Infinite derivative gravity

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.{{cite journal |last1=Krasnikov |first1=N. V. |title=Nonlocal gauge theories |journal=Theoretical and Mathematical Physics |date=November 1987 |volume=73 |issue=2 |pages=1184–1190 |doi=10.1007/BF01017588|bibcode=1987TMP....73.1184K |s2cid=122648433 }} Tomboulis (1997) later extended this work.{{cite arXiv |eprint=hep-th/9702146 |title=Superrenormalizable gauge and gravitational theories |last1=Tomboulis |first1=E. T |year=1997}} By looking at an equivalent scalar-tensor theory, Biswas, Mazumdar and Siegel (2005) looked at bouncing FRW solutions.{{cite journal |title=Bouncing Universes in String-inspired Gravity |journal=Journal of Cosmology and Astroparticle Physics |volume=2006 |issue=3 |pages=009 |arxiv=hep-th/0508194 |bibcode=2006JCAP...03..009B |doi=10.1088/1475-7516/2006/03/009 |year=2006 |last1=Biswas |first1=Tirthabir |last2=Mazumdar |first2=Anupam |last3=Siegel |first3=Warren|citeseerx=10.1.1.266.743 |s2cid=7445076 }} 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:{{Cite journal |arxiv=1110.5249 |title=Towards singularity and ghost free theories of gravity |journal=Physical Review Letters |volume=108 |issue=3 |pages=031101 |last1=Biswas |first1=Tirthabir |last2=Gerwick |first2=Erik |last3=Koivisto |first3=Tomi |last4=Mazumdar |first4=Anupam |year=2012 |doi=10.1103/PhysRevLett.108.031101 |pmid=22400725 |bibcode=2012PhRvL.108c1101B|s2cid=5517893 }}

:$S\; =\; \backslash int\; \backslash mathrm\{d\}^4x\; \backslash sqrt\{-g\}\; \backslash left(M^2\_P\; R+\; R\; F\_1\; (\backslash Box)\; R\; +\; R^\{\backslash mu\backslash nu\}\; F\_2\; (\backslash Box)\; R\_\{\backslash mu\backslash nu\}\; +\; C^\{\backslash mu\backslash nu\backslash lambda\backslash sigma\}\; F\_3\; (\backslash Box)\; C\_\{\backslash mu\backslash nu\backslash lambda\backslash sigma\}\; \backslash right)$

where the $F\_i\; (\backslash Box)=\backslash sum^\backslash infty\_\{n=0\}\; f\_\{i\_n\}\; \backslash left(\backslash Box/M^2\backslash right)^n$ are functions of the D'Alembert operator $\backslash Box=g^\{\backslash mu\backslash nu\}\; \backslash nabla\_\backslash mu\; \backslash nabla\_\backslash nu$ and a mass scale $M$, $R$ is the Ricci scalar, $R\_\{\backslash mu\backslash nu\}$ is the Ricci tensor and $C\_\{\backslash mu\backslash nu\backslash lambda\backslash sigma\}$ is the Weyl tensor.{{cite journal |arxiv=1308.2319 |title=Generalized ghost-free quadratic curvature gravity |journal=Classical and Quantum Gravity |volume=31 |issue=1 |pages=015022 |last1=Biswas |first1=Tirthabir |last2=Conroy |first2=Aindriú |last3=Koshelev |first3=Alexey S. |last4=Mazumdar |first4=Anupam |year=2013 |doi=10.1088/0264-9381/31/1/015022 |bibcode=2014CQGra..31a5022B |s2cid=119103482 }} In order to avoid ghosts, the propagator (which is a combination of the $F\_i\; (\backslash Box)$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,{{cite journal |title=Behavior of the Newtonian potential for ghost-free gravity and singularity free gravity |first1=James |last1=Edholm |first2=Alexey S. |last2=Koshelev |first3=Anupam |last3=Mazumdar |year=2016 |journal=Physical Review D |volume=94 |issue=10 |pages=104033 |doi=10.1103/PhysRevD.94.104033 |bibcode=2016PhRvD..94j4033E |arxiv=1604.01989|s2cid=118419505 }} as well as by using data on inflation{{cite journal |title=UV completion of the Starobinsky model, tensor-to-scalar ratio, and constraints on nonlocality |first=James |last=Edholm |date=6 February 2017 |journal=Physical Review D |volume=95 |issue=4 |pages=044004 |doi=10.1103/PhysRevD.95.044004 |bibcode=2017PhRvD..95d4004E |arxiv=1611.05062|s2cid=17258584 }} and on the bending of light around the Sun.{{cite journal |title=Light bending in infinite derivative theories of gravity |journal=Physical Review D |volume=95 |issue=8 |pages=084015 |doi=10.1103/PhysRevD.95.084015 |year=2017 |last1=Feng |first1=Lei |bibcode=2017PhRvD..95h4015F |arxiv=1703.06535|s2cid=119456666 }} The GHY boundary terms were found using the ADM 3+1 spacetime decomposition.{{cite journal |title=Generalised boundary terms for higher derivative theories of gravity |first1=Ali |last1=Teimouri |first2=Spyridon |last2=Talaganis |first3=James |last3=Edholm |first4=Anupam |last4=Mazumdar |date=1 August 2016 |journal=Journal of High Energy Physics |volume=2016 |issue=8 |pages=144 |doi=10.1007/JHEP08(2016)144 |bibcode=2016JHEP...08..144T |arxiv=1606.01911|s2cid=55220918 }} One can show that the entropy for this theory is finite in various contexts.{{cite journal |doi=10.1103/PhysRevD.95.106003 |title=Entropy of a black hole in infinite-derivative gravity |journal=Physical Review D |volume=95 |issue=10 |pages=106003 |year=2017 |last1=Myung |first1=Yun Soo |bibcode=2017PhRvD..95j6003M |arxiv=1702.00915|s2cid=119516555 }}{{cite journal |doi=10.1103/PhysRevLett.114.201101 |pmid=26047217 |title=Wald Entropy for Ghost-Free, Infinite Derivative Theories of Gravity |journal=Physical Review Letters |volume=114 |issue=20 |pages=201101 |year=2015 |last1=Conroy |first1=Aindriú |last2=Mazumdar |first2=Anupam |last3=Teimouri |first3=Ali |bibcode=2015PhRvL.114t1101C |arxiv=1503.05568|s2cid=7129585 }}

The effect of IDG on black holes and the propagator was examined by Modesto.{{Cite journal |title=Super-renormalizable Quantum Gravity |journal=Physical Review D |volume=86 |issue=4 |pages=044005 |arxiv=1107.2403 |last1=Modesto |first1=Leonardo |year=2011 |doi=10.1103/PhysRevD.86.044005|bibcode=2012PhRvD..86d4005M |s2cid=119310607 }}{{Cite journal |doi = 10.1007/JHEP12(2015)173|title = Exact solutions and spacetime singularities in nonlocal gravity|journal = Journal of High Energy Physics|volume = 2015|issue = 12|pages = 1–50|year = 2015|last1 = Li|first1 = Yao-Dong|last2 = Modesto|first2 = Leonardo|last3 = Rachwał|first3 = Lesław|bibcode = 2015JHEP...12..173L|arxiv = 1506.08619|s2cid = 117760918}}{{Cite journal |doi = 10.1088/1475-7516/2017/05/003|title = Spacetime completeness of non-singular black holes in conformal gravity|journal = Journal of Cosmology and Astroparticle Physics|volume = 2017|issue = 5|pages = 003|year = 2017|last1 = Bambi|first1 = Cosimo|last2 = Modesto|first2 = Leonardo|last3 = Rachwał|first3 = Lesław|arxiv = 1611.00865|bibcode = 2017JCAP...05..003B|s2cid = 119321606}} Modesto further looked at the renormalisability of the theory,{{Cite journal |title=Super-renormalizable & Finite Gravitational Theoriess |journal=Nuclear Physics B |volume=889 |pages=228–248 |arxiv=1407.8036 |last1=Modesto |first1=Leonardo |last2=Rachwal |first2=Leslaw |year=2014 |doi=10.1016/j.nuclphysb.2014.10.015|bibcode=2014NuPhB.889..228M |s2cid=119146778 }}{{Cite journal |title=Universally Finite Gravitational & Gauge Theories |journal=Nuclear Physics B |volume=900 |pages=147–169 |arxiv=1503.00261 |last1=Modesto |first1=Leonardo |last2=Rachwal |first2=Leslaw |year=2015 |doi=10.1016/j.nuclphysb.2015.09.006|bibcode=2015NuPhB.900..147M |s2cid=119282730 }} as well as showing that it could generate "super-accelerated" bouncing solutions instead of a big bang singularity.{{Cite journal |doi = 10.1140/epjc/s10052-014-2999-8|title = Super-accelerating bouncing cosmology in asymptotically free non-local gravity|journal = The European Physical Journal C|volume = 74|issue = 8|pages = 2999|year = 2014|last1 = Calcagni|first1 = Gianluca|last2 = Modesto|first2 = Leonardo|last3 = Nicolini|first3 = Piero|bibcode = 2014EPJC...74.2999C|arxiv = 1306.5332| s2cid=254107755 }} Calcagni and Nardelli investigated the effect of IDG on the diffusion equation.{{Cite journal |arxiv = 1004.5144|doi = 10.1103/PhysRevD.82.123518|title = Nonlocal gravity and the diffusion equation|journal = Physical Review D|volume = 82|issue = 12|pages = 123518|year = 2010|last1 = Calcagni|first1 = Gianluca|last2 = Nardelli|first2 = Giuseppe|bibcode = 2010PhRvD..82l3518C|s2cid = 54087795}} 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.{{cite journal |last1=Edholm |first1=James |date=28 August 2018 |title=Gravitational radiation in infinite derivative gravity and connections to effective quantum gravity |journal= Physical Review D |volume=98 |issue=4 |pages=044049 |doi= 10.1103/PhysRevD.98.044049 |bibcode=2018PhRvD..98d4049E |arxiv=1806.00845 |s2cid=52837779 }} This theory is shown to be stable and propagates finite number of degrees of freedom.{{cite arXiv|last1=Talaganis|first1=Spyridon|last2=Teimouri|first2=Ali|date=2017-05-22|title=Hamiltonian Analysis for Infinite Derivative Field Theories and Gravity|eprint=1701.01009|class=hep-th}}

This action can produce a bouncing cosmology, by taking a flat FRW metric with a scale factor $a(t)\; =\; \backslash cosh(\backslash sigma\; t)$ or $a(t)\; =\; e^\{\backslash lambda\; t^2\}$, thus avoiding the cosmological singularity problem.{{cite journal |title=On bouncing solutions in non-local gravity |first1=A. S. |last1=Koshelev |first2=S. Yu |last2=Vernov |date=1 September 2012 |journal=Physics of Particles and Nuclei |volume=43 |issue=5 |pages=666–668 |doi=10.1134/S106377961205019X |bibcode=2012PPN....43..666K |arxiv=1202.1289|s2cid=119152817 }}{{cite journal |title=On bouncing solutions in non-local gravity |journal=Physics of Particles and Nuclei |volume=43 |issue=5 |pages=666–668 |doi=10.1134/S106377961205019X |year=2012 |last1=Koshelev |first1=A. S |last2=Vernov |first2=S. Yu |bibcode=2012PPN....43..666K |arxiv=1202.1289|s2cid=119152817 }}{{Cite journal |arxiv=1802.09063 |title=Conditions for defocusing around more general metrics in Infinite Derivative Gravity |journal=Physical Review D |volume=97 |issue=8 |pages=084046 |first1=James |last1=Edholm |year=2018 |doi=10.1103/PhysRevD.97.084046 |bibcode=2018PhRvD..97h4046E |s2cid=119449377 }} The propagator around a flat space background was obtained in 2013.{{cite arXiv |title=Nonlocal theories of gravity: the flat space propagator |first1=Tirthabir |last1=Biswas |first2=Tomi |last2=Koivisto |first3=Anupam |last3=Mazumdar |date=3 February 2013 |eprint=1302.0532 |class=gr-qc}}

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. It also avoids the Hawking–Penrose singularity in this context.{{cite journal |doi=10.1088/1475-7516/2017/01/017 |title=Defocusing of null rays in infinite derivative gravity |journal=Journal of Cosmology and Astroparticle Physics |volume=2017 |issue=1 |pages=017 |year=2017 |last1=Conroy |first1=Aindriú |last2=Koshelev |first2=Alexey S |last3=Mazumdar |first3=Anupam |bibcode=2017JCAP...01..017C |arxiv=1605.02080|s2cid=115136697 }}{{cite journal |doi=10.1103/PhysRevD.96.044012 |title=Newtonian potential and geodesic completeness in infinite derivative gravity |journal=Physical Review D |volume=96 |issue=4 |pages=044012 |year=2017 |last1=Edholm |first1=James |last2=Conroy |first2=Aindriú |bibcode=2017PhRvD..96d4012E|arxiv=1705.02382 |s2cid=45816145 }}

It was shown that in non-local gravity, Schwarzschild singularities are stable to small perturbations.{{cite journal |title=Stability of Schwarzschild singularity in non-local gravity |journal=Physics Letters B |volume=773 |pages=596–600 |first1=Gianluca |last1=Calcagni |first2= Leonardo |last2= Modesto |date=4 July 2017 |arxiv=1707.01119 |doi=10.1016/j.physletb.2017.09.018 |bibcode=2017PhLB..773..596C |s2cid=119020924 }} Further stability analysis of black holes was carried out by Myung and Park.{{Cite journal | url=http://inspirehep.net/record/1636962/ | doi=10.1016/j.physletb.2018.02.023| title=Stability issues of black hole in non-local gravity| journal=Physics Letters B| volume=779| pages=342–347| year=2018| last1=Myung| first1=Yun Soo| last2=Park| first2=Young-Jai| bibcode=2018PhLB..779..342M| arxiv=1711.06411| s2cid=54665676}}

The equations of motion for this action are

: $\backslash begin\{align\}$

T^{\alpha\beta} &=P^{\alpha\beta} \\

&= G^{\alpha\beta} + 4 G^{\alpha\beta} F_1(\Box) R + g^{\alpha\beta}R F_1(\Box) R - 4 \left(\nabla^\alpha \nabla^\beta - g^{\alpha\beta} \Box\right)

F_1(\Box) R - 2 \Omega^{\alpha\beta}_1 + g^{\alpha\beta} \left(\Omega^\sigma_{1\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)^{;\alpha)}_{;\mu}+2 \Box \left(F_2(\Box) R^{\alpha\beta} \right)+ 2 g^{\alpha\beta} \left(F_2(\Box) R^{\mu\nu} \right)_{;\mu;\nu} - 2 \Omega^{\alpha\beta}_2 + g^{\alpha\beta} \left( \Omega^\sigma_{2\sigma} + \bar{\Omega}_2 \right) - 4 \Delta^{\alpha\beta}_2 \\

&\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^{\alpha\beta}_3 + g^{\alpha\beta} \left(\Omega^\gamma_{3\gamma} + \bar{\Omega}_3 \right) - 8 \Delta^{\alpha\beta}_3

\end{align}

where

:$\backslash begin\{align\}$

\Omega^{\alpha\beta}_1 &= \sum^\infty_{n=1} f_{1_n} \sum^{n-1}_{m=0} \nabla^\alpha \Box^m R \nabla^\beta \Box^{n-m-1} R, \\

\bar{\Omega}_1 &= \sum^\infty_{n=1} f_{1_n} \sum^{n-1}_{m=0} \Box^m R \Box^{n-m} R, \\

\Omega^{\alpha\beta}_2 &= \sum^\infty_{n=1} f_{1_n} \sum^{n-1}_{m=0} \nabla^\alpha \Box^m {R^\mu}_\nu \nabla^\beta \Box^{n-m-1} {R^\nu}_\mu, \\

\bar{\Omega}_2 &= \sum^\infty_{n=1} f_{1_n} \sum^{n-1}_{m=0} \Box^m {R^\mu}_\nu \Box^{n-m} {R^\nu}_\mu, \\

\Delta^{\alpha\beta}_2 &= \frac{1}{2} \sum^\infty_{n=1} f_{2_n} \sum^{n-1}_{\ell=0} \nabla_\nu \left[ \Box^\ell {R^\nu}_\sigma \nabla^{(\alpha}

\Box^{n-\ell-1} R^{\beta)\sigma} - \Box^\ell \nabla^{(\alpha} {R^nu}_\sigma \Box^{n-\ell-1} R^{\beta)\sigma} \right], \\

\Omega^{\alpha\beta}_3 &= \sum^\infty_{n=1} f_{3_n} \sum^{n-1}_{\ell=0} \nabla^\alpha \Box^\ell {C^\mu}_{\nu\lambda\sigma} \nabla^\beta \Box^{n-\ell-1}

{C_\mu}^{\nu\lambda\sigma}, \\

\bar{\Omega}_3 &= \sum^\infty_{n=1} f_{3_n} \sum^{n-1}_{\ell=0} \Box^\ell {C^\mu}_{\nu\lambda\sigma} \Box^{n-\ell} {C_\mu}^{\nu\lambda\sigma},\\

\Delta^{\alpha\beta}_3 &= \frac{1}{2} \sum^\infty_{n=1} f_{3_n} \sum^{n-1}_{\ell=0} \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^{\lambda\nu}_{\sigma\mu}

{C_\lambda}^{\beta)\sigma\mu} \right].

\end{align}

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Category:Theories of gravity

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