- #1

- 8,408

- 2,595

[itex]L = \sqrt{-g} ((\nabla_\mu \Phi)(\nabla^\mu \Phi) - m^2 \Phi^2)[/itex]

Is there a simple explanation for why this is scaled by [itex]\sqrt{-g}[/itex]?

- Thread starter stevendaryl
- Start date

- #1

- 8,408

- 2,595

[itex]L = \sqrt{-g} ((\nabla_\mu \Phi)(\nabla^\mu \Phi) - m^2 \Phi^2)[/itex]

Is there a simple explanation for why this is scaled by [itex]\sqrt{-g}[/itex]?

- #2

- 9,900

- 1,081

The simple explanation is the Lagrangian is a tensor density. You want to map a volume to a number (a Lorentz scalar), but representing the volume requires the factor of sqrt(g).

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html [Broken] (Ben Crowell's online book) talks a little about tensor densities in an informal way.

http://www.lightandmatter.com/html_books/genrel/ch04/ch04.html [Broken] (Ben Crowell's online book) talks a little about tensor densities in an informal way.

Last edited by a moderator:

- #3

- 31,656

- 10,389

- #4

- 260

- 1

The volume element in a set of arbitrary coordinates is given by dV=det(J)dx

You need the factor in front of d

- Replies
- 17

- Views
- 5K

- Last Post

- Replies
- 7

- Views
- 6K

- Last Post

- Replies
- 6

- Views
- 3K

- Last Post

- Replies
- 10

- Views
- 672

- Last Post

- Replies
- 3

- Views
- 2K

- Replies
- 8

- Views
- 6K

- Replies
- 4

- Views
- 3K

- Last Post

- Replies
- 4

- Views
- 540

- Replies
- 9

- Views
- 947

- Replies
- 5

- Views
- 3K