Category: Observation

E=mc^2 gets all the fame and press. That’s because this deceptively simple equation hides a rich spectrum of insights which still have ripple effects on the modern world. It tells us that a large input of energy is needed to generate a little mass; the flip-side of that statement is that if you can effect a small change in mass, you can get a LOT of energy. It was that realization, and a ton of hard work by both engineers and physicists, that led to the first atomic weapons. To this day, we feel the effects of those insights. The consequences of nuclear proliferation are still with us, the threat of loose nukes an omnipresent reality in modern geopolitics.

E=mc^2 has done more than destroy cities or create new political entanglements. Converting mass to energy, as through the annihilation of matter and anti-matter in the PET scan, saves lives. Nuclear power created a new industry, a new way to generate electricity, and new challenges both in engineering and politics.

But E=mc^2 is only one half of a greater story. The general equation relating energy, mass, AND motion is E^2 = (mc^2)^2 + (pc)^2 – that second part encodes the effect of motion on energy. Just as this equation is interesting when nothing is moving (p=0), it’s interesting when there is no mass (m=0).

No mass means all of a body’s energy arises from its motion. Without motion, it does not exist. E = pc is the beginning of something greater, though at first that may not be terribly obvious. This equation, the forgotten fraternal twin of E=mc^2, carries with it an important requirement: if you are going to have no mass, you must move at the speed of light. This revelation teaches us something about light itself – that regardless of its other properties, it has no mass-energy. It is all movement, yet it transmits inertia from one place to another.

The fact that E=pc requires mass-less things to move at the speed of light arises from a mathematical necessity; due to the definition of momentum, p, in relativity, the only way that a mass-less object can do anything but not exist at all is that it must move at the speed of light.

However, that’s all this equation can tell us. The mathematical requirement that E=pc means the object moves at the speed of light leaves us with an otherwise undefined equation; that’s because p depends on both the mass (which is zero) and a function of velocity that returns infinity when the speed is that of light. This zero and this infinity compete to keep energy from being trivially zero, but in doing so give us no further insight into light.

The parallel revolution of quantum physics and the discretization of energy in radiation was needed to make progress on light itself. Quantum physics gives us the framework to describe the energy and momentum of light, in terms that are definite.

E=pc is often forgotten in all the excitement about the benefits, powers, and geopolitics of E=mc^2. Yet, E=pc is a gateway into the very nature of light. Light is so fundamental to life, and interactions among the living, that it is no less important than matter. In the study of E=pc, and the nature of radiation, we eventually come full circle on the relationship between energy and matter. That’s a story for another time. For now, let’s just quietly adore E=pc.

The conductor of an orchestra is not for show. A conductor is not just a part of the social construct of the orchestra. A conductor is not just a means by which one person can be made more important than another. The reality is that the conductor of an orchestra is required, by the speed of sound and the speed of light, to exist.

What do I mean by this? Let’s consider the human head. The distance between our two ears is about 22cm (about 8.5 inches). This distance means that sounds that are not originating from directly in front of us reach our two ears at different times. The human auditory system is capable of discerning sounds that are no more than 0.000660 seconds apart from one another – that’s 0.660 milliseconds (ms) [1]. The result of this ability to process signals which arrive more than 0.660 ms apart is that we can localize sound in space. This process, called “Duplex theory,” means that if sounds are closer than 0.660 ms then they cannot be localized; we lose the ability to distinguish the sounds, and they seem to arrive at the same time.

So, what does this have to do with the conductor of an orchestra? Let’s imagine a situation where the orchestra, full of spite for the primadonna conductor, decides to go completely democratic and tosses the tyrant out onto the street. “We can synchronize ourselves!” they proclaim, and they turn to the oboe player.

“Oboe player, sit in the middle of the orchestra pit and blow a series of tones in the tempo of the song! That way, we can listen for the tempo and mark the passage of time, so that we, too, can stay on time.”

This seems like a great idea. Sadly, the universe has conspired against this democratic process. Sound travels at about 300 m/s. As a result, it takes 0.033s, or 33ms, for sound to travel even the 10 meters from the center of the pit to the outermost players (assuming a 20m-diameter pit). The result is that a player on the outskirts of the pit hears the oboe’s beat 33ms after a player next to the oboe hears it. Players on the outside of the pit are 33ms LATE in playing their instrument.

What does this mean? It means that the players on the outskirts of the pit become unsynchronised from their center-pit colleagues. Is this a bad thing? YES. Since the human ear is capable of hearing the difference between sounds that are at least 0.66ms apart, and the sounds from the inside and outside of the pit are 50 times further apart than that in time, the whole orchestra sounds like they’re playing out-of-time. The resulting cacophony will  surely cause ticket sales to plummet, and our democratic orchestra will go flat broke.

The conductor, on the other hand, standing in front of the orchestra, uses light, not sound, to synchronize the orchestra. Since light travels at about 300,000,000 m/s, it takes just 0.000067 ms for their hand gestures to be seen by players at the back of the pit, compared to those closest to the conductor. This means that players are just 0.000067 ms out-of-time with one another; this is vastly below the human ability to hear the difference between two sounds, saving the orchestra and insuring a flawless performance.

Before you cast out that tyrant of a conductor, remember this: the universe has conspired to make it necessary to use light, and not sound, to synchronise the orchestra. Instead of tossing the bum out on the street, remember that you need them more than you don’t. Or, at least, think about hiring a replacement.

[1] http://en.wikipedia.org/wiki/Interaural_time_difference

Sunday night, as we sat on the patio by the grill, we thought about thunder and lightning. The sky was overcast, and in the distance storms were brewing. Lightning reflected off the clouds above us, followed a short time later by thunder. We started to think about the relationship between the light and sound from the storm. We knew that it would be easy to come up with a quick relationship between the time between the sound of thunder and the flash of lightning.

The light and sound travel the same distance, d. The time the light requires to go from the storm to us is shorter than the time for the sound. The time for the light is t_l = d/c, where c is the speed of light (300,000,000 m/s). The time for sound is t_s = d/v_s, where v_s is the speed of sound (300 m/s). We are interested in the number of seconds between the sound and the light, so let’s compute the time difference using these formula:

t_s – t_l = d/v_s – d/c = d(1/v_s – 1/c)

We can rewrite the quantity (1/v_s – 1/c) = (1/300 – 1/300,000,000) = 1/300 (1-0.000001), which is (to a very good approximation) 1/300.

We arrive at our final relationship:

t_s – t_1 = d/300

where time is in seconds and distance is in meters. So, if we are sitting on our patio and we see the flash of lightning, and then 6 seconds later we hear the thunder, the storm is (6s * 300 m/s) = 1,800 m, or about 1.1 miles.