## Faster than light…why the lawyer is wrong

In a post well outside of his expertise, lawyer Michael D. Cicchini weighs in on the recent observation that some neutrinos have been observed traveling faster than the speed of light. In his own words “I try to stick to law-related topics, but every once in a while I’ll delve into matters that are way over my head, like college sports and even physics.” He should stick to law.

His post makes the following two claims:

1. physicists created the concept of “dark matter” simply to keep Einstein’s theory of general relativity alive
2. The speed of light, Einstein had said, was the fastest speed at which anything could move. If an object could move faster than that, then Einstein’s theories would be proved incorrect.

He concludes with

Hopefully scientists won’t continue to cling to Einstein’s theories if it’s not warranted. The hallmark of science has always been its willingness to discard what no longer works when new evidence comes along. And toward that end, here’s an admittedly uneducated suggestion: abandon Einstein’s quest of unifying the quantum with the cosmic. […] it appears that such an attempt is doomed to failure because one of the theories they’re attempting to unify is wrong.

Back to the drawing board.

This post is ill-informed for many reasons. Let me try to address a few.

Dark Matter

The concept of dark matter was not invented to keep Einstein’s theory alive, it was invented to keep Newton’s theory alive (which Einstein’s General Theory of Relativity is consistent with, and extends to new areas). Essentially, given the form of Newton’s law, we can estimate the mass of a galaxy from its spinning. It turns out that this method of measuring the mass yields 10 times the mass that we get by estimating it from the directly observed mass (i.e. from counting up the the light emitted by the mass). We then have several choices:

1. the counting up the light method underestimates mass by a factor of 10…this is highly unlikely, because the method has been consistent with many other measurements, and is a direct consequence of fundamental laws of physics
2. Newton’s laws are wrong
3. there is more mass than we can actually see…we give it a name, “dark matter”, just as a place-holder until we could determine what it is.

Guess what? Scientists have explored all these possibilities! The most relevant ones to this post have to do with MOND (Modified Newtonian Dynamics). For example, Newton’s law of gravity is of the form

$\displaystyle F=\frac{GMm}{r^2} = ma \ \ \ \ \ (1)$

The consequences of this have been checked out many decimal places, and explains things as diverse as the tides, the formation of planetary rings, and the development of galaxies. What would it take to displace this? You’d have to modify it in such a way as to be consistent with all of the things that it predicts well, as accurately as it currently predicts, and have your modifications be consistent with some other set of observations which are inconsistent with the current theory. This is why it took 200 years to significantly modify Newton’s Theory of Gravity, with Einsteins General Theory of Relativity. An example of MOND could be something like

$\displaystyle F=\frac{GMm}{r^2} = m\mu\left(\frac{a}{a_o}\right)a \ \ \ \ \ (2)$

where ${\mu(a/a_o)}$ is a function that needs to be close to 1 for normal situations (and thus look like Newton’s laws), but be different for the case of spinning galaxies. In this case there is a single parameter, ${a_o}$, which is a critical acceleration far above which the situation is “normal”. Typical values are around ${10^{-10} {\rm m/s^2}}$, and one possible form for ${\mu()}$ is

$\displaystyle \mu(x) = 1-e^{-x} \ \ \ \ \ (3)$

Near the Earth, this translates to

$\displaystyle \mu(x) = 0.\underbrace{99999999\cdots}_{10^{10} {\rm nines!}} \ \ \ \ \ (4)$

On the big scale, on the outskirts of galaxies, the accelerations are comparable to ${a_o}$ and the predictions can lead to consistent mass curves without introducing more mass.

The dark matter postulate has also been, and continues to be, explored. There are some theoretical reasons to suggest that dark matter exists in the form of new fundamental particles, and there are attempts now to directly measure them.

The point here is that the original post was completely wrong about the motivation to include dark matter. Further, it was wrong in implying that scientists don’t examine and try to replace well-established ideas. I tell my students that it is a good day if you can demonstrate that one of your colleagues is wrong. It is an even better day if you can overturn a well-established idea (you get phone calls from Sweden for this). However, the burden is on the scientist trying to overturn a well-established idea to put forward something that is both consistent with all of the observations of the previous theory, and to demonstrate consistency on new tests that yield a different (and thus wrong) result from the old theory. Thus, it gets harder and harder to overturn an idea that is well established in science.

The Speed of Light

First, let me point out that Einstein doesn’t say in his 1905 paper on Special Relativity that the speed of light is the maximum speed (note, this is different than General Relativity, which handles gravity…a distinction that Michael Cicchini doesn’t seem to grasp). He merely asserts that the speed of light is measured to be constant in all inertial reference frames (i.e. measurements made in situations of constant speed). The derivation of the maximum speed is a consequence of this, and results in the Lorentz transformation. What we see in these equations is a factor entering in throughout of the form

$\displaystyle \gamma = \frac{1}{\sqrt{1-v^2/c^2}} \ \ \ \ \ (5)$

where ${c}$ is the speed of light, and ${v}$ is the speed of an object. This factor goes to infinity as ${v\rightarrow c}$, which places the speed limit. However, in certain circumstances, it might be that the factor takes a different form and, say, looks like

$\displaystyle \gamma = \frac{1}{\sqrt{1-v^2/c^2+10^{-20}}} \ \ \ \ \ (6)$

which would allow speeds greater than the speed of light. This simplistic hack will break other parts of the theory, and so a more subtle modification would be needed, but it could certainly be done…and it wouldn’t require overturning Einstein’s theory. We must be particularly observant of the following two facts:

1. relativistic quantum mechanics (QED) makes predictions so accurate, it would be like predicting the width of North America to the width of a human hair…any replacement theory would need to do as well
2. one other significant observation of neutrinos comes from supernova explosions, many light years away. It is observed that the neutrinos reaches us at the time, or very slightly below the time that the light from the explosion reaches us. Thus, over far longer spans of space there is no observed violation of the speed of light limit.

Although the neutrino story is intriguing, it does not entail the overthrow of Einstein even if it turns out to be true. More likely it will not be true, however. The biggest lesson of this is that lawyers should not speak about physics (although to be fair, Edwin Hubble, the discoverer of the expansion of the universe, was a lawyer before he became an astronomer). My bet is still on Einstein, no matter what the result of the experiment is. This isn’t because I am dogmatically attached to Einstein’s ideas, but that his ideas have been tested to an astonishing degree, and any replacement will have to do just as well.