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Newton's law of gravitation is:

$$F = G m_1 m_2 \frac{1}{r^2}$$

It looks simple and natural.

But that's only in 3 dimensions. Let's look what happens in $n$ dimensions:

$$n=2 : F = 2 G m_1 m_2 \frac{1}{r}$$ $$n=4 : F = \frac{2}{\pi} G m_1 m_2 \frac{1}{r^3}$$ $$n=5 : F = \frac{3}{2 \pi^2} G m_1 m_2 \frac{1}{r^4}$$ $$n=6 : F = \frac{4}{\pi^2} G m_1 m_2 \frac{1}{r^5}$$

Oh no! Newton's force law becomes cluttered with unintuitive constants! But by defining $G^* = 4 \pi G$ Newton's law of gravitation can be reformulated as such:

$$F = G^* m_1 m_2 \frac{1}{4 \pi r^2}$$

Immediately we recognize that $4 \pi r^2$ is simply the surface area of a sphere of radius $r$.

But that's only in 3 dimensions. Let's look what happens in $n$ dimensions:

$$n=2 : F = G^* m_1 m_2 \frac{1}{2 \pi r}$$ $$n=4 : F = G^* m_1 m_2 \frac{1}{2 \pi^2 r^3}$$ $$n=5 : F = G^* m_1 m_2 \frac{1}{\frac{8}{3} \pi^2 r^4}$$ $$n=6 : F = G^* m_1 m_2 \frac{1}{\pi^3 r^5}$$

$2 \pi r$ is the surface area of a 2 dimensional sphere of radius $r$.

$2 \pi^2 r^3$ is the surface area of a 4 dimensional sphere of radius $r$.

$\frac{8}{3} \pi^2 r^4$ is the surface area of a 5 dimensional sphere of radius $r$.

$\pi^3 r^5$ is the surface area of a 6 dimensional sphere of radius $r$.

Newton's law of gravitation in $n$ dimensions is:

$$F = G^* m_1 m_2 \frac{1}{S_n}$$

Where $S_n$ is simply the surface area of a $n$ dimensional sphere of radius $r$. From this, it seems like $G^*$ would be a nicer definition for the gravitational constant.

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    $\begingroup$ You could add to the question that in cosmology we have so many equations with $4\pi G$ or $8\pi G$ in them that we often take one or the other to be the natural unit and equal to 1 $\endgroup$ – Jim May 16 '14 at 15:18
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    $\begingroup$ Possible duplicate $\endgroup$ – Ruslan May 16 '14 at 15:25
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    $\begingroup$ See also the difference between electrostatics expressed in SI and Gaussian units. $\endgroup$ – dmckee --- ex-moderator kitten May 17 '14 at 1:04
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Well, if it makes you feel better, in Einstein's equation, it's written down as:

$$R_{ab} - \frac{1}{2}Rg_{ab} = 8\pi G\,T_{ab}$$

and working physicists often get sick of carrying around the factor of 8$\pi$G, and will define $\kappa = 8\pi G$. (or, as Jim says, define the unit of mass such that $G =1$ or 8$\pi$G =1)

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