<p>Leckner's paper deals with the effect of induced polarization on the spheres. Electrons are redistributed, making the force different from what one would expect. The gravitational counterpart is tidal distortion: since the gravitational field is non-radial when you have two heavy masses close to each other, matter will move to make the surface an equipotential surface. This means that the gravitational acceleration at the surfaces will not be constant at all locations.</p>

<p>Doing an analytic solution of how two ellipsoids attract each other and deform seems to be tractable (e.g. <a href="https://physics.stackexchange.com/questions/16412/the-gravitational-potential-of-ellipsoid">see this question</a>) but algebraically very tedious and likely involves lots of special functions. See the addendum below for an approximate numeric model.</p>

<p>Black holes produce another complication: since spacetime nearby is curved and expanded the meaning of the distance in the inverse square law becomes problematic. The <a href="https://en.wikipedia.org/wiki/Paczy%C5%84ski%E2%80%93Wiita_potential" rel="nofollow noreferrer">Paczyński–Wiita potential</a> is an approximation of the potential, and it deviates from the <span class="math-container">$U=-GM/r$</span> as <span class="math-container">$U_{PW}=-GM/(r-R_S)$</span> (where <span class="math-container">$R_S$</span>  is the Schwarzschild radius). It makes the force increase faster than the classical potential as we approach <span class="math-container">$r=R_S$</span>.
<a href="https://i.sstatic.net/h6MYf.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/h6MYf.png" alt="enter image description here"></a></p>

<p>Addendum: I did a numeric exploration of the force between two ellipsoidal, self-gravitating masses with centres of mass separated by a given distance. To find the shape I started with spheres and adjusted the semi-major axis (while preserving volume) so the potential along the surface became more equal at the poles. After a few iterations this gives a self-consistent shape. Then I calculated the force (the derivative of the potential) due to this shape on the other mass.</p>

<p><a href="https://i.sstatic.net/Obwyc.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Obwyc.png" alt="Numerically estimated semi-major axis, force, and force times squared distance."></a></p>

<p>The result is indeed that the force increases faster than <span class="math-container">$1/r^2$</span>
as the bodies approach each other, since they elongate and eventually merge (a bit before this they will deviate from my ellipsoidal assumption). If one multiplies the force by the squared distance the product should be constant for pure <span class="math-container">$1/r^2$</span> forces, but it starts to increase as they approach enough. Note that this is a non-rotating model: with rotation the numbers will change and the ellipsoids will become tri-axial, but I suspect the qualitative behaviour stays the same. </p>

CC BY-SA 4.0
2020-03-15T22:50:53.007Z
That looks like a qualified yes to the question in the case of black holes with the caveat that distance becomes problematical near to the "surface". Right? And then less dense bodies having Schwarzschild radius smaller than surface radius need more math to evaluate. ? ?
DMac
CC BY-SA 4.0
2020-03-16T05:43:22.943Z
Very cool! Just the problem of two spheres of incompressible liquid orbiting their center of mass seems like a fascinating problem and one step above the one discussed in [How would a Maclaurin spheroid become a Jacobi ellipsoid? What happens?](https://math.stackexchange.com/q/3514142/284619) and also [What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?](https://hsm.stackexchange.com/q/11351/3869). I'll bet it's been addressed somewhere though.
uhoh
CC BY-SA 4.0
2020-03-15T17:42:15.157Z
2020-03-18T10:55:42.010Z
2020-03-18T10:55:42.010Z
Anders Sandberg
<p>Does the gravitational attraction near the surface of dense celestial objects (neutron star, white dwarf itc) diverge <strong>(to infinity)</strong> from inverse square?</p>

<p>This question is inspired by the similarity between EM and gravity (inverse square force.). The paper by John Lekner <a href="https://www.google.com/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=1&amp;cad=rja&amp;uact=8&amp;ved=2ahUKEwje7Kjl-JroAhVFRKwKHQ6HCFsQFjAAegQIChAC&amp;url=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2F10.1098%2Frspa.2012.0133&amp;usg=AOvVaw24AV-Crel31b0PFuEa7Jej" rel="nofollow noreferrer">here</a> (doi:10.1098/rspa.2012.0133) shows that there is an electrostatic attraction between charged spheres <strong><em>no matter the polarities of the charges</em></strong> and that it diverges at close separation until electrical short for almost all charge ratios. I am wondering if there is a similar kind of gravitational inverse square divergence for anything other than a black hole. </p>

<p>Actually make this for a black hole as well, although I know that a black hole is not thought to have a normal surface.</p>

CC BY-SA 4.0
2020-03-15T00:12:32.513Z
Lekner's paper deals with spheres where charges redistribute themselves on the surface. The corresponding gravity thing would redistribute mass and deform the objects. That definitely complicates things compared to 1/r^2
Anders Sandberg
CC BY-SA 4.0
2020-03-15T01:22:28.347Z
I'm not expert enough to answer properly, but I think it does, due to general relativity, but to describe how it does, you need to first define your radius and time coordinates, and there is no single particularly privileged choice.
Steve Linton
CC BY-SA 4.0
2020-03-15T01:25:09.353Z
Classical departure from inverse square happens when the "charges" (electrostatic or otherwise) are no longer spherically symmetric and Newton's [Shell theorem](https://en.wikipedia.org/wiki/Shell_theorem) no longer applies. Any time one or both gravitational bodies depart from spherical symmetric mass distribution the force between them can no longer be assumed to be inverse square. The bodies may naturally depart from spherical symmetry due to rotation, but they can also induce it. There will be a *very tiny* effect on the Moon's orbit due to the the motion of Earth's ocean tides it induces.
uhoh
CC BY-SA 4.0
2020-03-16T17:24:53.087Z
@uhoh I clarified the question above with the two words **(to infinity)**.
DMac
<gravity><newtonian-gravity>
Does the gravitational attraction near the surface of dense celestial objects diverge from inverse square?
DMac
# Does the gravitational attraction near the surface of dense celestial objects diverge from inverse square?

(Asked by: username_0 on 2020-03-14)

Does the gravitational attraction near the surface of dense celestial objects (neutron star, white dwarf itc) diverge **(to infinity)** from inverse square?

This question is inspired by the similarity between EM and gravity (inverse square force.). The paper by John Lekner [here](https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=2ahUKEwje7Kjl-JroAhVFRKwKHQ6HCFsQFjAAegQIChAC&url=https%3A%2F%2Froyalsocietypublishing.org%2Fdoi%2F10.1098%2Frspa.2012.0133&usg=AOvVaw24AV-Crel31b0PFuEa7Jej) (doi:10.1098/rspa.2012.0133) shows that there is an electrostatic attraction between charged spheres ***no matter the polarities of the charges*** and that it diverges at close separation until electrical short for almost all charge ratios. I am wondering if there is a similar kind of gravitational inverse square divergence for anything other than a black hole.

Actually make this for a black hole as well, although I know that a black hole is not thought to have a normal surface.

**Tags:** gravity, newtonian-gravity
> **Comments:**
> **username_1**: Lekner's paper deals with spheres where charges redistribute themselves on the surface. The corresponding gravity thing would redistribute mass and deform the objects. That definitely complicates things compared to 1/r^2  
> **username_2**: I'm not expert enough to answer properly, but I think it does, due to general relativity, but to describe how it does, you need to first define your radius and time coordinates, and there is no single particularly privileged choice.  
> **username_3**: Classical departure from inverse square happens when the "charges" (electrostatic or otherwise) are no longer spherically symmetric and Newton's [Shell theorem](https://en.wikipedia.org/wiki/Shell_theorem) no longer applies. Any time one or both gravitational bodies depart from spherical symmetric mass distribution the force between them can no longer be assumed to be inverse square. The bodies may naturally depart from spherical symmetry due to rotation, but they can also induce it. There will be a *very tiny* effect on the Moon's orbit due to the the motion of Earth's ocean tides it induces.  
> **username_0**: @uhoh I clarified the question above with the two words **(to infinity)**.  

---

## 1 answers

### Answer by username_1 on 2020-03-15. Score: 6

Leckner's paper deals with the effect of induced polarization on the spheres. Electrons are redistributed, making the force different from what one would expect. The gravitational counterpart is tidal distortion: since the gravitational field is non-radial when you have two heavy masses close to each other, matter will move to make the surface an equipotential surface. This means that the gravitational acceleration at the surfaces will not be constant at all locations.

Doing an analytic solution of how two ellipsoids attract each other and deform seems to be tractable (e.g. [see this question](https://physics.stackexchange.com/questions/16412/the-gravitational-potential-of-ellipsoid)) but algebraically very tedious and likely involves lots of special functions. See the addendum below for an approximate numeric model.

Black holes produce another complication: since spacetime nearby is curved and expanded the meaning of the distance in the inverse square law becomes problematic. The [Paczyński–Wiita potential](https://en.wikipedia.org/wiki/Paczy%C5%84ski%E2%80%93Wiita_potential) is an approximation of the potential, and it deviates from the $U=-GM/r$ as $U\_{PW}=-GM/(r-R\_S)$ (where $R\_S$ is the Schwarzschild radius). It makes the force increase faster than the classical potential as we approach $r=R\_S$.
[![enter image description here](https://i.sstatic.net/h6MYf.png)](https://i.sstatic.net/h6MYf.png)

Addendum: I did a numeric exploration of the force between two ellipsoidal, self-gravitating masses with centres of mass separated by a given distance. To find the shape I started with spheres and adjusted the semi-major axis (while preserving volume) so the potential along the surface became more equal at the poles. After a few iterations this gives a self-consistent shape. Then I calculated the force (the derivative of the potential) due to this shape on the other mass.

[![Numerically estimated semi-major axis, force, and force times squared distance.](https://i.sstatic.net/Obwyc.png)](https://i.sstatic.net/Obwyc.png)

The result is indeed that the force increases faster than $1/r^2$
as the bodies approach each other, since they elongate and eventually merge (a bit before this they will deviate from my ellipsoidal assumption). If one multiplies the force by the squared distance the product should be constant for pure $1/r^2$ forces, but it starts to increase as they approach enough. Note that this is a non-rotating model: with rotation the numbers will change and the ellipsoids will become tri-axial, but I suspect the qualitative behaviour stays the same.

> **Comments:**
> **username_0**: That looks like a qualified yes to the question in the case of black holes with the caveat that distance becomes problematical near to the "surface". Right? And then less dense bodies having Schwarzschild radius smaller than surface radius need more math to evaluate. ? ?  
> **username_3**: Very cool! Just the problem of two spheres of incompressible liquid orbiting their center of mass seems like a fascinating problem and one step above the one discussed in [How would a Maclaurin spheroid become a Jacobi ellipsoid? What happens?](https://math.stackexchange.com/q/3514142/284619) and also [What exactly was Lagrange's “grave mistake” with respect to rotating bodies under hydrostatic equilibrium?](https://hsm.stackexchange.com/q/11351/3869). I'll bet it's been addressed somewhere though.  

---

