Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Extra Dimensions

Hypothetical spatial dimensions beyond the three we observe, offering solutions to the hierarchy problem and influencing particle interactions, often represented as n+1n+1 dimensions.
📜

The statement of the theorem

Let the spacetime manifold be M=M4×Kn\mathcal{M} = \mathbb{M}^4 \times \mathcal{K}^n, where M4\mathbb{M}^4 is the observed 4D spacetime and Kn\mathcal{K}^n is the compact internal manifold. The higher-dimensional action is given by the Einstein-Hilbert action: SD=116πGD∫d4+nx−G(RD+Lmatter)S_{D} = \frac{1}{16\pi G_{D}} \int d^{4+n}x \sqrt{-G} \left( R_{D} + \mathcal{L}_{matter} \right) where GDG_{D} is the metric tensor in D=4+nD=4+n dimensions. Assuming a compactification radius RR for Kn\mathcal{K}^n, the effective 4D metric gμNg_{\mu N} and the resulting effective 4D action S4S_{4} are obtained via dimensional reduction, leading to the effective gravitational coupling G4=GD/(VK)G_{4} = G_{D}/(V_{\mathcal{K}}), where VK=Vol(Kn)V_{\mathcal{K}} = \text{Vol}(\mathcal{K}^n) is the volume of the internal space.
Source: Wikipedia