The form of the metric after a dimension is compactified Upon the compactifiation of one spatial dimension, it is said (as though  an axiom) that the 5 dimensional spacetime metric separates into a 4 dimensional metric, a vector, and a scalar, (4D gravity, electromagnetism, and a scalar). What  is the  reason  for  this? In general, how  can we tell  what  a metric  will break  up into? 
 A: Given a metric 
$$ \mathrm{d}s^2 = g_{MN}\mathrm{d}x^M\mathrm{d}x^N$$
in $n$ dimensions you find the decomposition down to $d$ dimensions by rewriting the $n$-dimensional metric in terms of objects with no, one, and two $d$-dimensional indices (in the following indicated by Greek letters):
$$ \mathrm{d}s^2 = g_{\mu\nu}\mathrm{d}x^\mu\mathrm{d}x^\nu + 2 A_{\mu N}\mathrm{d}x^\mu\mathrm{d}x^N + \phi_{MN}\mathrm{d}x^M\mathrm{d}x^N$$
where I have already suggestively renamed the metric components with less than 2 $d$-dimensional indices $A$ and $\phi$, and the sum over capital Latin indices only runs over the $n-d$ dimensions that are not labeled Greek. 
If you now just examine the transformation behaviour of the $g,A,\phi$ under $d$-dimensional transformations $x^\mu\mapsto x'^\mu(x^\nu)$ that leave the $n-d$ reduced/compactified dimensions $x^N$ alone, you directly see (from the transformation behaviour of $g_{MN}$) that each $A_{\mu N}$ for fixed $N$ transforms like a $d$-dimensional vector and each $\phi_{MN}$ transforms like a scalar, because transformations in the $d$ dimensions don't do anything to capital Latin indices at all. Therefore, a reduction from $n$ to $d$ dimensions generally gets you $n-d$ vectors and $(n-d)^2$ scalars.
