the unification of gravity and e&m via kaluza -klein theory
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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 37 895 (1926). - PowerPoint PPT PresentationTRANSCRIPT
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..
€
∇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:
€
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: