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Leibniz stated: "Thus committed to maintaining that if there were nothing more to motion than relative change of position, then, since motion could be ascribed with equal right to, say, Train A or Train B."

How does Leibniz's observation behave when one introduces a boundary condition? One can start thinking of bodies with respect to the boundary. Or even more interesting the boundary relative to itself. Have these thoughts been explored by philosophers? (Would be an interesting read in the context of general relativity)

P.S: I asked something similar previously

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  • What difference does a boundary make? It is just another object relative to which motion can be ascribed.
    – Conifold
    Jan 25, 2022 at 18:57
  • @conifold I don't think that's how one views the boundary in general relativity?
    – More Anonymous
    Jan 25, 2022 at 19:46
  • More specifically it becomes a feature of your geometry
    – More Anonymous
    Jan 26, 2022 at 3:23
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    In Lagrangian/Hamiltonian mechanics boundaries can act as Holonomic constraints. If you only have one rigid body in your system such as a ball falling along a moving slope on a frictionless ground, implicitly we need to mentally include earth or some other inertial frame to account for potential energy. So you'll never have only one object in any classical non-field mechanics theory, your only rigid body still have relative motion w.r.t. its inertial frame of reference (another earth object considered standing still only by convention).
    – Double Knot
    Jan 26, 2022 at 6:32
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    Boundary conditions in general relativity aren't usually timelike, are they? Usually they are on space like surfaces, meaning that under some definition of simultaneity they would be snapshots of the state of matter/energy at a particular instant in time. And there isn't any very obvious way to define velocity relative to a space like surface, though I suppose for any timelike worldline that crosses it you can talk about its instantaneous velocity in the local inertial frame whose definition of simultaneity is "parallel" to the surface at the crossing point.
    – Hypnosifl
    Jan 26, 2022 at 15:30

3 Answers 3

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Boundary conditions just mean plucking out a specific situation from very general math, like the +c in the solution to a quadratic equation, or initial conditions for a Lagrangian/Hamiltonian that could cover very general general circumstances without limiting to specifics. The potential at infinity just relates to definitions, needed in order to make fair comparisons with fields (it establishes zero or background, but actually can involve non-zero fields).

Frame Invariance shows us that Leibniz was right. It doesn't matter what boundary conditions apply within the system. But of course, the world outside the system is a boundary condition: if you ask 'How will this system impact the world?' you make the world part of the system.

But, there is no absolute frame of reference. Entropy, energy, linear and rotational momentum, are fundamentally relative. So within a system they cannot be evaluated except relative to the system. But when you say how will it impact something outside the system, you change the terms, you shift the question. So in your terms: Earth is a boundary condition, the damage train A or B can do to it arriving from space will depend on relative velocities of A & B to Earth, not just each other. We treat the universe as not rotating, but it wouldn't be detectable within it, if it was - rotational momentum is implicitly defined, as relative to the universe.

A really interesting case of this classical phenomena in tension with the quantum world, is the Unruh Effect. Clearly there are still matters to settle in both theory and experiment.

Spock says "Roses are red, roses are blue, depending on their relative velocity to you

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    lol Spock and redshift! Awesome.
    – J D
    Feb 21, 2023 at 0:15
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You appear to regard a boundary of spacetime (assuming spacetime is bounded) as if it were like a fence around a field. If that were the case, then yes you could treat the fence as providing an absolute frame relative to which you could define motion absolutely. However, you should consider instead that spacetime is bounded in a way that is analogous to the surface of a sphere. If you are moving over the surface of a sphere you never encounter a boundary, and your speed cannot be fixed in any absolute sense by reference to a boundary, even though the surface is finite and bounded.

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One of the more beautiful SEP articles is on boundaries ("limits")

https://plato.stanford.edu/entries/boundary/

how can one account for our ordinary and mathematical talk about boundaries if these are to be explained away as fictional abstractions

If boundaries do not exist, then how do we approach (etc.) them (for the metaphysicist), and does that mean they cannot move (for Leibniz)?

Of course the discussion is vulgar from me here.

We cross a boundary whenever we move, even if that "limit" is a methematical abstraction. If you want to think anything interesting about them and Leibniz, you'd have to find out what they think of abstractions, indivisibles, etc.

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