classical mechanics - Lagrangian for free particle in special relativity

From definition of Lagrangian: $L = T - U$. As I understand for free particle ($U = 0$) one should write $L = T$.

In special relativity we want Lorentz-invariant action thus we define free-particle Lagrangian as follows:

$$L = -\frac{ m c^2}{\gamma} - U$$

At the same point, we have that definition of 4-momentum implies the kinetic energy is:

$$T = (\gamma - 1) m c^2.$$

As you might guess, 1) question is how to relate all these formulas?

2) I do not understand why there is no $1/\gamma$ near $U$ in relativistic Lagrangian?

3) What is meaning of the first term in $L$ for relativistic case?

Answer

It helps to write the full action:

$$S = \int \frac{-mc^2}{\gamma}dt - \int U dt$$

The first term can be put in a much better form by noting that $d\tau = \frac{dt}{\gamma}$ represents the proper time for the particle. The action is then:

$$S = -mc^2\int d\tau - \int U dt$$ The first term is Lorentz invariant, being only the distance between two points given by the Minkowski metric, and is good in relativity. The second term however, isn't (assuming that $U$ is a scalar); there is no way it can be a relativistic action.

There are two easy ways out:

1. The first is simply to change the term to $\frac{U}{\gamma}$. This gives the action: $$S = -\int (mc^2+U)d\tau$$


2. The second is to "promote" the term (a terminology used in Zee's Einstein Gravity in a Nutshell) to a relativistic dot product, giving the action: $$S = -mc^2\int d\tau - \int U_\mu dx^{\mu}$$

The former has no real world classical analog (that I know of), and the latter is more or less the interaction of a particle with a static electromagnetic field. But the original form is recovered from the latter when the spatial components of $U_\mu$ vanish, leaving only $U_0$.

The kinetic energy is obtained by transforming the Lagrangian to the Hamiltonian (see here).