# On Unification

I presume you're asking whether just classical gravity & classical EM can be unified.

They sure can!

Classical General Relativity and Classical Electromagnetism are unified in Kaluza-Klein-Theory, which proves that 5-dimensional general relativity is equivalent to 4-dimensional general relativity plus 4-dimensional maxwell equations. Rather interesting, isn't it? A byproduct is the scalar "Radion" or "Dilaton" which appears due to the "55" component of the metric tensor. In other words, the Kaluza-Klein metric tensor equals the GR metric tensor with maxwell stuff on the right and at the bottom; BUT you have an extra field down there.

$${g_{\mu \nu }} = \left[ {\begin{array}{*{20}{c}} {{g_{11}}}&{{g_{12}}}&{{g_{13}}}&{{g_{14}}}&{{g_{15}}} \\ {{g_{21}}}&{{g_{22}}}&{{g_{23}}}&{{g_{24}}}&{{g_{25}}} \\ {{g_{31}}}&{{g_{32}}}&{{g_{33}}}&{{g_{34}}}&{{g_{35}}} \\ {{g_{41}}}&{{g_{42}}}&{{g_{43}}}&{{g_{44}}}&{{g_{45}}} \\ {{g_{51}}}&{{g_{52}}}&{{g_{53}}}&{{g_{54}}}&{{g_{55}}} \end{array}} \right]$$

Imagine 2 imaginary lines now.

$${g_{\mu \nu }} = \left[ {\begin{array}{*{20}{cccc|c}} {{g_{11}}}&{{g_{12}}}&{{g_{13}}}&{{g_{14}}} & {{g_{15}}} \\ {{g_{21}}}&{{g_{22}}}&{{g_{23}}}&{{g_{24}}} & {{g_{25}}} \\ {{g_{31}}}&{{g_{32}}}&{{g_{33}}}&{{g_{34}}} & {{g_{35}}} \\ {{g_{41}}}&{{g_{42}}}&{{g_{43}}}&{{g_{44}}} & {{g_{45}}} \\ \hline {{g_{51}}}&{{g_{52}}}&{{g_{53}}}&{{g_{54}}} & {{g_{55}}} \end{array}} \right]$$

So the stuff on the top-left is the GR metric for gravity, and the stuff on the edge ($g_{j5}$ and $g_{5j}$) is for electromagnetism and you have an additional component on the bottom right. This is the radion/dilaton.

An extension to kaluza - klein is supergravity, which also talks about the weak and strong forces, and requires supersymmetry.

# On Geometry

In quantum-electrodynamics, the gauge group for electromagnetism is $U(1)$.

Now, the key thing here is that Electromagnetism is then *The Curvature of the $U(1)$ bundle*.

This is not the only geometric connection between General Relativity and Quantum Field Theory. In the same context, the covariant derivatives is general relativity are such that $\nabla_\mu-\partial_\mu$ sort-of measures the gravity, in a certain way, while this is also true in QFT, where to some constants, $\nabla_\mu-\partial_\mu=ig_sA_\mu$.

It is to be noted that both are in similiar context.