observe that the total amount of the angular momentum per unit mass of the particles decreases with time and oscillatory with the polar angle \(\phi\). Therefore, there is a transfer of energy and angular momentum between the gravitational field and the free particles.

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References

  1. D. Kramer, H. Stephani, M.A.H. MacCallum, H. Herlt, Exact Solutions of the Einstein’s Field Equations (Cambridge University Press, Cambridge, 1980)

    MATH  Google Scholar 

  2. J. Ehlers, W. Kundt, Exact Solutions of the Gravitational Field Equations, in Gravitation: an Introduction to Current Research, edited by L (Witten, New York, 1962)

    Google Scholar 

  3. P. Jordan, J. Ehlers and W. Kundt, Republication of Exact solutions of the field equations of the general theory of relativity. Gen. Relativ. Grav., 9(41), (2009)

  4. Flores, L. José, and M. Sánchez, The Ehlers-Kundt conjecture about Gravitational Waves and Dynamical Systems. J. of Differ. Eqs., 268(12) (2020)

  5. W.C. Kilminster, Gravitation-an introduction to current research. Edited by Louis Witten. John Wiley & Sons, London. 1962. 481 pp. Illustrated. 113s, Aeronauti. J. 67(630), (1963)

  6. A. Peres, Some gravitational waves, Phy. Rev. Lett.3(12) (1959)

  7. H. Bondi, F. Pirani, and I. Robinson, Gravitational waves in general relativity III. Exact Plane Waves, Proc. R. Soc. Lond. A, 251(1267), (1959)

  8. K. Andrzejewski and S. Prencel, Memory effect, conformal symmetry and gravitational plane waves, Phy. Lett. B, (782), (2018)

  9. A. Fuster, C. Pabst, and C. Pfeifer, Berwald spacetimes and very special relativity. Phy. Rev. D 98(8), (2018)

  10. K. Andrzejewski and S. Prencel, Niederer’s transformation, time-dependent oscillators and polarized gravitational waves, Clas. Quantum Gravity 36(15) (2019)

  11. J. B. Formiga, The energy-momentum tensor of gravitational waves, Wyman spacetime, and freely falling observers, Annalen. der Physik. 530(12), (2018)

  12. C. W. Misner and J. A. Wheeler, Classical physics as geometry, Ann. Phys.2(6), (1957)

  13. V. P. Frolov and D. V. Fursaev, Gravitational field of a spinning radiation beam pulse in higher dimensions, Phy. Rev. D, 71(10), (2005)

  14. V. P. Frolov, W. Israel, and A. Zelnikov, Gravitational field of relativistic gyratons, Phy. Rev. D,72(8), (2005)

  15. J. Podolsky, R. Steinbauer, and R. Svarc, Gyratonic pp-waves and their Impulsive Limit, Phy. Rev. D. 90(4), (2014)

  16. W. B. Bonnor, Charge moving with the speed of light in Einstein-Maxwell theory, Inter. Jour. of Theor. Phy. 3(1), (1970)

  17. J. B. Griffiths, Some physical properties of neutrino-gravitational fields, Inter. J. Theor. Phy. 5(3), (1972)

  18. H. W. Brinkmann, On Riemann spaces conformal to Euclidean space, Proceedings of the National Academy of Sciences of the United States of America, (1923)

  19. E. Noether, Invariant variations problems, Nachr. Konig. Gissell. Wissen., Gottingen, Math. Phys. Kl. (1918), (1918)

  20. I. Ibragimov, N. Khaĭrullovich, and N. Khaĭrullovich Ibragimov, Elementary Lie group Analysis and Ordinary Differential Equations, Vol. 197. New York: Wiley, (1999)

  21. l. Ovsiannikov, Group analysis of differential equations, Academic Press, (2014)

  22. H. Bondi, F. A. E. Pirani and I. Robinson, Gravitational waves in general relativity. III. Exact Plane Waves. Proc. Roy. Soc. Lond. A, 251(1267), (1959)

  23. A. Einstein and N. Rosen, On gravitational waves, J. Frank. Inst. 58(9), (1937)

  24. J. Ehlers and W. Kundt, Exact solutions of the gravitational field equations gravitation: an introduction to current problems, eds. L. Witten, John Wiley, (1962)

  25. S. Siddhant, I. Chakraborty, and S.Kar, Kundt geometries and memory effects in the Brans-Dicke theory of gravity, The Europ. Phys. Jour. C, 81(4), (2021)

  26. G. M. Shore, Memory, Penrose limits and the geometry of gravitational shockwaves and gyratons. J. High Energy Phys 2018(12), (2018)

  27. J. W. Maluf , J. F. da Rocha-Neto, S. C. Ulhoa and F. L. Carneiro, Energy-momentum and angular-momentum of a gyratonic pp-waves spacetime. Phys. Rev. D., 100(2), (2019)

  28. J. Maluf, J. F. da Rocha-Neto, S. C. Ulhoa, and F. L. Carneiro, Kinetic energy and angular momentum of free particles in the gyratonic pp-waves spacetimes. Clas. and Quan. Grav. 35(11), (2018)

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Acknowledgements

The authors acknowledge Prof. Asghar Qadir for his useful discussions on the work.

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Authors and Affiliations

  1. School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, 44000, Pakistan

    Kamran Qadir Abbasi

  2. School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, 44000, Pakistan

    Ibrar Hussain

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  1. Kamran Qadir Abbasi
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  2. Ibrar Hussain
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Correspondence to Ibrar Hussain.

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Appendix A:

Appendix A:

The components of angular momentum in each case of H and J are given by Eqs. (48–53)

  • Case-I (i) When \(H(t,z, \rho , \phi )= \rho \sin \phi\) or \(\rho \cos \phi\) and \(J(t,z)= t\).

    $$\begin{aligned} \begin{aligned} M_{x}&= {} \frac{e^{-\phi /2} z \rho \cos \phi \left( -\rho f(\phi ) -2 \rho _0+2 e^{\phi /2} \rho _0\right) }{4t \rho _0} +\sin \phi \left( \frac{1}{4} e^{-\phi /2} \rho -A\right) ,\\ M_{y}&= {} \frac{e^{-\phi /2} z \rho \sin \phi \left( -\rho f(\phi ) -2 \rho _0+2 e^{\phi /2} \rho _0\right) }{4t \rho _0} -\cos \phi \left( \frac{1}{4} e^{-\phi /2} \rho -A\right) ,\\ M_{z}&= -\frac{e^{-\phi /2} \rho ^2 \left( -\rho f(\phi )-2 \rho _0+2 e^{\phi /2} \rho _0\right) }{4 t \rho _0},\\ \end{aligned} \end{aligned}$$
    (48)

    where \(A=\frac{z \sqrt{e^{-\phi } \left( -\rho ^4 \sin \phi ^2+\rho \left( t^2+4 \left( -1+e^{\phi /2}\right) \rho ^2\right) \sin \phi \rho _0+2 \left( t^2-2 \rho ^2-2 e^{\phi /2} \left( t^2-2 \rho ^2\right) +2 e^{\phi } \left( -4 t^2-\rho ^2\right) \right) \rho _0^2\right) }}{4 t \rho _0}.\)

  • Case-I (ii) When \(H(t,z, \rho , \phi )= \rho (\cos \phi +\sin \phi )\) and \(J(t,z)= t\)

    $$\begin{aligned} \begin{aligned} M_{x}&= -\frac{e^{-\phi /2} \rho \cos \phi \left( \rho \cos \phi +2 e^{\phi /2} \rho _0\right) }{4 \rho _0}+\sin \phi \left( \frac{1}{4} e^{-\phi /2} \rho -B\right) ,\\ M_{y}&=-\frac{e^{-\phi /2} \rho \sin \phi \left( \rho \cos \phi +2 e^{\phi /2} \rho _0\right) }{4 \rho _0}-\cos \phi \left( \frac{1}{4} e^{-\phi /2} \rho -B\right) ,\\ M_{z}&= \frac{e^{-\phi /2} \rho ^2 \left( \rho \cos \phi +2 e^{\phi /2} \rho _0\right) }{4 z \rho _0},\\ \end{aligned} \end{aligned}$$
    (49)

    where \(B=\frac{\sqrt{-e^{-\phi } \rho ^4 \cos \phi ]^2+e^{-\phi } z^2 \rho \cos \phi \rho _0-4 e^{-\phi /2} \rho ^3 \cos \phi \rho _0-16 z^2 \rho _0^2-2 e^{-\phi } z^2 \rho _0^2+4 e^{-\phi /2} z^2 \rho _0^2-4 \rho ^2 \rho _0^2}}{4 \rho _0}.\)

  • Case-I (iii)When \(H(t,z, \rho , \phi )= \rho ^{2}(\cos \phi ^{2}-\sin \phi ^{2})~ \text {and}~ J(t,z)= t.\)

    $$\begin{aligned} \begin{aligned} M_{x}&= -z \rho \cos \phi {\dot{\phi }}+\sin \phi \left( \frac{1}{4} e^{-\phi /2} \rho -C\right) ,\\ M_{y}&={-\frac{e^{-\phi /2} z \rho \sin \phi \left( -\rho \cos \phi -\rho \sin \phi -2 \rho _0+2 e^{\phi /2} \rho _0\right) }{4 t \rho _0}+} \cos \phi \left( -\frac{1}{4} e^{-\phi /2} \rho +C\right) ,\\ M_{z}&=\frac{e^{-\phi /2} \rho ^2 \left( -\rho \cos \phi -\rho \sin \phi -2 \rho _0+2 e^{\phi /2} \rho _0\right) }{4 t \rho _0},\\ \end{aligned} \end{aligned}$$
    (50)

    where \(C=\frac{e^{-\phi /2} z \sqrt{\rho ^4 (1+\sin 2 \phi )+\rho \left( 3 t^2-4 \left( -1+e^{\phi /2}\right) \rho ^2\right) (\cos \phi +\sin \phi ) \rho _0+2 \left( \left( 3-2 e^{\phi /2}+8 e^{\phi }\right) t^2+2 \left( -1+e^{\phi /2}\right) ^2 \rho ^2\right) \rho _0^2}}{4 t \rho _0}\)

  • Case-II (i) When \(H(t,z, \rho , \phi )= \rho \sin \phi\) or \(\rho \cos \phi\) and \(J(t,z)= z\).

    $$\begin{aligned} \begin{aligned} M_{x}&=-\frac{e^{-\phi /2} \rho \cos \phi \left( \rho ^2 f(\phi )+2 e^{\phi /2} \rho _0^2\right) }{4 \rho _0^2}+\sin \phi \left( \frac{1}{4} e^{-\phi /2} \rho -D\right) ,\\ M_{y}&=-\frac{e^{-\phi /2} \rho \sin \phi \left( \rho ^2 f(\phi )+2 e^{\phi /2} \rho _0^2\right) }{4 \rho _0^2}+\text {Cos}[\phi ] \left( -\frac{1}{4} e^{-\phi /2} \rho +D\right) ,\\ M_{z}&=\frac{e^{-\phi /2} \rho ^2 \left( \rho ^2 f(\phi )+2 e^{\phi /2} \rho _0^2\right) }{4 z \rho _0^2},\\ \end{aligned} \end{aligned}$$
    (51)

    where \(D= \frac{\sqrt{e^{-\phi } \left( -\rho ^6 \cos 2 \phi ^2+\rho ^2 \left( z^2-4 e^{\phi /2} \rho ^2\right) \cos 2 \phi \rho _0^2-2 \left( \left( 1-2 e^{\phi /2}+8 e^{\phi }\right) z^2+2 e^{\phi } \rho ^2\right) \rho _0^4\right) }}{4 \rho _0^2}.\)

  • Case-II (ii) When \(H(t,z, \rho , \phi )=\rho ^{2}(\cos \phi ^{2}-\sin \phi ^{2})\) and \(J(t,z)= z.\)

    $$\begin{aligned} \begin{aligned} M_{x}&= \frac{e^{-\phi /2} z \rho \cos \phi \left( -\rho ^2 \cos 2 \phi -2 \rho _0^2+2 e^{\phi /2} \rho _0^2\right) }{4 t \rho _0^2}+\sin \phi \left( \frac{1}{4} e^{-\phi /2} \rho -E\right) ,\\ M_{y}&= \frac{e^{-\phi /2} z \rho \sin \phi \left( -\rho ^2 \cos 2 \phi -2 \rho _0^2+2 e^{\phi /2} \rho _0^2\right) }{4 t \rho _0^2}+\cos \phi \left( -\frac{1}{4} e^{-\phi /2} \rho +E\right) ,\\ M_{z}&= -\frac{e^{-\phi /2} \rho ^2 \left( -\rho ^2 \cos 2 \phi -2 \rho _0^2+2 e^{\phi /2} \rho _0^2\right) }{4 t \rho _0^2}, \end{aligned} \end{aligned}$$
    (52)

    where \(E= \frac{e^{-\phi /2} z \sqrt{\rho ^6 \cos 2 \phi ^2-\rho ^2 \left( t^2+4 \left( -1+e^{\phi /2}\right) \rho ^2\right) \cos 2 \phi \rho _0^2+2 \left( \left( -1+2 e^{\phi /2}+8 e^{\phi }\right) t^2+2 \left( -1+e^{\phi /2}\right) ^2 \rho ^2\right) \rho _0^4}}{4 t \rho _0^2}.\)

  • Case-III When, \(H(t,z, \rho , \phi )= \ln ( \frac{\rho }{\rho _{0}})^2,~ J(t,z)= t.\)

    $$\begin{aligned} \begin{aligned} M_{x}&= -\frac{e^{-\phi /2} \left( 2 e^{\phi /2}+t\right) z \cos \phi }{4 \rho }+\left( \frac{1}{4} e^{-\phi /2} \rho -\frac{e^{-\phi /2} z \sqrt{4 e^{\phi }-t^2+2 \rho ^2+16 e^{\phi } \rho ^2+\rho ^2 \ln \left[ \frac{\rho ^2}{\rho _0^2}\right] }}{4 \rho }\right) \sin \phi \\ M_{y}&= \cos \phi \left( -\frac{1}{4} e^{-\phi /2} \rho +\frac{e^{-\phi /2} z \sqrt{4 e^{\phi }-t^2+2 \rho ^2+16 e^{\phi } \rho ^2+\rho ^2 \ln \left[ \frac{\rho ^2}{\rho _0^2}\right] }}{4 \rho }\right) -\frac{e^{-\phi /2} \left( 2 e^{\phi /2}+t\right) z \sin \phi }{4 \rho }\\ M_{z}&=\frac{1}{4} e^{-\phi /2} \left( 2 e^{\phi /2}+t\right) \end{aligned} \end{aligned}$$
    (53)

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Abbasi, K.Q., Hussain, I. Study of gyratonic pp-waves by using the Noether symmetry approach. Eur. Phys. J. Plus 137, 1359 (2022). https://doi.org/10.1140/epjp/s13360-022-03567-w

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