The Unification of Gravity and E&M via Kaluza-Klein Theory

The Unification of Gravity and E&M via Kaluza-Klein Theory

Chad A. MiddletonMesa State CollegeSeptember 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).

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

Maxwell’s equations in differential form (in vacuum)

€

r∇ ⋅

rE =

ρ

ε0

€

r∇ ×

rE +

∂r B

∂t= 0

Gauss’ Law for E-field

Gauss’ Law for B-field

Faraday’s Law

Ampere’s Law with Maxwell’s Correction

€

r∇ ⋅

rB = 0

€

r∇ ×

rB − μ0ε0

∂r E

∂t= μ0

r J

€

rF = q

r E +

r v ×

r B ( )

these plusthe Lorentz force completely describe

classical Electromagnetic Theory

Taking the curl of the 3rd & 4th eqns (in free space when = J = 0) yield..

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∇2 −1

c 2

∂ 2

∂t 2

⎡

⎣ ⎢

⎤

⎦ ⎥r E = 0

The wave equations for theE-, B-fields with

predicted wave speed

€

∇2 −1

c 2

∂ 2

∂t 2

⎡

⎣ ⎢

⎤

⎦ ⎥r B = 0

Light = EM wave!

€

c =1

μ0ε0

≅ 3.0 ×108 m /s

Notice the similarity between the treatment of space & time.

Maxwell’s equations…

€

r∇ ⋅

rE =

ρ

ε0

€

r∇ ×

rE +

∂r B

∂t= 0

Gauss’ Law for E-field

Gauss’ Law for B-field

Faraday’s Law

Ampere’s Law with Maxwell’s Correction

€

r∇ ⋅

rB = 0

€

r∇ ×

rB − μ0ε0

∂r E

∂t= μ0

r J

Q: Can we write the Maxwell eqns in terms of potentials?

E, B in terms of A, Φ…

€

rB =

r ∇ ×

r A

€

rE = −

r ∇φ −

∂r A

∂t Φ is called the Scalar Potential is called the Vector Potential

€

rA

Write the Maxwell equations in terms of the potentials.

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

€

∇2φ +∂

∂t

r ∇ ⋅

r A ( ) = −

ρ

ε0

r ∇

r ∇ ⋅

r A ( ) − ∇ 2

r A − μ0ε0

∂ 2r A

∂t 2

⎡

⎣ ⎢

⎤

⎦ ⎥+ μ0ε0

∂

∂t

r ∇φ( ) = μ0

r J

Gauss’ Law

Ampere’s Law

Gauge Invariance of A, Φ..

€

rB =

r ∇ ×

r A

Notice:E & B fields are invariant under the transformations:

€

φ→ ′ φ =φ−∂Λ∂t

for any function

€

Λ=Λ(r r , t)

€

rE = −

r ∇φ −

∂r A

∂t

€

rA →

r ′ A =

r A +

r ∇Λ

Show gauge invariance of E & B.

Introducing 4-vector calculus..

Define the 4-vector potential, Aα, as…

Define the 4-vector current density, Jα, as…

Define the 4-vector operator…

€

Aα ≡ (A0,r A ) = φ /c,

r A ( )

€

Jα ≡ (J 0,r J ) = ρc,

r J ( )

€

∂α ≡(∂0,r

∇) =1

c

∂

∂t,r

∇ ⎛

⎝ ⎜

⎞

⎠ ⎟ & ∂α ≡ (∂ 0,

r ∇) = −

1

c

∂

∂t,r

∇ ⎛

⎝ ⎜

⎞

⎠ ⎟

Relativistic form of the Maxwell Eqns..

€

∂αF αβ = μ0Jβ

where is called the EM field-strength tensor.

€

F αβ = ∂α Aβ −∂ β Aα

Notice:The gauge invariance of the 4-vector potential becomes

€

Aα → ′ A α = Aα + ∂α Λ(x μ )

Calculate β=0 component of the Maxwell equation

In 1915, Einstein gives the world his General Theory of Relativity

describes the curvature of spacetime

describes the matter & energy in spacetime

€

Gαβ = 8πGTαβ

€

Gαβ

€

Tαβ

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

Line element in 4D curved spacetime

€

ds2 = gαβ dxα dx β

is the metric tensor

€

gαβ

defines the geometry of spacetime

Know , know geometry

€

gαβ

€

gαβ

i.e. In flat space:

€

ds2 = −c 2dt 2 + dx 2 + dy 2 + dz2

Assumptions of Kaluza…

1. Nature = pure gravity

2. Mathematics of 4D GR can be extended to 5D

3. No dependence on the 5th coordinate

Assumptions of Kaluza…

1. Nature = pure gravity

2. Mathematics of 4D GR can be extended to 5D

3. No dependence on the 5th coordinate

O. Klein discovers a way to drop this assumption.

GR in 5D..

€

ˆ g AB =ˆ g αβ

ˆ g α 5

ˆ g 5βˆ g 55

⎛

⎝ ⎜

⎞

⎠ ⎟

The 5D metric tensor can be expressed as..

€

A,B = 0,1,2,3,5 & α ,β = 0,1,2,3

Notice:• from a 4D viewpoint, these are a tensor, a vector, and a scalar• where the indicies range over the values

Parameterize the 5D metric tensor as..

where ,

Notice: Aα is a 4-vector. Q: Is Aα the 4-vector potential?

GR in 5D..

€

ˆ g AB =gαβ + κ 2Aα Aβ κAα

κAβ 1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

gαβ = gαβ (x μ )

€

Aα = Aα (x μ )

Parameterize the 5D metric tensor as..

where ,

Notice: Aα is a 4-vector. Q: Is Aα the 4-vector potential?A: Only if it satisfies the Maxwell Equations!

GR in 5D..

€

ˆ g AB =gαβ + κ 2Aα Aβ κAα

κAβ 1

⎛

⎝ ⎜

⎞

⎠ ⎟

€

gαβ = gαβ (x μ )

€

Aα = Aα (x μ )

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

This metric ansatz yields the 5D line element..

€

ds2 = gαβ dxα dx β + (κAα dxα + dy)2

€

Aα → ′ A α = Aα + ∂α Λ(x μ )

€

y → ′ y = y −κΛ(x μ )

According to Kaluza-Klein theory: Gauge invariance arises from translational invariance in y!

where

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

€

Gαβ =1

2κ 2T EM

αβ

∇α Fαβ = 0

€

T EMαβ =

1

4gαβ Fμν F μν − F μ

α Fβμ

where

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

€

Gαβ =1

2κ 2T EM

αβ

∇α Fαβ = 0

€

T EMαβ =

1

4gαβ Fμν F μν − F μ

α Fβμ

The 4D Einstein equations with matter (radiation) from Einstein eqns in 5D w/out matter!

where

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

€

Gαβ =1

2κ 2T EM

αβ

∇α Fαβ = 0

€

T EMαβ =

1

4gαβ Fμν F μν − F μ

α Fβμ

The Maxwell equations in 4D in the absence of a current!

where

Plugging our metric ansatz into the 5D GR eqns yield..

€

Gαβ =1

2κ 2T EM

αβ

∇α Fαβ = 0

€

T EMαβ =

1

4gαβ Fμν F μν − F μ

α Fβμ

The 4D EM stress-energy tensor!

ConclusionsAccording 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?

Kaluza-Klein Compactification

R(5) R(4 ) 1

4F F

Consider a 5D theory, w/ the 5th dimension periodic…

F A A

A' A where

€

y = y + 2πR

•Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math. Phys. 1921: 966–972•Klein, Oskar (1926) Zeitschrift für Physik, 37 (12): 895–906

http://images.iop.org/objects/physicsweb/world/13/11/9/pw1311091.gif

The Maxwell & GR equations of are derivable from an action, just like the Lagrange eqns.

€

δ Ldt = 0∫ ⇒∂L

∂x−

d

dt

∂L

∂˙ x = 0

Classical Dynamics:

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δ −1

4F αβ Fαβ + μ0Aα Jα ⎛

⎝ ⎜

⎞

⎠ ⎟d4 x = 0∫ ⇒ ∂α F αβ = μ0J

β

€

δ −gR(4 )d4 x = 0∫ ⇒ Gαβ = 8πGTαβ

Electromagnetic Theory:

General Relativity: