- #1
etotheipi
Well, okay, I should say: what does Newtonian gravitation look like in a ##2+1## dimensional Newtonian universe? Consider a flat Earth, i.e. a region ##\mathcal{E} = \{ (x,y): x^2 + y^2 \leq R \}## with mass density ##\rho##, then for ##r > R## a natural guess for the gravitational field seem like it might be$$\begin{align*}
\mathbf{g} &= - \frac{GM}{r} \mathbf{e}_r \\
\implies 2\pi GM&= - 2\pi rg_r = \int_0^{2\pi} -r g_r d\varphi = -\int_0^{2\pi} [r g_r \cos^2{\varphi} + rg_r \sin^2{\varphi}] d\varphi
\end{align*}$$where ##\mathbf{X}(\varphi) = (r\cos{\varphi}, r\sin{\varphi})## is a parameterisation of ##\partial \Omega##; then by Green's theorem$$2\pi G \int_{\Omega} \rho dS = 2\pi GM = - \oint_{\partial \Omega} g_1 dy - g_2 dx = - \int_{\Omega} \partial_i g_i dS= - \int_{\Omega} \nabla \cdot \mathbf{g} dS$$which leads to the identification $$\nabla \cdot \mathbf{g} = - 2\pi G\rho \implies \nabla^2 \phi = 2 \pi G \rho$$with a potential ##\phi(r) - \phi(r_0) = GM \ln{(r/r_0)}##. The equations of motion can be derived pretty easily in principle from that.
But I was wondering if this is the correct modification of Poisson's equation? In other words, is the assumption that ##\mathbf{g} = - (GM/r) \mathbf{e}_r## correct for a flat Earth in a 2d universe?
\mathbf{g} &= - \frac{GM}{r} \mathbf{e}_r \\
\implies 2\pi GM&= - 2\pi rg_r = \int_0^{2\pi} -r g_r d\varphi = -\int_0^{2\pi} [r g_r \cos^2{\varphi} + rg_r \sin^2{\varphi}] d\varphi
\end{align*}$$where ##\mathbf{X}(\varphi) = (r\cos{\varphi}, r\sin{\varphi})## is a parameterisation of ##\partial \Omega##; then by Green's theorem$$2\pi G \int_{\Omega} \rho dS = 2\pi GM = - \oint_{\partial \Omega} g_1 dy - g_2 dx = - \int_{\Omega} \partial_i g_i dS= - \int_{\Omega} \nabla \cdot \mathbf{g} dS$$which leads to the identification $$\nabla \cdot \mathbf{g} = - 2\pi G\rho \implies \nabla^2 \phi = 2 \pi G \rho$$with a potential ##\phi(r) - \phi(r_0) = GM \ln{(r/r_0)}##. The equations of motion can be derived pretty easily in principle from that.
But I was wondering if this is the correct modification of Poisson's equation? In other words, is the assumption that ##\mathbf{g} = - (GM/r) \mathbf{e}_r## correct for a flat Earth in a 2d universe?
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