Physics Asked by kevin012 on July 26, 2020
Poincare 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 with translation or rotation. In this case, space has symmetry.
But from Einstein’s field equation,
$$
R_{munu} – frac{1}{2}Rg_{munu} + Lambda g_{munu} = frac{8pi G}{c^4} T_{munu}
$$
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 from the Poincare symmetry and still the experiment on earth verifies our theory.
Why does the Poincare symmetry work to describe the particles even though space doesn’t seem homogeneous?
The quick and easy answer is that the amount of spacetime curvature created by particles is so small compared to the effects of the electromagnetic, strong, and weak interactions that gravity can be ignored when studying particle interactions in the lab.
Answered by niels nielsen on July 26, 2020
The homogeneous and isotropic property of the space holds in appoximate sense, or when looking at large enough distances. Definitely, the world around us is not a homogeneous mass, we have stars, planets, galaxies, or on shorter distances - mountains, lakes, trees.
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 $sim 10^{-5}$ ).
Answered by spiridon_the_sun_rotator on July 26, 2020
By analogy, one can consider Euclidean planar geometry as a useful model approximating the geometry of your table, which is intermediate between the microscopic scale that sees the irregularities in the wood and the larger scale that sees the curvature of the non-flat earth.
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 $R^4$).
(It may be that "the very-very small and smaller" or "the very-very large and larger" shows Poincare symmetry... but, for at least the intermediate range of scales that we work with right now, there are lower and upper bounds.)
As others have noted, the effect of gravitation (curvature of spacetime) is relatively small at typical particle-physics scales (in space and time).
Answered by robphy on July 26, 2020
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