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Is there any relationship between Gravity and Electromagnetism?

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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?

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Hossam Mohamed
asked Sep 3, 2013 in Theoretical Physics by Hossam Mohamed (10 points) [ no revision ]
Possible duplicates: physics.stackexchange.com/q/944/2451 and links therein.

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Qmechanic
ncatlab.org/nlab/show/Kaluza-Klein+mechanism

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Urs Schreiber
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?

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user user29224
@UrsSchreiber: Oh my god, that's an Excellent article!

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Dimensio1n0
@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/… .

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Urs Schreiber
@DImension10AbhimanyuPS: Concerning retagging, I would say that the QED tag is not appropriate here, since the relationship between GR and EM is mainly at the classical level.

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Qmechanic
@Qmechanic: I added it in, because the U(1) bundle thing is relevant to QED. However, feel free to remove it, of course.

This post imported from StackExchange Physics at 2014-03-24 03:25 (UCT), posted by SE-user Dimensio1n0

1 Answer

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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.

answered Sep 3, 2013 by dimension10 (1,950 points) [ revision history ]
edited Jan 31, 2015 by dimension10

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