I am struggling with a concept in Classical Mechanics/Special Relativity.

I want to find the relativistic Lagrangian of a free particle, the method for which I have found in a few dfferent places, but I specifically want my Lagrangian to be Lorentz invariant. The only source that I could find provided this as a relativistic Lorentz invariant Lagrangian of a free particle:

$$ L = -mc \sqrt{-g_{\mu\nu}\dot x^{\mu}\dot x^{\nu}} $$

My real question, I suppose, is "How do I know that the Lagrangian is/isn't Lorentz invariant?"

Edit: I do not think the suggested link answers my question. I may not have expressed myself very well, but what I really don't understand is how we can check if the Lagrangian is Lorentz invariant, as I have read in one or two different places that just because the acton is invariant, this does not necessarily mean the Lagrangian is.

  • $\begingroup$ OK, I've reopened the question. $\endgroup$ – John Rennie Mar 31 '17 at 11:55
  • $\begingroup$ What is the definition of Lorentz invariance that you know? $\endgroup$ – DanielC Mar 31 '17 at 11:59

Since $x^\mu$ is a vector while $\tau$ is a scalar, $\dot{x}^\mu:=\frac{dx^\mu}{d\tau}$ is a vector. The contraction $g_{\mu\nu}\dot{x}^\mu\dot{x}^\nu$ is therefore a scalar, i.e. invariant under general coordinate transformations. Thus $L$ as defined is also a scalar. So is the action $\int d^4xL\sqrt{|g|}$. (Jargon: the scalar $L$ is the scalar Lagrangian density, whereas the scalar density $L\sqrt{|g|}$ is the Lagrangian density.) Scalars are de Sitter invariant in de Sitter space, Lorentz invariant in Minkowski space etc.


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