Dr. Christopher S. Baird

# Why is physics scale invariant?

Category: Physics      Published: December 6, 2013

By: Christopher S. Baird, author of The Top 50 Science Questions with Surprising Answers and Associate Professor of Physics at West Texas A&M University

These salmonella bacteria would cease to function if scaled up to the size of humans. Public Domain Image, source: CDC.

In general, physics is not scale invariant. There are a few reasons for this:

1. The universe is quantum on small scales. When an interaction field is quantized, it means that it is composed of a collection of fundamental units that can not be further subdivided. These units (quanta) have fixed properties. For instance, a beam of electromagnetic waves (such as light) is composed ultimately of photon particles, and each photon carries a fixed energy E = hf. This means that a radio wave with frequency f = 2 GHz is composed of photons that each have an energy of 1.3×10-24 Joules. The smallest radio beam you can have is composed of one photon, so the lowest energy radio beam possible at 2 GHz is a beam of 1.3×10-24 Joules. You cannot get a radio beam weaker than this at 2 GHz. So you can never scale a 2 GHz system beyond this limit. This quantized, discrete nature exists everywhere in the universe if you look small enough. For instance, a clock is made out of atoms. It is therefore impossible to make a clock that is smaller than 10-11 meters because this size is smaller than even the tiniest atoms. Every tangible object is made out of atoms and therefore has this atomic limit to its scaling.

2. The fundamental forces vary differently with distance. The fundamental forces are gravity, electromagnetism (which includes all chemical forces), the weak nuclear force and the strong nuclear force. The weak and strong nuclear force are very short-ranged and do not extend much beyond the size of atomic nuclei. The gravitational and electromagnetic force both die off with respect to distance r according to 1/r2. But, electric charges usually exist as electric dipoles, and the electromagnetic force due to a dipole dies off much quicker, at a rate of 1/r3. This means that on astronomical scales, gravity dominates; on human scales, gravity and electromagnetism dominate; on microscopic and atomic scales, electromagnetism dominates; and on nuclear scales, the nuclear forces dominate. If you took a machine that relied primarily on gravity, such as an hourglass sand timer, and scaled it down to the size of bacteria, it would cease to function. At this scale, the electromagnetic forces would overwhelm the gravitational force and the sand would stick to the walls rather than flow down. If you took a single human cell or an ant and scaled it up to the size of an elephant, it would collapse under its own weight, because gravity would become stronger at this large scale than the electromagnetic forces holding the cell or ant intact.

3. The surface area to volume ratio of an object changes as you change the scale. An object mostly interacts with its environment through its surface. An object with a larger surface relative to its volume will interact with its environment more quickly. For instance, smaller objects cool more quickly than larger objects of the same shape because they have proportionally more surface area to cool from. Bigger objects rust more slowly than smaller objects with the same shape. A sphere has a surface area of 4πr2 and a volume of (4/3)πr3, giving a surface area to volume ratio of 3/r. As the scale r of the sphere gets bigger, its surface-to-volume ratio gets smaller. This fact means that bigger spheres have proportionally less surface through which to interact with the environment. Chemical reactions between substances that are not atomically mixed occur at the surface interface between the lumps or particles of material. The more surface area available, the more atoms there are available to react, and the faster the reaction occurs. Therefore, chemical reaction rates depend on the scales involved. Whereas a bucket of grain will hardly burn at all, a mist of fine grain particles sprayed into the air will explode. Salt that has large grains will dissolve in water more slowly than salt with fine grains. Animals and organs with greater surface area can absorb nutrients more quickly from their surroundings.

For the most part, all other scale-dependent physical effects are due to a combination of the three effects listed above. For example, water will climb up narrow channels in trees using capillary action but will not climb up large plumbing pipes. This discrepancy is due to the fact that a pipe at a large scale has gravity dominate over intermolecular electromagnetic forces in general, and because water in the larger pipe has a smaller surface area to volume ratio, and therefore proportionally less surface tension to aid in capillary action.