Preamble: When absolute space and absolute time were falsified in the
early 20th century, the concept of absolute spacetime scale became
very suspect. Note that the source-free equations of general
relativity (Einstein's equations) and electromagnetism (Maxwell's
equations) are both scale invariant. On the other hand, absolute size
scale seemed to be required empirically. After all, galaxies are very
big, neutron stars are middle-sized and atomic nuclei are very small.
Are they not? Here I will propose a possible resolution to this
paradox.

Context: If our global spacetime has a new symmetry principle called
discrete scale invariance (synonymous with discrete dilation
invariance or discrete self-similarity), then the group structure of
this global spacetime is the Relativistic Similarity Group (aka the
Weyl Group: = Poincare Group + dilation invariance), but it is
globally symmetric under *discrete* transformations of scale, as
opposed to the more familiar and scale-restricted examples of
continuous scale invariance.

The physical embodiment of the new discrete symmetry principle would
be an infinite hierarchical universe of physical systems manifesting
discrete self-similarity. When one observes nature objectively,
emphasizing empirical knowledge and treating most theoretical
assumptions as open to question, then that is precisely what we appear
to have: a discrete hierarchy of self-similar systems
(i.e., ...galactic systems, composed of stellar systems, composed of
atomic systems, ...). There is considerable evidence for discrete self-
similarity among analogues on the different cosmological scales. A
veritable cornucopia of observational and theoretical support for this
paradigm can be found at www.amherst.edu/~rloldershaw .

In such a universe, a galaxy could be viewed as a galaxy, a neutron
star, or an atomic nucleus depending on the cosmological reference
scale chosen by the observer.

An empirically-based scientific idea this radical has not come along
in a very long time. Absolute scale, beyond the restricted "absolute"
scale that applies only *within* individual cosmological scales would
be relegated to the dustbin of history.

Knecht

www.amherst.edu/~rloldershaw