**The Relativistic Point Particle**

We want to calculate the action of a point particle under a boost. It is natural to begin with

where is the path of the worldline, and the Lagrangian,

which retrieves our classical interpretation at low order approximation. Using the Euler-Lagrange equation, we then find that

and thus

which is precisely what we expect as a cursory approach.

The relativistic point particle requires more than simple boosts, but we will still remain in flat spacetime. As such, we have the metric , however, for the action we want this in terms of some timelike value, , so we can instead say that , where Our action will then be

If we add in the electromagnetic field, it can be shown that the relativistic action of a charged particle is

Next, it must be noted that

and so

and thus

which gives us reparameterization invariance. A good choice for that allows us to solve several equations is the equality . This is known as **static gauge** because, physically, we are examining slices of space at static moments.

**The Relativistic String**

If we construct the matrix

for arbitrary parameters , we find that is the induced metric on , where

Furthermore, we find that for , the area

is also invariant under reparameterization.

We therefore define our two-dimensional worldsheet for a string by the parameters and , where is timelike, and is spacelike.

Furthermore, we can define proper area on the worldsheet to be

Here, the dot is the relativistic dot product. Now the action has units of (the same as , coincidentally), so by multiplying the proper area by we obtain the correct units and thus our action becomes

This is called the **Nambu-Goto action** of a relativistic string.

Our reparameterization invariance allows us to construct the matrix

such that the Nambu-Goto action becomes

where

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Ok, this is good, this is all in a familiar language! Once I’m done my telescope proposals (I’m an astrophysicist), I’ll read through this a little more carefully. Thanks!