This paper calculates the Kaluza field equations with the aid of a computer package for tensor algebra, xAct. The xAct file is provided with this paper. We find that Thiry’s field equations are correct, but only under limited circumstances. The full five-dimensional field equations under the cylinder condition are provided here, and we see that most of the other references miss at least some terms from them. We go on to establish the remarkable Kaluza Lagrangian, and verify that the field equations calculated from it match those calculated with xAct, thereby demonstrating self-consistency of these results. Many of these results can be found scattered throughout the literature, and we provide some pointers for historical purposes. But our intent is to provide a definitive exposition of the field equations of the classical, five-dimensional metric ansatz of Kaluza, along with the computer algebra data file to verify them, and then to recover the unique Lagrangian for the theory. In common terms, the Kaluza theory is an “

In 1921, Kaluza [

Moreover, there are diverse expressions in the literature for the associated Kaluza Lagrangian density, and constructions of ever-more-elaborate scalar field Lagrangians [

The field equations are difficult to calculate, and neither Thiry nor any of the other references show their calculations. It is difficult to make sense of the differing results and contradictions in the literature. The results presented here are perhaps found in German or French papers from the 1940s and 1950s, as suggested by [

In the following, we use Greek indices for spacetime components and the index

The Kaluza field equations follow from applying the machinery of general relativity to the ansatz for the 5D metric

The terms quadratic in

Along with the form (

Consider the 5D Ricci tensor

Since

It is customary to set

Since

For the spacetime part of the 5D Ricci tensor, xTensor obtains

For the 5D scalar curvature, xTensor obtains

Let us now assemble the 5D Einstein tensor

For the 55-component, xTensor obtains

For the mixed component of the Einstein tensor, xTensor obtains

For the spacetime part of the Einstein tensor, we construct analytically from the foregoing and verify with xTensor:

It is important and illustrative to understand the assumptions and limitations in Thiry’s results. Kaluza’s original paper contained the result, indeed, the compelling aesthetic feature of the theory, that the gravitational equations with electromagnetic sources could be understood as vacuum equations in a higher-dimensional space. Einstein himself was always aware that the curvature terms of the Einstein equations had a deep and profound beauty that was not reflected in the rather ad hoc stress-energy terms they were equated to, and it was his vision to discover how the matter sources might actually spring from the fields, uniting matter and fields.

So it is sufficient to set the 5D Ricci scalar

While Thiry does construct the 5D Einstein tensor

As for the 5D Ricci tensor expressions, the Thiry expressions for the 5D Einstein tensor also drop terms. The expression given for

Interestingly, during his discussion of the 5D Einstein tensor

Another problem with the Thiry expressions for the vacuum equations is that they are not self-consistent. Because of the missing terms, one cannot construct Thiry’s expression for

Among more recent review articles, Overduin and Wesson [

We now turn to the Lagrange density of the 5D theory. This will provide an independent check on the field equations just derived and allow us to determine the Lagrangian that corresponds to the field equations. But it will also allow us to evaluate the 5D theory in the context of other scalar field theories. We will find that unnecessary parameterizations of the scalar field have polluted the 5D Lagrangian, which actually needs no kinetic term, contrary to what one commonly sees in the literature, even going back to the work of Jordan in the 1940s [

There is of course a freedom in the choice of Lagrangian. The foregoing development was based on the ansatz (

We can combine (

Let us consider the field equations implied by (

In their review article, Overduin and Wesson [

Yet we see that, for the Kaluza Lagrangian (

It is an expression of the elegance of the Kaluza hypothesis that the scalar field has no tunable coupling parameter.

Now we proceed to evaluation of the Lagrangian (

Consider now the field equations for

Consider now the field equations for the inverse metric

Varying the first term in the Lagrangian (

Varying the second term in the Lagrangian (

Now put together the pieces (

So this shows that the unique 5D Hilbert Lagrangian (

This appendix gathers the connections calculated from (

All the connections are conveniently written as the six connections that distinguish the fifth coordinate,

There are a couple of convenient identities:

The author declares that there is no conflict of interests regarding the publication of this paper.