The Unification of Gravity and E

1
The Unification of Gravity and EM 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 37 895 (1926).
• O. Klein, Nature 118 516 (1926).

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Outline

• Electromagnetic Theory
• Differential form of the Maxwell equations
• Scalar and vector potentials in EM
• Maxwells equations in terms of the potentials
• Relativistic form of the Maxwell equations
• Intro to Einsteins General Relativity
• Kaluza-Klein metric ansatz in 5D
• Einstein field equations in 5D

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Maxwells equations in differential form (in
vacuum)

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells 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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Maxwells equations

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells Correction
Q Can we write the Maxwell eqns in terms of
potentials?
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E, B in terms of A, F

• F is called the Scalar Potential
• is called the Vector Potential

? Write the Maxwell equations in terms of the
potentials.
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Maxwells equations in terms of the Scalar
Vector Potentials

Gauss Law
Amperes Law
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Gauge Invariance of A, F..

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, Aa, asDefine the
4-vector current density, Ja, asDefine 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 ß0 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

• Nature pure gravity
• Mathematics of 4D GR can be extended to 5D
• No dependence on the 5th coordinate

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Assumptions of Kaluza

• Nature pure gravity
• Mathematics of 4D GR can be extended to 5D
• No dependence on the 5th 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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GR in 5D..

• Parameterize the 5D metric tensor as..
• where ,
• Notice Aa is a 4-vector.
• Q Is Aa the 4-vector potential?

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GR in 5D..

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

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This metric ansatz yields the 5D line element..

• Notice
• The line element is invariant under translations
  in y

According to Kaluza-Klein theory Gauge
invariance arises from translational invariance
in y!
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Plugging our metric ansatz into the 5D GR eqns
yields..

where
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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!

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

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

The 4D EM stress-energy tensor!
where
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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
• 5th dimension is not observed!
• Why does the metric tensor the vector
  potential not depend on the 5th dimension?

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Kaluza-Klein Compactification
Consider a 5D theory, w/ the 5th dimension
periodic
http//images.iop.org/objects/physicsweb/world/13/
11/9/pw1311091.gif
where

• Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math.
  Phys. 1921 966972
• Klein, Oskar (1926) Zeitschrift für Physik, 37
  (12) 895906

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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