GR Point Masses: Seeking Help from Relativists

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In summary, the author claims that Einstein's equation does not allow point masses as a source of gravitational energy. There is a difficulty with defining energy at a point in general relativity, just as there is a difficulty in defining global energy; it's the notion of a quasi-local definition of energy which GR seems to lack. Laszlo Szabados has many good papers on the ArXiv on this subject. Apparently it's somewhat possible to define energy in an extended but finite region of spacetime.
  • #1
humanino
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Dear relativists,

I have problems trying to understand the following statement in Forces from Connes' geometry
2.2 said:
Einstein’s equation is nonlinear and therefore does not allow point masses as source
I would appreciate if somebody with a better understanding of GR could elaborate. I know there is no proper general local definition of gravitational energy but I always had difficulties on this aspect.

Thanks in advance for your comments.
 
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  • #2
The answer to your question is subtle, and it's too late here for me to come up with a convincing explanation. Nonetheless, this claim

humanino said:
I know there is no proper general local definition of gravitational energy

does deserve comment. There's no difficulty with defining energy at a point in general relativity, just as there's no difficulty in defining global energy; it's the notion of a quasi-local definition of energy which GR seems to lack, i.e., energy in an extended but finite region of spacetime.

Laszlo Szabados has many good papers on the ArXiv on this subject.
 
  • #3
Apparently it's somewhat possible: Poisson, http://relativity.livingreviews.org/Articles/lrr-2004-6/ .
 
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  • #4
shoehorn said:
energy in an extended but finite region of spacetime.
Yes, that seems more accurate even to me :smile:
Thanks for the comment

atyy said:
Apparently it's somewhat possible: Poisson, http://relativity.livingreviews.org/Articles/lrr-2004-6/ .
I am not too sure, here he deals with point masses on a background fixed spacetime. Intuitively, I would guess, if one does not fix the background, then point masses will be black holes in GR. Maybe that is what the author meant. But black holes are "allowed", at least several authors in the past have tried to describe fundamental particles as "sort of" black holes.
 
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  • #5
It is not that you can't have a point source for gravity (theoretically) but that with non-linear equations you can't "add" solutions. That is you cannot treat an extended mass as being a "bunch of point sources" as you could with Newton's theory.
 
  • #6
HallsofIvy said:
It is not that you can't have a point source for gravity (theoretically) but that with non-linear equations you can't "add" solutions. That is you cannot treat an extended mass as being a "bunch of point sources" as you could with Newton's theory.
Ah, yes, sure, that definitely makes sense of the sentence. Thank you very much.
 
  • #8
George Jones said:
Read Stingray's comments in

https://www.physicsforums.com/showthread.php?t=111148

and maybe the first page of the reference that Stingray gives in post #10.

Thanks a lot (and to Stingray too !) for this reference. :smile:
Even the MTW does not go into those considerations.
 
  • #9
humanino said:
I am not too sure, here he deals with point masses on a background fixed spacetime. Intuitively, I would guess, if one does not fix the background, then point masses will be black holes in GR. Maybe that is what the author meant. But black holes are "allowed", at least several authors in the past have tried to describe fundamental particles as "sort of" black holes.

Yes, he deals with a point mass as a black hole, and the point mass does perturb the background. However, reading Stingray's comments, there is no "source" here, since everything is a vacuum solution.
 
  • #10
atyy said:
Yes, he deals with a point mass as a black hole, and the point mass does perturb the background. However, reading Stingray's comments, there is no "source" here, since everything is a vacuum solution.
I understand. Stingray's comment address my question exactly. I downloaded the reference he provided Phys. Rev. D 36, 1017 (1987).
 
  • #11

What is the concept of "GR Point Masses"?

GR Point Masses refers to the application of the theory of General Relativity (GR) to point-like masses. In this context, a point mass is a hypothetical object with no spatial extent, but with a mass value. By using GR, we can understand how these point masses interact with each other in the context of gravity.

How is "GR Point Masses" different from Newtonian gravity?

Newtonian gravity is a theory of gravity that was developed by Sir Isaac Newton in the 17th century. It is based on the idea that there is a force of attraction between masses, and this force is proportional to the masses and inversely proportional to the square of the distance between them. In contrast, "GR Point Masses" uses the theory of General Relativity, which views gravity as a curvature of spacetime caused by the presence of mass and energy.

What are some examples of "GR Point Masses" in real life?

While point masses do not exist in reality, we can use this concept to model and understand the behavior of objects with very small dimensions, such as subatomic particles. We can also use it to study the motion of celestial bodies, such as planets, stars, and galaxies. Additionally, GR Point Masses can be applied to the behavior of black holes, which are objects with infinite density and therefore can be considered as point masses in GR.

What is the significance of studying "GR Point Masses"?

Studying GR Point Masses allows us to understand the behavior of objects in extreme conditions, such as near black holes or during the early stages of the universe. It also helps us to make more accurate predictions and calculations in astrophysics and cosmology. Additionally, by using GR Point Masses, we can bridge the gap between the theories of gravity and quantum mechanics, which can lead to a more complete understanding of the fundamental laws of the universe.

What are some challenges in using "GR Point Masses" to study gravity?

One of the challenges in using GR Point Masses is that it assumes that the objects being studied have no spatial extent. In reality, all objects have some physical size, which can affect their gravitational interactions. Additionally, solving the equations of GR for point masses can be complex and requires advanced mathematical techniques. Moreover, the theory of GR is still incomplete and may not fully explain certain phenomena, such as the behavior of gravity at the quantum level.

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