Symmetric and conserved energymomentum tensors in moving media
Finn Ravndal, Institute of Physics,
University of Oslo, Norway.
• Photon momenta in media
• Minkowski and Abraham tensors
• A scalar field in the Gordon metric
• The new energy-momentum tensor
• Conclusion
Tuesday, 23 August 2011
Photon in medium with refractive index n:
Energy
E = hν
with
νλ = c = 1/n
i.e.
E = p/n
Abraham momentum:
and momentum
p = h/λ
Minkowski momentum
E = np
Quantum mechanics??
Photon mass
Tuesday, 23 August 2011
ntum
Minkowski:
Abraham:
E 2 − p2 = (1 − n2 )E 2 < 0
E 2 − p2 = (n2 − 1)p2 > 0
tachyon!
Maxwell equations:
∂B
∇×E+
= 0,
∂t
∇·B=0
∂D
∇×H−
= 0,
∂t
∇·D=0
Constitutive equations
D = εE
with index of refraction
Energy density
n=
√
B = µH
εµ
"
1!
E = E·D+B·H
2
and Poynting vector
S=E×H
Energy conservation:
∂E
+∇·S=0
∂t
Tuesday, 23 August 2011
Momentum density
G=D×B
and conservation
∂G
+∇·T=0
∂t
where
1
Tij = −(Ei Dj + Bi Hj ) + δij (E · D + B · H)
2
are Maxwell stresses.
Covariant formulation
Minkowski energy-momentum tensor:
µν
TM
=
�
D × B = n2 E × H
E
E×H
D×B
Tij
�
E
E×H
�
Abraham energy-momentum tensor:
µν
TA
Tuesday, 23 August 2011
=
�
E×H
Tij
Electromagnetic Lagrangian in dielectrikum:
1 2 2 1 2
L= n E − B
2
2
with
E = −∂A/∂t
and
B=∇×A
Instead consider scalar, massless field:
1 2 � ∂φ �2 1
L= n
− (∇φ)2
2
∂t
2
Can now make covariant formulation by considering field in
external metric:
η̂ µν = (n2 , −1, −1, −1) → η µν + (n2 − 1)uµ uν
=⇒
L = (1/2)η̂ µν ∂µ φ∂ν φ
Equation of motion: η̂ µν ∂µ ∂ν φ = [∂ µ ∂µ + (n2 − 1)(uµ ∂µ )2 ]φ = 0
Tuesday, 23 August 2011
Gordon, 1923:
Action
1
S[φ] =
2
η̂ µν → ĝ µν (x)
�
d4 x
�
Equivalent
gravitational
problem
−ĝ ĝ µν ∂µ φ ∂ν φ
y
x
z
Tuesday, 23 August 2011
Energy-momentum tensor from
ĝµν → ĝµν + δĝµν
Using
δĝ
and
�
�
δ −ĝ = (1/2) −ĝĝ αβ δĝαβ
=⇒
µν
= −ĝ
1
δS[φ] = −
2
µα νβ
�
ĝ
δĝαβ
�
d4 x −ĝ [ĝ αµ ĝ βν ∂µ φ ∂ν φ − ĝ αβ L]δĝαβ
with the result:
T̂
µν
= η̂
µα νβ
η̂
∂α φ ∂β φ − η̂
Symmetric:
T̂
Conserved:
∂µ T̂ µν = 0
Tuesday, 23 August 2011
µν
= T̂
νµ
µν
L
Electromagnetic fields
Field tensor
Fµν = ∂µ Aν − ∂ν Aµ
=
�
0
E
−E −Bij
�
Equation of motion:
∂λ Fµν + ∂ν Fλµ + ∂µ Fνλ = 0
Material fields
H µν = η̂ µα η̂ νβ Fαβ
=
Lagrangian
Tuesday, 23 August 2011
�
0
D
−D
−Hij
1
L = − Fµν H µν
4
Equation of motion:
�
∂µ H µν = 0
Energy-momentum tensor derived by same
procedure
=⇒
T̂
Symmetric
and conserved
µν
= η̂
µα
Fαβ H
βν
− η̂
µν
L
T̂ µν = T̂ νµ
∂µ T̂ µν = 0
Minkowski energy-momentum tensor:
T µν = η µα Fαβ H βν − η µν L
Tuesday, 23 August 2011
In medium rest frame:
T̂
µν
=
�
2
n E
D×B
D×B
T̂ij
�
D × B = n2 E × H
Extra factor in energy components from red shift ĝ00
NB!
1
= 2
n
There is another energy-momentum tensor that couples to
the metric gµν
This turns out to be exactly the Abraham tensor
that describes the coupled field-medium system!
F. Ravndal:
Tuesday, 23 August 2011
arXiv: 1107.5074