Thursday, 27 April 2017

energy - What is wrong with my argument to derive the Hamiltonian in relativity?


In General Relativity (and special too) the Lagrangian for a particle of mass m in the absence of forces other than gravity is


L=mgμνUμUν


where Uμ is the four-velocity. In that case we can derive the momentum pμ by


pμ=LUμ=UμmgαβUαUβ


pμ=mgαβ2gαβUαUβ(δαμUβ+δβμUα)=mgμαUαgαβUαUβ


If we parametrize the worldline by proper time τ then L(γ(τ),γ(τ))=m and we get of the square root on the denominator which is just 1. Then


pμ=mgμαUα,


and these are the components of a covector. This directly leads to the momentum four-vector


pμ=mUμ.



Everything works here. Now I want to compute the energy. Well the Hamiltonian as always should be


H=pμUμmgμνUμUν=mgμνUμUνmgμνUμUν.


But if things are parametrized by propertime, when we compute H on the path, that is H(γ(τ),γ(τ)) we get zero!


What I expected was to get H=p0.


What am I doing wrong here? Why am I getting zero?



Answer





  1. The problem is that the Legendre transformation from 4-velocity to 4-momentum is singular: The 4 components of the 4-momentum pμ are constrained to live on the mass-shell pμgμνpν = ±m2.

    Here the ± refers to the choice of Minkowski sign convention (±,,,). Therefore it is a constrained system. The 4-momentum has only 3 independent components.





  2. How to perform the singular Legendre transformation for a relativistic point particle is explained in e.g. this Phys.SE post.




  3. It turns out that the appearance of the constraint (A) and the vanishing energy/Hamiltonian reflect the worldline reparametrization invariance of the model. See also e.g. this Phys.SE post.




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