A shifted clock can preserve different things¶
Cross-Domain EchoesShared pattern · Symmetry
Moving every event time forward by the same interval can leave a physical law unchanged. Moving every index of a strictly stationary random sequence by the same integer can leave its joint distribution unchanged. Both are symmetry claims under a named group of translations, but the object declared unchanged differs. A physical trajectory can keep moving, and a random sequence can keep fluctuating. The diagrams separate the transformation from the invariant so that “the same over time” does not quietly become “the same value at every time.”
Choose a role to see its counterpart in both examples. The diagrams show relationships, not measured quantities.
Physics
Shift the clock, preserve the law
Read Time-translation symmetryDomain-specific abstraction
The same physical laws apply after all event times are shifted by a common interval.
In this example: This is invariance of governing laws, not a claim that an individual trajectory or state repeats.
Probability
Shift the indices, preserve the joint distribution
Read Stationary sequenceDomain-specific abstraction
For a two-sided strictly stationary sequence, every finite tuple has the same joint distribution after a common integer index shift.
In this example: Strict stationarity concerns a law, not identical sample values, independent observations, or guaranteed time-average convergence.
The equality claim concerns the stated object, not a particular observed value.
Written comparison
What is being compared
Physics
A governing physical law
Probability
A finite-dimensional joint probability law
The equality claim concerns the stated object, not a particular observed value.
A group of common translations
Physics
Shift all times by one interval
Probability
Shift all indices by one integer
In the selected unbounded-time and two-sided-index settings, shifts compose, include zero, and have inverse shifts.
The preserved structure
Physics
Form of the physical law
Probability
Complete joint distribution
The common algebraic structure is translation invariance; the physical and probabilistic consequences remain separate.
What carries across
Name both the transformation and the object it preserves. Unchanged laws or distributions do not imply an unchanging realization.
Where the comparison stops
Time-translation symmetry of physical laws does not make every evolving physical state stationary. Strict stationarity does not by itself establish physical energy conservation.
- The stochastic case selects strict stationarity on a two-sided index set; equal mean and covariance alone are the weaker convention.
- Stationarity does not imply independence or ergodicity, and it does not make one fluctuating sample path repeat.
Conditions for this comparison
- The transformation acts by the same shift on every event time or index.
- Use a two-sided sequence so integer shifts have inverses; do not silently extend a one-sided finite record beyond its domain.
- This is a mathematical comparison, not a claim that an observed time series has passed a stationarity test.
Source entries
Shared pattern
Symmetry
Prime
Core Idea
(1) Symmetry is invariance under a specified group of transformations: a system is symmetric with respect to an action when applying the action leaves the system unchanged in a specified sense (identical, equivalent, isomorphic, or indistinguishable-for-the-operations-of-interest); the defining commitment is not the loose "looks balanced" but the precise algebraic claim that a stated transformation, applied to the object, yields the same object back.
Physics
Time-translation symmetry
Domain-specific abstraction
Core Idea
Time-translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time-translation symmetry is the law that the laws of physics are unchanged (i.e. invariant) under such a transformation. Time-translation symmetry is a rigorous way to formulate the idea that the laws of physics are the same throughout history.
Probability
Stationary sequence
Domain-specific abstraction
Core Idea
Strict stationarity preserves the complete joint law, while weak stationarity preserves only finite mean and lag-dependent covariance; ergodicity is a separate time-average property. A shift acts on every finite index tuple, and the corresponding joint distribution remains unchanged for every admissible displacement.