Einstein's General Theory of Relativity | Lecture 8

TL;DR

Curvature is a commutator of two operators: carry a vector around a small closed loop and it does not quite return to itself, and that failure is computed as a commutator. The covariant derivative combines an ordinary derivative operator with matrix multiplication by the Christoffel symbol gamma, treated as four matrices, one per coordinate direction.

Transcript

this program is brought to you by Stanford University please visit us at stanford.edu we talked a little bit about curvature last time curvature these are hard concepts they're not easy and partly they require in order to do them in an elegant way requires a degree of notational notation that's a little bit sophisticated I told you what the answer ... Read More

Key Insights

• Curvature is detected by transporting a vector around a small closed loop: the vector does not quite come back to itself, and that discrepancy is an indication of curvature that can be expressed as a commutator of two operators.
• A linear operator is an operation applied to a function to produce another function, and the space it acts on can be a space of functions, as seen when a derivative operates on functions in quantum mechanics.
• Three example operators act on a vector field V-alpha of X: the derivative with respect to a coordinate X-mu, multiplication by a function f of X, and multiplication by a numerical matrix M whose entries M-alpha-beta multiply V-beta summed over beta.
• Operators can be combined so that a matrix depending on position multiplies V-beta of X, giving an operator that is simultaneously a matrix and a function of X rather than just one or the other.
• The covariant derivative del-mu is defined as the ordinary derivative of V-alpha with respect to X-mu plus the Christoffel term, making it an operator applied to a vector field just like the ordinary derivative but more interesting.
• The Christoffel symbol gamma-mu should be pictured as a set of four matrices, one for each direction mu, where alpha and beta are the matrix indices and mu selects which matrix applies to the vector.
• In shorthand, del-mu equals the derivative with respect to X-mu plus gamma-mu, meaning you differentiate the vector and then add gamma-mu, a position-dependent matrix, applied to that same vector.
• The commutator of two operators A and B is AB minus BA, where AB means first act with B on the object, then act with A on the result, and this ordering matters because the operators generally do not commute.

Questions & Answers

Q: How do you detect curvature using a vector?

You carry a vector around a small closed loop and check whether it returns to itself. If the vector does not quite come back to its original state after traversing the loop and returning to the origin, that discrepancy is an indication of curvature. Susskind explains that this process of going around a small closed loop and asking how something changes when you come back to the origin is, in effect, calculating a commutator of two operators, which gives the fast way to derive the curvature formula.

Q: What is a linear operator in this context?

A linear operator is a particular kind of operation you can perform on a function to get another function. The vector space it acts on can itself be a space of functions, as commonly seen in quantum mechanics where a derivative is a linear operator acting on functions. In the lecture, operators act on V-alpha of X, a collection of functions corresponding to the components of a vector field. Examples include the derivative with respect to a coordinate, multiplication by a function, and multiplication by a matrix.

Q: What are the three example operators given in the lecture?

The first is the derivative with respect to any coordinate X-mu, which differentiates all four components of a vector. The second is multiplication by a function f of X, which simply multiplies the starting function by that function. The third is multiplication by a numerical matrix M, where the alpha component is obtained from M-alpha-beta times V-beta summed over beta. These can be combined into a matrix that itself depends on position, giving an operator that is both a matrix and a function of X.

Q: How is the covariant derivative defined?

The covariant derivative del-mu with respect to coordinate X-mu, applied to a vector with components V-alpha, is by construction the ordinary derivative of V-alpha with respect to X-mu plus the Christoffel term gamma. It is an operator applied to a vector field, just like the ordinary derivative but more interesting. It combines a differential derivative operator with matrix multiplication, and gamma-mu is a numerical matrix that typically depends on position rather than being purely numerical.

Q: Why does the lecture describe gamma as a set of four matrices?

The Christoffel symbol gamma-mu-alpha-beta has three indices, and Susskind asks you to imagine it as a set of four matrices where alpha and beta are the matrix entries and mu tells you which of the four matrices you are using. There is one matrix for each direction mu. In four-dimensional space-time there are four such matrices, and the number would differ for a space of different dimension. This picture keeps two hidden matrix indices in mind while gamma-mu also remains a function of position X.

Q: What does the shorthand del-mu equals derivative plus gamma-mu mean?

It is a notational device summarizing the full covariant derivative expression. When del-mu acts on a vector, you differentiate the vector with respect to X-mu and then add the matrix gamma-mu applied to that vector. The gamma-mu here is the same set of four matrices, carrying two hidden indices that act as matrix indices on the vector. Although the shorthand looks trivial, it means exactly the fuller written-out definition when applied to an actual vector.

Q: What is a commutator of two operators?

Given two operators A and B, which may contain derivatives, matrix multiplications, or matrix functions of position, the product AB means you first act with B on whatever object is being operated on, producing a new function, and then act with A on that result. The commutator is the difference AB minus BA. Because the order of operations generally changes the outcome, this difference is not zero, and it is the quantity used to compute curvature.

Q: What is the commutator of a derivative with a function f of X?

The commutator of the derivative with respect to X and a function f of X equals df by dX, though Susskind notes the sign still needs to be worked out. Here f of X acts as an operator by multiplication, and both terms act on some function V of X. The first term d by dX applied to f times V means multiply by f then differentiate the whole product, while the second term f times d by dX multiplies after differentiating, and their difference isolates the derivative of f.

Summary

In this video, the concept of curvature and the derivation of the curvature formula are discussed. The curvature of a vector is represented by the commutator of two covariant derivatives. The Riemann tensor is introduced as the expression for the curvature, and it is shown how to calculate the Riemann tensor using the formulas for covariant derivatives and the Christoffel symbols.

Questions & Answers

Q: What is the curvature of a vector and how is it represented mathematically?

The curvature of a vector is a measure of how the vector changes when going around a closed loop. Mathematically, it is represented by the commutator of two covariant derivatives.

Q: What is a covariant derivative and how does it relate to curvature?

A covariant derivative is an operator applied to a vector that takes into account the curvature of the space. It combines derivative operations and matrix multiplication. The commutator of two covariant derivatives gives the curvature of a vector.

Q: What are some examples of linear operators?

Some examples of linear operators are derivatives with respect to the coordinates and multiplication by matrices or functions of the coordinates.

Q: How does the commutator of derivatives with respect to the coordinates relate to curvature?

The commutator of derivatives with respect to the coordinates gives the change in a function when going around a closed loop. It is equal to the derivative of the function multiplied by the derivative of the function.

Q: What is the covariant derivative of a vector and how is it calculated?

The covariant derivative of a vector is an operator applied to the vector that takes into account the curvature of the space. It is calculated by taking the derivative of the vector with respect to the coordinates and adding the matrix multiplication with the Christoffel symbols.

Q: How does the covariant derivative of a vector relate to the Riemann tensor?

The covariant derivative of a vector can be expressed in terms of the Riemann tensor, which represents the curvature of the space.

Q: What are the properties of the Riemann tensor?

The Riemann tensor has four indices and is anti-symmetric under the interchange of two indices. It is also anti-symmetric under the interchange of two other indices and symmetric under the interchange of the remaining two indices.

Q: How can the Riemann tensor be calculated from the Christoffel symbols?

The Riemann tensor can be calculated by taking the commutator of two covariant derivatives and substitution of the Christoffel symbols.

Q: What are the components of the Riemann tensor made up of?

The components of the Riemann tensor are made up of second derivatives and quadratic combinations of first derivatives of the metric tensor.

Q: How can the Riemann tensor be expressed in a more elegant form?

The Riemann tensor can be expressed as the commutator of two covariant derivatives, without explicitly writing out the matrix indices.

Takeaways (in one paragraph)

The concept of curvature and the calculation of the Riemann tensor were discussed in this video. The Riemann tensor represents the curvature of a space and is calculated using the commutator of two covariant derivatives. The Riemann tensor has several properties, including anti-symmetry and symmetry under index interchange. The components of the Riemann tensor are made up of second derivatives and quadratic combinations of first derivatives of the metric tensor. Understanding and calculating the Riemann tensor is essential in understanding the curvature of a space and its implications in general relativity.

Summary & Key Takeaways

• The lecture sets out to derive the curvature formula previously stated, where going around a small closed loop leaves a vector not quite returned to itself. Susskind notes this quantity is a commutator, so he introduces abstract definitions of linear operators before showing the fast way to derive curvature.

• A linear operator acts on functions or vectors to yield new functions. Examples include differentiation with respect to a coordinate, multiplication by a function f of X, and multiplication by a numerical matrix M whose entries M-alpha-beta act on V-beta. These can be combined into position-dependent matrix operators.

• The covariant derivative del-mu equals the ordinary derivative of a vector plus gamma-mu, where gamma-mu is understood as four matrices, one per direction. This is summarized as del-mu equals derivative plus gamma-mu. The commutator of A and B, defined as AB minus BA, becomes the tool for computing curvature.