(Fall, 2012)

General relativity is the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics. General relativity generalises special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. (Source: Wikipedia) This course uses the physics of black holes extensively to develop and illustrate the concepts of general relativity and curved spacetime.

The principle of equivalence of gravity and acceleration, or gravitational and inertial mass is the fundamental basis of general relativity.   This was Einstein's key insight.  Professor Susskind begins the first lecture of the course with Einstein's derivation of this equivalence principle.  He then moves on to the mathematics of general relativity, including generalized coordinate transformations and tensor analysis.  This topic includes the important point that the determination as to whether a spatial geometry is flat (i.e. Euclidean) is equivalent in some respects to the determination of whether an object is in a gravitational field, or merely an accelerated reference frame.

Topics

• The equivalence principle

• Accelerated reference frames

• Curvilinear coordinate transformations

• Effect of gravity on light

• Tidal forces

• Euclidean geometry

• Riemannian geometry

• Metric tensor

• Distance measurement in a curved geometry

• Intrinsic geometry

• Flat spacetime

• Einstein summation convention

• Covariant and contravariant vectors and tensors

References

• Equivalence principle

• Non-inertial reference frame

• Curvilinear coordinates

• Riemannian geometry

• Metric tensor

• Minkowski space

• Einstein notation

• Covariance and contravariance of vectors

Source: http://theoreticalminimum.com/courses/general-relativity/2012/fall

This lecture focuses on the mathematics of tensors, which represent the core concepts of general relativity. Professor Susskind opens the lecture with a brief review the geometries of flat and curved spaces. He then develops the mathematics of covariant and contravariant vectors, their coordinate transformations, and their relationship to tensors. In the second half of the lecture, Professor Susskind defines tensor operations including addition, multiplication, and contraction, and discusses the properties of the metric tensor.

Topics

• Flat space

• Metric tensor

• Scalar and tensor fields

• Tensor analysis

• Tensor mathematics: addition, multiplication, contraction

References

• Scalar field

• Vector field

• Tensor

• Tensor field

Source: http://theoreticalminimum.com/courses/general-relativity/2012/fall

In this lecture, Professor Susskind presents the mathematics required to determine whether a spatial geometry is flat or curved. The method presented is to find a diagnostic quantity which, if zero everywhere, indicates that the space is flat. This method is simpler than evaluating all possible metric tensors to determine whether the space is flat. 

The diagnostic that we are looking for is the curvature tensor. The curvature tensor is computed using covariant derivatives which require the computation of the Christoffel symbols. The Christoffel symbols are computed using the equation for covariant derivative of the metric tensor for Gaussian normal coordinates. We take the second covariant derivative of a vector using two different orders for the indices, and subtract these two derivatives to get the curvature tensor. If the curvature tensor is equal to zero everywhere, the space is flat. Professor Susskind demonstrates the intuitive picture of this computation using a cone, which is a flat two-dimensional space everywhere except at the tip. 

Topics

• Riemannian geometry

• Metric tensor

• Gaussian normal coordinates

• Covariant derivatives

• Christoffel symbols

• Curvature tensor

• Cones

References

• Riemannian geometry

• Covariant derivative

• Christoffel symbols

• Riemann curvature tensor

Professor Susskind begins the lecture with a review of covariant and contravariant vectors and derivatives, and the method for determining whether a space is flat. He then introduces the concept of geodesics, which are the straightest paths between two points in a given space. A geodesic is a path that is locally as straight as possible, which means that the derivative of the tangent vector is equal to zero at every point.

Professor Susskind then moves on to relate the mathematics of Riemannian geometry (which we have been studying so far) to spacetime. Spacetime is represented by Minkowski space, which has a different metric from that of flat Riemannian space in that the coefficient of the time dimension is negative. Minkowski space is the geometry of special relativity.

The rest of the lecture presents uniformly accelerated reference frames and how they transform under special relativity. Professor Susskind shows how uniformly accelerated reference frames produce the same equations of motion as those for a uniform gravitational field, thereby beginning to establish the basis for the equivalence principle which is at the heart of general relativity.

Topics

• Parallel transport

• Tangent vectors

• Geodesics

• Spacetime

• Special relativity

• Uniform acceleration

• Uniform gravitational fields

References

• Parallel transport

• Geodesic

• Spacetime

• Special relativity

In this lecture, Professor Susskind derives the metric for a gravitational field, and introduces the relativistic mathematics that describe a black hole. He begins by reviewing the concept of light cones and space- and time-like intervals from special relativity. He then moves on to review the flat space-time metric and geodesics, and the connection between the mathematics of geodesics and the Lagrangian formulation of classical mechanics. This leads to the mechanics of a particle moving in a gravitational field, and then to the derivation of the metric for a gravitational field, also known as the Schwarzschild metric. These are the fundamental mathematics that show the equivalence of a gravitational field and curved space-time.

The metric for a gravitational field has an undefined value at a particular radius from the center of a gravitating body. Where this radius occurs outside of the body, the body is a black hole, and the radius defines the location of the event horizon. The lecture concludes with an introduction to some of the very strange properties of a black hole, including that, to an outside observer, the velocity of light slows and light rays become stuck at the horizon.

Topics

• Space-like, time-like, and light-like intervals

• Light cone

• Black holes

• Schwarzschild metric

• Event horizon

References

• Light cone

• Black hole

• Schwarzschild metric

• Event horizon

Professor Susskind continues the discussion of black hole physics. He begins by reviewing the Schwarzschild metric, and how it results in the event horizon of a black hole. Light rays can orbit a black hole. Professor Susskind derives the equations of motion for such an orbit using classical mechanics and the conservation of energy and angular momentum. This derivation yields the photon sphere at the orbital radius of a light ray around a black hole.

Professor Susskind then moves on to the physics of the event horizon of a black hole. An in-falling observer experiences nothing unusual at the event horizon, but to an outside observer, it takes an infinite amount of time for the in-falling observer to reach the horizon. The physics of the horizon are analyzed using the hyperbolic coordinates of a uniformly accelerated reference frame. One inside the horizon, in-falling objects cannot avoid the singularity at the center of a black hole because the radial dimension effectively becomes a time dimension and the singularity is a point in the future of every event.

Topics

• Schwarzschild metric

• Schwarzschild Radius

• Black hole event horizon

• Light ray orbiting a black hole

• Photon sphere

• Hyperbolic coordinates

• Black hole singularity

References

• Schwarzschild metric

• Schwarzschild radius

• Event horizon

• Photon sphere

• Gravitational singularity

Professor Susskind continues the in-depth discussion of the physics of black holes. He begins with the Schwarzschild metric and then applies coordinate transformations to demonstrate that spacetime is nearly flat in the vicinity of the event horizon of a large black hole. In other words, nothing special happens at the event horizon for an observer falling towards the black hole.

Professor Susskind then uses spacetime diagrams with hyperbolic coordinates to describe the physics of falling through the event horizon and into the black hole. The students' have many questions about the unusual properties of black holes.

Topics

• Schwarzschild metric

• Event horizon

• Singularity

References

• Event horizon

• Gravitational singularity

Professor Susskind begins the lecture with a review of Kruskal coordinates, and how they apply to the study of black holes.  He then moves on to develop a coordinate system which allows the depiction of all of spacetime on a finite blackboard.  This results in a Penrose diagram for flat spacetime.  The Penrose diagram for black holes leads to an understanding of wormholes, also known as Einstein-Rosen bridges.

Professor Susskind then describes the process of black hole formation through the simplest possible mechanism: an infalling sphere of radiation.  This process is studied by marrying a Penrose diagram for the flat spacetime inside the sphere, with a Penrose diagram for the black hole under formation outside the sphere of radiation.  The boundary between the two diagrams is the radiation sphere itself, and this approach demonstrates how the black hole horizon develops and begins to expand even before the black hole itself forms.

Topics

• Kruskal–Szekeres coordinates

• Penrose diagrams

• Wormholes

• Formation of a black hole

• Newton's shell theorem

References

• Kruskal–Szekeres coordinates

• Penrose diagram

• Wormhole

• Newton's shell theorem

Professor Susskind derives the Einstein field equations of general relativity.  Beginning with Newtonian gravitational fields, an analogy with the four-current, and the continuity equation, he develops the stress-energy tensor (also known as the energy momentum tensor).  Putting these concepts together and generalizing the Newtonian field equation leads to the definition of the Ricci tensor, the Einstein tensor, and ultimately the Einstein field equations.  These equations equate curvature of spacetime as expressed by the Einstein tensor, with the energy and momentum within that spacetime as expressed by the stress–energy tensor.

Topics

• Newtonian gravitational field

• Continuity equation

• Stress–energy tensor (also known as the energy-momentum tensor)

• Curvature scalar

• Ricci tensor

• Einstein tensor

• Einstein field equations

References

• Gravitational field

• Continuity equation

• Stress–energy tensor

• Curvature scalar

• Ricci curvature

• Einstein tensor

• Einstein field equations

Professor Susskind demonstrates how Einsteins's equations can be linearized in the approximation of a weak gravitational field. The linearized equation is a wave equation, and the solution to these equations create the theory of gravitational radiation and gravity waves. Gravity waves represent waves in the curvature of spacetime and thus are effectively tidal forces that change over time. Gravity waves propagate at the speed of light. A rotating binary pulsar is the most likely source of detectable gravity waves.

Professor Susskind closes the final lecture of the course by developing the Einstein-Hilbert action for general relativity, and discussing how minimizing this action leads to the Einstein field equations.

Topics

• Weak gravitational fields

• Gravitational radiation

• Gravity waves

• Einstein-Hilbert action for general relativity

References

• Gravitational wave

• Einstein–Hilbert action