# Time as coordinate in many-body relativistic mechanics

In pre-relativistic mechanics a system of N particles would be described by 3N coordinates $$x_i,y_i,z_i$$ parametrized by time $$t$$.

Would a relativistic system be properly described by 4N coordinates $$x_i,y_i,z_i,t_i$$, with a time variable for each particle? If so, how can we ever speak of the Lagrangian of the system as a whole or the Hamiltonian of the system as a whole? Or the action of the system as a whole?

In general, can we find a single quantity to parametrize the state of a system of N particles?

Yes we need 4n coords in SR as the notation of simultaneity is not well defined. If we are dealing with non-interacting particles (or interacting with some external field) this is not problem and we can still solve our system. However if the particles in our system are interacting then our system may not be solvable.

I had never thought about the implications of this before, but doesn't this mean that the concept of the state of a system breaks down since, without simultaneity, there is no way to refer to all the particles collectively. All we can say is that particle 1 is at $$x_1$$ when its clock reads $$\tau_1$$ and that particle two is at $$x_2$$ when its clock reads $$\tau_2$$.

Or is there? Can we synchronize all the clocks in the system via light signals? In general they will never agree with each again after synchronization (the "Twin Paradox") but at least we can then choose a single time and refer to position and momentum of each particle when its clock reads that specific time.

That is, if we can indeed synchronize the clocks so that we choose positions $$x_1(\tau_1=0),...,x_N(\tau_N=0)$$, then thereafter we can specify any $$t$$ and refer to the "state" of the system associated with $$t$$ as the positions $$x_1(\tau_1=t),...,x_N(\tau_N=t)$$ and the momenta $$p_1(\tau_1=t),...,p_N(\tau_N=t)$$.

That's just off the top of my head. Can an expert please clarify?

Dale
Mentor
2020 Award
There is no problem with defining the state of the system. With GR you can use any arbitrary synchronization procedure that you wish and thus use arbitrary hypersurfaces of simultaneity to define your state.

Thanks. I'm pondering it.

After some more thought I realize it is still not clear to me. Let me be a little more explicit. For the sake of this discussion I assume that though our spacetime may be curved, we are dealing with "test particles" and the metric is fixed.

Let $$x_j^\mu$$ denote the four-vector of the jth particle. The question amounts to, I think, can we in general parametrize the system with a single parameter s such that the action

$$S=\int{L(x_1^\mu(s),...,x_N^\mu(s),\frac{dx_1^\mu}{ds},...,\frac{dx_N^\mu}{ds})ds}$$

determines the correct dynamics?

Dale
Mentor
2020 Award
I believe that you are looking for the Einstein-Hilbert action, but I'm afraid that I am at the limit of my GR (or maybe a little beyond).