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Homogeneity of Space

Space is homogeneous means one part (or point) of space is identical or equivalent to any other part (or point) of space. Any physical process will occur the same way (or identically) under identical conditions, no matter where (i.e. in whatever part of space) it occurs; a particular experiment, whatever performed in Kanpur or New York, will yield same result. (Incidently Kanpur and New York represent same inertial frame).

Let us apply the basic property of homogeneity of space to an isolated system of two interacting particles; if both the particles of the system are displaced by same amount δ r, there should be no change in the state of the system or in its internal motion. That is, the total work done by internal forces when the system is displaced by δ r must be zero:

F₁₂ . δ r + F₂₁ . δ r = 0

Space-Time Structure, Newton’s Laws and Gallilean Transformation

Since δ r is arbitrary, we get

F₁₂ + F₂₁ = 0

This is Newton’s III law which leads to law of momentum conservation for the isolated system of two particles. Thus, momentum conservation emerges from the fundamental property of homogeneity of space.

(The above argument can be extended to an isolated system of n-interacting particles. No work done by internal forces imply:

δ r . (F₁₂ + F₂₁) + δ r . (F₁₃ + F₃₁) + …. = 0

where the series includes all pairs of two particles out of n-particle system.

That is, in short, we can write

δ r . Σ_i≠_j Σ_j (F_ij + F_ji) = 0 i, j = 1 …. n

Since δ r is arbitrary, we find

Σ_i Σ_j (F_ij + F_ji) = 0 i, j = 1, …., n ; i ≠ j

The comments made on Newton’s III law also apply to the above proof of momentum conservation (of particles) from homogeneity of space. If we consider the forces propagating with finite speed, homogeneity of space leads to conservation of momentum of complete system, viz. both particles and the carrier of force.

Since space is homogeneous, we can choose the origin of our co-ordinate system anywhere we wish. Shifting the origin means displacing the system; and it does not affect the processes.