Inhomogeneous Wave Eq. & Minkowski Spacetime Interval

[Prez Cannady]

Answer is probably not, but is there some connection between the inhomogeneous wave equation with a constant term and the spacetime interval in Minkowski space?

$$
1) ~~ \nabla^2 u - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = \sum_{i=0}^2 \frac{\partial^2 u}{\partial x_i^2} - \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} = S
$$
$$
2) ~~ \sum_{i=0}^2 dx_i^2 - c^2 dt^2 = ds^2
$$

 

[haushofer]

The only connection I can think of, is that you use the very same metric in both equations. But I suspect you're looking for connections which aren't to be found.

 

[robphy]

Tensorially, these are written as
$g^{ab}\nabla_a\nabla_b u=S$
$g_{ab}ds^a ds^b=ds^2$

Using the shorthand
$\nabla^b\equiv g^{ab} \nabla_a$
$ds_{b}\equiv g_{ab}ds^a$
known as raising and lowering indices,
we have
$\nabla^b\nabla_b u=S$
$ds_b ds^b=ds^2$

Using more index-gymnastics,
we can write
$\nabla^b\nabla_b u=S$
$ds^b ds_b=ds^2$

As @haushofer said, both equations use the same metric.
The first equation is an equality of scalar fields, the d’Alembertian of a scalar field $u$ and a scalar field $S$ .
The second equation is an infinitesimal scalar [at a point] using the RHS as an abbreviation of the LHS.