- #1

madness

- 815

- 70

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

In summary, there seems to be a discrepancy in how the volume element is defined in Ray d'Inverno's book on general relativity. While other sources state it should transform with the Jacobian, d'Inverno defines it as a scalar density of weight -1. This may be due to a difference in notation or terminology, but it is unclear where his definition has come from. His argument involves the generalised kronecker delta symbol and the levi-cevita tensor, but it is still unclear why he has chosen this definition. Further clarification is needed to understand this discrepancy.

- #1

madness

- 815

- 70

Physics news on Phys.org

- #2

shoehorn

- 424

- 2

- #3

madness

- 815

- 70

His definition of a volume element doesn't appear to include the metric, and is actually stated to be a scalar density of weight -1 and not +1 as you have stated. This is my problem, I don't understand where his definition has come from.

He uses this fact to demand that the Lagrangian is a scalar of weight +1, and include the square root of the determinant here.

Can anyone explain this discrepancy with volume definitions?

- #4

madness

- 815

- 70

Can nobody help with this?

- #5

- 6,986

- 2,460

madness said:Can nobody help with this?

You may need to transcribe the relevant passages here.

It may simply be a confusion in notation or terminology.

(Unfortunately, there are no book previews from Google or Amazon.)

- #6

madness

- 815

- 70

The volume element is a mathematical concept used to measure the volume of a three-dimensional object. It is important in scientific calculations because it allows us to accurately measure and compare the sizes of objects, as well as calculate important physical quantities such as density and pressure.

The volume element is a scalar density of weight -1 because it represents the reciprocal of the density of a substance. In other words, it is a measure of the amount of volume occupied by a unit mass of a substance. This relationship is important in many scientific fields, including physics, chemistry, and engineering.

The volume element is a scalar quantity because it only has magnitude and no direction. Unlike vectors, which have both magnitude and direction, scalar quantities are only concerned with the size or amount of a physical quantity. Therefore, the volume element, which is simply a measure of volume, is considered a scalar quantity.

In calculus, the volume element is used to represent the infinitesimal volume of a three-dimensional object. This allows for the integration of functions over three-dimensional regions, which is essential in many mathematical applications. The volume element is also used in other mathematical fields, such as differential geometry and topology, to study the properties of three-dimensional spaces.

Yes, the volume element and its scalar density have many real-world applications. For example, in fluid mechanics, the volume element is used to calculate the velocity and pressure fields of a fluid. In thermodynamics, the volume element is used to determine the internal energy and heat transfer of a system. Additionally, the volume element is used in medical imaging to measure and analyze the size and shape of organs and tissues in the human body.

- Replies
- 10

- Views
- 1K

- Replies
- 10

- Views
- 2K

- Replies
- 7

- Views
- 2K

- Replies
- 11

- Views
- 4K

- Replies
- 21

- Views
- 2K

- Replies
- 4

- Views
- 842

- Replies
- 17

- Views
- 2K

- Replies
- 14

- Views
- 5K

- Replies
- 4

- Views
- 650

- Replies
- 4

- Views
- 2K

Share: