215 Space
A three-dimensional Euclidean set that is used to locate where objects and events sit in classical physics, where a Euclidean set is a flat space whose distances obey the Pythagorean rule.
Absolute space. Space is an independent motionless arena that is not changed by the matter inside it. Absolute space is that distinguished background against which acceleration is judged. This principle is used to tell real forces from fictitious forces in accelerating frames.
A Galilean shift of origin at constant velocity is
\[ \mathbf{x}' = \mathbf{x} - \mathbf{v}t \]
where
- \(\mathbf{x}'\) is the position in the moving frame.
- \(\mathbf{x}\) is the position in the original frame.
- \(\mathbf{v}\) is the constant relative velocity.
- \(t\) is time.
Euclidean space. Classical space is flat, so the shortest distance between two points is given by the Pythagorean rule and does not depend on how the axes are labeled. This principle is used to define lengths, angles, and the kinetic energy of motion.
The Euclidean line element is
\[ ds^{2} = dx^{2} + dy^{2} + dz^{2} \]
where
- \(ds\) is an infinitesimal distance.
- \(dx\), \(dy\), and \(dz\) are infinitesimal Cartesian displacements.
Homogeneity of space. Space is the same at every location, so the laws of physics do not depend on where an experiment is done. Homogeneity is that uniformity under a shift of position. This principle is used to guarantee that a closed system keeps its linear momentum.
Invariance of the Lagrangian under a constant shift is
\[ \mathbf{q}\rightarrow\mathbf{q}+\mathbf{a} \implies \delta L = 0 \]
where
- \(\mathbf{q}\) is the position of the system.
- \(\mathbf{a}\) is a constant displacement.
- \(L\) is the Lagrangian.
Isotropy of space. Space has no preferred direction, so the laws of physics do not change when a system is rotated. Isotropy is that sameness in every direction. This principle is used to guarantee that a closed system keeps its angular momentum.
The separation of space from time. Time is kept as a separate variable, not as a fourth geometric coordinate of space. This principle is used to write classical motion as a path in three-dimensional space parameterized by time.
Note: Homogeneity is also called spatial translation symmetry.
215.1 References
- Hassani, S. Mathematical Physics: A Modern Introduction to Its Foundations. Springer, 2013. §28.1, §30.3 — classical space as three-dimensional Euclidean space.
- Schwichtenberg, J. Physics from Symmetry. Springer, 2018. §2.1, §2.4, §4.5 — Euclidean space, homogeneity, isotropy, and translation symmetry.