## PHY450H1S. Relativistic Electrodynamics Lecture 23 (Taught by Prof. Erich Poppitz). Energy Momentum Tensor.

Posted by peeterjoot on April 10, 2011

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# Reading.

FIXME: Covering chapter X material from the text [1].

Covering lecture notes pp. 173-178: the force on a surface element of a body (177-178); a plane wave example (179-180).

pp. 181-195: [read on your own: on diagonalizability of energy-momentum tensor (181)]; the Lagrangian for a system of nonrelativistic charged particles to zeroth order in : electrostatic energy of a system of charges and .mass renormalization. (182-189) [Tuesday, Mar. 29]; the EM potentials to order (190-193); the “Darwin Lagrangian. and Hamiltonian for a system of nonrelativistic charged particles to order and its many uses in physics (194-195) [Wednesday, Mar. 30]

Next week (last topic): attempt to go to the next order – radiation damping, the limitations of classical electrodynamics, and the relevant time/length/energy scales.

# Recap.

Last time we found that spacetime translation invariance led to the four conservation relations

where

last time we found for

Here

# Spatial components of

Now for we write

so we write

Recall that we argued that

(it also comes from Noether’s theorem).

or

Integrating over we have

We write this as

and describe our spatial tensor components as

latex \alpha$-th momentum through a unit area },\end{aligned} \hspace{\stretch{1}}(3.11)$

where

We define

This is the Maxwell stress tensor. Maxwell apparently derived this without any use of four vectors or symmetry arguments. I’d be curious what his arguments were and how he related this to the Lorentz force?

In gigantic matrix form, where symmetric opposites are omitted, we have for

In words this matrix is

latex \hat{\mathbf{x}}$}) & \frac{1}{{c}} (\text{energy flux in }) & \frac{1}{{c}} (\text{energy flux in }) \\ c \times (\text{momentum density})^x& (\text{momentum})^x \text{flux in }& (\text{momentum})^x \text{flux in }& (\text{momentum})^x \text{flux in } \\ c \times (\text{momentum density})^y& (\text{momentum})^y \text{flux in }& (\text{momentum})^y \text{flux in }& (\text{momentum})^y \text{flux in } \\ c \times (\text{momentum density})^z& (\text{momentum})^z \text{flux in }& (\text{momentum})^z \text{flux in }& (\text{momentum})^z \text{flux in } \\ \end{bmatrix}\end{aligned} \hspace{\stretch{1}}(3.14)$

# On the geometry.

PICTURE: rectangular area with normal , and area .

is the amount of that goes through unit area in unit time. (no sum) is the amount of through in unit time.

For a general surface element

PICTURE: normal decomposed into perpendicular components , , with respective area elements and .

PICTURE: triangulated area element decomposed into three perpendicular areas with their respective normals.

We have

Write

where . The amount of momentum that goes through in unit time is

If this is greater than zero, this is a flow in the direction, whereas if less than zero the momentum flows in the direction.

If is at the surface of the body, the rate of flow of through is the that acts on this element.

PICTURE: arbitrary surface depicted with an inwards normal .

For this surface with the inwards normal we can write

The acting on the surface element. With an outwards normal we can write this in terms of the Maxwell stress tensor, which has an inverted sign

To find the force on the body we want

We can calculate the EM force on any body. We need to know on the surface, so we need the EM field on this boundary.

## Example. Wall absorbing all radiation hitting it.

With propagation direction along the direction, and mutually perpendicular and .

latex P^x$ going in unit area in unit time}\end{aligned} \hspace{\stretch{1}}(4.21)$

with , our fields are

The off diagonal components vanish since we have no non-zero pair of or . Our other two diagonal terms are also zero

For non-perpendicular reflection we have the same deal.

PICTURE: reflection off of a wall, with reflection coefficient .

# References

[1] L.D. Landau and E.M. Lifshitz. *The classical theory of fields*. Butterworth-Heinemann, 1980.

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