# Is there any relationship between Gravity and Electromagnetism? [duplicate]

We all know that the universe is governed by four Fundamental Forces which are The strong force , The weak force , The electromagnetic force and The gravitational force .

Now, is there any relationship between Electromagnetism and gravity?

• Possible duplicates: physics.stackexchange.com/q/944/2451 and links therein. Sep 3, 2013 at 15:46
• ncatlab.org/nlab/show/Kaluza-Klein+mechanism Sep 4, 2013 at 18:51
• I have no formal physics training, but it seems to me that they are intrinsically connected. If gravity effects mass and mass & energy (electromagnetic waves or light) are related through Einsteins equation, then gravity and light possess a very clear relationship. It seems logical to me, but can someone more qualified chime in?
– user29224
Sep 4, 2013 at 22:43
• @UrsSchreiber: Oh my god, that's an Excellent article! Sep 7, 2013 at 8:43
• @DImension10, thanks for the feedback. I just went through that entry again and expanded a bit more here and there. For instance the Examples-section now has a new subsection "Cascades of KK-reductions from holographic boundaries" ncatlab.org/nlab/show/… . Sep 7, 2013 at 12:50

# 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 , which also talks about the weak and strong forces, and requires .

# On Geometry

In , the gauge group for 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.