In General Relativity (and special too) the Lagrangian for a particle of mass m in the absence of forces other than gravity is
L=m√gμνUμUν
where Uμ is the four-velocity. In that case we can derive the momentum pμ by
pμ=∂L∂Uμ=∂∂Uμm√gαβUαUβ
pμ=mgαβ2√gαβUαUβ(δαμUβ+δβμUα)=mgμαUα√gαβUαUβ
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=pμUμ−m√gμνUμUν=mgμνUμUν−m√gμνUμUν.
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
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.How to perform the singular Legendre transformation for a relativistic point particle is explained in e.g. this Phys.SE post.
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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