Inertial mass and gravitational mass of 5 dimensional stars

Question

Consider the following metric which is 5 dimensional (2-parameter) spherically symmetric Kaluza-Klein solution

$$-\left(\frac{1-m/r}{1+m/r}\right)^{2/\alpha}dt^2+(1+\frac{m}{r})^4\left(\frac{1-m/r}{1+m/r}\right)^{2(\alpha-\beta-1)/\alpha}(dr^2+r^2d\Omega^2)+\left(\frac{1-m/r}{1+m/r}\right)^{2\beta/\alpha}dx_5^2$$

where $x_5$ is the periodic fifth coordinate. In this paper (page 15) I read

the inertial mass of the star can be determined by the asymptotic behavior of this metric, assuming no interior singularities, and is equal to $M_{in}=\frac{(1+\beta)m}{\alpha}$

What should I understand as "inertial mass" of a star here? How can I see that this formula represents inertial mass?

In the same page, just a little below it says

the gravitational mass of the stars can be determined by the asymptotic form of $g_{00}$ and is given by $M_g=m/\alpha$

In this case, what should I understand as "gravitational mass" here? Again, how can I see that this formula represents gravitational mass?

Answer 1

Are they identifying inertial mass with the ADM mass?

There is an old style way of writing $g_{00}$ of a stationary metric, far from the matter source, as $-\left(1 - \frac{1}{2}\phi({\vec x})\right)$, where $\phi$ is the potential function for the metric, and then you can monopole expand this and identify the numerator of the $\frac{1}{r}$ term of the expansion with the "mass" of the body.${}^{1}$. This interpretation would explain the rest of their statements (as well as the dependence of the "gravitational mass" on $\alpha$ but not $\beta$). This is difficult to answer without more context, though.

${}^{1}$Myself, this metric is simple enough that I"d just look for geodesics with $\dot x_{5} = 0$, and do ordinary physics to find the "gravitational mass" of the object.

Answer 2

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...