Oct 18, 2012

Coulomb's Law in $d$-Dimension

In 3+1 dimension, Coulomb's law and Newton's law of gravity takes the form of distance inverse-squared,
\[ f = \frac{1}{r^2} \] with proper definition of the source and distance. What is Coulomb's law and Newton's law look like in high dimensions?

To answer this question, we have to make assumptions. We assume the Lagrangian stays the same form in $d+1$ dimension. It means, the Maxwell equations hold; or equivalently, Poisson equation holds \[ \nabla^2 \varphi(\mathbf{x}) = 0. \]
Solving $\varphi$ in free space will produce the potential hence the force. By doing Fourier transform, \[ \varphi(\mathbf{x}) = \int \frac{\mathrm{d}^d k}{(2\pi)^d} \frac{ e^{i \mathbf{k}\cdot \mathbf{x}}}{k^2}. \]

Solving $\varphi(r)$

\[\varphi = \frac{1}{(2\pi)^d}\int \mathrm{d} k \; k^{d-3} \mathrm{d}^{d-1}\Omega {e^{i k r \cos \theta_1}} \], where $\mathrm{d}^{d-1} \Omega $ is the $(d-1)$-D angular element. Now this integral involves the integral of one azimuthal angle $\theta$. We can parametrize the coordinate in $d$-D spherical coordinate as: \[ x_1 = r \cos\theta_1; x_2 = r \sin\theta_1 \cos\theta_2; \cdots; x_d = r \sin\theta_1 \sin\theta_2\cdots \sin\theta_{d-2}\cos\phi; \] Then the surface element becomes: $\mathrm{d}^{d-1} \Omega = \sin^{d-2}\theta_1 \sin^{d-3}\theta_2 \cdots \sin \theta_{d-2} \mathrm{d}\theta_1 \mathrm{d}\theta_2 \cdots \mathrm{d}\theta_{d-2} \mathrm{d} \phi $. The integral over angles except $\theta_1$ is just the surface area of a $(d-2)$-D hypersphere (See appendix for a derivation) \[ S_{d-2} = \frac{2 \pi^{\frac{d-1}{2}}}{\Gamma\left( \frac{d-1}{2} \right) } \]

So $\varphi = \frac{ S_{d-2}}{ (2\pi)^d r^{d-2}} I_{d} $, where \[ I_d = \int_0^\infty \mathrm{d}\xi \; \int_0 ^\pi \mathrm{d}\theta \; \xi^{d-3} \sin^{d-2}\theta \exp\left[ i \xi \cos\theta \right]. \]
It's tempted to do the $\xi$ integral first, because it gives gamma function and leaves the rest a integral over $\tan\theta$: $ \int_0^{\pi/2} \mathrm{d}\theta \tan^{d-2}\theta + (-1)^{d-2}\int^{\pi/2}_\pi \mathrm{d}\theta \tan^{d-2}\theta$. The problem is that integral of $\tan$ function at $\pi/2$ is singular . We can do the $\theta$ integral first, which yields (using mathematica):\[ I_d = \int_0^\infty \mathrm{d}\xi \; \sqrt{\pi} \Gamma\left( \frac{d-1}{2} \right) \frac{{ }_0F_1\left( \frac{d}{2}, -\frac{\xi^2}{4} \right)}{\Gamma\left( \frac{d}{2} \right)} \xi^{d-3} = 2^{d-3} \sqrt{\pi} \Gamma\left( \frac{d-2}{2} \right) \Gamma\left( \frac{d-1}{2} \right). \]
Therefore, \[ \varphi = \frac{ S_{d-2}}{ (2\pi)^d r^{d-2}} 2^{d-3} \sqrt{\pi} \Gamma\left( \frac{d-2}{2} \right) \Gamma\left( \frac{d-1}{2} \right) = \frac{\Gamma\left( \frac{d-2}{2}\right)}{4 \pi^\frac{d}{2} }\frac{1}{r^{d-2}} \]
Coulomb potential in higher dimensions

Gauss Law

There is a much easier method to solve this problem. We note that Gauss theorem (in mathematics) hence Gauss law (in physics) still holds. \[ E(r) \cdot S_{d-1} = 1 \] where $S_{d-1}$ is the area of a $d-1$ D hypersphere. So we get Coulomb's law in $d+1$ dimension as:\[ f = \frac{\Gamma\left( \frac{d}{2} \right) }{2 \pi^\frac{d}{2}} \frac{1}{r^{d-1}}. \]

It can be checked, $-\frac{\partial}{\partial r} \varphi(r) = E(r)$, Just as we expected. Of course, the direct integration can be used in where Gauss law does not hold.

Coulomb's law for massive boson exchange

Another interesting result is the Coulomb's law for classical theories with massive intermediate boson in higher dimentions. The Poisson equation becomes \[ (\nabla^2 - m^2) \varphi(\mathbf{x}) = 0. \]
By doing Fourier transform, \[ \varphi(\mathbf{x}) = \int \frac{\mathrm{d}^d k}{(2\pi)^d} \frac{ e^{i \mathbf{k}\cdot \mathbf{x}}}{k^2+m^2}. \] Apply the same technique again (except the Gauss Law), \[
\varphi(r) = \frac{ (m r)^{\frac{d}{2}-1} K_{\frac{d}{2}-1} (mr)}{(2 \pi)^\frac{d}{2}}\frac{1}{ r^{d-2}} \] where $K_n(x)$ is the Bessel function of the second kind.
Comparison of field potential of massless and massive boson exchange in higher dimensions
Comparison of field potential of massless and massive boson exchange in higher dimensions at large $r$

Appendix: the surface area of a $(d-1)$-D hypersphere

Consider the following Gaussian integeral: \[ \int \mathrm{d}^n x \exp\left( - \mathbf{x}^2 \right) = \left( \int \mathrm{d}x \exp\left[ - x^2 \right] \right)^n = \pi ^{\frac{n}{2}} \]
The left hand side can be written as $ \int \mathrm{d}r \; r^{n-1} \exp\left[ -r^2 \right] S_{n-1}$. So $ \frac{1}{2} \Gamma\left(\frac{n}{2}\right) S_{n-1} = \pi^\frac{n}{2} $. \[ S_{n-1} = \frac{2 \pi^\frac{n}{2} }{\Gamma\left(\frac{n}{2}\right)}. \]

See also: 


update, March 21, 2014:

  • I was solving the Green's function with free space boundary condition. The Poisson equation should have been    

\[ \nabla^2 \varphi(\mathbf{x}) = \delta(\mathbf{x}) \]

  • This is not a the only generalization. It may not even be the natural generalization, from the point view the Newtonian approximation in general relativity. In GR, one should write down the Einstein equation in $d+1$ D and do the linearization there, as our commentator explained. (That means one fix G, which is not always appreciated.)

1 comment:

  1. This essentially finds the Green's function of Delta in arbitrary dimension. One word of warning is that for some problems, the operator you want to use is actually Delta times a prefactor. For example, if you demand that Einstein's equation look the same in all dimensions, the correct Newtonian limit comes from (d-1)/(2(d-2)) Delta.