# Schwarzschild’s equations & Newtonian Potential

• TheSicilianSa
In summary, the conversation discusses the Newtonian gravitational potential and the concept of effective mass within a sphere of uniform density and radius. It also explores the idea of using the sphere's density to eliminate the singularity at the center and how this relates to black holes. The possibility of quantum gravity models preventing a singularity and the concept of high density in a small region are also mentioned.
TheSicilianSa

I was taught that the Newtonian gravitational potential was given as -Gm/r; and when examining a sphere of uniform density ρ and radius r, then at any point x ≤ r (within the sphere), m would refer to the effective mass of an equivalent sphere of radius x. -The logic given was that the effective mass of all points > x would “cancel each other out”.

I was content with that explanation, accepting that at the center of the sphere the gravitational potential would be zero. However, I was always uneasy with the “indeterminacy” at r=0 | m=0.

However, if we were to rephrase the “effective mass” (at any point x) in terms of the sphere’s density, would we not get :

(1) m(x) = 4π ρ { r δ(x – r) + x δ(r – x) } /3; where δ = delta function

And, were we to replace “m” with (1) in Schwarzschild’s equations (i.e. in lieu of m in the standard expression for the Newtonian potential), would this not eliminate the singularity at x=0?

And, further, does this not imply that for a black hole, the gravitational effect of its mass must go from a maximum at the event horizon to zero at the body’s center?

Ordinarily different people would answer. So you get a chance to try discussing with a variety. Just patiently keep asking good questions until a range of people (George Jones, Cristo, Russ Watters, Hellfire, Matt.O...to mention only a few) respond.

I'm probably not the best to reply to BH questions. But this has been on the board for a few hours and nobody has responded, so I'll take a shot.

What goes on inside the eventhorizon of a BH is necessarily model dependent.
What you think it looks like and what the falling-in experience is like depends on what model.

I respect and am interested by the fact that you set up your own mathematical model.

I'm not sure I understand but I think you have like a continuous density inside the eventhorizon (EH).

The prevailing model doesn't. It has a great emptiness inside the EH, all except for a tiny dot at the center which has very very high density.

The classical 1918 Schwarzschild model actually breaks down right at the center and can't say anything about conditions---it suffers from infinities

Vintage 2008 modifications of that model don't break down, they just achieve a small region of very very high density, but not infinite density.

Various things happen. It's my bedtime and I'm sleepy. Maybe I can discuss this tomorrow.
In a number of papers I've looked at, there is a runaway collapse which gets down to where the density is a substantial fraction of Planck density.

Some models of quantum gravity predict that as Planck density is approached certain correction terms become important and cancel out the attractiveness of gravity, the fundamental degrees of freedom (constituents) of geometry and matter fight back and halt the collapse. It's just a math model that happens to exhibit this strange feature where quantum corrections in the equations make gravity repell when the density gets high enough. An actual singularity (a math breakdown) is avoided.

In those models you don't get infinite density (which seems unreal so not to consider seriously) but you get high density in a very small region. Maybe they have something about this at the Einstein Online website. (link in sig at end of post). More about this tomorrow, hopefully from more besides myself.

## 1. What are Schwarzschild's equations and how are they related to Newtonian Potential?

Schwarzschild's equations are a set of mathematical equations that describe the curvature of spacetime around a spherically symmetric body, such as a planet or star. They were developed by German physicist Karl Schwarzschild in 1916, based on Einstein's theory of general relativity. These equations are used to calculate the gravitational potential, which is closely related to Newtonian Potential, a concept used in classical mechanics to describe the potential energy of an object in a gravitational field.

## 2. How do Schwarzschild's equations differ from Newton's law of gravitation?

Schwarzschild's equations are a more accurate and comprehensive description of gravity compared to Newton's law of gravitation. While Newton's law only works for weak gravitational fields and low speeds, Schwarzschild's equations take into account the effects of strong gravitational fields and high speeds, as well as the curvature of spacetime. Additionally, Newton's law is a purely classical theory, while Schwarzschild's equations are based on Einstein's theory of general relativity, which incorporates concepts from quantum mechanics.

## 3. What is the significance of the Schwarzschild radius in these equations?

The Schwarzschild radius, also known as the gravitational radius, is a key parameter in Schwarzschild's equations. It represents the distance from the center of a spherically symmetric body at which the escape velocity exceeds the speed of light. This means that anything within this radius, including light, will be unable to escape the body's gravitational pull. The Schwarzschild radius is also used to calculate the event horizon of black holes.

## 4. How are Schwarzschild's equations used in modern physics?

Schwarzschild's equations have a wide range of applications in modern physics. They are used to study the behavior of objects in strong gravitational fields, such as black holes and neutron stars. They are also used in cosmology to understand the evolution of the universe and the formation of large-scale structures. Additionally, Schwarzschild's equations have been used in the development of technologies such as GPS navigation, which relies on precise measurements of time and space.

## 5. Are there any limitations or criticisms of Schwarzschild's equations?

While Schwarzschild's equations have been extremely successful in explaining many phenomena in modern physics, they are not without limitations and criticisms. Some scientists have argued that the equations are incomplete and do not fully capture the complexities of gravity. Others have pointed out that the equations break down at extremely small scales, where quantum effects become significant. Furthermore, the equations are based on the assumption of a spherically symmetric body, which may not always be the case in reality.

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