2.2 Center of Mass Dynamics

The dynamics of an extended body can be formulated in general as the composition of the center of mass motion and the motion relative to the center of mass.
A $p$-brane is by definition a spatially extended object. Thus we expect to be able to separate the motion of its center of mass from the shape-shifting about the center of mass. However, given the point like nature of the center of mass, its spacetime coordinates depend on one parameter only, say, the proper time $\tau$. Thus, the factorization of the center of mass motion automatically breaks the general covariance of the action in parameter space since it breaks the symmetry between the temporal parameter $\tau$ and the spatial coordinates $s ^i$. We can turn ``needs into virtue'' by choosing a coordinate mesh on $\Sigma_{p+1}$ that reflects the breakdown of general covariance in parameter space. Indeed, we can choose the model manifold $\Sigma_{p+1}$ of the form

$$\Sigma_{p+1}=I\otimes \Sigma_p\quad , \quad \partial I=\{ P_0 , P\}\quad ,\quad \partial \Sigma_p=\emptyset,$$ (3)

where $I$ is an open interval of the real axis, which has two points, say $P _{0}$ and $P$, as its boundary and $\Sigma_p$ is a finite volume, $p$-dimensional manifold, without boundary. Thus, $\partial \Sigma_{p+1}= P_0\otimes\Sigma_p \cup P\otimes\Sigma_p$ and the spacetime image of $\partial \Sigma_{p+1}$ under the embedding $Y$ represents the initial and final brane configuration in target spacetime.

In terms of coordinates, the above factorization of $\Sigma_{p+1}$ amounts to defining $\tau$ as the center of mass proper time and the $s_i$'s as spatial coordinates of $\Sigma_p$. Accordingly, the invariant line element reads:

$$dl^2= \overline g_{\, mn} \, d\sigma^m\, d\sigma^n= -e^2(\, \tau\, )\,\, d\tau^2 +h_{ij}(\, \vec s\, )\, \, ds^i\, \, ds^j$$ (4)

where $\tau$ plays the role of ``cosmological time'', that is, all clocks on $\Sigma_p$ are synchronized with the center of mass clock.
Now, we are in a position to introduce the center of mass coordinates $x^\mu(\tau)$ and the relative coordinates $Y^\mu(\, \tau , s^i\, )$:

$$X^\mu(\, \tau\ , \vec s\, ) \equiv x^\mu(\tau) +{1\over\sqrt{m_{p+1}}}\, Y^\mu(\, \tau\ , \vec s \,)\ ,$$ (5)

$$x^\mu(\tau) \equiv { 1\over V_p} \int_{{ {S}}_p} d^p s\, \sqrt{h(\,\vec s\, )}\, X^\mu(\, \tau\ , \vec s\, )$$ (6)

$$V_p \equiv \int_{{ {S}}_p} d^p s\, \sqrt{h(\, \vec s\, )}\ , \qquad h(\, \vec s\, )\equiv det\left(\, h_{ij}\,\right)\ .$$ (7)

Using the above definitions in the action (2) and replacing $g_{ m n}$ with $\overline g_{\, mn}$ as indicated in Eq.(4), we find

$$S = - {1\over 2}\int_\Sigma d^{p+1}\sigma\, \sqrt{\, \overline g\, }\... ...^\mu\, \partial_n\, Y_\mu - m_{p+1}\, (p-1)\, \right)\, \right]\ ,\qquad p\ge 1$$

$$= -{1 \over 2}\, m_{p+1}\, V_p \,\int_0^T \!\!\!\!\!d\tau \left[\, ... ...}^{\, m n}\, \partial_m\, Y^\mu\, \partial_n \, Y_\mu -m_{p+1}\, p\, \right]\ .$$ (8)

The first term describes the free motion of the bulk center of mass. The absence of a mixed term, one that would couple the center of mass to the bulk oscillation modes, is due to the vanishing of the metric component $\overline g_{\, 0i}$ in the adopted coordinate system (4). The last term represents the usual bulk modes free action for a covariant ``scalar field theory'' in parameter space.
Finally, if we define the brane volume mass, $M_0 \equiv V_p \, m_{p+1}$, representing the brane inertia under volume variation, then, from the above expression, we can read off the center of mass action and the corresponding Lagrangian

$$S_{cm}=-{M_0\over 2} \int _0^T d\tau\, \left[\, -{\dot x^\mu... ..._0^T d\tau\, L_{cm}\left(\, \dot x^\mu\ ; e(\tau)\, \right)\ ,$$ (9)

where the einbein $e(\tau)$ ensures $\tau$ reparametrization invariance along the center of mass world-line.
Summarizing, the final result of this subsection is that, in the adopted coordinate frame where the center of mass motion is separated from the bulk and boundary dynamics, we can write the total action as the sum of two terms

$$S=S_{cm} -{1\over 2}\, \int_{\Sigma_{p+1}} d^{ p+1}\sigma\, ... ...l_m \, Y^\mu\, \partial_n\, Y_\mu -m_{p+1}\, p\, \right]\ .$$ (10)

We emphasize that, in order to derive the expression (10), it was necessary to break the full invariance under general coordinate transformations of the initial theory, preserving only the more restricted symmetry under independent time and spatial coordinate reparametrizations.