In this paper the author derives a "Polyakov style" $p$-brane action which is given by
\begin{equation} S_{p}=-\frac{T_{p}}{2} \int d^{p+1} \xi \sqrt{-g}\left(g^{A B} h_{A B}-(p-1)\right) \tag{1} \end{equation}
where $h_{AB} = \partial_AX \cdot \partial_B X $ is the induced metric on the world volume of the $p$-brane, $g_{AB}$ is the intrinsic world volume metric, and $T_p$ is the $p$-brane tension. The sign convention is $(-,+,\ldots,+)$
A $0$-brane is a point particle with a non-negative mass $m = T_0$. So, substituting $p=0$ in the above equation we have
\begin{equation} \begin{aligned} S_{PP} &= -\frac{m}{2} \int d\tau \sqrt{-g_{00}} \left( g^{00} \dot{X}^2 +1 \right), \ h_{00} =\dot{X}^2 \equiv \partial_0X \cdot \partial_0 X\\ &= \frac{1}{2} \int d\tau \left( \frac{m}{\sqrt{-g_{00}}} \dot{X}^2 - m \sqrt{-g_{00}} \right) \ \end{aligned} \tag{2} \end{equation}
where I have used $g^{00} = 1/g_{00}$.
This form is similar to the famous einbein action for a point particle
\begin{equation} S_{einbein} = \frac{1}{2} \int d\tau \left( \frac{\dot{X}^2}{e} - e m^2 \right) \tag{3} \end{equation} In order to make (2) have the same form of (3), I defined
$$e \equiv \frac{\sqrt{-g_{00}}}{m} \tag{4}$$
I'm not sure if this definition is consistent, because the einbein action is supposed to work for both massive and massless particles and my $e$ is ill defined for $m=0$. Where's my mistake?
