Deriving Gauss' Law for Gravity from Newton's Law

In summary, to derive Gauss' law for gravity from Newton's law, you will need to use the divergence theorem and the equation \nabla\cdot\frac{\vec{r}}{r^2} = 4\pi\delta^3({\vec{r}}). This can be found in the applications section of the Wikipedia page for the divergence theorem.
  • #1
Varnick
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0

Homework Statement


How would I derive Gauss' law for gravity from Newton's law?

Homework Equations



[tex]\mathbf{g}(\mathbf{r}) &=& -G\frac{m_1}{{\vert \mathbf{r}\vert^2}}\hat{\mathbf{r}}[/tex]

to

[tex]\nabla\cdot\mathbf{g} = -4\pi G \rho[/tex]

The Attempt at a Solution


I have no reference material outside the wide and bountiful internet, and wikipedia gives this equation as the first step of the derivation, which is where I'm really stuck.

[tex]\mathbf{g}(\mathbf{r}) = -G\int_V \frac{\rho(\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^3} dV(\mathbf{s})[/tex]

I'm just not sure how on Earth I'd get to this equation, any help appreciated.

V
 
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  • #2
I think that you'll need to use the divergence theorem (look it up on wikipedia). On the wikipedia webpage you can actually look at this problem under the applications part ('Gauss's law for gravity').
 
  • #3
It may help at some point if you know that
[tex]\nabla\cdot\frac{\vec{r}}{r^2} = 4\pi\delta^3({\vec{r}})[/tex]
 

1. What is Gaussian Gravity?

Gaussian Gravity is a theory proposed by physicist Theodor Kaluza in the early 20th century, which attempts to unify the fundamental forces of nature by incorporating an additional fifth dimension into Einstein's theory of general relativity.

2. How does Gaussian Gravity differ from traditional gravity?

Traditional gravity, as described by Newton's law of gravitation, is a force that acts between two massive objects. Gaussian Gravity, on the other hand, views gravity as a consequence of the curvature of spacetime caused by the presence of matter and energy.

3. What is the significance of the Gaussian curvature in this theory?

The Gaussian curvature is a measure of the curvature of a surface, and in Gaussian Gravity it is linked to the distribution of matter and energy in the universe. This allows for a more comprehensive understanding of gravity and its effects on the universe.

4. How does Gaussian Gravity explain the other fundamental forces?

In Gaussian Gravity, the other fundamental forces (electromagnetism, strong nuclear force, and weak nuclear force) are seen as manifestations of the curvature of spacetime in different ways. This unification of forces is a major goal of the theory.

5. What evidence supports the concept of Gaussian Gravity?

There is currently no direct evidence for Gaussian Gravity, as it is a theoretical concept that has yet to be fully tested. However, it is a promising theory that has gained attention in the scientific community and has potential to explain many unanswered questions in physics.

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