Polyakov action, reparameterisation q, string theory

[binbagsss]

Homework Statement

i am stuck on part d , see below

Homework Equations

parts a to c are fine

polyakov action:
## \frac{1}{2} \int \frac{1}{e(t)} \frac{dX^u}{dt}\frac{dX_u}{dt}-m^2 e(t) dt ##

EoM of ##e(t)##:
##\frac{-1}{(e(t))^2} \frac{dX^u}{dt}\frac{dX_u}{dt}-m^2=0## [1]you plug the EoM of ##e(t)## (which is equivalent to the mass-shell constraint) into the polykov action to recover the Nambu-Goto action.

The Attempt at a Solution

[/B]
More than anything, I am confused as to why we are given the mass, since the dimension of the space-time is not given, ##u=0,1...d##, ##d## is not specified. If it were ##d=1## I guess we would use it as something like computing the other space-time coordinate.

Whilst ##e(t)## has a transformation rule, ##X^u(t)## just acts as a scalar on the world-sheet in string theory? or in this case on the world-line, and as such has no transformation rule and so it is just a case of plugging in.

I interpret the trajectory in terms of ##t## as (apologies i have done ##\tau=t## and will also do ##\tau'=t'##) ##X^0(t)=t## and ##X^i(t)=0 ## for ##i=0,1,..,d##

So that the mass-shell constraint reads ##\frac{\partial X^0}{\partial t}\frac{\partial X_0}{\partial t}=1/2 ## and via the chain rule of t' and t it is easy to check that this is consistent.

So simply plugging in I get ## X^0(t'(t))=t' ^2##

However I'm guessing this is wrong since I have no idea why we are given the mass, any help much appreciated,ta.

 

[kurt.physics]

d) Show that the Polyakov action can be written as ##\int \sqrt{\dot{x}^2-m^2}dt##Also if I am wrong about any of the above please inform me.