Is the determinant of metric tensor stationary wrt. proper time for a particle moving along its world line? While writing the expression for stress energy tensor of a free massive particle moving along its world-line some authors take out of the integral sign, the $\sqrt{-g}$ where $g$ is the metric tensor determinant. It is as if $\sqrt{-g}$ is independent of proper time $\tau$. Is it true and can it be proved both mathematically and intuitively? Refer for example M.P. Hobson’s book eq. (8.25).
 A: The fact that the determinant of the metric is constant is false even with Euclidean metrics (however the text you point out may have another intepretation as discussed in Mmeent's answer).
Consider the polar coordinates $r, \theta$ in $R^2$ and a radial geodesic $r=s$, $\theta =$ constant. Its length parameter $s$ (corresponding to the proper time in Lorentzian geometry) is the radial coordinate itself. The determinant of the Euclidean metric is $g(r, \theta) = r^2$. Along the said geodesic it evidently varies.
The Lorenzian version of the same phenomenon is immediately obtained if dealing with the Lorentzian  metric
$$ds^2 = -dt^2 + t^2 d\theta^2\:.$$
ADDENDUM. If working in a normal coordinate system adapted to the geodesic (also known as Fermi coordinates system), and the coordinate alongo the curve is just the proper time if dealing with a timelike geodesic, the components of the metric turn out to be  constant along the curve, so that the determinant is also constant with respect to the proper time. 
A: (I don't have Hobson's book here, but I guess this is what is happening:)
It depends on whether $\sqrt{-g}$ is being evaluated at the "field point" or "particle position". The integrand will look something like this:
$$ \int d\tau f(x) \delta^4( x-x_0(\tau)) = \int d\tau f(x_0(\tau)) \delta^4( x-x_0(\tau)),$$
where $f$ is some appropriate function (invloving $\sqrt{-g}$). The left and right are equal because of the usual identities of delta functions. In the left version $f(x)$ does not depend on $\tau$ and can be written outside the integral.
