Tensors for Physics

18.5 Force Density and Stress Tensor 387

f^4 =jνEν/c=ρvνEν/c=vνkν/c. (18.75)

By analogy to the 3D description, where the force density is expressed as a spatial
derivative of the Maxwell stress tensorTμνaccording tokμ=∇νTμν, cf. Sect.8.5.4,
the force densityfiis related to the 4D stress tensorTkiby

fi=∂kTki. (18.76)

18.5.2 Maxwell Stress Tensor

The explicit expression for the stress tensor in terms of the field tensor is obtained
from (18.74) with the help of the Maxwell equations (18.58), (18.62). The derivation
is deferred to the Exercise18.3. For a linear medium, the result is

Tki=gmFmiHk−

1

4

gikFmHm. (18.77)

For a comparison with the 3D tensorTμνnotice thatgik,in(18.77) plays the role of
δμνin (8.120) and (8.121), furthermore one has

1

4

FmHm=

1

2

(E·D−B·H).

The top-left 3 by 3 part of the 4D tensor, i.e. elements in the which do not involve the
component 4, are equal to the components of the 3D Maxwell stress tensor (8.122).
The componentsT^14 ,T^24 ,T^34 are linked with the Poynting vectorS=E×H,cf.
(8.110), while the componentsT^41 ,T^42 ,T^43 are proportional to the density of the
linear momentum of the electromagnetic fields, cf. (8.125), viz.

jS=D×B. (18.78)

TheT^44 component is essentially the energy densityu=^12 (E·D+B·H).Inmatrix
notation, one has

Tik=

⎛

⎜

⎜

⎜

⎝

T 11 Max T 12 Max T 13 Max −^1 cS 1

T 21 Max T 22 Max T 23 Max −^1 cS 2

T 31 Max T 32 Max T 33 Max −^1 cS 3

−cjS1−cjS2−cjS3 −u

⎞

⎟

⎟

⎟

⎠

. (18.79)

Here the components of the 3D Maxwell tensor are denoted byT..Max. For electro-
magnetic fields in vacuum and for linear isotropic media one hasTik =Tki.In
general, however, the stress tensorTikis not symmetric.