Dirac delta in curved spacetime

In summary: He does use differential form notation and various other mathematical concepts. He also references various sources. But I think it would be helpful if there were a more comprehensive and accessible presentation of his derivations.
  • #1
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Does anyone know what the Dirac delta function would look like in a space with curvature and torsion? The Dirac delta function is a type of distribution. But that distribution might look differently in curved spacetime than in flat spacetime. I wonder what it would look like in curved spacetime. Any help? Thanks.
 
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  • #2
The Dirac delta function is a distribution as you say. The definition of any function or distribution has nothing to do with "curved" or flat space. Perhaps you mean something else.
 
  • #3
HallsofIvy said:
The Dirac delta function is a distribution as you say. The definition of any function or distribution has nothing to do with "curved" or flat space. Perhaps you mean something else.

I suppose that being a distribution really doesn't matter. I was thinking that a distribution might help with visualizing physically on the situation.

Where I am going is that I have seen where the path integral of quantum mechanics is developed from the definition of the Dirac delta functin. This development is in flat spacetime. So I wonder if a similar development can be done in curved spacetime. Any help would be appreciated. Thanks.
 
  • #4
Try the book on Path Integrals by Hagen Kleinert. He treats path integrals in spaces with curvature and torsion.
 
  • #5
Pere Callahan said:
Try the book on Path Integrals by Hagen Kleinert. He treats path integrals in spaces with curvature and torsion.

Yes, Kleinert has an on-line presentation of this at:

http://www.physik.fu-berlin.de/~kleinert/kleiner_re252/node11.html

I can follow it until about equation 115 where he got and complex exponential for the Jacobian.
 
  • #6
I cannot follow Kleinert's presentation at all. :smile: I have his book and I once took two courses on path integrals taught by him, but his way of doing things is just too different from mine.
Still the book is a great resource if you're prepared for pages and pages of weird and seemingly unmotivated/unjustified calculations (only my opinion)
 
  • #7
Pere Callahan said:
I cannot follow Kleinert's presentation at all. :smile: I have his book and I once took two courses on path integrals taught by him, but his way of doing things is just too different from mine.
Still the book is a great resource if you're prepared for pages and pages of weird and seemingly unmotivated/unjustified calculations (only my opinion)


Does he use a lot of differential form notation, etc? Hagen Kleinert's name comes up a lot when researching path integrals in curved spacetime. But I wish there were more agreement that his treatment is acceptable. And it would be nice if there were an expanded version of his derivations for us dummies. As it is, he mentions things like "nonholomorphic" and expects everyone to automatically know what he is talking about when there is not even any reference to it on Wikipedia, etc.
 

1. What is the Dirac delta in curved spacetime?

The Dirac delta in curved spacetime is a mathematical concept that describes a point-like distribution of matter or energy in a curved space. It is a generalization of the Dirac delta function, which is used to model point-like particles in flat space.

2. How is the Dirac delta in curved spacetime different from the Dirac delta in flat space?

The main difference between the Dirac delta in curved spacetime and the Dirac delta in flat space is that the former takes into account the curvature of space. In curved spacetime, the delta function is defined in terms of the metric tensor, which describes the curvature of space, whereas in flat space, it is defined simply as a point-like distribution.

3. What are the applications of the Dirac delta in curved spacetime?

The Dirac delta in curved spacetime has various applications in theoretical physics, particularly in general relativity. It is used to model the gravitational effects of point-like objects in a curved space, such as black holes. It is also used in quantum field theory to study the behavior of particles in curved spacetime.

4. How is the Dirac delta in curved spacetime related to the Einstein field equations?

The Einstein field equations, which describe the relationship between the curvature of spacetime and the distribution of matter and energy, can be expressed using the Dirac delta in curved spacetime. The delta function is used to represent the mass-energy density of a point-like object, and its inclusion in the equations accounts for the effects of this object on the curvature of space.

5. Are there any limitations or criticisms of using the Dirac delta in curved spacetime?

One criticism of using the Dirac delta in curved spacetime is that it is a highly idealized concept that does not accurately represent the physical reality of point-like objects in curved space. Additionally, the delta function is not well-defined in certain cases, and its use can lead to mathematical inconsistencies. However, it remains a useful tool for modeling and understanding the behavior of objects in curved spacetime.

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