Some progress towards something
On Thursday night I was reading Polyakov’s contribution to the book 50 years of Yang-Mills theory. This is my version of a bed time story. 
Polyakov mentions some of his earlier work and how he solved different problems. I actually got tired after a while, and decided to work and solve my own problem. I had been sort of running away from it with feelings of overwhelming difficulty. There is also a chance that reading this post on the (now dead) string coffee table provided some motivation. In any case, here we go…
Let us consider the following metric in 5-dimensional spacetime:
We have coordinates 
Then taking derivatives of the coordinate vector we have:
Then, the ingredients that go into the Nambu-Goto lagrangian are:
![\left(\partial_{1}C\right)^2 = -A\left[\left(\partial_{1}x_{0}\right)^2 - 1\right]+B\left(\partial_{1}r\right)^2](http://l.wordpress.com/latex.php?latex=%5Cleft%28%5Cpartial_%7B1%7DC%5Cright%29%5E2+%3D+-A%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E2+-+1%5Cright%5D%2BB%5Cleft%28%5Cpartial_%7B1%7Dr%5Cright%29%5E2+&bg=fff&fg=222&s=0 "\left(\partial_{1}C\right)^2 = -A\left[\left(\partial_{1}x_{0}\right)^2 - 1\right]+B\left(\partial_{1}r\right)^2")
![\left(\partial_{2}C\right)^2 = -A\left[\left(\partial_{2}x_{0}\right)^2 - 1\right]+B\left(\partial_{2}r\right)^2 .](http://l.wordpress.com/latex.php?latex=%5Cleft%28%5Cpartial_%7B2%7DC%5Cright%29%5E2+%3D+-A%5Cleft%5B%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E2+-+1%5Cright%5D%2BB%5Cleft%28%5Cpartial_%7B2%7Dr%5Cright%29%5E2+.&bg=fff&fg=222&s=0 "\left(\partial_{2}C\right)^2 = -A\left[\left(\partial_{2}x_{0}\right)^2 - 1\right]+B\left(\partial_{2}r\right)^2 .")
The cross term goes as:
The determinant of the induced metric will be denoted by:
We see that since we have:
and
![\left(\partial_{1}C\right)^{2} \left(\partial_{2}C\right)^{2} = A^{2}\left[\left(\partial_{1}x_{0}\right)^2 - 1\right] \left[\left(\partial_{2}x_{0}\right)^2 - 1\right] + B^{2} (\partial_1 r)^2 (\partial_2 r)^2 - AB\left[\left(\partial_{1}r\right)^{2}\left[\left(\partial_{2}x_{0}\right)^{2}-1\right]+\left(\partial_{2}r\right)^{2}\left[\left(\partial_{1}x_{0}\right)^{2}-1\right]\right].](http://l.wordpress.com/latex.php?latex=%5Cleft%28%5Cpartial_%7B1%7DC%5Cright%29%5E%7B2%7D+%5Cleft%28%5Cpartial_%7B2%7DC%5Cright%29%5E%7B2%7D+%3D+A%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E2+-+1%5Cright%5D+%5Cleft%5B%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E2+-+1%5Cright%5D+%2B+B%5E%7B2%7D+%28%5Cpartial_1+r%29%5E2+%28%5Cpartial_2+r%29%5E2+-+AB%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dr%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D-1%5Cright%5D%2B%5Cleft%28%5Cpartial_%7B2%7Dr%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D-1%5Cright%5D%5Cright%5D.&bg=fff&fg=222&s=0 "\left(\partial_{1}C\right)^{2} \left(\partial_{2}C\right)^{2} = A^{2}\left[\left(\partial_{1}x_{0}\right)^2 - 1\right] \left[\left(\partial_{2}x_{0}\right)^2 - 1\right] + B^{2} (\partial_1 r)^2 (\partial_2 r)^2 - AB\left[\left(\partial_{1}r\right)^{2}\left[\left(\partial_{2}x_{0}\right)^{2}-1\right]+\left(\partial_{2}r\right)^{2}\left[\left(\partial_{1}x_{0}\right)^{2}-1\right]\right].")
we get some cancellations in 
![-G = -A^{2} - 2AB\partial_{1}x_{0} \partial_{2}x_{0} \partial_{1}r \partial_{2}r + A^{2}\left[\left(\partial_{1}x_{0}\right)^{2} + \left(\partial_{2}x_{0}\right)^{2} \right] + AB\left[\left(\partial_{1}r\right)^{2}\left[\left(\partial_{2}x_{0}\right)^{2}-1\right]+\left(\partial_{2}r\right)^{2}\left[\left(\partial_{1}x_{0}\right)^{2}-1\right]\right].](http://l.wordpress.com/latex.php?latex=-G+%3D+-A%5E%7B2%7D+-+2AB%5Cpartial_%7B1%7Dx_%7B0%7D+%5Cpartial_%7B2%7Dx_%7B0%7D+%5Cpartial_%7B1%7Dr+%5Cpartial_%7B2%7Dr+%2B+A%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D+%2B+%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D+%5Cright%5D+%2B+AB%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dr%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D-1%5Cright%5D%2B%5Cleft%28%5Cpartial_%7B2%7Dr%5Cright%29%5E%7B2%7D%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D-1%5Cright%5D%5Cright%5D.&bg=fff&fg=222&s=0 "-G = -A^{2} - 2AB\partial_{1}x_{0} \partial_{2}x_{0} \partial_{1}r \partial_{2}r + A^{2}\left[\left(\partial_{1}x_{0}\right)^{2} + \left(\partial_{2}x_{0}\right)^{2} \right] + AB\left[\left(\partial_{1}r\right)^{2}\left[\left(\partial_{2}x_{0}\right)^{2}-1\right]+\left(\partial_{2}r\right)^{2}\left[\left(\partial_{1}x_{0}\right)^{2}-1\right]\right].")
We can write the lagrangian as:
with
![J = A\left[\left(\partial_{1}x_{0}\right)^{2} + \left(\partial_{2}x_{0}\right)^{2} - 1 \right] + B \left[\right]^{2}](http://l.wordpress.com/latex.php?latex=J+%3D+A%5Cleft%5B%5Cleft%28%5Cpartial_%7B1%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D+%2B+%5Cleft%28%5Cpartial_%7B2%7Dx_%7B0%7D%5Cright%29%5E%7B2%7D+-+1+%5Cright%5D+%2B+B+%5Cleft%5B%5Cright%5D%5E%7B2%7D&bg=fff&fg=222&s=0 "J = A\left[\left(\partial_{1}x_{0}\right)^{2} + \left(\partial_{2}x_{0}\right)^{2} - 1 \right] + B \left[\right]^{2}")
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