# Why are particles described with Poincaré symmetry even though space seems inhomogeneous?

Poincaré transformation consists of translation, rotation, and boosting. And by assuming the physical quantities are invariant and equations are covariant under the transformations, we build the models on particles. The invariance and covariance make sense if space is the same under the transformations. Space has symmetry in the case.

But from Einstein's field equation,

$$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$

space is curved when energy is placed on it.

So the very existence of particles would make the space irregular by making dents randomly. And with this process, space would not keep the symmetry anymore. The earth itself would distort the space by a large amount. But the particles are described with the Poincaré symmetry and the experiments on earth verify the theories constructed with the symmetry.

Why does the Poincaré symmetry work to describe the particles even though space doesn't seem homogeneous?

P.S. The several answers point that the symmetry holds approximately. But I feel difficulty with the approximation view. For example, there is the spin-statistics theorem. The theorem relies on the exchange symmetry:

$$\phi(x)\phi(y) = \pm \phi(y)\phi(x)$$

The equation represents equality between the left and the right side and there is no room for approximation. If the particles are approximate entities, I guess the spin-statistics theorem would not work.

• Concerning your statement "the earth itself would distort the space by a large amount": to within a few orders of magnitude, the deviation from flatness of an object of mass $M$ and radius $R$ is $\mathcal{O}(GM/R c^2)$. For the Earth, this works out to be $\approx 10^{-9}$. So it's not surprising that flat-spacetime approximations still work pretty well on Earth. – Michael Seifert Jul 26 '20 at 19:43
• Just a food for thought: due to equivalence principle, the structures defined locally need to retain the flat spacetime symmetries. If you have some vector equation it needs to retain symmetry under Poincare group acting in the tangent space, as this is where the vector lives. Thus it seems to me, that invariance and covariance still makes sense even in curved spacetime. The problem I guess arises only with nonlocal structures, as is the exchange symmetry. – Umaxo Jul 27 '20 at 6:02

Poincare symmetry should probably be thought similarly of as a useful model approximating symmetries seen at a particular scale, certainly intermediate between the very small (where spacetime might not make sense and/or the spacetime might not be like $$R^4$$) and the very large (where significant spacetime curvature at extreme-astrophysical or cosmological scales might arise and/or the spacetime might not be like $$R^4$$).
At these scales, the existence of localized density clusters breaks the translational and rotatitonal invariance. However, when investigating the properties of universe on cosmological scales, larger than the galaxy clusters, the matter would be distibuted almost uniformly and isotropically (however, the CMB has an anisotropy of $$\sim 10^{-5}$$ ).