Einstein's General Theory of Relativity | Lecture 4 | Summary and Q&A

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February 19, 2009
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Stanford
Einstein's General Theory of Relativity | Lecture 4

Summary

In this video, the instructor discusses the theoretical minimum of tensor analysis that one needs to understand before moving on to more advanced topics such as general relativity and cosmology. He starts by explaining the transformation properties of covariant and contravariant components of a tensor. He then introduces the concept of the metric tensor, which is related to the distance between neighboring points in a space. He shows how the metric tensor can be used to raise and lower indices of tensors, allowing for different representations of the same object. He also explains how the metric tensor can be used to write the distance between points in different ways. The instructor emphasizes that tensors are essential for understanding the geometry of curved spaces.

Q: What is the purpose of teaching the "theoretical minimum" in these courses?

The purpose of teaching the theoretical minimum is to provide students with the minimum amount of knowledge required to progress to more advanced topics, such as general relativity and cosmology. It is meant to give students the necessary tools to understand these topics without overwhelming them with unnecessary information and applications.

Q: What are the transformation properties of covariant and contravariant components of a tensor?

Covariant components of a tensor transform in a certain way when changing between different coordinate systems, while contravariant components transform differently. The transformation property of a covariant component involves multiplying by the appropriate transformation factor, while the transformation property of a contravariant component involves dividing by the transformation factor.

Q: How can the metric tensor be used to raise and lower indices of tensors?

The metric tensor can be used to raise and lower indices of tensors by multiplying the tensor with the appropriate components of the metric tensor and then summing over the raised or lowered index. Raising an index changes the tensor from having contravariant components to having covariant components, while lowering an index does the opposite. This allows for different representations of the same object.

Q: What is the relationship between tensors with upper and lower indices?

Tensors with upper and lower indices can be thought of as different representations of the same object. The components with upper indices are covariant components, while the components with lower indices are contravariant components. The relationship between the components is defined by the metric tensor, which allows for raising and lowering indices.

Q: How does the metric tensor relate to the distance between neighboring points in a space?

The metric tensor is related to the distance between neighboring points in a space. It can be used to calculate the distance between two points by contracting the indices of the metric tensor with the differentials of the coordinate variables. The resulting expression represents the square of the distance between the points.

Q: What does it mean for a tensor to be zero in one coordinate system?

If a tensor is zero in one coordinate system, it is zero in every coordinate system. This is because tensors have well-defined transformation properties, and if all the components of a tensor are zero in one coordinate system, they will be zero in every coordinate system.

Takeaways

Tensors play a crucial role in understanding the geometry of curved spaces and are essential for topics such as general relativity and cosmology. The transformation properties of covariant and contravariant components of tensors, as well as the use of the metric tensor to raise and lower indices, enable different representations of the same object. The metric tensor also allows for the calculation of distances between neighboring points in a space. Overall, understanding tensors is a fundamental part of studying advanced topics in physics.