Lecture 1: Introduction to gravitational-wave science

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 We begin our journey with a broad overview of the science of Gravitational Waves (GWs). In this lecture, we will address fundamental questions such as, "What are gravitational waves?" and "What astrophysical events in the Universe can produce them?" We will also review the key discoveries of the past decade, following the first direct detection of GWs. This first lecture begins with a brief overview of the limitations of Newtonian gravity and its successor, Einstein's General Theory of Relativity. We then introduce the astrophysical sources of gravitational waves and trace the key historical developments leading to their discovery.

Lecture 2: Introduction to gravitational-wave science (II)

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 In this second lecture, we introduce the basic principles of gravitational wave detection, the characteristic waveforms produced by compact binary systems, and their important physical properties. We conclude the lecture with a discussion on the future prospects of Gravitational-Wave Astronomy and its potential to revolutionize our understanding of the cosmos.

Lecture 3: Basic review on special and general relativity

• Gabriel Luz Almeida (Shing-Tung Yau center of the Southeast University)

Room: 逸夫建筑馆1502 In the first lecture this Saturday, we will review the elementary tools of Special and General Relativity that are essential for introducing Linearized Gravity. We will begin by covering the foundational principles of both theories, then reviewing basic definitions of their geometric descriptions, before moving on to the basics of these theories' symmetry groups.

Lecture 4: Introduction to the linearized theory of gravity

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 In this second lecture, we will study the Linearized Theory of Gravity, starting from the expansion of the Einstein field equations. As we will see, the symmetry group of General Relativity plays a crucial role. In particular, by exploring its properties, we will arrive at a wave equation that gives rise to the notion of Gravitational Waves. This lecture covers the textbook's §1.1.

Lecture 5: The transverse-traceless gauge

• Gabriel Luz Almeida (Shing-Tung Yau center of the Southeast University)

Room: 逸夫建筑馆1502 In this lecture, we will study the wave equation in a vacuum. We will show how a suitable gauge choice allows us to write the wave solutions explicitly in terms of only two physical modes. As we will see, this is accomplished by adopting the so-called TT (transverse-traceless) gauge. This lecture covers the textbook's §1.2.

Lecture 6: Geodesic equation and geodesic deviation

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 This lecture will prepare us for next week's topic—the interaction of gravitational waves with test masses. To that end, we will study the geodesic equation and the equation of geodesic deviation in detail, in particular, going through the derivation of the latter starting from the former. These two equations are important to understand how one can detect the passage of a GW. This lecture covers the textbook's §1.3.1.

Lecture 7: Interaction of GWs with test masses - The TT frame

• Gabriel Luz Almeida (Shing-Tung Yau center of the Southeast University)

Room: 逸夫建筑馆1502 In this lecture, we will cover how GWs interact with test masses, focusing on the specific frame of the TT gauge. In this part, we will learn how to understand physically the coordinate system related to the TT gauge. Today's lectures cover the textbook's §1.3.

Lecture 8: Interaction of GWs with test masses - The detector frame

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 In this lecture, we will cover how GWs interact with test masses, focusing on another relevant frame, called the detector frame. Similarly to the previous lecture, we will understand in physical terms what does this reference frame corresponds to. With this, we finish the textbook's §1.3.

Lecture 9: Separation of scales on generic curved background

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 Today's pair of lectures will be devoted to the study and derivation of the energy-momentum tensor of GWs based on the geometrical approach. In particular, in this first part, corresponding to the ninth lecture, we will introduce the concept of separation of scales, which will be important to understand how, in general, GWs can be meaninfully defined on curved backgrounds.

Lecture 10: The energy-momentum tensor of GWs

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 For the second part, lecture number ten, we will separate the Einstein's equation into its low- and a high-frequency parts. As we will see, this separation is grounded on the separation of scales introduced in the previous lecture, each having a deep physical meaning. In particular, we will be able to explicitly derive the energy-momentum tensor of GWs from the low-frequency part.

Lecture 11: The energy flux of GWs

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 Having derived the energy-momentum tensor carried by the GWs in the previous lecture, today we will obtain important relations out of it. In particular, we will derive the energy flux, showing that it can be written in terms of the two physical modes of the GW, i.e., the plus and cross polarizations. Also in this lecture, we will discuss to what extent an expansion around flat spacetime makes sense.

Lecture 12: A brief introduction to the Mathematica package xAct

• Gabriel Luz Almeida (Shing-Tung Yau center, Southeast University)

Room: 逸夫建筑馆1502 During today's second lecture, we will introduce xAct, a Mathematica package for efficient tensor calculation. Our focus on this brief introduction will be the derivation of important formulae used along the first chapter of our textbook. In particular, used in the derivation of the energy-momentum tensor of GWs, we will derive the expression for the Ricci tensor up to second order in the perturbation.

Lecture 13: A brief review of the classical field theory

• Hongbin Wang (Southeast university)

Room: 逸夫建筑馆1502 In the first lecture, I would like to give a brief review of the classical field theory, including the derivation of Euler-Lagrange equation of fields, symmetry transformation and Noether's theorem in a general form. After that we will discuss a particular symmetry transformation, the translation under which we will get the expression of the energy-momentum tensor as the conservative current.

Lecture 14: The energy-momentum tensor

• Hongbin Wang (Southeast university)

Room: 逸夫建筑馆1502 In this lecture, I will derive the expression of energy-momentum tensor of the classical electromagnetic field and the gravitational field, from the field-theoretical point of view. We will talk about the electromagnetic field as a good example to understand the basic concepts and techniques in field theory, and then derive the expression of energy-momentum tensor which should be consistent with the one derived in chapter 1. Finally we will discuss some difference between the electromagnetic field and the gravitational field.

Lecture: 15

• Hongbin Wang (Southeast university)

Room: 1502 In this lecture, we derived the angular momentum expression for gravitational waves from the Noether's theorem, by the symmetry of the action under rotations in three dimensions.

Lecture: 16

• Hongbin Wang (Southeast university)

Room: 1502 In this lecture, we talked about why the gravitational field is the spin-2 fields. We examine different cases, from spin-0, spin-1 and spin more than 3. We found all of these cases except spin-2 are ruled out by some physical prediction contradictory to the facts.

Lecture 17

• Hongbin Wang (Southeast university)

Room: 1502 In this lecture, we derived the Pauli-Fierz action, which is gauge-invariant and leads to the correct equation for linear order of gravitional field.

Lecture 18

• Hongbin Wang (Southeast university)

Room: 1502 From the action leading to the linear equation of motion, we derived the propagator of gravitons in De Donder gauge. From the propagator we derived the expression of the gravitional interaction between two static point particles, which showed the consistency with Newton's gravity.

Lecture 19

• Hongbin Wang (Southeast university)

Room: 1502 In this lecture, we explained the nonlinearity of gravity, and how construct higher-order terms in the action. From the dimensions, we saw how to determine the higher-order interacting terms in the action.

Lecture 20

• Hongbin Wang (Southeast university)

Room: 1502 In the low-energy case, we saw we need only consider the leading terms in the expansion, from which point of view we explained the general relativity as a totally adequate low-energy field theory.

Lecture 21

• Hongbin Wang (Southeast university)

Room: 1502 In this lecture, we considered the theory of massive gravitons. From some basic observations, we could determine the range of mass of gravitons.

Lecture 22

• Hongbin Wang (Southeast university)

Room: 1502 By analogy with the massive electromagnetic field, we considered the massive gravitational field, deriving the propagators of massive gravitons. From some contradiction, we haven't established a "correct" massive theory of graviton.