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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Black Holes, Mercury, and Einstein: The Role of Dimensional Analysis

In last week’s posts we looked at basic astronomy and Einstein’s famous E=mc2 through the lens of the secret weapon of theoretical physicists, “dimensional analysis”, which imposes a simple consistency check on any known or proposed physics equation.  For instance, E=mc2 (with E being some kind of energy, m some kind of mass, and c the cosmic speed limit [also the speed of light]) passes this consistency condition.

But what about E=mc or E=mc4 or E=m2c3 ? These equations are obviously impossible! Energy has dimensions of mass * length2 / time2. If an equation sets energy equal to something, that something has to have the same dimensions as energy. That rules out m2c3, which has dimensions of mass2 * length3 / time3. In fact it rules out anything other than E = # mc2 (where # represents an ordinary number, which is not necessarily 1). All other relations fail to be consistent.

That’s why physicists were thinking about equations like E = # mc2 even before Einstein was born. 

The same kind of reasoning can teach us (as it did Einstein) about his theory of gravity, “general relativity”, and one of its children, black holes.  But again, Einstein’s era wasn’t first to ask the question.   It goes back to the late 18th century. And why not? It’s just a matter of dimensional analysis.

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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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General Relativity, Tides, and Who Orbits Whom

Why have I been debunking Professor Muller’s claim that “the Sun orbits the Earth just as much as the Earth orbits the Sun”? Understanding why he’s wrong makes it easier to appreciate some central but subtle concepts in general relativity, Einstein’s conception of gravity.

What I want to do today is look at the notion of tides. Tides take on more importance in general relativity than in Newton’s theory of gravity. They can tell you which objects are gravitationally dominant in a coordinate-independent way.

A few posts ago, some of the commenters attempting to refute Professor Muller focused on showing the Sun is gravitationally dominant over the Earth. They were on a correct path! But nobody quite completed the argument, so I’ll do it here.

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Coordinate Independence, Kepler, and Planetary Orbits

Could you, merely by changing coordinates, argue that the Sun gravitationally orbits the Earth?  And could Einstein’s theory of gravity, which works equally well in all coordinate systems, allow you to do that?  

Despite some claims to the contrary — that all Copernicus really did was choose better coordinates than the ancient Greek astronomers — the answer is: No Way. 

How badly does the Sun’s path, nearly circular in Earth-centered (geocentric) coordinates, violate the Earth’s version of Kepler’s law?  (Kepler’s third law is the relation T=R3/2 between the period T of a gravitational orbit and the distance R, which is half the long axis of the ellipse that the orbit forms.)   Since the Moon takes about a month to orbit the Earth, and the Sun is about 400 = 202 times further from Earth than the Moon, the period of the Sun would be 4003/2 = 8000 times longer than the Moon’s, i.e. about 600 years, not 1 year. 

But is this statement coordinate-independent? Can it serve to prove, even in Einstein’s theory, that the Earth orbits the Sun and the Sun does not orbit the Earth? Yes, it is, and yes, it does. That’s what I claimed last time, and will argue more carefully today.

Of course the question of “Does X orbit Y?” is already complicated in Newtonian gravity.  There are many situations in which the question could be ambiguous (as when X and Y have almost equal mass), or when they form part of a cluster of large mass made from many objects of small mass (as with stars within a galaxy.)  But this kind of ambiguity is not what’s in question here.  Professor Muller of the University of California Berkeley claimed that what is uncomplicated in Newtonian gravity is ambiguous in Einsteinian gravity.  And we’ll see now that this is false.

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Even in Einstein’s General Relativity, the Earth Orbits the Sun (& the Sun Does Not Orbit the Earth)

Back before we encountered Professor Richard Muller’s claim that “According to [Einstein’s] general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun,” I was creating a series of do-it-yourself astronomy posts. (A list of the links is here.) Along the way, we rediscovered for ourselves one of the key laws of the planets: Kepler’s third law, which relates the time T it takes for a planet to orbit the Sun to its distance R from the Sun. Because we’ll be referring to this law and its variants so often, let me call it the “T|R law”. [For elliptical orbits, the correct choice of R is half the longest distance across the ellipse.] From this law we figured out how much acceleration is created by the Sun’s gravity, and concluded that it varies as 1/R2.

That wasn’t all. We also saw that objects that orbit the Earth — the Moon and the vast array of human-built satellites — satisfy their own T|R law, with the same general relationship. The only difference is that the acceleration created by the Earth’s gravity is less at the same distance than is the Sun’s. (We all secretly know that this is because the Earth has a smaller mass, though as avid do-it-yourselfers we admit we didn’t actually prove this yet.)

T|R laws are indeed found among any objects that (in the Newtonian sense) orbit a common planet. For example, this is true of the moons of Jupiter, as well as the rocks that make up Jupiter’s thin ring.

Along the way, we made a very important observation. We hadn’t (and still haven’t) succeeded in figuring out if the Earth goes round the Sun or the Sun goes round the Earth. But we did notice this:

This was all in a pre-Einsteinian context. But now Professor Muller comes along, and tells us Einstein’s conception of gravity implies that the Sun goes round the Earth just as much (or just as little) as the Earth goes round the Sun. And we have to decide whether to believe him.

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Earth Around the Sun, or Not? The Earth-Centered Coordinates You Should Worry About

We’re more than a week into a discussion of Professor Richard Muller’s claim that “According to the general theory of relativity, the Sun does orbit the Earth. And the Earth orbits the Sun. And they both orbit together around a place in between. And both the Sun and the Earth are orbiting the Moon.” Though many readers have made interesting and compelling attempts to prove the Earth orbits the Sun, none have yet been able to say why Muller is wrong.

A number of readers suggested, in one way or another, that we go far from the Sun and Earth and use the fact that out there, far from any complications, Newtonian physics should be good. From there, we can look back at the Sun and Earth, and see what’s going on in an unbiased way. Although Muller would say that you could still claim the Sun orbits the Earth by using “geocentric” coordinates centered on the Earth, these readers argued that such coordinates would not make sense in this distant, Newtonian region.

Are they correct about this?

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Is it Meaningful to Say that Earth Goes Round the Sun, or Not? (And Why Is This So Hard…?)

Is the statement “The Sun Orbits the Earth” false? Not according to professor Richard Muller of the University of California, Berkeley, as I discussed yesterday. Muller argues that Einstein’s theory of general relativity implies that you can view the Sun as orbiting the Earth if you like, or that both the Sun and Earth orbit Venus, or a random point in space, or anything else for that matter. Meanwhile, every science textbook in our kids’ classrooms says that “The Earth Orbits the Sun“. But for all of our discussions yesterday on this subject, we did not yet collectively come to any conclusions as to whether Muller is right or wrong. And we can’t hope to find evidence that the Earth orbits the Sun if the reverse is equally true!

When we’re trying to figure out whether a confusing statement is really true or not, we have to speak precisely. Up to this stage, I haven’t been careful enough, and in this post, I’m going to try to improve upon that. There are a few small but significant points of clarification to make first. Then we’ll look in detail at what it means to “change coordinates” in such a way that would put the Sun in orbit around the Earth, instead of the other way round.

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