# The correlation between scales of time and space

I guess it is quite well known that there is an empirical correlation between scales of time and space.

That means that small things are usually move / rotate relatively (compared to its sizes) faster then the large ones.

For example ants move relatively really fast, compared to large animals. The bacterias are much much faster. Atoms and electrons live in quite short times, in about 10e-7 seconds scale.

On the other side of scales planets move relatively slow (day or year), stars and galaxies are quite static for our time scale.

In general it looks like the scale of time is almost proportional to the scale of space.

How to explain that using standard physics ideas ?

Is it a kind of inertia properties or something like that ?

• conservation of energy/momentum, Newton's laws. To accelerate/move large mass requires large energies.
– Kosm
Mar 14, 2019 at 12:29

Let's take a look at these examples:

For example ants move relatively really fast, compared to large animals.

This idea can be justified in a very, very crude way using the following argument:

Let's assume:

1. We can define a species-independent length scale $$L$$ (this would be easy if all animals were the same shape; the fact that they are not is the main reason this is a very crude argument).
2. All animals have similar density (again, not strictly true, but probably within an order of magnitude).
3. The strength of muscles (i.e. their maximum power output) is proportional to their cross-sectional area (in reality, this depends on both geometric and biochemical factors that cause large differences between unrelated animal body types).
4. To run at a given speed relative to body size, the muscles of an animal must provide a certain amount of power per unit mass. Hence, the maximum running speed is proportional to the power per unit mass able to be exerted by the muscles (this doesn't factor in things like air resistance, different locomotion methods, etc.).

Since the length scale is species-independent, the muscle strength is proportional to their cross-sectional area, which is, in turn, proportional to $$L^2$$, and since the densities of the animals are roughly the same, the mass is proportional to the volume $$L^3$$. The maximum running speed is proportional to the muscle strength per unit mass, which is, in turn, proportional to $$L^2/L^3=1/L$$. Therefore, smaller animals run faster relative to their size.

Atoms and electrons live in quite short times, in about 10e-7 seconds scale.

I honestly don't know where you got this number. Atoms and electrons can both exist indefinitely, as both are stable objects. Even if we assume you're talking about an interaction that takes a time of roughly $$10^{-7}$$ seconds, such an interaction time would correspond to an interaction with electromagnetic waves of a frequency of 10 MHz, which is between AM radio and FM radio waves in frequency. This is a very low-energy interaction, and as such, shouldn't correspond to any of the common atomic transitions. It may correspond to a rotational mode of a large molecule, but not a single atom. You might have gotten confused by the fact that the typical wavelength of atomic transitions is smaller than $$10^{-7}$$ meters.

On the other side of scales planets move relatively slow (day or year)

Not always. The exoplanet K2-137b, for example, orbits its star once every 4.31 hours. In any case, a planet that orbited close enough to a star to obtain a much shorter orbital period would almost certainly be destroyed by the star. The minimum period of a gravitational orbit is set by two things: the strength of the gravitational force itself, and the minimum distance the planet can be from a star before it is destroyed. Gravity is quite weak, and stars are pretty large (in fact, they have a maximum density due to the fact that you can't shove two electrons into the same quantum state), so the minimum period is pretty large.

stars and galaxies are quite static for our time scale.

Stars actually move quite fast (for example, the Sun orbits the galactic center moving at hundreds of km/sec, and the fastest-moving stars can reach speeds of 1200 km/s). It's just that the distance from them to us is incredibly large, so the apparent angular motion (proportional to the ratio between their lateral speed and the distance to us) is quite slow. The reason the distance between objects in space is so large relative to their size is partly due to the action of gravity: if objects get too close to each other, they interact on a much shorter timescale and get either flung apart at high speeds, destroyed, or merge. So gravity selects against objects being too close.

In any case, not all processes involving stars are "quite static." Binary stars can orbit quite close to each other, revolving with a period of only a few hours. Variable stars can change their brightness on a scale of hours, days, or even minutes. The bulk of the energy of a supernova is released within a few seconds. Pulsars rotate once every few milliseconds, or even faster. A black hole merger releases most of its gravitational waves in a few-millisecond interval.

Overall, your examples don't have much in common. There are reasons why each of them (except one) exists the way it does, but the reasons are different in each case. The fact that the gravitational force is quite weak and stars have a maximum density doesn't really have a whole lot to do with the idea that muscle strength is proportional to cross-sectional area, nor does it have much to do with the wavelengths of atomic transitions.

• Well, a rotation of common planet is about a day. When looking at star-planet systems, the scale is about a year. This are just an average values from our common G2 solar system, I guess these values are quite common in galaxies. Mar 14, 2019 at 14:19
• The orbital speed of sun in wilky way galaxy is about 200km/sec, but compared to its size (100x Earth's) the relative speed is about 100 minutes for 1 radius. The overall galactic year is 250M years - this is almost static in our time scale. Mar 14, 2019 at 14:23
• @PPrivalov Even within the Solar System, that's not really true. Jupiter has a rotation period of only 10 hours, and Venus has a rotation period of 243 days. Our star is certainly a common one, but the abundance of the particular characteristics of the planets in our solar system is not yet known. We don't know much about exoplanet rotation periods, but I wouldn't be surprised if we found even more extreme values, especially given that tidal locking tends to happen, which could bring the rotation period of K2-137b down to 4.31 hours. Mar 14, 2019 at 14:24
• @PPrivalov We are in the outer region of a rather large galaxy - the stars closer in can have much faster orbital periods. For example, the star S0-102 orbits the galactic center with a period of just 11.5 years (granted, it's basically directly orbiting Sagittarius A*, but the point still stands). Mar 14, 2019 at 14:28
• In the world of molecules - take a look at Rotational correlation time. It took about 2 picoseconds (*10e-12 seconds) for a water molecule to rotate one radian. Mar 14, 2019 at 14:45