Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the particles?

Baladdaa9 2022-07-22 Answered
Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the particles?
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Answers (1)

Mireya Hoffman
Answered 2022-07-23 Author has 14 answers
A counterexample:
1) Take two particles in a laboratory S
2) Set the clocks attached to the particles so that at laboratory time t = 0 both clocks show τ 1 = 0, τ 2 = 0
3) Leave the particles to evolve under the following lorentz-invariant lagrangian:
L = L 1 + L 2 + L i n t L i n t = u 1   μ ( τ 1 ) u 2 μ ( τ 2 ) ,
where L 1 , L 2 are free-particle lagrangians. The total lagrangian is lorenz-invariant, but the physical system is not due to its initial setup, which allows the lagrangian to transmit the interactions in a space-like manner.
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