9780192806727.pdf

330 THE LATER JOURNEY

in 1921 [K2], but he already had this idea in 1919, for in April of that year
Einstein wrote to him, 'The idea of achieving [a unified theory] by means of a
five-dimensional cylinder world never dawned on me. ... At first glance I like
your idea enormously' [E16]. Still very much in the Machian mood, Einstein
added that one ought to examine whether this new theory would allow for a sen-
sible solution of the cosmological problem.* A few weeks later, he wrote to him
again, 'The formal unity of your theory is startling'[E18]. In 1921 he communi-
cated Kaluza's work to the Prussian Academy. (I do not know why this publica-
tion was delayed so long.)
Kaluza's well-written paper contains nearly all the main points of the five-
dimensional approach:

1. The introduction of an invariant line element

in which the metric tensor 7^ satisfies two constraints. First, the 7^ shall

depend only on the space-time coordinates x', i' = 1,.. ,4:

Secondly, 755 is assumed to be a positive numerical invariant that may be nor-

malized such that

Thus we deal with a cylinder world, the fifth axis is preferred, the fifth direc-

tion is space-like. Equation 17.3 has become known as the cylinder condition.

2. 7,5, a 4-vector field relative to the Riemannian space-time submanifold R 4 , is
   assumed to be proportional to the electromagnetic potential.


3. The field equations are

where R^ and R are the familiar functions of the F^ and their first derivatives

and Tw is the energy-momentum tensor exclusive of the purely electromag-

netic contribution. Kaluza considered only the case where the source is a single

point particle with mass m and charge e, T^1 " = mu^u", u" = dx^/ds, and

showed that for ;u, v = i, j, one obtains the gravitational field equations; /n, v

= i, 5, yield the Maxwell equations; /*, v — 5, 5, reduces to a trivial identity.

The identification of the Maxwell equations requires that w^5 be proportional

to e/m. Thus mil* is the 5-vector of 'momentum-energy-charge.'

4. A geodesic in the cylinder world can be identified with the trajectory of a
   charged test particle moving in a combined gravitational-electromagnetic field.
   *As was mentioned in Section 15e, Einstein had used the cosmological term in 1919 for the purpose
   of linking electromagnetism to gravitation [E17].