# Lightning Fast Math for Neutrinos versus the Speed of Light

Albert Michelson calculated the speed of light in 1879

On Sept. 22, 2011, scientists at the European Organization for Nuclear Research reported that they had measured neutrinos traveling faster than light. This discovery has taken about two years, and is based on the OPERA experiment at CERN.

## The Speed of Light and of Neutrinos

Let There be Light

Since light is the “cosmic speed limit,” the speed of light should be measured “in a vacuum.” Light slows down as it passes through various materials. This change in speed causes refraction, but that whole subject is ignored when discussing the relative speed of neutrinos.

The speed of light in a vacuum is 299,792,458 m/s (meters per second). The speed of light is symbolized as the English letter c.

Light may be considered as a set of fast-moving particles with no “rest mass” (no weight). The particles are called “photons”. Light may also be thought of as a wave; we see different wavelengths as different colors.

## Speeding Neutrinos

A neutrino is a “little neutral” particle with almost no “rest mass”, and no electrical charge. It is very rare for a neutrino to interact with any other matter. However, the sheer number of neutrinos created by some quantum reactions allows some interactions to be observed.

The news reports agree that the neutrinos were 60 ns (nanoseconds, or billionths of a second) faster than light over a distance of 730 km (kilometers). The scientists measured the time to within 10 ns, and the distance to within 30 cm (centimeters).

Edward Morley collaborated with Albert Michelson

## Math Problems: the Hazards of Reporting Numbers

Various articles give similar reports, but lead to somewhat different numbers.

## Reporting the Measured Velocity

According to the “Scientists stunned…” article, the newly measured speed of neutrinos through the earth is 300,006,000 m/s.

By that account, the neutrinos go 100.071,229,944,016,803,785% of the speed of light, with some rounding from my Windows Vista Microsoft Calculator program. The difference, as a rounded percentage, is 0.07123%.

Over the 730 km distance, it would take neutrinos 0.0024332846676399805337226588801557 versus photons 0.0024350178949465099619017100156669 seconds.

That leaves a difference of 0.000,001,733 seconds, which is much more than the 60 nanoseconds ( 0.000,000,060 ) as reported elsewhere.

## Another Version of the Traffic Report

As noted above, the reported difference was 60 ns over a distance of 730 km (kilometers), with a measurement error of perhaps 10 ns. This result was detected some 1,600 times over a period of about two years.

Light should take  0.002,435,017,894,946,509,961,901,710,015,667 seconds to travel 730 km.

Subtract 60 ns, or 0.000,000,060 for a time of 0.002,434,958 seconds, rounded, for the neutrinos.

Therefore the neutrino traveled at 100.002,459,793,824,368,301% of the speed of light. That difference is 0.0025% faster; but much less than the 0.07123% from the first report.

## A Third Version of the Report

The “Particles Appear…” article in Science Daily saved me from doing any arithmetic. There the reported difference is 20 parts in a million, or 0.002%.

Although most human beings will not notice a change in the way the universe works, scientists have some work to do.

The first task is for other scientists to review the data and methodologies used in the OPERA/CERN tests. The second is to replicate the results at some other facilities.

## Why would these Results Matter?

Relativity theory is one of the pillars of modern physics. The speed of light is extremely important to this theory.

#### Nothing is so Constant as the Speed of Light in a Vacuum

By 1879, Albert Michelson measured the speed of light to be very close to 186,350 miles per second, plus or minus 30. At that time, light was known to have wave-like properties, and therefore must be a wave in some substance, a propagating medium, just as sound travels through air, water or railway tracks. Light’s unknown medium was called the “aether”.

James Maxwell: photo by Fergus of Greenstock

In addition, James Maxwell theorized that light’s speed should be about 186,300 miles per second. Maxwell and Michelson were in very close agreement.

Michelson decided to measure how quickly the earth is flying through this aether by measuring the difference in speed of light traveling back-and-forth across the direction of motion, versus traveling back-and-forth along the direction of motion. (The details of this test must await another article, if there is popular demand).

If light behaved as a wave propagating in aether, there should be an interference pattern in the light traveling in different directions relative to the earth’s motion. The earth spins on its axis, orbits around the sun, and follows the sun in its orbit around the center of the Milky Way galaxy.

Edward Morley collaborated with Michelson; the definitive Michelson-Morley experiment was performed in 1887. It demonstrated that there was no interference, and therefore no aether to propagate light as a wave.

This experiment also implied, however, that light was not “shot like a bullet” with extra speed based on the speed of the “gun”. This was only fully demonstrated in 1964 by Alvager et al. However, it seemed clear even in 1887 that the speed of light in a vacuum was constant, regardless of the motion of the emitter.

In 1905, Einstein published “Special Relativity.” His mathematics demonstrated that any observer who was not accelerating would measure the speed of light as exactly the same c as any other observer moving at any other constant speed.

In part, his reasoning was built on Maxwell’s equations, as well as the concept that physics “works the same” for any observer moving at any constant speed. That, in itself, would dictate that the speed of light should be the same regardless of how fast the observer is traveling.

As well, Einstein stated that it takes a larger force to accelerate an object as its speed approaches c. In fact, if the object were accelerated to exactly the speed of light, it would effectively have infinite mass. Perhaps more accurately, it would require an infinitely strong force to cause any additional acceleration. For a velocity ‘v’, the factor, called gamma or ‘γ’ is equal to “1 / (  ( 1 – (v^2/c^2 ) )^(1/2) )”.

At low speeds, this factor is effectively equal to 1. At nearly the speed of light, γ approaches infinity as “v approaches c“.

1. says

I’m here in the “math” more than “physics” side.
Direct tests on this planet involve Michelson-Morley style multiple reflections to get some distance.
If memory serves, speed of light and/or distance to moon were tested using a reflector planted on the lunar surface.

Anyway, from what I understand of the OPERA/CERN neutrino test, they know the through-the-earth distance. There’s no tunnel, so they don’t race the photons directly. They just calculate using speed of light in a vacuum.

A nice side-by-side race from, say, earth-to-mono would be fun. Expensive but fun.

Actually that was done in larger scale. One supernova’s neutrinos were spotted 4 hours before the light. That was explained as the neutrinos got past the star’s gas before the gas moved out of the way. I just read (but didn’t note the URL…sorry) that if the OPERA/CERN result held for the interstellar distance, the time gap would have been 4 years.

The jury is still out on this one. I just reported on a bit of the math.

2. russell_of_clan_gunn says

Thanks for the great overview of the controversy.
I am curious how we can be sure of the speed of light we are comparing the neutrino speed to is actually correct. Clearly light is affected by gravity and space is not a perfect vacuum I think given the slippery wave/partial behavior of light that frequently interacts with other matter the speed might be difficult to get perfect.