EM Wave Equation in Higher Dimensions: Gravitation Text

[snowstorm69]

Caution: I'm new at this stuff.
On page 573 of Gravitation (Misner, Thorne, Wheeler), they write down what I think is the electromagnetic wave equation for a discussion on Optics, "Next insert the vector potential (22.25) into the source-free wave equation (22.19d):"
I am wondering if the equations mentioned, 22.19d, and 22.25, are applicable to any dimension, for example 4+1 dimensions as in Kaluza-Klein theory, and higher dimensions?

I ask this because I was wondering how to write the good ole equation for a propagating electromagnetic wave in higher dimensions, such as 4+1 or higher. I have read that the curl cannot be written in some higher dimensions in the customary way at least, without going to differential geometry and topology. And so I was wondering if these 2 equations, written with tensors, hold for higher dimensions?
Thanks.

 

[Markus Hanke]

Mathematically and notationally, the form of the de Rham wave equation 22.19d would make sense in 4+1 dimensions ( or more generally in D+1 with D>3 ) just as well as it does in 3+1 dimensions; all that changes is the range of values that the indices cover. The altogether more interesting and much harder to answer question is whether this wave equation is physically meaningful in anything else but (3+1) dimensional space-time, i.e. whether it is the correct model for the propagation of electromagnetic waves in curved space-times of that dimensionality. But I will leave this to the experts here to address.