Given that space is expanding, why is it that it is still '3,000 miles' from New York to San Francisco? Why is it that the volume of the space in a crystal lattice of, say, NaCl, still X cubic Angstroms? Doesn't 'local' space expand too?

Asked by: Jim Hamby

Answer

In the examples you give, there are forces that interact between the points in question to maintain their mutual distances, even though the universe as a whole is expanding.

The usual analogy used to describe the expanding universe is to consider a balloon on which dots have been painted to represent, say, the galaxies. As the balloon is inflated, the distance between any two 'galaxies' increases. Furthermore, the farther apart the two galaxies, the greater the speed with which they separate as the balloon is inflated. Because the balloon is spherical, these observations hold true for all pairs of galaxies.

The observed increase of the rate of separation of galaxies with the increased distance between them is known as Hubble's constant [Hubble's constant is estimated to range between 70 and 90 km/s'Mpc, where Mpc = megaparsec = 3.26 x 10^6 light years].

The key observation in the balloon analogy is that there are no forces constraining the distances between the 'galaxies' on the balloon. They are free to increase as the balloon is inflated simply because there is more 'space' between them. However, in the cases of the distance between New York and San Francisco and the lattice spacing in an NaCl crystal, there are very real forces that act to maintain the distances so that they do not increase with the overall increase of volume of the universe.

It should also be noted that, even in the absence of constraining forces, the rate of separation in the examples would be very small indeed. Note that the rate of separation is proportional to the distance between the objects. Assuming that Hubble's constant has the value 80 km/s'Mpc, I calculate that two objects 3000 miles apart will recede from each other at the rate of 1.25 x 10^-9 cm/sec. Thus, it would take a billion seconds, or about 31.7 years, for New York and San Francisco to separate by 1.25 cm due to the expansion of the universe (assuming no constraints). The effect on the NaCl crystal would be proportionally and, hence, extraordinarily smaller.

An interesting and relevant, if somewhat technical, discussion dealing with the question of what happens to rigid measuring rods in the presence of a gravitational wave, which causes a local expansion of space, can be found in:

'If light waves are stretched by gravitational waves, how can we use light as a ruler to detect gravitational waves?', Peter R. Saulson, American Journal of Physics, Vol. 65, No. 6, June 1997, pp. 501-505.
Answered by: Warren Davis, Ph.D., President, Davis Associates, Inc., Newton, MA USA