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it will sweep out some \(p + 1\) dimensional surface (a line for \(p = 0\), area for \(p = 1\), volume for \(p > 1\)).
The world line, area, or (most generally) volume, can be determined by integrating the change in each of the \(D\) coordinates of \(X^{\mu}\) in relation to each of its parameters. I'll return to this shortly.
The most general form of the pbrane action is given by \(S_{p} = T_{p}V\) where \(V\) is the worldvolume to be minimized by the least action principal and \(T_{p}\) is described as a tension or energy density. Since time is standardly written as the first coordinate of a Ddimensional spacetime, we can consider the example of a morphed string shown before with an added time axis in the vertical direction which would produce a morphing column over time similar to the examples shown in the image taken from Wikipedia at the start of this post.
The generalized volume factor in \(S_{p} = T_{p}V\) for any pbrane in \(D\) dimensions is given by \[ V = \int\sqrt{det(g_{\mu\nu}(X)\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}}d^{p + 1}\sigma\] where \(X^{0}\) corresponds to the axis \(\tau\), \(X^{1}\) to \(\sigma^{1}\), and \(X^{2}\) to \(\sigma^{2}\). We can note that \(X^{0} = \sigma^{0} = \tau\) thus \(\partial_{\sigma}X^{0} = 0\) and likewise, ...(CHECK). \(\alpha\) and \(\beta\) range over the values \(0, ..., p\) and \(\mu\), \(\nu\) range over the values \(0, ..., D1\). The determinant serves to permute all nonzero combinations into a "single line" (or nonmatrix form) equation and \(d^{p + 1}\sigma\) indicates the infinitesimal elements of the parameters \(\sigma\) and \(\tau\).
Specifically, for a string (p = 1) the action \(S_{1} = TV\) simplifies to the NambuGoto action: \[S_{NG} = T\int\sqrt{(\dot{X}\cdot X')^{2}  \dot{X}^{2}\cdot X'^{2}}d\sigma d\tau\] where \(\dot{X}^{\mu} = \frac{\partial X^{\mu}}{\partial\tau}\) and \(X'^{\mu} = \frac{\partial X^{\mu}}{\partial\sigma}\) with \(A\cdot B = \eta_{\mu\nu}A^{\mu}B^{\nu}\) for a flat spacetime.
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