8. Lie transport, Killing vectors, tensor densities.

TL;DR

Tensor densities, such as the Levi-Civita symbol and the determinant of the metric, play a crucial role in general relativity by allowing the definition of covariant volume elements and preserving tensor relations despite coordinate transformations.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] SCOTT HUGHES: Last Thursday, we began the work of moving from special relativity to general relativity, and we spent a lot of time unpacking two formulations of the principle of equivalence. So one, which goes under the name "weak equivalence principle"-- a simpler way of saying that is that, at least over a suffic... Read More

Key Insights

• 🧑‍🏭 Tensor densities differ from tensors in their transformation properties, involving an additional factor related to the determinant of the coordinate transformation matrix.
• 🤬 The Levi-Civita symbol and the determinant of the metric are two important tensor densities in general relativity.
• 🖐️ Tensor densities play a crucial role in defining covariant volume elements and preserving tensor relations in curved spacetimes.

Questions & Answers

Q: What is the main difference between tensors and tensor densities?

Tensors are mathematical objects that transform in a specific way under coordinate transformations, while tensor densities transform similarly but with an additional factor related to the determinant of the coordinate transformation matrix.

Q: How does the Levi-Civita symbol transform under coordinate transformations?

When the Levi-Civita symbol is applied to a matrix representing a coordinate transformation, it is equal to the determinant of the matrix multiplied by the Levi-Civita symbol in the original coordinates.

Q: What is the role of tensor densities in general relativity?

Tensor densities, such as the determinant of the metric, allow for the definition of covariant volume elements and preserve tensor relationships despite coordinate transformations. They are critical in general relativity for maintaining consistency in calculations.

Q: Can you provide an example of a tensor density in the context of spherical coordinates?

In spherical coordinates, the determinant of the metric is r^4 sin^2(theta), which is a tensor density of weight 2. By taking the square root, we obtain a tensor density of weight 1, allowing for the proper conversion of differentials into volume elements.

Summary & Key Takeaways

• Tensor densities are quantities that transform almost like tensors but with an additional factor related to the determinant of the coordinate transformation matrix.

• The Levi-Civita symbol is an example of a tensor density, and it has a special transformation property when applied to a matrix.

• The determinant of the metric is another important tensor density in general relativity, and it corrects the transformation of the metric to preserve tensor relationships.