Is the Peculiar Velocity a Tensor?

By Steven J. Grisafi, PhD.

In this age of computer search engines I suppose that most persons, when they come upon a concept with which they are not familiar, will use a search engine, such as Google, to obtain a brief summary regarding their curiosity. Hence, I suspect that most readers of this forum post will follow this rule and submit the words peculiar velocity to a computer search engine. If the reader were to do so, one would see a list a many references, all of which leading entries on the list would pertain to astronomy. Any such person unfamiliar with the concept of the peculiar velocity would invariably come away from the Internet search with a somewhat distorted understanding regarding the peculiar velocity. One would think that it is a quantity associated with astronomy and the motions of celestial objects only. Consequently, I can understand such a newly introduced person to the concept of the peculiar velocity as having a somewhat limited perception of its full character. But I cannot understand any such person so strikingly ignorant as having earned a PhD in any of the natural sciences.

Yet, this has happened. The reader should understand my dismay at having been confronted by a referee for an article that I submitted for publication to a science journal who asserted to me that celestial peculiar velocities are not vectors but are (second order) tensors. For the reader who may not be familiar with either vector or tensor analysis, a vector is a first order tensor. The dyadic product of two vectors forms a special type of second order tensor, this special type of which is called a dyad. While I can understand that empirical data evaluated by astronomers may possess the characteristics of a second order tensor field, I cannot understand their confusion as not recognizing that all velocities are vectors. Whether the velocity is of a terrestrial or celestial object, all velocities are directed motions possessing one direction and a magnitude. A dyad, or more generally a second order tensor, possesses two directions: the flow direction and the direction of the gradient. If astronomers have measured a field possessing two directed quantities, and not merely one, then what they have measured is perhaps not merely a velocity but a velocity field possessing a gradient.

My consternation towards the confusion of astronomers with regard to the peculiar velocity results from my knowledge that the concept of the peculiar velocity originated within my special field of study, the kinetic theory of gases. The development of the concept of the peculiar velocity began within the study of the statistical motions of gas molecules, which was then extended to the motions of all fluid particles. At no time did the concept of velocity lose its character as a vector being speed with direction. Yet, once the concept has left the laboratory reference frame on earth, astronomers assert that velocity is no longer a vector but now a (second order) tensor. Ugh!

How does one earn a PhD in physics and still confuse such basic tenets of mechanics? I suppose that such things happen because of the increasing need to specialize the education of a science professional. Educators must move quickly through fundamentals to be able to present the advanced material that students will need within their specialty. It could be that astronomers have confused the peculiar velocity with a flow field possessing both a flow direction and the direction of its gradient.Yet, I suspect the matter could be the simple confusion of multiplying two unit vectors instead of adding them vectorially. When one has only a flimsy understanding of the basic tenets of mechanics, we ought to look for the simplest explanation first.