Infinite derivative gravity

Infinite derivative gravity is a theory of gravity which attempts to remove cosmological and black hole singularities by adding extra terms to the Einstein–Hilbert action, which weaken gravity at short distances.

History

In 1987, Krasnikov considered an infinite set of higher derivative terms acting on the curvature terms and showed that by choosing the coefficients wisely, the propagator would be ghost-free and exponentially suppressed in the ultraviolet regime.^[1] Tomboulis (1997) later extended this work.^[2] By looking at an equivalent scalar-tensor theory, Biswas, Mazumdar and Siegel (2005) looked at bouncing FRW solutions.^[3] In 2011, Biswas, Gerwick, Koivisto and Mazumdar demonstrated that the most general infinite derivative action in 4 dimensions, around constant curvature backgrounds, parity invariant and torsion free, can be expressed by:^[4]

S=\int \mathrm {d} ^{4}x{\sqrt {-g}}\left(M_{P}^{2}R+RF_{1}(\Box )R+R^{\mu \nu }F_{2}(\Box )R_{\mu \nu }+C^{\mu \nu \lambda \sigma }F_{3}(\Box )C_{\mu \nu \lambda \sigma }\right)

where the $F_{i}(\Box )=\sum _{n=0}^{\infty }f_{i_{n}}\left(\Box /M^{2}\right)^{n}$ are functions of the D'Alembert operator $\Box =g^{\mu \nu }\nabla _{\mu }\nabla _{\nu }$ and a mass scale $M$ , $R$ is the Ricci scalar, $R_{\mu \nu }$ is the Ricci tensor and $C_{\mu \nu \lambda \sigma }$ is the Weyl tensor.^[5] In order to avoid ghosts, the propagator (which is a combination of the $F_{i}(\Box )$ s) must be the exponential of an entire function. A lower bound was obtained on the mass scale of IDG using experimental data on the strength of gravity at short distances,^[6] as well as by using data on inflation^[7] and on the bending of light around the Sun.^[8] The GHY boundary terms were found using the ADM 3+1 spacetime decomposition.^[9] One can show that the entropy for this theory is finite in various contexts.^[10]^[11]

The effect of IDG on black holes and the propagator was examined by Modesto.^[12]^[13]^[14] Modesto further looked at the renormalisability of the theory,^[15]^[16] as well as showing that it could generate "super-accelerated" bouncing solutions instead of a big bang singularity.^[17] Calcagni and Nardelli investigated the effect of IDG on the diffusion equation.^[18] IDG modifies the way gravitational waves are produced and how they propagate through space. The amount of power radiated away through gravitational waves by binary systems is reduced, although this effect is far smaller than the current observational precision.^[19] This theory is shown to be stable and propagates finite number of degrees of freedom.^[20]

Avoidance of singularities

This action can produce a bouncing cosmology, by taking a flat FRW metric with a scale factor $a(t)=\cosh(\sigma t)$ or $a(t)=e^{\lambda t^{2}}$ , thus avoiding the cosmological singularity problem.^[3]^[21]^[22]^[23] The propagator around a flat space background was obtained in 2013.^[24]

This action avoids a curvature singularity for a small perturbation to a flat background near the origin, while recovering the $1/r$ fall of the GR potential at large distances. This is done using the linearised equations of motion which is a valid approximation because if the perturbation is small enough and the mass scale $M$ is large enough, then the perturbation will always be small enough that quadratic terms can be neglected.^[4] It also avoids the Hawking–Penrose singularity in this context.^[25]^[26]

Stability of black hole singularities

It was shown that in non-local gravity, Schwarzschild singularities are stable to small perturbations.^[27] Further stability analysis of black holes was carried out by Myung and Park.^[28]

Equations of motion

The equations of motion for this action are^[5]

{\begin{aligned}T^{\alpha \beta }&=P^{\alpha \beta }\\&=G^{\alpha \beta }+4G^{\alpha \beta }F_{1}(\Box )R+g^{\alpha \beta }RF_{1}(\Box )R-4\left(\nabla ^{\alpha }\nabla ^{\beta }-g^{\alpha \beta }\Box \right)F_{1}(\Box )R-2\Omega _{1}^{\alpha \beta }+g^{\alpha \beta }\left(\Omega _{1\sigma }^{\sigma }\right)\\&\qquad +4{R^{\beta }}_{\mu }R^{\mu \alpha }-g^{\alpha \beta }R^{\mu \nu }F_{2}(\Box )R_{\mu \nu }-4\left(F_{2}(\Box )R^{\mu (\beta }\right)_{;\mu }^{;\alpha )}+2\Box \left(F_{2}(\Box )R^{\alpha \beta }\right)+2g^{\alpha \beta }\left(F_{2}(\Box )R^{\mu \nu }\right)_{;\mu ;\nu }-2\Omega _{2}^{\alpha \beta }+g^{\alpha \beta }\left(\Omega _{2\sigma }^{\sigma }+{\bar {\Omega }}_{2}\right)-4\Delta _{2}^{\alpha \beta }\\&\qquad -g^{\alpha \beta }C^{\mu \nu \lambda \sigma }F_{3}(\Box )C_{\mu \nu \lambda \sigma }+4{C^{\alpha }}_{\rho \theta \psi }F_{3}(\Box )C^{\beta \rho \theta \psi }-4\left[2\nabla _{\mu }\nabla _{\nu }+R_{\mu \nu }\right]F_{3}(\Box )C^{\beta \mu \nu \alpha }-2\Omega _{3}^{\alpha \beta }+g^{\alpha \beta }\left(\Omega _{3\gamma }^{\gamma }+{\bar {\Omega }}_{3}\right)-8\Delta _{3}^{\alpha \beta }\end{aligned}}

where

{\begin{aligned}\Omega _{1}^{\alpha \beta }&=\sum _{n=1}^{\infty }f_{1_{n}}\sum _{m=0}^{n-1}\nabla ^{\alpha }\Box ^{m}R\nabla ^{\beta }\Box ^{n-m-1}R,\\{\bar {\Omega }}_{1}&=\sum _{n=1}^{\infty }f_{1_{n}}\sum _{m=0}^{n-1}\Box ^{m}R\Box ^{n-m}R,\\\Omega _{2}^{\alpha \beta }&=\sum _{n=1}^{\infty }f_{1_{n}}\sum _{m=0}^{n-1}\nabla ^{\alpha }\Box ^{m}{R^{\mu }}_{\nu }\nabla ^{\beta }\Box ^{n-m-1}{R^{\nu }}_{\mu },\\{\bar {\Omega }}_{2}&=\sum _{n=1}^{\infty }f_{1_{n}}\sum _{m=0}^{n-1}\Box ^{m}{R^{\mu }}_{\nu }\Box ^{n-m}{R^{\nu }}_{\mu },\\\Delta _{2}^{\alpha \beta }&={\frac {1}{2}}\sum _{n=1}^{\infty }f_{2_{n}}\sum _{\ell =0}^{n-1}\nabla _{\nu }\left[\Box ^{\ell }{R^{\nu }}_{\sigma }\nabla ^{(\alpha }\Box ^{n-\ell -1}R^{\beta )\sigma }-\Box ^{\ell }\nabla ^{(\alpha }{R^{n}u}_{\sigma }\Box ^{n-\ell -1}R^{\beta )\sigma }\right],\\\Omega _{3}^{\alpha \beta }&=\sum _{n=1}^{\infty }f_{3_{n}}\sum _{\ell =0}^{n-1}\nabla ^{\alpha }\Box ^{\ell }{C^{\mu }}_{\nu \lambda \sigma }\nabla ^{\beta }\Box ^{n-\ell -1}{C_{\mu }}^{\nu \lambda \sigma },\\{\bar {\Omega }}_{3}&=\sum _{n=1}^{\infty }f_{3_{n}}\sum _{\ell =0}^{n-1}\Box ^{\ell }{C^{\mu }}_{\nu \lambda \sigma }\Box ^{n-\ell }{C_{\mu }}^{\nu \lambda \sigma },\\\Delta _{3}^{\alpha \beta }&={\frac {1}{2}}\sum _{n=1}^{\infty }f_{3_{n}}\sum _{\ell =0}^{n-1}\nabla _{\nu }\left[\Box ^{\ell }{C^{\lambda \nu }}_{\sigma \mu }\Box ^{n-\ell -1}{C_{\lambda }}^{(\beta |\sigma \mu |;\alpha )}-\Box ^{\ell }\nabla ^{(\alpha }C_{\sigma \mu }^{\lambda \nu }{C_{\lambda }}^{\beta )\sigma \mu }\right].\end{aligned}}

References

^ Krasnikov, N. V. (November 1987). "Nonlocal gauge theories". Theoretical and Mathematical Physics. 73 (2): 1184–1190. Bibcode:1987TMP....73.1184K. doi:10.1007/BF01017588. S2CID 122648433.
^ Tomboulis, E. T (1997). "Superrenormalizable gauge and gravitational theories". arXiv:hep-th/9702146.
^ ^a ^b Biswas, Tirthabir; Mazumdar, Anupam; Siegel, Warren (2006). "Bouncing Universes in String-inspired Gravity". Journal of Cosmology and Astroparticle Physics. 2006 (3): 009. arXiv:hep-th/0508194. Bibcode:2006JCAP...03..009B. CiteSeerX 10.1.1.266.743. doi:10.1088/1475-7516/2006/03/009. S2CID 7445076.
^ ^a ^b Biswas, Tirthabir; Gerwick, Erik; Koivisto, Tomi; Mazumdar, Anupam (2012). "Towards singularity and ghost free theories of gravity". Physical Review Letters. 108 (3): 031101. arXiv:1110.5249. Bibcode:2012PhRvL.108c1101B. doi:10.1103/PhysRevLett.108.031101. PMID 22400725. S2CID 5517893.
^ ^a ^b Biswas, Tirthabir; Conroy, Aindriú; Koshelev, Alexey S.; Mazumdar, Anupam (2013). "Generalized ghost-free quadratic curvature gravity". Classical and Quantum Gravity. 31 (1): 015022. arXiv:1308.2319. Bibcode:2014CQGra..31a5022B. doi:10.1088/0264-9381/31/1/015022. S2CID 119103482.
^ Edholm, James; Koshelev, Alexey S.; Mazumdar, Anupam (2016). "Behavior of the Newtonian potential for ghost-free gravity and singularity free gravity". Physical Review D. 94 (10): 104033. arXiv:1604.01989. Bibcode:2016PhRvD..94j4033E. doi:10.1103/PhysRevD.94.104033. S2CID 118419505.
^ Edholm, James (6 February 2017). "UV completion of the Starobinsky model, tensor-to-scalar ratio, and constraints on nonlocality". Physical Review D. 95 (4): 044004. arXiv:1611.05062. Bibcode:2017PhRvD..95d4004E. doi:10.1103/PhysRevD.95.044004. S2CID 17258584.
^ Feng, Lei (2017). "Light bending in infinite derivative theories of gravity". Physical Review D. 95 (8): 084015. arXiv:1703.06535. Bibcode:2017PhRvD..95h4015F. doi:10.1103/PhysRevD.95.084015. S2CID 119456666.
^ Teimouri, Ali; Talaganis, Spyridon; Edholm, James; Mazumdar, Anupam (1 August 2016). "Generalised boundary terms for higher derivative theories of gravity". Journal of High Energy Physics. 2016 (8): 144. arXiv:1606.01911. Bibcode:2016JHEP...08..144T. doi:10.1007/JHEP08(2016)144. S2CID 55220918.
^ Myung, Yun Soo (2017). "Entropy of a black hole in infinite-derivative gravity". Physical Review D. 95 (10): 106003. arXiv:1702.00915. Bibcode:2017PhRvD..95j6003M. doi:10.1103/PhysRevD.95.106003. S2CID 119516555.
^ Conroy, Aindriú; Mazumdar, Anupam; Teimouri, Ali (2015). "Wald Entropy for Ghost-Free, Infinite Derivative Theories of Gravity". Physical Review Letters. 114 (20): 201101. arXiv:1503.05568. Bibcode:2015PhRvL.114t1101C. doi:10.1103/PhysRevLett.114.201101. PMID 26047217. S2CID 7129585.
^ Modesto, Leonardo (2011). "Super-renormalizable Quantum Gravity". Physical Review D. 86 (4): 044005. arXiv:1107.2403. Bibcode:2012PhRvD..86d4005M. doi:10.1103/PhysRevD.86.044005. S2CID 119310607.
^ Li, Yao-Dong; Modesto, Leonardo; Rachwał, Lesław (2015). "Exact solutions and spacetime singularities in nonlocal gravity". Journal of High Energy Physics. 2015 (12): 1–50. arXiv:1506.08619. Bibcode:2015JHEP...12..173L. doi:10.1007/JHEP12(2015)173. S2CID 117760918.
^ Bambi, Cosimo; Modesto, Leonardo; Rachwał, Lesław (2017). "Spacetime completeness of non-singular black holes in conformal gravity". Journal of Cosmology and Astroparticle Physics. 2017 (5): 003. arXiv:1611.00865. Bibcode:2017JCAP...05..003B. doi:10.1088/1475-7516/2017/05/003. S2CID 119321606.
^ Modesto, Leonardo; Rachwal, Leslaw (2014). "Super-renormalizable & Finite Gravitational Theoriess". Nuclear Physics B. 889: 228–248. arXiv:1407.8036. Bibcode:2014NuPhB.889..228M. doi:10.1016/j.nuclphysb.2014.10.015. S2CID 119146778.
^ Modesto, Leonardo; Rachwal, Leslaw (2015). "Universally Finite Gravitational & Gauge Theories". Nuclear Physics B. 900: 147–169. arXiv:1503.00261. Bibcode:2015NuPhB.900..147M. doi:10.1016/j.nuclphysb.2015.09.006. S2CID 119282730.
^ Calcagni, Gianluca; Modesto, Leonardo; Nicolini, Piero (2014). "Super-accelerating bouncing cosmology in asymptotically free non-local gravity". The European Physical Journal C. 74 (8): 2999. arXiv:1306.5332. Bibcode:2014EPJC...74.2999C. doi:10.1140/epjc/s10052-014-2999-8. S2CID 254107755.
^ Calcagni, Gianluca; Nardelli, Giuseppe (2010). "Nonlocal gravity and the diffusion equation". Physical Review D. 82 (12): 123518. arXiv:1004.5144. Bibcode:2010PhRvD..82l3518C. doi:10.1103/PhysRevD.82.123518. S2CID 54087795.
^ Edholm, James (28 August 2018). "Gravitational radiation in infinite derivative gravity and connections to effective quantum gravity". Physical Review D. 98 (4): 044049. arXiv:1806.00845. Bibcode:2018PhRvD..98d4049E. doi:10.1103/PhysRevD.98.044049. S2CID 52837779.
^ Talaganis, Spyridon; Teimouri, Ali (22 May 2017). "Hamiltonian Analysis for Infinite Derivative Field Theories and Gravity". arXiv:1701.01009 [hep-th].
^ Koshelev, A. S.; Vernov, S. Yu (1 September 2012). "On bouncing solutions in non-local gravity". Physics of Particles and Nuclei. 43 (5): 666–668. arXiv:1202.1289. Bibcode:2012PPN....43..666K. doi:10.1134/S106377961205019X. S2CID 119152817.
^ Koshelev, A. S; Vernov, S. Yu (2012). "On bouncing solutions in non-local gravity". Physics of Particles and Nuclei. 43 (5): 666–668. arXiv:1202.1289. Bibcode:2012PPN....43..666K. doi:10.1134/S106377961205019X. S2CID 119152817.
^ Edholm, James (2018). "Conditions for defocusing around more general metrics in Infinite Derivative Gravity". Physical Review D. 97 (8): 084046. arXiv:1802.09063. Bibcode:2018PhRvD..97h4046E. doi:10.1103/PhysRevD.97.084046. S2CID 119449377.
^ Biswas, Tirthabir; Koivisto, Tomi; Mazumdar, Anupam (3 February 2013). "Nonlocal theories of gravity: the flat space propagator". arXiv:1302.0532 [gr-qc].
^ Conroy, Aindriú; Koshelev, Alexey S; Mazumdar, Anupam (2017). "Defocusing of null rays in infinite derivative gravity". Journal of Cosmology and Astroparticle Physics. 2017 (1): 017. arXiv:1605.02080. Bibcode:2017JCAP...01..017C. doi:10.1088/1475-7516/2017/01/017. S2CID 115136697.
^ Edholm, James; Conroy, Aindriú (2017). "Newtonian potential and geodesic completeness in infinite derivative gravity". Physical Review D. 96 (4): 044012. arXiv:1705.02382. Bibcode:2017PhRvD..96d4012E. doi:10.1103/PhysRevD.96.044012. S2CID 45816145.
^ Calcagni, Gianluca; Modesto, Leonardo (4 July 2017). "Stability of Schwarzschild singularity in non-local gravity". Physics Letters B. 773: 596–600. arXiv:1707.01119. Bibcode:2017PhLB..773..596C. doi:10.1016/j.physletb.2017.09.018. S2CID 119020924.
^ Myung, Yun Soo; Park, Young-Jai (2018). "Stability issues of black hole in non-local gravity". Physics Letters B. 779: 342–347. arXiv:1711.06411. Bibcode:2018PhLB..779..342M. doi:10.1016/j.physletb.2018.02.023. S2CID 54665676.

[1] Krasnikov, N. V. (November 1987). "Nonlocal gauge theories". Theoretical and Mathematical Physics. 73 (2): 1184–1190. Bibcode:1987TMP....73.1184K. doi:10.1007/BF01017588. S2CID 122648433.

[2] Tomboulis, E. T (1997). "Superrenormalizable gauge and gravitational theories". arXiv:hep-th/9702146.

[ReferenceA-3] Biswas, Tirthabir; Mazumdar, Anupam; Siegel, Warren (2006). "Bouncing Universes in String-inspired Gravity". Journal of Cosmology and Astroparticle Physics. 2006 (3): 009. arXiv:hep-th/0508194. Bibcode:2006JCAP...03..009B. CiteSeerX 10.1.1.266.743. doi:10.1088/1475-7516/2006/03/009. S2CID 7445076.

[ReferenceB-4] Biswas, Tirthabir; Gerwick, Erik; Koivisto, Tomi; Mazumdar, Anupam (2012). "Towards singularity and ghost free theories of gravity". Physical Review Letters. 108 (3): 031101. arXiv:1110.5249. Bibcode:2012PhRvL.108c1101B. doi:10.1103/PhysRevLett.108.031101. PMID 22400725. S2CID 5517893.

[Ghostfree-5] Biswas, Tirthabir; Conroy, Aindriú; Koshelev, Alexey S.; Mazumdar, Anupam (2013). "Generalized ghost-free quadratic curvature gravity". Classical and Quantum Gravity. 31 (1): 015022. arXiv:1308.2319. Bibcode:2014CQGra..31a5022B. doi:10.1088/0264-9381/31/1/015022. S2CID 119103482.

[6] Edholm, James; Koshelev, Alexey S.; Mazumdar, Anupam (2016). "Behavior of the Newtonian potential for ghost-free gravity and singularity free gravity". Physical Review D. 94 (10): 104033. arXiv:1604.01989. Bibcode:2016PhRvD..94j4033E. doi:10.1103/PhysRevD.94.104033. S2CID 118419505.

[7] Edholm, James (6 February 2017). "UV completion of the Starobinsky model, tensor-to-scalar ratio, and constraints on nonlocality". Physical Review D. 95 (4): 044004. arXiv:1611.05062. Bibcode:2017PhRvD..95d4004E. doi:10.1103/PhysRevD.95.044004. S2CID 17258584.

[8] Feng, Lei (2017). "Light bending in infinite derivative theories of gravity". Physical Review D. 95 (8): 084015. arXiv:1703.06535. Bibcode:2017PhRvD..95h4015F. doi:10.1103/PhysRevD.95.084015. S2CID 119456666.

[9] Teimouri, Ali; Talaganis, Spyridon; Edholm, James; Mazumdar, Anupam (1 August 2016). "Generalised boundary terms for higher derivative theories of gravity". Journal of High Energy Physics. 2016 (8): 144. arXiv:1606.01911. Bibcode:2016JHEP...08..144T. doi:10.1007/JHEP08(2016)144. S2CID 55220918.

[10] Myung, Yun Soo (2017). "Entropy of a black hole in infinite-derivative gravity". Physical Review D. 95 (10): 106003. arXiv:1702.00915. Bibcode:2017PhRvD..95j6003M. doi:10.1103/PhysRevD.95.106003. S2CID 119516555.

[11] Conroy, Aindriú; Mazumdar, Anupam; Teimouri, Ali (2015). "Wald Entropy for Ghost-Free, Infinite Derivative Theories of Gravity". Physical Review Letters. 114 (20): 201101. arXiv:1503.05568. Bibcode:2015PhRvL.114t1101C. doi:10.1103/PhysRevLett.114.201101. PMID 26047217. S2CID 7129585.

[12] Modesto, Leonardo (2011). "Super-renormalizable Quantum Gravity". Physical Review D. 86 (4): 044005. arXiv:1107.2403. Bibcode:2012PhRvD..86d4005M. doi:10.1103/PhysRevD.86.044005. S2CID 119310607.

[13] Li, Yao-Dong; Modesto, Leonardo; Rachwał, Lesław (2015). "Exact solutions and spacetime singularities in nonlocal gravity". Journal of High Energy Physics. 2015 (12): 1–50. arXiv:1506.08619. Bibcode:2015JHEP...12..173L. doi:10.1007/JHEP12(2015)173. S2CID 117760918.

[14] Bambi, Cosimo; Modesto, Leonardo; Rachwał, Lesław (2017). "Spacetime completeness of non-singular black holes in conformal gravity". Journal of Cosmology and Astroparticle Physics. 2017 (5): 003. arXiv:1611.00865. Bibcode:2017JCAP...05..003B. doi:10.1088/1475-7516/2017/05/003. S2CID 119321606.

[15] Modesto, Leonardo; Rachwal, Leslaw (2014). "Super-renormalizable & Finite Gravitational Theoriess". Nuclear Physics B. 889: 228–248. arXiv:1407.8036. Bibcode:2014NuPhB.889..228M. doi:10.1016/j.nuclphysb.2014.10.015. S2CID 119146778.

[16] Modesto, Leonardo; Rachwal, Leslaw (2015). "Universally Finite Gravitational & Gauge Theories". Nuclear Physics B. 900: 147–169. arXiv:1503.00261. Bibcode:2015NuPhB.900..147M. doi:10.1016/j.nuclphysb.2015.09.006. S2CID 119282730.

[17] Calcagni, Gianluca; Modesto, Leonardo; Nicolini, Piero (2014). "Super-accelerating bouncing cosmology in asymptotically free non-local gravity". The European Physical Journal C. 74 (8): 2999. arXiv:1306.5332. Bibcode:2014EPJC...74.2999C. doi:10.1140/epjc/s10052-014-2999-8. S2CID 254107755.

[18] Calcagni, Gianluca; Nardelli, Giuseppe (2010). "Nonlocal gravity and the diffusion equation". Physical Review D. 82 (12): 123518. arXiv:1004.5144. Bibcode:2010PhRvD..82l3518C. doi:10.1103/PhysRevD.82.123518. S2CID 54087795.

[19] Edholm, James (28 August 2018). "Gravitational radiation in infinite derivative gravity and connections to effective quantum gravity". Physical Review D. 98 (4): 044049. arXiv:1806.00845. Bibcode:2018PhRvD..98d4049E. doi:10.1103/PhysRevD.98.044049. S2CID 52837779.

[20] Talaganis, Spyridon; Teimouri, Ali (22 May 2017). "Hamiltonian Analysis for Infinite Derivative Field Theories and Gravity". arXiv:1701.01009 [hep-th].

[21] Koshelev, A. S.; Vernov, S. Yu (1 September 2012). "On bouncing solutions in non-local gravity". Physics of Particles and Nuclei. 43 (5): 666–668. arXiv:1202.1289. Bibcode:2012PPN....43..666K. doi:10.1134/S106377961205019X. S2CID 119152817.

[22] Koshelev, A. S; Vernov, S. Yu (2012). "On bouncing solutions in non-local gravity". Physics of Particles and Nuclei. 43 (5): 666–668. arXiv:1202.1289. Bibcode:2012PPN....43..666K. doi:10.1134/S106377961205019X. S2CID 119152817.

[23] Edholm, James (2018). "Conditions for defocusing around more general metrics in Infinite Derivative Gravity". Physical Review D. 97 (8): 084046. arXiv:1802.09063. Bibcode:2018PhRvD..97h4046E. doi:10.1103/PhysRevD.97.084046. S2CID 119449377.

[24] Biswas, Tirthabir; Koivisto, Tomi; Mazumdar, Anupam (3 February 2013). "Nonlocal theories of gravity: the flat space propagator". arXiv:1302.0532 [gr-qc].

[25] Conroy, Aindriú; Koshelev, Alexey S; Mazumdar, Anupam (2017). "Defocusing of null rays in infinite derivative gravity". Journal of Cosmology and Astroparticle Physics. 2017 (1): 017. arXiv:1605.02080. Bibcode:2017JCAP...01..017C. doi:10.1088/1475-7516/2017/01/017. S2CID 115136697.

[26] Edholm, James; Conroy, Aindriú (2017). "Newtonian potential and geodesic completeness in infinite derivative gravity". Physical Review D. 96 (4): 044012. arXiv:1705.02382. Bibcode:2017PhRvD..96d4012E. doi:10.1103/PhysRevD.96.044012. S2CID 45816145.

[27] Calcagni, Gianluca; Modesto, Leonardo (4 July 2017). "Stability of Schwarzschild singularity in non-local gravity". Physics Letters B. 773: 596–600. arXiv:1707.01119. Bibcode:2017PhLB..773..596C. doi:10.1016/j.physletb.2017.09.018. S2CID 119020924.

[28] Myung, Yun Soo; Park, Young-Jai (2018). "Stability issues of black hole in non-local gravity". Physics Letters B. 779: 342–347. arXiv:1711.06411. Bibcode:2018PhLB..779..342M. doi:10.1016/j.physletb.2018.02.023. S2CID 54665676.

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