General Relativity Lecture 10

Transcript

Stanford University so the questions that were asked to me tonight are more or less by accident they exactly the ones that I want to address tonight weak gravitational fields linearity versus nonlinearity um and gravitational waves let's again working out the equations of general relativity is always unpleasant and we're not going to do it on the B... Read More

Summary

In this video, the lecturer discusses weak gravitational fields, linearity versus non-linearity, and gravitational waves. They start by explaining that although working out the equations of general relativity is complicated and not suitable for the blackboard, the principles are straightforward. The lecturer goes on to describe weak gravitational waves as small amplitude fluctuations in the gravitational field, and explains how to make approximations when solving for these waves. They then discuss equilibrium situations and how the metric of flat space can be chosen and represented in different coordinates. Moving on to the equations of motion, the lecturer presents a schematic view of the equations, explaining that they have a relatively simple form and resemble wave equations. Finally, they discuss the nature of gravitational waves, the transversality of the fields, and the effect of these waves on the metric tensor.

Questions & Answers

Q: How can the equations of general relativity be summarized?

The equations of general relativity are typically unpleasant to work with on the blackboard, but they can be summarized by the principles and solutions obtained from solving the equations.

Q: What are weak gravitational waves?

Weak gravitational waves refer to small amplitude fluctuations in the gravitational field. They are characterized by the amplitudes of the waves being small enough to make approximations.

Q: Explain the concept of equilibrium situations.

Equilibrium situations describe solutions with no time dependence and no matter on the right-hand side of the equations. In other words, it is a situation of empty space with no curvature or interesting gravitational fields.

Q: How does the metric of flat space depend on coordinates?

The metric of flat space depends on the coordinates used. While one commonly used metric is the Kronecker Delta, there can be other metrics depending on the choice of coordinates. The special feature of flat space is that there are coordinates in which the metric has a simple form.

Q: What are the components of the metric tensor of flat space?

The metric tensor of flat space can be written as a matrix of components: 1 -1 -1 -1 for the first row and column, followed by 0 0 0 0, 0 0 1 0, and 0 0 0 1.

Q: What is the significance of weak gravitational waves being small?

The smallness of weak gravitational waves allows for approximations to be made, such as ignoring higher order terms. This simplifies the equations and makes them more manageable to work with.

Q: How can the equations of motion for weak gravitational waves be summarized?

The equations of motion for weak gravitational waves have a relatively simple form and resemble wave equations. They can be written using second derivatives of the metric tensor and involve the Christoffel symbol, Ricci tensor, and more.

Q: How does the metric tensor change with gravitational waves?

The metric tensor of flat space is modified by gravitational waves, represented by a small correction term called H_mu_nu. This term depends on the position and time coordinates, and describes the wave field.

Q: What are the constraints on gravitational waves derived from Einstein's field equations?

The constraints on gravitational waves require the transversality of the fields, meaning that the time and Z components of H must be zero. Additionally, the trace of H_ij must be zero.

Q: Can you provide an example of the components of the metric tensor for a gravitational wave?

For gravitational waves propagating along the Z axis, the components of the metric tensor that are allowed to be nonzero are H_ij times sine(KX) times sine(KZ-T), where K represents the wave number. The other components are set to zero.

Takeaways

Gravitational waves are small amplitude fluctuations in the gravitational field. They can be approximated as weak gravitational waves, allowing for simplifications in the equations of motion. The metric tensor of flat space can be modified by gravitational waves, characterized by a small correction term called H_mu_nu. The constraints on gravitational waves include transversality of the fields and the trace of H_ij being zero. Gravitational waves can cause tidal forces and deformations in physical objects, making them an interesting field of study.