Is Gravitational Force conserved at the origin (r=0)?

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Discussion Overview

The discussion revolves around the nature of the gravitational force field at the origin (r=0), particularly whether it can be considered conservative in the presence of a singularity. Participants explore theoretical implications, mathematical considerations, and the behavior of point masses in gravitational fields.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that the gravitational force field is not defined at r=0 due to the presence of a singularity, which complicates the evaluation of line integrals.
  • Others suggest examining curves that enclose the singularity, like the unit circle, to better understand the conservative or non-conservative nature of the gravitational field.
  • There is a discussion about whether the gravitational field of an ideal point mass leads to singularities, while real spherically symmetric masses do not present this issue.
  • One participant questions the predictability of positions for two point masses in a radial elliptic trajectory after passing through a singularity, raising concerns about the implications of undefined integrals.
  • Another participant notes that elliptical orbits of two non-trivial masses approximated as point particles do not pass through singularities, suggesting that the problem may not arise in practical scenarios.
  • There is speculation about the behavior of masses at the singularity, with one participant mentioning that a point mass could be treated as a black hole, introducing additional complexities to the discussion.
  • Participants express uncertainty about whether masses can pass through the singularity and the potential impacts of such interactions on gravitational dynamics.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the nature of gravitational force at the origin, with no consensus reached on whether the gravitational force can be considered conservative in this context.

Contextual Notes

Limitations include the dependence on definitions of singularities, the mathematical challenges of evaluating integrals through singular points, and the assumptions made about ideal versus real gravitational sources.

chi_rho
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I know gravity is a conservative force field and can be treated as such for all intents and purposes, but I was just thinking that in order to show that a vector field is conservative that vector field must be defined everywhere (gravitational force field is not defined at r=0).

I was thinking that you may need to examine a curve that encloses the singularity (r=0) like the unit circle, but I'm not sure this is the best way to think about this problem.

Any additional insight into how to treat/explain the conservative/non-conservative nature of the gravitational force field at the origin would be appreciate.
 
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chi_rho said:
gravitational force field is not defined at r=0
Why not?
 
chi_rho said:
I was thinking that you may need to examine a curve that encloses the singularity (r=0) like the unit circle, but I'm not sure this is the best way to think about this problem.

We're talking about the gravitational field of an ideal point mass here, right? If we're dealing with a real planet or other spherically symmetric mass with non-zero size, there won't be a singularity at r=0 and there won't be any problem.

But with that said, yes, you have the right idea. If there is a singularity at a given point, you can't evaluate a line integral that passes through that point and you must consider only paths that pass arbitrarily close to it. In practice, this is a seldom a concern because we never encounter gravitating point masses in real life; when we model a real planet as a point mass, we're only working with the space outside the surface of the planet so we don't have trajectories that pass through r=0.
 
Nugatory said:
If there is a singularity at a given point, you can't evaluate a line integral that passes through that point and you must consider only paths that pass arbitrarily close to it.

What does this mean for a radial elliptic trajectory, e.g. for two point masses? Is their position still predictable after they passed the singularity? That shouldn't be possible if the integral can't be evaluated through that point.
 
DrStupid said:
What does this mean for a radial elliptic trajectory, e.g. for two point masses? Is their position still predictable after they passed the singularity? That shouldn't be possible if the integral can't be evaluated through that point.
Would a mass, point or otherwise, pass through the singularity? No impact?
 
tfr000 said:
Would a mass, point or otherwise, pass through the singularity? No impact?
If you want to take it that far, you are into GR - no?
 
Even worse - a point mass will be a black hole.
 
DrStupid said:
What does this mean for a radial elliptic trajectory, e.g. for two point masses? Is their position still predictable after they passed the singularity? That shouldn't be possible if the integral can't be evaluated through that point.

I'm sorry, but I don't understand what you're asking here. The elliptical orbit of two non-trivial masses approximated as point particles doesn't pass through either singularity, so the problem doesn't arise.
 
  • #10
tfr000 said:
Would a mass, point or otherwise, pass through the singularity?

I don't know if they pass through the singularity. In the limit of zero angular momentum a classical orbit would result in a degenerate ellipse with a 180° turn in the center. In this case the point masses would bounce off the center. But I do not think it is that easy. The singularity changes the situation.

tfr000 said:
No impact?

That would result in additional complications. Thus we should limit the interaction to gravity only.
 

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