Relatively Confused: Is It True That Nothing Can Exceed Light Speed?

A post for general readers:

Einstein’s relativity. Everybody’s heard of it, many have read about it, a few have learned some of it.  Journalists love to write about it.  It’s part of our culture; it’s always in the air, and has been for over a century.

Most of what’s in the air, though, is in the form of sound bites, partly true but often misleading.  Since Einstein’s view of relativity (even more than Galileo’s earlier one) is inherently confusing, the sound bites turn a maze into a muddled morass.

For example, take the famous quip: “Nothing can go faster than the speed of light.”  (The speed of light is denoted “c“, and is better described as the “cosmic speed limit”.) This quip is true, and it is false, because the word “nothing” is ambiguous, and so is the phrase “go faster”. 

What essential truth lies behind this sound bite?

Faster Than Light? An Example.

Let’s first see how it can lead us astray.

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E = m c-Squared: The Simple Dimensions of a Discovery

In my last post I introduced you to dimensional analysis, an essential trick for theoretical physicists, and showed you how you could address and sometimes solve interesting and important problems with it while hardly doing any work. Today we’ll look at it differently, to see its historical role in Einstein’s relativity.

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Dimensional Analysis: A Secret Weapon in Physics

It’s not widely appreciated how often physicists can guess the answer to a problem before they even start calculating. By combining a basic consistency requirement with scientific reasoning, they can often use a heuristic approach to solving problems that allows them to derive most of a formula without doing any work at all. This week I want to introduce this to you, and show you some of its power.

The trick, called “dimensional analysis” or “unit analysis” or “dimensional reasoning”, involves requiring consistency among units, sometimes called “dimensions.” For instance, the distance from the Earth to the Sun is, obviously, a length. We can state the length in kilometers, or in miles, or in inches; each is a unit of length. But for today’s purposes, it’s irrelevant which one we use. What’s important is this: the Earth-Sun distance has to be expressed in some unit of length, because, well, it’s a length! Or in physics-speak, it has the “dimensions of length.”

For any equation in physics of the form X = Y, the two sides of the equation have to be consistent with one another. If X has dimensions of length, then Y must also have dimensions of length. If X has dimensions of mass, then Y must also. Just as you can’t meaningfully say “I weigh twelve meters” or “I am seventy kilograms old”, physics equations have to make sense, relating weights to weights, or lengths to lengths, or energies to energies. If you see an equation X=Y where X is in meters and Y is in Joules (a measure of energy), then you know there’s a typo or a conceptual mistake in the equation.

In fact, looking for this type of inconsistency is a powerful tool, used by students and professionals alike, in checking calculations for errors. I use it both in my own research and when trying to figure out, when grading, where a student went wrong.

That’s nice, but why is it useful beyond checking for mistakes?

Sometimes, when you have a problem to solve involving a few physical quantities, there might be only one consistent equation relating them — only one way to set an X equal to a Y. And you can guess that equation without doing any work.

Well, that’s pretty abstract; let’s see how it works in a couple of examples.

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Physics is Broken!!!

Last Thursday, an experiment reported that the magnetic properties of the muon, the electron’s middleweight cousin, are a tiny bit different from what particle physics equations say they should be. All around the world, the headlines screamed: PHYSICS IS BROKEN!!! And indeed, it’s been pretty shocking to physicists everywhere. For instance, my equations are working erratically; many of the calculations I tried this weekend came out upside-down or backwards. Even worse, my stove froze my coffee instead of heating it, I just barely prevented my car from floating out of my garage into the trees, and my desk clock broke and spilled time all over the floor. What a mess!

Broken, eh? When we say a coffee machine or a computer is broken, it means it doesn’t work. It’s unavailable until it’s fixed. When a glass is broken, it’s shattered into pieces. We need a new one. I know it’s cute to say that so-and-so’s video “broke the internet.” But aren’t we going a little too far now? Nothing’s broken about physics; it works just as well today as it did a month ago.

More reasonable headlines have suggested that “the laws of physics have been broken”. That’s better; I know what it means to break a law. (Though the metaphor is imperfect, since if I were to break a state law, I’d be punished, whereas if an object were to break a fundamental law of physics, that law would have to be revised!) But as is true in the legal system, not all physics laws, and not all violations of law, are equally significant.

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Did BICEP2 Detect Gravitational Waves Directly or Indirectly?

A few weeks ago there was (justified) hullabaloo following the release of results from the BICEP2 experiment, which (if correct as an experiment, and if correctly interpreted) may indicate the detection of gravitational waves that were generated at an extremely early stage in the universe (or at least in its current phase)… during a (still hypothetical but increasingly plausible) stage known as cosmic inflation.  (Here’s my description of the history of the early universe as we currently understand it, and my cautionary tale on which parts of the history are well understood (and why) and which parts are not.)

During that wild day or two following the announcement, a number of scientists stated that this was “the first direct observation of gravitational waves”.  Others, including me, emphasized that this was an “indirect observation of gravitational waves.”  I’m sure many readers noticed this discrepancy.  Who was right?

No one was wrong, not on this point anyway.  It was a matter of perspective. Since I think some readers would be interested to understand this point, here’s the story, and you can make your own judgment.

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The Amazing Feat of Quantum Tunneling

Our quantum world has many odd and counter-intuitive features.  One of these is “tunneling” — the ability of objects to pass through walls, escape from traps, and slip under mountains into the next valley.   We don’t encounter this effect in daily life; objects we’re used to are so incredibly unlikely to tunnel from one … Read more

Dog Brains and Fishing Line: 2 Fun Articles

Nothing about quantum physics today, but … wait, everything is made using quantum physics… — Could you imagine getting a dog to sit absolutely still, while fully awake and listening to voices, for as much as 8 minutes? Researchers trained dogs to do it, then put them in an MRI [Magnetic Resonance Imaging] machine to … Read more

Happy (Chilly) New Year

Welcome 2014! And quite a start to the year, with a cold snap that rivals anything we’ve seen in two decades. I don’t remember cold like this since the horrid winter of 1994, when the Northeastern U.S. saw snowstorms and extreme cold that alternated back and forth for weeks. Of course, when I was a child in the 1970s, such chills happened a lot more often; I remember a number of New England mornings where I awoke to a thermometer reading of -20ºFahrenheit (-29ºCelsius) [244 Kelvin].

The scariest negative temperature numbers that one hears about from the media are associated with the “wind chill”, which is a number that is supposed to measure how cold the air “feels” to your skin.  But “wind chill” is a rather subjective and controversial measure — there’s no unique way to define it, since you’ll feel differently depending on how much exposed skin you have, on your body weight, on your age and conditioning, etc.  By contrast, the temperature measured by a thermometer is defined independent of how humans feel, and experts agree on what it is and means. Oh sure, people use different scales to measure it: Fahrenheit (F), Centigrade or Celsius (C), and Kelvin (K).  But the differences are no more than the distinction between meters and feet, or between kilograms and pounds; it’s straightforward, if a bit annoying, to convert from one to the other.

So everyone agrees the temperature is and feels extremely cold, But is it, from the point of nature, really that much colder than usual? To say it another way: it was 84ºF (29ºC) in southern Florida yesterday.  How much warmer is that than the -40ºF (-40ºC) that was registered in the cold Minnesota morning?

Well, you might first think: wow, it’s a difference of 124ºF (69ºC), which sounds like a huge difference.  But is it really so huge?

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