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### Twentyfifth School: Heigenbrücken, 2. – 13.09.2019

funded by the Wilhelm and Else Heraeus Foundation

### Lecture Program:

E. Bjerrum-Bohr(Copenhagen) |
General relativity from scattering amplitudes |

C. de Rham(London) |
Gravity at large distances |

H. Godazgar(Potsdam) |
Asymptotic symmetries, charges and soft theorems |

D. Giulini(Hannover & Bremen) |
Hamiltonian general relativity |

M. Perry(Cambridge) |
Asymptotic symmetries in gravity |

The instruction language is English.

Some lecture notes may appear later and will be made available here

### Recommended Literature: (to be provided)

**Bjerrum-Bohr**:

Lecture 1 General Relativity as an Effective Field Theory

1. R. P. Feynman, Lectures on Gravitation

2. J. F. Donoghue, “Introduction to the effective field theory description of gravity,” gr-qc/9512024.

3. N. E. J. Bjerrum-Bohr, “Quantum gravity, effective fields and string theory,” hep-th/0410097.

4. B. R. Holstein, “Graviton Physics,” Am. J. Phys. 74, 1002 (2006) [gr-qc/0607045].

Lecture 2 Modern Methods for Particle Scattering

1. L. J. Dixon, “Calculating Scattering Amplitudes Efficiently,” In *Boulder 1995, QCD and beyond* 539-582 [hep-ph/9601359].

2. L. J. Dixon, “A brief introduction to modern amplitude methods,” arXiv:1310.5353 [hep-ph].

Lecture 3 On-Shell Gravity Amplitude Computations

1. H. Kawai, D. C. Lewellen and S. H. H. Tye, “A Relation Between Tree Amplitudes of Closed and Open Strings,” Nucl. Phys. B 269 (1986) 1.

2. N. E. J. Bjerrum-Bohr, P. H. Damgaard, T. Sondergaard and P. Vanhove, “The Momentum Kernel of Gauge and Gravity Theories,” JHEP 1101 (2011) 001 [arXiv:1010.3933 [hep-th]].

3. N. E. J. Bjerrum-Bohr, J. F. Donoghue and P. Vanhove, “On-Shell Techniques and Universal Results in Quantum Gravity,” JHEP 1402 (2014) 111 [arXiv:1309.0804 [hep-th]].

Lecture 4 General Relativity from Amplitudes

1. N. E. J. Bjerrum-Bohr, P. H. Damgaard, G. Festuccia, L. Planté and P. Vanhove, “Gen
eral Relativity from Scattering Amplitudes,” Phys. Rev. Lett. 121, no. 17, 171601 (2018)
[arXiv:1806.04920 [hep-th]].

2. Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon and M. Zeng, “Scattering Ampitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order” Phys. Rev. Lett. 122,
no. 20, 201603 (2019) [arXiv:1901.04424 [hep-th]].

3. A. Cristofoli, N. E. J. Bjerrum-Bohr, P. H. Damgaard and P. Vanhove, ``On Post-Minkowskian Hamiltonians in General Relativity,” arXiv:1906.01579 [hep-th].

**de Rham:**

“The Cosmological Constant Problem: Why it’s hard to get Dark Energy

from Micro-physics”, Burgess, arXiv:1309.4133

“Massive Gravity”, Living Rev.Rel. 17 (2014) 7, arXiv:1401.4173

Graviton Mass Bounds ” target=”_blank” rel=”noopener”>http://inspirehep.net/record/1472784>,

Rev.Mod.Phys. 89 (2017) no.2, 025004,

arXiv:1606.08462

The gravitational rainbow beyond Einstein gravity, Int.J.Mod.Phys.D28 (2019) no.05, 1942003,

DOI:10.1142/S0218271819420033<https://doi.org/10.1142/S0218271819420033>

Exercises for the lecture are found here:

https://www.dropbox.com/s/f8xmo3rng0997rj/Questions_Set1.pdf?dl=0

https://www.dropbox.com/s/jfl7y6movxxm4wg/Questions_Set2.pdf?dl=0

**Godazgar**:

–Introductory reading: General relativity, Wald: chapter 11 on asymptotic flatness

References on what the course will cover:

–Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory; arXiv:1703.05448

These are lectures on the broad theme of the lectures

— Compere and Fiorucci, Advanced Lectures on General Relativity; arXiv:1801.07064

Lectures on charges and how they are constructed.

–Ashtekar, Asymptotic quantization 1987

Lectures on asymptotic flatness and construction of phase space

–Geroch, Asymptotic structure of spacetime, 1977

More mathematical treatment of asymptotic flatness

Papers for more advanced reading:

—Barnich and Troessaert; arxiv:1106.0213

BMS charges and algebra are constructed

He, Lysov, Mitra, Strominger; arxiv:1401.7026

In these papers the phase space for gravitons is constructed and the BMS symmetry related to the soft graviton theorem.

— Godazgar, Godazgar, Pope; arxiv:1812.01641

Godazgar, Godazgar, Pope; arxiv:1812.06935

Dual BMS charges are found and extended

Godazgar, Godazgar, Pope; arxiv:1908.01164

Phase space for gravitational scattering is constructed without requiring boundary conditions at spacelike infinity leading to more general soft theorem.

**D. Giulini:** “Dynamical and Hamiltonian Formulation of General Relativity”

**Perry**: