Disclaimer: while I have a good grasp of GR fundamentals, it is not my area of expertise.

The gravitational mass is distinct from the inertial mass as follows. The gravitational mass defines how strongly the body curves space-time, qualitatively it answers the question "how strong is the gravitational force from this object?". The inertial mass defines how the body reacts to forces, qualitatively it answers the question "how will the body react to the application of a given force".

The metric you give is approximated to first order near $r\rightarrow\infty$ by:

$${\rm d}s^2 \approx -\left(1-\frac{4m}{\alpha r}\right){\rm d}t^2 + \left(1+\frac{4(\beta+1)m}{\alpha r}\right)\left({\rm d}r^2+r^2{\rm d}\Omega^2\right) + \left(1-\frac{4\beta m}{\alpha r}\right){\rm d}{x_5}^2$$

For the gravitational mass, there is a similarity with the Schwarzschild metric. Begin with the approximate (at $r\rightarrow\infty$) $g_{00}$:

$$g_{00} \approx -\left(1-\frac{4m}{\alpha r}\right)$$

Compare this to the Schwarzschild solution in isotropic coordinates, also expanded at $r\rightarrow\infty$:

$$ds^2_{\rm Scwarzschild} \approx -\left(1-\frac{2M}{r}\right){\rm d}t^2 + \left(1+\frac{2M}{r}\right)\left({\rm d}r^2+r^2{\rm d}\Omega^2\right)$$

$$g_{00,\rm Schwarzschild} = -\left(1-\frac{2M}{r}\right)$$

Notice that $M\leftrightarrow 2m/\alpha$. I'm not sure why there's a difference of a factor of $2$, but I think it may have to do with what Jerry Schirmer mentioned in his answer, with $g_{00} = -\left(1-\frac{1}{2}\phi(\vec{x})\right)$; they do mention (about the Schwarzschild metric):

[...] and the gravitational potential is $\frac{1}{2}(g_{00}+1)=\frac{M}{R}$.

For the inertial mass, there's a $4(\beta+1)m/\alpha r$ (the inertial mass they quote, and again a factor of $2$) in the approximation to the metric that I gave where you'd expect to find a $2M$ in the Schwarzschild case, this time in the spatial coordinate term.

If anyone can explain that factor of $2$, I'm curious...

This post imported from StackExchange Physics at 2014-07-11 16:28 (UCT), posted by SE-user Kyle