A Modified Gravity Theory: Null Aether
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摘要
General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the "aether". In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory(NAT) — a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordstr?m-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT:We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions — AdS-plane waves and pp-waves — in this theory in any dimension D ≥ 3. Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a "mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.
General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the "aether". In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory(NAT) — a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordstr?m-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT:We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions — AdS-plane waves and pp-waves — in this theory in any dimension D ≥ 3. Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a "mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.
General quantum gravity arguments predict that Lorentz symmetry might not hold exactly in nature. This has motivated much interest in Lorentz breaking gravity theories recently. Among such models are vector-tensor theories with preferred direction established at every point of spacetime by a fixed-norm vector field. The dynamical vector field defined in this way is referred to as the "aether". In this paper, we put forward the idea of a null aether field and introduce, for the first time, the Null Aether Theory(NAT) — a vector-tensor theory. We first study the Newtonian limit of this theory and then construct exact spherically symmetric black hole solutions in the theory in four dimensions, which contain Vaidya-type non-static solutions and static Schwarzschild-(A)dS type solutions, Reissner-Nordstr?m-(A)dS type solutions and solutions of conformal gravity as special cases. Afterwards, we study the cosmological solutions in NAT:We find some exact solutions with perfect fluid distribution for spatially flat FLRW metric and null aether propagating along the x direction. We observe that there are solutions in which the universe has big-bang singularity and null field diminishes asymptotically. We also study exact gravitational wave solutions — AdS-plane waves and pp-waves — in this theory in any dimension D ≥ 3. Assuming the Kerr-Schild-Kundt class of metrics for such solutions, we show that the full field equations of the theory are reduced to two, in general coupled, differential equations when the background metric assumes the maximally symmetric form. The main conclusion of these computations is that the spin-0 aether field acquires a "mass" determined by the cosmological constant of the background spacetime and the Lagrange multiplier given in the theory.
引文
[1] D. Mattingly, Living Rev. Relativity 8(2005)5.
[2] T. Jacobson and D. Mattingly, Phys. Rev. D 64(2001)024028.
[3] T. Jacobson, Proceedings of From Quantum to Emergent Gravity:Theory and Phenomenology, Trieste, Itly, 11–15June 2007, Proc. Sci.(2007)020, arXiv:qg-ph/0801.1547.
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[6] D. Garfinkle, C. Eling, and T. Jacobson, Phys. Rev. D 76(2007)024003.
[7] T. Tamaki and U. Miyamoto, Phys. Rev. D 77(2008)024026.
[8] E. Barausse, T. Jacobson, and T. P. Sotiriou, Phys. Rev.D 83(2011)124043.
[9] P. Berglund, J. Bhattacharyya, and D. Mattingly, Phys.Rev. D 85(2012)124019.
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[12] C. Ding, A. Wang, and X. Wang, Phys. Rev. D 92(2015)084055.
[13] E. Barausse, T. P. Sotiriou, and I. Vega, Phys. Rev. D 93(2016)044044.
[14] M. Gu¨rses, Gen. Rel. Grav. 41(2009)31.
[15] M. Gu¨rses and C?. S?entu¨rk, Gen. Rel. Grav. 48(2016)63.
[16] S. M. Carroll and E. A. Lim, Phys. Rev. D 70(2004)123525.
[17] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys.Rev. D 75(2007)044017.
[18] C. Bonvin, R. Durrer, P. G. Ferreira, et al., Phys. Rev. D77(2008)024037.
[19] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys.Rev. D 77(2008)084010.
[20] J. Zuntz, T. G. Zlosnik, F. Bourliot, et al., Phys. Rev. D81(2010)104015.
[21] A. B. Balakin and J. P. S. Lemos, Ann. Phys. 350(2014)454.
[22] T. Y. Alpin and A. B. Balakin, Int. J. Mod. Phys. D 25(2016)1650048.
[23] T. G. Rizzo, JHEP 09(2005)036.
[24] L. Ackerman, S. M. Carroll, and M. B. Wise, Phys. Rev.D 75(2007)083502.
[25] S. M. Carroll and H. Tam, Phys. Rev. D 78(2008)044047.
[26] A. Chatrabhuti, P. Patcharamaneepakorn, and P.Wongjun, JHEP 08(2009)019.
[27] S. M. Carroll, T. R. Dulaney, M. I. Gresham, and H. Tam,Phys. Rev. D 79(2009)065011.
[28] W. Donnelly and T. Jacobson, Phys. Rev. D 82(2010)081501.
[29] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 97(2006)021601.
[30] T. Jacobson and D. Mattingly, Phys. Rev. D 70(2004)024003.
[31] J. Oost, M. Bhattacharjee, and A. Wang, Gen. Rel. Grav.50(2018)124.
[32] Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos,Phys. Rev. D 97(2018)084040.
[33].I. Gu¨llu¨, M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys.Rev. D 83(2011)084015.
[34] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 86(2012)024009.
[35] M. Gu¨rses, S. Hervik, T. C?. S?i?sman, and B. Tekin, Phys.Rev. Lett. 111(2013)101101.
[36] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 90(2014)124005.
[37] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 92(2015)084016.
[38] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Class. Quantum Grav. 34(2017)075003, arXiv:1603.06524.
[39] P. Breitenlohner and D. Z. Freedman, Phys. Lett. B 115(1982)197.
[40] C. Eling and T. Jacobson, Phys. Rev. D 69(2004)064005.
[41] J. W. Elliot, G. D. Moore, and H. Stoica, J. High Energy Phys. 0508(2005)066.
[42] B. Z. Foster and T. Jacobson, Phys. Rev. D 73(2006)064015.
[43] T. Jacobson, Einstein-Aether Gravity:Theory and Observational Constraints, in the Proceedings of the Meeting on CPT and Lorentz Symmetry(CPT 07), Bloomington Indiana, 8-11 Aug. 2007, arXiv:gr-qc/0711.3822.
[44] J. A. Zuntz, P. G. Ferreira, and T. G. Zlosnik, Phys. Rev.Lett. 101(2008)261102.
[45] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Phys. Rev.D 89(2014)084067.
[46] J. Oost, S. Mukohyama, and A. Wang, Phys. Rev. D 97(2018)124023.
[47] H. Stephani, D. Kramer, M. MacCallum, et al., Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge(2003).
[48] M. Gu¨rses and E. Sermutlu, Class. Quantum Grav. 12(1995)2799.
[49] P. C. Mannheim and D. Kazanas, Astrophys. J. 342(1989)635.
[50] R. P. Kerr and A. Schild, Proc. Symp. Appl. Math. 17(1965)199; G. C. Debney, R. P. Kerr, and A. Schild, J.Math. Phys. 10(1969)1842.
[51] M. Gu¨rses and F. Gu¨rsey, J. Math. Phys. 16(1975)2385.
? In the context of Einstein-Aether theory, black hole solutions were considered in Refs.[4–13] and plane wave solutions were studied in Refs.[30–32].
∥The case withε=+1 is associated with Einstein-Aether theory.[2-3]
**In Einstein-Aether theory, these parameters are constrained by some theoretical and observational arguments.[2-3,16,32,40-46]
??At this point, it is worth mentioning that, although the Null Aether Theory being discussed here is inherently different from the Einstein-Aether theory[2-3]with a unit timelike vector field, the constraint c3≤0 in Eq.(186)is not in conflict with the range given in the latter theory. Indeed, imposing that the PPN parameters of Einstein-Aether theory are identical to those of general relativity, the stability against linear perturbations in Minkowski background, vacuum- ■erenkov, and nucleosynthesis constraints require that(see, e.g., Ref.[42])0the range 2/3 tricted to the range -c+(3c+-2)/6(1-c+)there is always a region where c3 is negative; for example, when c+=4/5, we have-4/15
[2] T. Jacobson and D. Mattingly, Phys. Rev. D 64(2001)024028.
[3] T. Jacobson, Proceedings of From Quantum to Emergent Gravity:Theory and Phenomenology, Trieste, Itly, 11–15June 2007, Proc. Sci.(2007)020, arXiv:qg-ph/0801.1547.
[4] C. Eling and T. Jacobson, Class. Quantum Grav. 23(2006)5625.
[5] C. Eling and T. Jacobson, Class. Quantum Grav. 23(2006)5643.
[6] D. Garfinkle, C. Eling, and T. Jacobson, Phys. Rev. D 76(2007)024003.
[7] T. Tamaki and U. Miyamoto, Phys. Rev. D 77(2008)024026.
[8] E. Barausse, T. Jacobson, and T. P. Sotiriou, Phys. Rev.D 83(2011)124043.
[9] P. Berglund, J. Bhattacharyya, and D. Mattingly, Phys.Rev. D 85(2012)124019.
[10] C. Gao and Y. G. Shen, Phys. Rev. D 88(2013)103508.
[11] E. Barausse and T. P. Sotiriou, Class Quantum Grav. 30(2013)244010.
[12] C. Ding, A. Wang, and X. Wang, Phys. Rev. D 92(2015)084055.
[13] E. Barausse, T. P. Sotiriou, and I. Vega, Phys. Rev. D 93(2016)044044.
[14] M. Gu¨rses, Gen. Rel. Grav. 41(2009)31.
[15] M. Gu¨rses and C?. S?entu¨rk, Gen. Rel. Grav. 48(2016)63.
[16] S. M. Carroll and E. A. Lim, Phys. Rev. D 70(2004)123525.
[17] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys.Rev. D 75(2007)044017.
[18] C. Bonvin, R. Durrer, P. G. Ferreira, et al., Phys. Rev. D77(2008)024037.
[19] T. G. Zlosnik, P. G. Ferreira, and G. D. Starkman, Phys.Rev. D 77(2008)084010.
[20] J. Zuntz, T. G. Zlosnik, F. Bourliot, et al., Phys. Rev. D81(2010)104015.
[21] A. B. Balakin and J. P. S. Lemos, Ann. Phys. 350(2014)454.
[22] T. Y. Alpin and A. B. Balakin, Int. J. Mod. Phys. D 25(2016)1650048.
[23] T. G. Rizzo, JHEP 09(2005)036.
[24] L. Ackerman, S. M. Carroll, and M. B. Wise, Phys. Rev.D 75(2007)083502.
[25] S. M. Carroll and H. Tam, Phys. Rev. D 78(2008)044047.
[26] A. Chatrabhuti, P. Patcharamaneepakorn, and P.Wongjun, JHEP 08(2009)019.
[27] S. M. Carroll, T. R. Dulaney, M. I. Gresham, and H. Tam,Phys. Rev. D 79(2009)065011.
[28] W. Donnelly and T. Jacobson, Phys. Rev. D 82(2010)081501.
[29] A. G. Cohen and S. L. Glashow, Phys. Rev. Lett. 97(2006)021601.
[30] T. Jacobson and D. Mattingly, Phys. Rev. D 70(2004)024003.
[31] J. Oost, M. Bhattacharjee, and A. Wang, Gen. Rel. Grav.50(2018)124.
[32] Y. Gong, S. Hou, D. Liang, and E. Papantonopoulos,Phys. Rev. D 97(2018)084040.
[33].I. Gu¨llu¨, M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys.Rev. D 83(2011)084015.
[34] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 86(2012)024009.
[35] M. Gu¨rses, S. Hervik, T. C?. S?i?sman, and B. Tekin, Phys.Rev. Lett. 111(2013)101101.
[36] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 90(2014)124005.
[37] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Phys. Rev. D 92(2015)084016.
[38] M. Gu¨rses, T. C?. S?i?sman, and B. Tekin, Class. Quantum Grav. 34(2017)075003, arXiv:1603.06524.
[39] P. Breitenlohner and D. Z. Freedman, Phys. Lett. B 115(1982)197.
[40] C. Eling and T. Jacobson, Phys. Rev. D 69(2004)064005.
[41] J. W. Elliot, G. D. Moore, and H. Stoica, J. High Energy Phys. 0508(2005)066.
[42] B. Z. Foster and T. Jacobson, Phys. Rev. D 73(2006)064015.
[43] T. Jacobson, Einstein-Aether Gravity:Theory and Observational Constraints, in the Proceedings of the Meeting on CPT and Lorentz Symmetry(CPT 07), Bloomington Indiana, 8-11 Aug. 2007, arXiv:gr-qc/0711.3822.
[44] J. A. Zuntz, P. G. Ferreira, and T. G. Zlosnik, Phys. Rev.Lett. 101(2008)261102.
[45] K. Yagi, D. Blas, E. Barausse, and N. Yunes, Phys. Rev.D 89(2014)084067.
[46] J. Oost, S. Mukohyama, and A. Wang, Phys. Rev. D 97(2018)124023.
[47] H. Stephani, D. Kramer, M. MacCallum, et al., Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge(2003).
[48] M. Gu¨rses and E. Sermutlu, Class. Quantum Grav. 12(1995)2799.
[49] P. C. Mannheim and D. Kazanas, Astrophys. J. 342(1989)635.
[50] R. P. Kerr and A. Schild, Proc. Symp. Appl. Math. 17(1965)199; G. C. Debney, R. P. Kerr, and A. Schild, J.Math. Phys. 10(1969)1842.
[51] M. Gu¨rses and F. Gu¨rsey, J. Math. Phys. 16(1975)2385.
? In the context of Einstein-Aether theory, black hole solutions were considered in Refs.[4–13] and plane wave solutions were studied in Refs.[30–32].
∥The case withε=+1 is associated with Einstein-Aether theory.[2-3]
**In Einstein-Aether theory, these parameters are constrained by some theoretical and observational arguments.[2-3,16,32,40-46]
??At this point, it is worth mentioning that, although the Null Aether Theory being discussed here is inherently different from the Einstein-Aether theory[2-3]with a unit timelike vector field, the constraint c3≤0 in Eq.(186)is not in conflict with the range given in the latter theory. Indeed, imposing that the PPN parameters of Einstein-Aether theory are identical to those of general relativity, the stability against linear perturbations in Minkowski background, vacuum- ■erenkov, and nucleosynthesis constraints require that(see, e.g., Ref.[42])0