506 ELECTRODYNAMICS OF MOVING MEDIA

sion

of

the

conditions

for

the field

vectors at

the

boundary

between two media, and

derived

a

different

set

of

boundary

conditions.[23] In

a supplementary

note to

their

paper,[24]

Ein-

stein and Laub

acknowledged

Laue's

criticism and

gave

their

own

derivation of

his

bound-

ary

conditions.

Their

first

paper

had rederived and

applied

Minkowski's

version

of

the

field

equations.

In

their

second

paper,[25]

Einstein and Laub criticized his

expression

for

the

ponderomotive

force

density on a

conduction current element

in

a

magnetizable

medium.

Minkowski

gave

this force

density

as

[s,B],

the

vector

product

of

the current vector

density s

and the

mag-

netic induction

B.

Using arguments

based

on

the electron

theory, they rejected

this

expres-

sion and

proposed

another:

[s,H],

where H

is

the

magnetic

force, which

they supported

by a simple

example.[26]

Minkowski had shown

that,

from

the four-dimensional

point

of

view,

the

energy

den-

sity,

momentum

density,

and

Maxwell stress tensor

are

unified in

a

four-dimensional

ten-

sor.

The

tensor he

proposed

is

now

often referred to

as

Minkowski's

form

of

the stress–

momentum-energy

tensor

of

the

electromagnetic

field in

a

material

medium.[27]

By taking

the four-dimensional

divergence

of

this

tensor,

he

derived

an expression

for the total

pon-

deromotive force

density

exerted

on a magnetizable

medium

by

the

electromagnetic

field.

Starting

from

the

standpoint

of

the electron

theory,

Einstein and Laub derived

a

different

expression

for the total

ponderomotive

force

density

that includes

a

term

[s,H],

which

they interpreted as

the force

density per

unit volume

on a

conduction current element.[28]

They

also

gave

expressions

for the

components

of

the

stress-momentum-energy

tensor

of

the field in

a medium,

differing

from those

of

Minkowski,

from which

their

expression

for

the total

ponderomotive

force

density can

be

derived.[29]

Soon after

he

returned

to

Würzburg,

Laub wrote to Einstein

setting

forth

Wien's

objec-

tions to their

expression

for

the

ponderomotive

force

density.[30]

These

objections were

presumably

discussed in letters

now

missing,

for

two

months later Einstein

wrote

Laub:

"I

am

also

quite

firmly

convinced that

our expressions

for the

ponderomotive

forces

are

the correct

ones"

("Ich

bin auch

ganz

fest

überzeugt,

dass

unsere

Ausdrücke

für

die

pon-

deromotorischen Kräfte die

richtigen

sind").[31]

In

1910,

Einstein

gave a

talk in which he

[23]

For

their

original

boundary conditions,

see

Einstein

and

Laub 1908a

(Doc. 51), p.

535.

Laue's

letter

with his comments is mentioned in

Einstein

and Laub 1909

(Doc. 54), p.

445. Laue

published

a

discussion

of

these

boundary

condi-

tions in

Laue

1911b, pp.

127-129.

[24]

See

Einstein and

Laub 1909

(Doc. 54).

[25]

See Einstein

and

Laub 1908b

(Doc. 52).

[26]

See Einstein

and

Laub 1908b

(Doc. 52),

pp.

545-546.

[27]

See Minkowski

1908,

pp.

92-93, 97. Min-

kowski,

rather than

using

tensor notation,

actu-

ally

wrote

out the matrix

corresponding

to this

tensor. Abbreviated

and variant forms

of

the

name

"stress-momentum-energy

tensor"

are

common.

[28]

See

Einstein and

Laub 1908b

(Doc. 52),

§2.

[29]

See Einstein

and

Laub 1908b

(Doc. 52),

§

3.

Since

they

did

not

use

four-dimensional

no-

tation in this

paper,

Einstein and Laub did

not

combine these

expressions

into

a

four-dimen-

sional

tensor,

nor

take its four-dimensional di-

vergence.

[30]

See Jakob

Laub to

Einstein,

30

May

1908.

[31]

Einstein to Jakob

Laub,

30

July

1908. The

letter also indicates that

they were contemplating

additional calculations

involving

the

pondero-

motive

forces,

but Einstein

was

not enthusiastic

about the

prospect.