In relativistic physics , the electromagnetic stress–energy tensor is the contribution to the stress–energy tensor due to the electromagnetic field .[1] The stress–energy tensor describes the flow of energy and momentum in spacetime . The electromagnetic stress–energy tensor contains the negative of the classical Maxwell stress tensor that governs the electromagnetic interactions.
Definition [ edit ]
SI units [ edit ]
In free space and flat space–time, the electromagnetic stress–energy tensor in SI units is[2]
T
μ
ν
=
1
μ
0
[
F
μ
α
F
ν
α
−
1
4
η
μ
ν
F
α
β
F
α
β
]
.
{\displaystyle T^{\mu \nu }={\frac {1}{\mu _{0}}}\left[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right]\,.}
where
F
μ
ν
{\displaystyle F^{\mu \nu }}
is the electromagnetic tensor and where
η
μ
ν
{\displaystyle \eta _{\mu \nu }}
is the Minkowski metric tensor of metric signature (− + + +) . When using the metric with signature (+ − − −) , the expression on the right of the equation will have opposite sign.
Explicitly in matrix form:
T
μ
ν
=
[
1
2
(
ϵ
0
E
2
+
1
μ
0
B
2
)
S
x
/
c
S
y
/
c
S
z
/
c
S
x
/
c
−
σ
xx
−
σ
xy
−
σ
xz
S
y
/
c
−
σ
yx
−
σ
yy
−
σ
yz
S
z
/
c
−
σ
zx
−
σ
zy
−
σ
zz
]
,
{\displaystyle T^{\mu \nu }={\begin{bmatrix}{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)&S_{\text{x}}/c&S_{\text{y}}/c&S_{\text{z}}/c\\S_{\text{x}}/c&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\S_{\text{y}}/c&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\S_{\text{z}}/c&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}},}
where
S
=
1
μ
0
E
×
B
,
{\displaystyle \mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,}
is the Poynting vector ,
σ
i
j
=
ϵ
0
E
i
E
j
+
1
μ
0
B
i
B
j
−
1
2
(
ϵ
0
E
2
+
1
μ
0
B
2
)
δ
i
j
{\displaystyle \sigma _{ij}=\epsilon _{0}E_{i}E_{j}+{\frac {1}{\mu _{0}}}B_{i}B_{j}-{\frac {1}{2}}\left(\epsilon _{0}E^{2}+{\frac {1}{\mu _{0}}}B^{2}\right)\delta _{ij}}
is the Maxwell stress tensor , and c is the speed of light . Thus,
T
μ
ν
{\displaystyle T^{\mu \nu }}
is expressed and measured in SI pressure units (pascals ).
CGS units [ edit ]
The permittivity of free space and permeability of free space in cgs-Gaussian units are
ϵ
0
=
1
4
π
,
μ
0
=
4
π
{\displaystyle \epsilon _{0}={\frac {1}{4\pi }},\quad \mu _{0}=4\pi \,}
then:
T
μ
ν
=
1
4
π
[
F
μ
α
F
ν
α
−
1
4
η
μ
ν
F
α
β
F
α
β
]
.
{\displaystyle T^{\mu \nu }={\frac {1}{4\pi }}[F^{\mu \alpha }F^{\nu }{}_{\alpha }-{\frac {1}{4}}\eta ^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }]\,.}
and in explicit matrix form:
T
μ
ν
=
[
1
8
π
(
E
2
+
B
2
)
S
x
/
c
S
y
/
c
S
z
/
c
S
x
/
c
−
σ
xx
−
σ
xy
−
σ
xz
S
y
/
c
−
σ
yx
−
σ
yy
−
σ
yz
S
z
/
c
−
σ
zx
−
σ
zy
−
σ
zz
]
{\displaystyle T^{\mu \nu }={\begin{bmatrix}{\frac {1}{8\pi }}(E^{2}+B^{2})&S_{\text{x}}/c&S_{\text{y}}/c&S_{\text{z}}/c\\S_{\text{x}}/c&-\sigma _{\text{xx}}&-\sigma _{\text{xy}}&-\sigma _{\text{xz}}\\S_{\text{y}}/c&-\sigma _{\text{yx}}&-\sigma _{\text{yy}}&-\sigma _{\text{yz}}\\S_{\text{z}}/c&-\sigma _{\text{zx}}&-\sigma _{\text{zy}}&-\sigma _{\text{zz}}\end{bmatrix}}}
where Poynting vector becomes:
S
=
c
4
π
E
×
B
.
{\displaystyle \mathbf {S} ={\frac {c}{4\pi }}\mathbf {E} \times \mathbf {B} .}
The stress–energy tensor for an electromagnetic field in a dielectric medium is less well understood and is the subject of the unresolved Abraham–Minkowski controversy .[3]
The element
T
μ
ν
{\displaystyle T^{\mu \nu }\!}
of the stress–energy tensor represents the flux of the μ th-component of the four-momentum of the electromagnetic field,
P
μ
{\displaystyle P^{\mu }\!}
, going through a hyperplane (
x
ν
{\displaystyle x^{\nu }}
is constant). It represents the contribution of electromagnetism to the source of the gravitational field (curvature of space–time) in general relativity .
Algebraic properties [ edit ]
The electromagnetic stress–energy tensor has several algebraic properties:
T
μ
ν
=
T
ν
μ
{\displaystyle T^{\mu \nu }=T^{\nu \mu }}
The tensor
T
ν
α
{\displaystyle T^{\nu }{}_{\alpha }}
is traceless :
T
α
α
=
0
{\displaystyle T^{\alpha }{}_{\alpha }=0}
.
Proof
Starting with
T
μ
μ
=
η
μ
ν
T
μ
ν
{\displaystyle T_{\mu }^{\mu }=\eta _{\mu \nu }T^{\mu \nu }}
Using the explicit form of the tensor,
T
μ
μ
=
1
4
π
[
η
μ
ν
F
μ
α
F
ν
α
−
η
μ
ν
η
μ
ν
1
4
F
α
β
F
α
β
]
{\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}[\eta _{\mu \nu }F^{\mu \alpha }F^{\nu }{}_{\alpha }-\eta _{\mu \nu }\eta ^{\mu \nu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }]}
Lowering the indices and using the fact that
η
μ
ν
η
μ
ν
=
δ
μ
μ
{\displaystyle \eta ^{\mu \nu }\eta _{\mu \nu }=\delta _{\mu }^{\mu }}
T
μ
μ
=
1
4
π
[
F
μ
α
F
μ
α
−
δ
μ
μ
1
4
F
α
β
F
α
β
]
{\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}[F^{\mu \alpha }F_{\mu \alpha }-\delta _{\mu }^{\mu }{\frac {1}{4}}F^{\alpha \beta }F_{\alpha \beta }]}
Then, using
δ
μ
μ
=
4
{\displaystyle \delta _{\mu }^{\mu }=4}
,
T
μ
μ
=
1
4
π
[
F
μ
α
F
μ
α
−
F
α
β
F
α
β
]
{\displaystyle T_{\mu }^{\mu }={\frac {1}{4\pi }}[F^{\mu \alpha }F_{\mu \alpha }-F^{\alpha \beta }F_{\alpha \beta }]}
Note that in the first term, μ and α and just dummy indices, so we relabel them as α and β respectively.
T
α
α
=
1
4
π
[
F
α
β
F
α
β
−
F
α
β
F
α
β
]
=
0
{\displaystyle T_{\alpha }^{\alpha }={\frac {1}{4\pi }}[F^{\alpha \beta }F_{\alpha \beta }-F^{\alpha \beta }F_{\alpha \beta }]=0}
T
00
≥
0
{\displaystyle T^{00}\geq 0}
The symmetry of the tensor is as for a general stress–energy tensor in general relativity . The trace of the energy–momentum tensor is a Lorentz scalar ; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon .[4]
Conservation laws [ edit ]
The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is:
∂
ν
T
μ
ν
+
η
μ
ρ
f
ρ
=
0
{\displaystyle \partial _{\nu }T^{\mu \nu }+\eta ^{\mu \rho }\,f_{\rho }=0\,}
where
f
ρ
{\displaystyle f_{\rho }}
is the (4D) Lorentz force per unit volume on matter .
This equation is equivalent to the following 3D conservation laws
∂
u
e
m
∂
t
+
∇
⋅
S
+
J
⋅
E
=
0
{\displaystyle {\frac {\partial u_{\mathrm {em} }}{\partial t}}+\mathbf {\nabla } \cdot \mathbf {S} +\mathbf {J} \cdot \mathbf {E} =0\,}
∂
p
e
m
∂
t
−
∇
⋅
σ
+
ρ
E
+
J
×
B
=
0
{\displaystyle {\frac {\partial \mathbf {p} _{\mathrm {em} }}{\partial t}}-\mathbf {\nabla } \cdot \sigma +\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} =0\,}
(or equivalently
f
+
ϵ
0
μ
0
∂
S
∂
t
=
∇
⋅
σ
{\displaystyle \mathbf {f} +\epsilon _{0}\mu _{0}{\frac {\partial \mathbf {S} }{\partial t}}\,=\nabla \cdot \mathbf {\sigma } }
with
f
{\displaystyle \mathbf {f} }
being the Lorentz force density),
respectively describing the flux of electromagnetic energy density
u
e
m
=
ϵ
0
2
E
2
+
1
2
μ
0
B
2
{\displaystyle u_{\mathrm {em} }={\frac {\epsilon _{0}}{2}}E^{2}+{\frac {1}{2\mu _{0}}}B^{2}\,}
and electromagnetic momentum density
p
e
m
=
S
c
2
{\displaystyle \mathbf {p} _{\mathrm {em} }={\mathbf {S} \over {c^{2}}}}
where J is the electric current density and ρ the electric charge density .
See also [ edit ]
References [ edit ]
^ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
^ Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
^ however see Pfeifer et al., Rev. Mod. Phys. 79, 1197 (2007)
^ Garg, Anupam. Classical Electromagnetism in a Nutshell , p. 564 (Princeton University Press, 2012).