Introduction to modified gravity

Quantum Gravity

General lectures on quantum gravity.

An elementary introduction to loop quantum gravity

An introduction to loop quantum gravity is given, focussing on the fundamental aspects of the theory, different approaches to the dynamics, as well as possible future directions. It is structured in five lectures, including exercises, and requires only little prior knowledge of quantum mechanics, gauge theory, and general relativity. The main aim of these lectures is to provide non-experts with an elementary understanding of loop quantum gravity and to evaluate the state of the art of the field. Technical details are avoided wherever possible.

贝纳苏克关于高斯玻色子系统和类引力的讲座

Benasque Lectures on Gaussian Bosonic Systems and Analogue Gravity

本文介绍了高斯框架用于描述线性玻色子系统，并探讨其在半经典黑洞和类引力中的应用。

Instabilities in field theories: Lecture notes with a view into modified gravity

Modified theories of gravity usually present new degrees of freedom, as well as higher order derivatives, wrong signs in certain terms and complicated couplings already present in the Lagrangian from the beginning or originated by the field redefinitions needed to reach an Einstein frame. As a consequence, they are very prone to present dynamical instabilities that could spoil any attempt to construct viable models within these frameworks. In these three lectures we introduce the most common types of instabilities that appear in field theory as well as some techniques to detect them, and suppl…

Modave讲义：量子场论和半经典引力中的能量条件

Modave lectures on energy conditions in quantum field theory and semi-classical gravity

本文回顾了经典能量条件及其对引力解的影响，包括霍金和彭罗斯奇点定理，然后讨论了量子场与引力耦合时能量密度约束的方向，包括时空区域的平均和能量与量子信息的关系，探讨了这些约束对量子场论与引力的有效理论的影响，以及如何指导对量子引力的理解。

环量子引力讲义。LN2：初值问题与预量子态

Lecture Notes in Loop Quantum Gravity. LN2: Cauchy problems and pre-quantum states

本文探讨了环量子引力中初值问题与预量子态的建立，通过讨论主符号和初值气泡来解决相对论模型的确定性问题。

关于自旋泡沫路径积分的莱斯胡什讲座

Les Houches lectures on Spinfoam Path Integrals

该文档是2025年莱斯胡什圈量子引力学校的讲义，对量子引力路径积分的自旋泡沫框架基础进行教学性回顾，从二维BF理论引入自旋泡沫路径积分，逐步构建到三维量子引力及四维广义相对论的量子化。

Gravity & Hydrodynamics: Lectures on the fluid-gravity correspondence

We discuss recent developments in the hydrodynamic description of strongly coupled conformal field theories using the AdS/CFT correspondence. In particular, we review aspects of the fluid-gravity correspondence which provides a map between a class of inhomogeneous, dynamical, black hole solutions in asymptotically AdS spacetimes and arbitrary fluid flows in the strongly interacting boundary field theory. We explain how the geometric duals to the fluid dynamics are constructed in a boundary derivative expansion and use the construction to extract the hydrodynamic transport coefficients. In addi…

洛斯霍ches关于二维引力与全息的讲义

Les Houches lectures on two-dimensional gravity and holography

本文介绍了二维引力与全息的理论方法，探讨了引力路径积分技术及在黑洞研究中的应用。

Lectures on Screened Modified Gravity

The acceleration of the expansion of the Universe has led to the construction of Dark Energy models where a light scalar field may have a range reaching up to cosmological scales. Screening mechanisms allow these models to evade the tight gravitational tests in the solar system and the laboratory. I will briefly review some of the salient features of screened modified gravity models of the chameleon, dilaton or symmetron types using $f(R)$ gravity as a template.

Two lectures on Two-Dimensional Gravity

(from the talk:) I shall here speak on gravity in (1+1)-dimensional space-time --- lineal gravity. The purpose of studying lower dimensional theories, and specifically lower dimensional gravity, is to gain insight into difficult conceptional issues, which are present and even more opaque in the physical (3+1)-dimensional world. Perhaps lessons learned in the lower-dimensional setting can be used to explicate physical problems. Moreover, if we are lucky, the lower-dimensional theories can have a direct physical relevance to modelling phenomena that is actually dynamically confined to the lower …

Lecture on Gauge Gravitation Theory. Gravity as a Higgs Field

Gravitation theory is formulated as gauge theory on natural bundles with spontaneous symmetry breaking where gauge symmetries are general covariant transformations, gauge fields are general linear connections, and Higgs fields are pseudo-Riemannian metrics.

Lectures on Gravity and Entanglement

The AdS/CFT correspondence provides quantum theories of gravity in which spacetime and gravitational physics emerge from ordinary non-gravitational quantum systems with many degrees of freedom. Recent work in this context has uncovered fascinating connections between quantum information theory and quantum gravity, suggesting that spacetime geometry is directly related to the entanglement structure of the underlying quantum mechanical degrees of freedom and that aspects of spacetime dynamics (gravitation) can be understood from basic quantum information theoretic constraints. In these notes, we…

Lectures on the Infrared Structure of Gravity and Gauge Theory

This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at https://goo.gl/3DJdOr. Please send typos or corrections to strominger@physics.harvard.edu.

量子引力讲座

Lectures in Quantum Gravity

本讲义集基于Nordita博士学校课程，系统介绍量子引力从微扰理论、有效场论到渐近安全引力和弦理论，并讨论黑洞时空中的量子效应，旨在提供连贯的研究图景。

Loop Representations

The loop representation plays an important role in canonical quantum gravity because loop variables allow a natural treatment of the constraints. In these lectures we give an elementary introduction to (i) the relevant history of loops in knot theory and gauge theory, (ii) the loop representation of Maxwell theory, and (iii) the loop representation of canonical quantum gravity. (Based on lectures given at the 117. Heraeus Seminar, Bad Honnef, Sept. 1993)

Three-dimensional gravity and instability of $\text{AdS}_{3}$

This is an extended version of my lecture at the LIII Cracow School of Theoretical Physics in Zakopane in which I presented the results of joint work with Piotr Bizoń concerning (in)stability of the three-dimensional anti-de Sitter spacetime.

Advanced Lectures on General Relativity

These lecture notes are intended for starting PhD students in theoretical physics who have a working knowledge of General Relativity. The 4 topics covered are (1) Surface charges as conserved quantities in theories of gravity; (2) Classical and holographic features of three-dimensional Einstein gravity; (3) Asymptotically flat spacetimes in 4 dimensions: BMS group and memory effects; (4) The Kerr black hole: properties at extremality and quasi-normal mode ringing. Each topic starts with historical foundations and points to a few modern research directions.

Thermodynamical aspects of black holes in modified gravity

We discuss some thermodynamical definitions for black holes in modified theories of gravity.

Dust-shell Universe in the modified gravity scenario

The dynamics of the dust-shell model of universe is exactly solved for the modified Schwarzschild solution. This solution is used to derive the cosmology corresponding to the modified gravity.

Lecture on Quantum Gravity with Perimeter Action and Gravitational Singularities

Motivated by quantum-mechanical considerations we earlier suggested an alternative action for discretised quantum gravity which measures the perimeter of the space-time and has a dimension of length. It is the so called perimeter action, since it is a "square root" of the area action in gravity and has a new constant of dimension one in front. The physical reason to introduce the perimeter/linear action was to suppress singular configurations "spikes" in the quantum-mechanical integral over geometries. Here we shall consider the continuous limit of the discretised perimeter/linear action. We s…

Lectures on LQG/LQC

A School on Loop Quantum Gravity was held at the IMSc during Sept 8 -- 18, 2009. In the first week a basic introduction to LQG was provided while in the second week the focus was on the two main application, to cosmology (LQC) and to the black hole entropy. These notes are an expanded written account of the lectures that I gave. These are primarily meant for beginning researchers.

Modified gravity from the quantum part of the metric

It is shown that if a metric in quantum gravity can be decomposed as a sum of classical and quantum parts then Einstein quantum gravity looks approximately like modified gravity with a nonminimal interaction between gravity and matter.

Gif Lectures on Cosmic Acceleration

These lecture notes cover some of the theoretical topics associated with cosmic acceleration. Plausible explanations to cosmic acceleration include dark energy, modified gravity and a violation of the Copernican principle. Each of these possibilities are briefly described.

The fluid/gravity correspondence: Lectures notes from the 2008 Summer School on Particles, Fields, and Strings

This is a paper compiled by students of the 2008 Summer School on Particles, Fields, and Strings held at the University of British Columbia on lectures given by Veronika Hubeny as understood and interpreted by the authors. We start with an introduction to the AdS/CFT duality. More specifically, we discuss the correspondence between relativistic, conformal hydrodynamics and Einstein's theory of gravity. Within our framework the Einstein equations are an effective description for the string theory in the bulk of AdS_5 spacetime and the hydrodynamic fluid equations represent the conformal field t…

Stability of the Einstein static universe in modified theories of gravity

We present a brief overview of the stability analysis of the Einstein static universe in various modified theories of gravity, like f(R) gravity, Gauss-Bonnet or f(G) gravity, and Horava-Lifshitz gravity.

圈量子引力论导论：Rovelli的LQG讲义

Introduction to Loop Quantum Gravity: Rovelli's lectures on LQG

本文是Carlo Rovelli于2018年在马赛举行的圈量子引力论（LQG）讲座的LaTeX转录笔记，按原讲座顺序整理并补充了图表与方程，旨在为视频版本提供辅助学习材料。

Three lectures on Poincare gauge theory

In these lectures we review the basic structure of Poincare gauge theory of gravity, with emphasis on its fundamental principles and geometric interpretation. A specific limit of this theory, defined by the teleparallel geometry of spacetime, is discussed as a viable alternative for the description of macroscopic gravitational phenomena.

Canonical Quantum Gravity and the Problem of Time

This is the write-up of my lectures at the NATO Summer School held in Salamanca in June 1992. The paper deals with the problem of time in quantum gravity. All the major schemes are reviewed. Please note that the paper is in two parts for ease of email transmission; this is part 1. The mailer from gr-qc may further subdivide these two sections.

Horizon Entropy in Modified Gravity

We present an observation about the proposal that four-dimensional modification of general relativity may explain the observed cosmic acceleration today. Assuming that the thermodynamical nature of gravity theory continues to hold in modified gravity theories, we derive the modified horizon entropy formula from the modified Friedmann equation. We argue that our results imply that there are conceptual problems in some models of four-dimensional modification of general relativity.

Connecting noncommutative geometry to f(R) modified gravity

It is shown in this note that a noncommutative-geometry background determines the modified-gravity function $f(R)$ for modeling dark matter.

Introduction to Loop Quantum Gravity

This article is based on the opening lecture at the third quantum geometry and quantum gravity school sponsored by the European Science Foundation and held at Zakopane, Poland in March 2011. The goal of the lecture was to present a broad perspective on loop quantum gravity for young researchers. The first part is addressed to beginning students and the second to young researchers who are already working in quantum gravity.

Reconstruction of Einstein-Aether Gravity from other Modified Gravity Models

We briefly describe the modified Friedmann equations for Einstein-Aether gravity theory and we find the effective density and pressure. The purpose of our present work is to reconstruction of Einstein-Aether Gravity from other modified gravities like $f(T)$, $f(R)$, $f(G)$, $f(R,T)$ and $f(R,G)$ and check its viability. The scale factor is chosen in power law form. The free function $F(K)$ for Einstein-Aether gravity (where $K$ is proportional to $H^{2}$) have been found in terms for $K$ by the correspondence between Einstein-Aether gravity and other modified gravities and the nature of $F(K)$…

From Big to Little Rip in modified F(R,G) gravity

We discuss the cosmological reconstruction in modified Gauss-Bonnet (GB) gravity. It is demonstrated that the modified GB gravity may describe the most interesting features of late-time cosmology. We derive explicit form of effective phantom cosmological models ending by the finite-time future singularity (Big Rip) and without singularities in the future (Little Rip).

String-Inspired Chern-Simons Modified Gravity In 4-Dimensions

Chern-Simons modified gravity models in 4-dimensions are shown to be special cases of low energy effective string models to first order in the string constant.

Stability in Generalized Modified Gravity

The stability issue of generalized modified gravitational models is discussed with particular emphasis to de Sitter solutions. Two approaches are briefly presented.

Restrictions on curved cosmologies in modified gravity from metric considerations

This study uses very simple symmetry and consistency considerations to put constraints on possible Friedmann equations for modified gravity models in curved spaces. As an example, it is applied to loop quantum cosmology.

Lectures on Loop Quantum Gravity

Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past fifteen years to a mathematically rigorous candidate quantum field theory of the gravitational field. The features that distinguish it from other quantum gravity theories are 1) background independence and 2) minimality of structures. Background independence means that this is a non-perturbative approach in which one does not perturb around a given, distinguished, classical background metric, rather arbitrary fluctuations are allowed, thus precisely encoding the quantum version of Einstein's radi…

Accelerating cosmology in modified gravity with scalar field

The modified gravity with 1/R term (R being scalar curvature) and the Einstein-Hilbert term is studied by incorporating the phantom scalar field. A number of cosmological solutions are derived in the presence of the phantom field in the perfect fluid background. It is shown the current inflation obtained in the modified gravity is affected by the existence the phantom field.

Solar System experiments do not yet veto modified gravity models

The dynamical equivalence between modified and scalar-tensor gravity theories is revisited and it is concluded that it breaks down in the limit to general relativity. A gauge-independent analysis of cosmological perturbations in both classes of theories lends independent support to this conclusion. As a consequence, the PPN formalism of scalar-tensor gravity and Solar System experiments do not veto modified gravity, as previously thought.

Modified Newton's gravity in Finsler Space as a possible alternative to dark matter hypothesis

A modified Newton's gravity is obtained as the weak field approximation of the Einstein's equation in Finsler space. It is found that a specified Finsler structure makes the modified Newton's gravity equivalent to the modified Newtonian dynamics (MOND). In the framework of Finsler geometry, the flat rotation curves of spiral galaxies can be deduced naturally without invoking dark matter.

A simple argument for dark matter as an effect of slightly modified gravity

This note presents a simple argument showing that dark matter is an effect of $f(R)$ gravity based on the definition of slightly modified gravitational theories previously proposed by the author.

Exact Spherically Symmetric Solutions in Modified Teleparallel gravity

Finding spherically symmetric exact solutions in modified gravity is usually a difficult task. In this paper we use the Noether's symmetry approach for a modified Teleparallel theory of gravity labelled as $f(T,B)$ gravity where $T$ is the scalar torsion and $B$ the boundary term. Using the Noether's theorem, we were able to find exact spherically symmetric solutions for different forms of the function $f(T,B)$ coming from the Noether's symmetries.

Fermi gas and modified gravity

We derive the equations of state of Fermi gas by maximizing the Fermi-Dirac entropy in modified gravity in relativistic and non-relativistic case. It will be demonstrated that the microphysics must depend on a given theory of gravity in order to consistently describe a physical system at a statistical equilibrium state.

Matter stability in modified teleparallel gravity

We study the matter stability in modified teleparallel gravity or $f(T)$ theories. We show that there is no Dolgov-Kawasaki instability in these types of modified teleparallel gravity theories. This give the $f(T)$ theories a great advantage over their $f(R)$ counterparts because from the stability point of view there isn't any limit on the form of functions that can be chosen.

A viability criterion for modified gravity with an extra force

A recently proposed theory of modified gravity with an explicit ``anomalous'' coupling of the Ricci curvature to matter is discussed, and an inequality is derived which expresses a necessary and sufficient condition to avoid the notorius Dolgov-Kawasaki instability.

Stability analysis in Modified Non-Local Gravity

In this paper we consider FRW cosmology in modified non-local gravity. The stability analysis shows that there is only one stable critical point for the model and the universe undergoes a quintessence dominated era.

Generalized modified gravity models: the stability issue

A brief introduction on the issue of stability in generalized modified gravity is presented and the dynamical system methods are used in the investigation of the stability of spatially flat homogeneous cosmologies within a large class of generalized modified gravity models in the presence of a relativistic matter-radiation fluid.

Modified gravity with vacuum polarization

A brief review of cosmology in some generalized modified gravity theories with vacuum polarization is presented. Stability question of de Sitter solution is investigated.

修正引力框架下多流体类型宇宙学的二元相互作用

Binary Interaction with Multiple Fluid Type Cosmology Under Modified Gravity Frame

本文在修正引力框架下建立了三种流体通过二元相互作用耦合的多流体宇宙学模型，并引入可变引力常数与可变宇宙学常数，其中修正效应等效于可变宇宙学常数。

Stability in Generalized Modified Gravity

The stability issue of a large class of modified gravitational models is discussed with particular emphasis to de Sitter solutions. Three approaches are briefly presented and the generalization to more general cases is mentioned.

Rotating black hole and entropy for modified theories of gravity

In this paper we obtain the entropy of the Kerr black hole for a number of modified theories of gravity. We show that as long as the deviation from Einstein Hilbert term consists purely of terms involving scalar curvature and Ricci tensor, the entropy is solely given by the area law. However, the area law will be modified by appropriate corrections when we consider terms involving Riemann tensor explicitly.

Modified $f(R)$ Gravity and Thermodynamics of Time-Dependent Wormholes at Event Horizon

In the context of modified $f(R)$ gravity theory, we study time-dependent wormhole spacetimes in the radiation background. In this framework, we attempt to generalize the thermodynamic properties of time-dependent wormholes in $f(R)$ gravity. Finally, at event horizon, the rate of change of total entropy has been discussed.

Test particle motion in modified gravity theories

We derive the equations of motion of an electrically neutral test particle for modified gravity theories in which the covariant divergence of the ordinary matter energy-momentum tensor dose not vanish (i.e. $\nabla_μT^{μν}\neq0$). In fact, we generalize the Mathisson-Papapetrou equations by deriving a general form for the equations of motion of a test particle. Furthermore, using the generalized Mathisson-Papapetrou equations, we investigate the equations of motion of a pole-dipole (spinning) particle in the context of Modified Gravity (MOG).