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The Unification of Gravity and E&M via Kaluza-Klein Theory Chad A. Middleton Mesa State College September 16, 2010 Th. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Klasse 996 (1921). O. Klein, Z.F. Physik (1926). O. Klein, Nature (1926).

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Outline… Electromagnetic Theory Differential form of the Maxwell equations Scalar and vector potentials in E&M Maxwell’s equations in terms of the potentials Relativistic form of the Maxwell equations Intro to Einstein’s General Relativity Kaluza-Klein metric ansatz in 5D Einstein field equations in 5D

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Maxwell’s equations in differential form (in vacuum) Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction these plus the Lorentz force completely describe classical Electromagnetic Theory

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Taking the curl of the 3rd & 4th eqns (in free space when = J = 0) yield.. The wave equations for the E-, B-fields with predicted wave speed Light = EM wave! Notice the similarity between the treatment of space & time.

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Maxwell’s equations… Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction Q: Can we write the Maxwell eqns in terms of potentials?

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E, B in terms of A, Φ … Φ is called the Scalar Potential is called the Vector Potential Write the Maxwell equations in terms of the potentials.

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Maxwell’s equations in terms of the Scalar & Vector Potentials Gauss’ Law Ampere’s Law

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Gauge Invariance of A, Φ.. Notice: E & B fields are invariant under the transformations: for any function Show gauge invariance of E & B.

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Introducing 4-vector calculus.. Define the 4-vector potential, A α, as… Define the 4-vector current density, J α, as… Define the 4-vector operator…

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Relativistic form of the Maxwell Eqns.. where is called the EM field-strength tensor. Notice: The gauge invariance of the 4-vector potential becomes Calculate β =0 component of the Maxwell equation

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In 1915, Einstein gives the world his General Theory of Relativity describes the curvature of spacetime describes the matter & energy in spacetime

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When forced to summarize the general theory of relativity in one sentence; time and space and gravity have no separate existence from matter - Albert Einstein Matter tells space how to curve Space tells matter how to move

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Line element in 4D curved spacetime is the metric tensor defines the geometry of spacetime Know, know geometry i.e. In flat space:

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Assumptions of Kaluza… 1. Nature = pure gravity 2. Mathematics of 4D GR can be extended to 5D 3. No dependence on the 5 th coordinate

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Assumptions of Kaluza… 1. Nature = pure gravity 2. Mathematics of 4D GR can be extended to 5D 3. No dependence on the 5 th coordinate O. Klein discovers a way to drop this assumption.

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GR in 5D.. The 5D metric tensor can be expressed as.. Notice: from a 4D viewpoint, these are a tensor, a vector, and a scalar where the indicies range over the values

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Parameterize the 5D metric tensor as.. where, Notice: A α is a 4-vector. Q: Is A α the 4-vector potential? GR in 5D..

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Parameterize the 5D metric tensor as.. where, Notice: A α is a 4-vector. Q: Is A α the 4-vector potential? A: Only if it satisfies the Maxwell Equations! GR in 5D..

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Notice: The line element is invariant under translations in y: This metric ansatz yields the 5D line element.. According to Kaluza-Klein theory: Gauge invariance arises from translational invariance in y!

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where Plugging our metric ansatz into the 5D GR eqns yields..

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where Plugging our metric ansatz into the 5D GR eqns yields.. The 4D Einstein equations with matter (radiation) from Einstein eqns in 5D w/out matter!

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where Plugging our metric ansatz into the 5D GR eqns yields.. The Maxwell equations in 4D in the absence of a current!

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where Plugging our metric ansatz into the 5D GR eqns yield.. The 4D EM stress-energy tensor!

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Conclusions According to Kaluza-Klein theory: 5D Einstein equations in vacuum induce 4D Einstein equations with matter (EM radiation) Electromagnetic theory is a product of pure geometry Gauge invariance arises from translational invariance in the extra dimension. Shortcomings: 5 th dimension is not observed! Why does the metric tensor & the vector potential not depend on the 5 th dimension?

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Kaluza-Klein Compactification Consider a 5D theory, w/ the 5 th dimension periodic… where Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math. Phys. 1921: 966–972 Klein, Oskar (1926) Zeitschrift für Physik, 37 (12): 895–906

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The Maxwell & GR equations of are derivable from an action, just like the Lagrange eqns. Classical Dynamics: Electromagnetic Theory: General Relativity:

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