Einstein's General Theory of Relativity | Lecture 6

Transcript

this program is brought to you by Stanford University please visit us at stanford.edu today I want to do two things one is to discuss geodesics a little more and to explain a little more thoroughly the connection between gravitational forces and geodesics or the motion of geodesics uh through space time so for the first part of the lecture tonight ... Read More

Summary

In this video, the speaker discusses geodesics and the connection between gravitational forces and geodesics in space-time. They explain the mathematics of space-time, including the metric and proper time. They also introduce the concept of curved coordinates and the analogy between general relativity and special relativity.

Questions & Answers

Q: What is the metric in space-time called?

The metric in space-time is called the generalized metric tensor.

Q: How is the metric different in special relativity compared to general relativity?

In special relativity, the metric is constant everywhere and is usually taken to be the symbol "atmu nu," which is given by a 4x4 matrix with the elements 1, 0, 0, -1. In general relativity, the metric can vary with position and is generally a more complex tensor.

Q: What is the proper time along a trajectory?

The proper time along a trajectory is the time as recorded by a clock moving with the object. It is defined in terms of the metric and represents the time experienced by the object as it follows its trajectory through space-time.

Q: What is the significance of the speed of light in the formula for proper time?

The speed of light can be included in the formula for proper time by dividing the space components of the metric by the square of the speed of light. This ensures that the proper time has units of time and not space.

Q: How are coordinates in space-time chosen when dealing with curved space?

Curved coordinates are chosen when dealing with curved space-time. They are introduced when working with accelerated frames of reference or when there is a real gravitational field that cannot be removed by changing coordinates.

Q: Can the number of dimensions in a space-time matrix account for the extra dimensions in string theory?

Yes, the number of dimensions in the space-time matrix can be related to the number of dimensions in the physical theory being studied. In string theory, for example, the number of entries in the matrix corresponds to the number of dimensions in the theory.

Q: Why do we introduce curved coordinates if they complicate the description of things?

Curved coordinates are introduced because they are necessary when working with accelerated frames of reference or when there is a real gravitational field present. While they may complicate the description of things, they are required to accurately describe the geometry of space-time.

Q: How can we determine if a space or space-time is flat?

A space or space-time is considered flat if the metric can be chosen to be constant everywhere or if the derivatives of the metric with respect to coordinates can be chosen to be zero. This means that the space is the same at every point and does not have any curvature.

Q: How many independent entries are there in a symmetric matrix in d dimensions?

In d dimensions, a symmetric matrix has d times (d+1)/2 independent entries. In three dimensions, for example, there are six independent entries in a symmetric matrix.

Q: What is a singularity in space or space-time?

A singularity is a place in space or space-time where the metric or the coordinates are not well-behaved. It is a point where it is not possible to choose coordinates that make the metric continuous and differentiable. A singularity can indicate a real physical singularity or a breakdown in the coordinates being used.

Takeaways

In this video, we learned about geodesics and the connection between gravitational forces and geodesics in space-time. We explored the mathematics of space-time, including the metric and proper time. We also discussed the use of curved coordinates and the relationship between general relativity and special relativity. It is important to note that a singularity in space or space-time can indicate a point where the metric or coordinates become problematic and may not accurately describe the underlying physics.