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.

Simple Example: Velocity, Radius and Period

First, a super-easy one just to illustrate the point. Suppose we want to find the velocity v of the Earth as it travels round the Sun. If we call the average radius of the Earth’s roughly circular orbit R, and we call T the time it takes to orbit the Sun, then what formula should we write down? Well, there’s only one possibility that’s consistent. Velocity v is a length traveled per time; it has dimensions of length over time. R is a length. T is a time. And so the equation that relates them must be of the form

  • v = # R / T

where “#” is an unknown number that this argument doesn’t specify.

Since we don’t know this number, have we really learned anything? Yes we have! The formula cannot set v equal to R2/T, or T2/R, or R T, or R1.4 /T2.6. Any formula other than v = # R/T would relate a length per time to something that isn’t a length per time… and would therefore be nonsensical. Just by demanding sense, we have mostly solved the problem without doing any work at all, except for one unknown #.

If we want to be precise, we’ll still have to calculate the unknown #. If the orbit were circular that would be easy; # = 2π. For a realistic, elliptical orbit, you have to actually calculate it. Still, for a nearly-circular orbit like the Earth’s, this # it’s not going to be a billion for a nearly-circular orbit, nor is it going to be a billionth. It will be a number that’s not far from 2π, which in turn is not too, too far from 1. (2π is approximately 6.) So we can make an estimate without doing much, if any, calculation.

Interesting Example: Kepler’s Law, In Detail

Now let’s take a less trivial example, though still easy to do using other methods. Recently, using do-it-yourself techniques, I showed you how you yourself could derive Kepler’s third law, which relates the radius of a planet’s orbit R to its orbital period T, specifically that R3 is proportional to T2. We found this was true for objects that orbit the Sun. We also found it was true for objects that orbit the Earth, but with details that were different. Can we find a formula which is true both for the Sun and the Earth — one that explains the difference?

Well, under an assumption — that Newton’s gravity is involved somehow — we can. This is where physics reasoning and some experience comes in.

First, if gravity is at work, an experienced physicist knows that Newton’s constant G always appears, because this constant characterizes the overall strength of gravity. The dimensions of G have to be consistent with Newton’s gravitational force equation

  • F = G M m / r2

which gives the force of gravity between two objects of mass M and m that are separated by a distance r. Rearranging for convenience, we can write this as

  • G = F r2 /(M m)

In first-year physics we learn that force has dimensions of mass times length divided by time squared. M and m have dimensions of mass, and r has dimensions of length. From the above equation G = F r2 /(M m), we find the dimensions of G itself:

  • dimensions of G = dimensions of F r2/(M m) [for consistency!]
    • = (dimensions of F) * (dimensions of r2)/(dimensions of M m)
    • = (mass * length / time2) * (length2)/(mass2) = (length3/time2/mass)

Moreover, under our assumption that gravity is at work, and since we are considering objects that all orbit the same central body, such as the Sun, we can guess that the mass of that central body comes in somehow. Let’s refer to that mass as “M”.

So now to Kepler’s law: might there be an equation that relates gravity’s ever-present constant, G, the mass of a central body, M, the period T of an object orbiting that central body, and the radius R of that orbit? Well, how about

  • G / M = # R4/T2


  • G M5 = # R9 T2



  • G M3/2 = # T5/R7/3 ?

No Way! Each of these possible equations is nonsense! because the dimensions of the left hand side are not equal to the dimensions of the right hand side!

But there is in fact (as I’ll convince you in a moment) one and only one possible answer that could make sense! That’s this one:

  • G M = # R3/T2

And this confirms that for a particular central object of mass M, all objects that orbit it have the same ratio for R3 to T2. In other words, you can guess Kepler’s third law of orbits simply by using dimensional analysis. No complicated equations are required.

Again, # is an unknown number that we would have to calculate. But even though we don’t know it yet, we’re most of the way to finding the formula we want, and we haven’t done any work other than checking dimensions!

Why is this the only possible formula? One can be systematic about it, but here’s a quick way to see it. R and T have no units of mass, but G and M do. So to relate G and M to R and T, there must be some combination of G and M in which the dimensions of mass cancel. Since, as we just saw above, G has dimensions of something divided by mass, G M is the only combination where the dimensions of mass cancel, leaving only dimensions of length and time! In fact GM has dimensions of length3 divided by time2 — and that means G M can only be related to R3/T2. That’s all there is to it!

A little calculation shows that the unknown # is (2π)2, approximately 39.5, which is not too, too far from 1. (It’s not that close, admittedly. But remember that this unknown # could have been 483,248,342,198 or 0.000000000000932 — and so, relative to what it could have been, it’s still pretty close to 1.) This tendency for these unknown #’s to be not to so far from 1 is one we need to keep an eye on.

Yet we don’t even need to know the unknown # to learn something extremely important! Suppose we have studied R3/T2 for the Moon and satellites moving around the Earth, and R3/T2 for the planets orbiting the Sun. We have

  • G Msun = # R3/T2 for objects orbiting the Sun

For instance we could focus on the Earth’s orbit, so we would take R to be the Earth-Sun distance RES and T to be one Earth year TE. Meanwhile,

  • G Mearth = # R3/T2 for objects orbiting the Earth

Here we could focus on the Moon’s orbit, and take R to be the Moon-Earth distance RME and T to be one Moon month TM. Now we can take the ratio of these two expressions! The G cancels, and so does the unknown #! That leaves us with

  • Msun/Mearth = (R3/T2)earth_around_sun/(R3/T2)moon_around_earth = (RES3/RME3)(TM2/TE2)

Since the Earth-Sun distance RES is about 388 times the Moon-Earth distance, and the Earth-Year is about 13.4 times the Moon-Month, we find

  • Msun/Mearth = (388)3/(13.4)2 = 325,000

which is correct, to within a few percent. Look at that! We calculated the ratio of the Sun’s mass to the Earth’s mass just using dimensional analysis! All we needed to know was the distances and times relevant to the orbits of the Earth and Moon.

We never had to solve a gravity equation to figure this out! We just had to assume that gravity was involved somehow.

I hope that this convinces you that if you use this reasoning well, it can be immensely powerful, with the potential to simplify difficult problems dramatically. Next time we’ll look at how dimensional analysis can be (and was) used to learn things about relativity and black holes, and then we’ll look at atomic physics and beyond.

News Flash: Has a New Axial Higgs Boson (Possibly Dark Matter) Been Discovered?


No, no, no.

I was tempted to blame the science journalists for the incredibly wrong articles about this, but in fact it seems entirely the fault of the scientists involved.

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A Big Think Made of Straw: Bad Arguments Against Future Colliders

Here’s a tip.  If you read an argument either for or against a successor to the Large Hadron Collider (LHC) in which the words “string theory” or “string theorists” form a central part of the argument, then you can conclude that the author (a) doesn’t understand the science of particle physics, and (b) has an absurd caricature in mind concerning the community of high energy physicists.  String theory and string theorists have nothing to do with whether such a collider should or should not be built.

Such an article has appeared on Big Think. It’s written by a certain Thomas Hartsfield.  My impression, from his writing and from what I can find online, is that most of what he knows about particle physics comes from reading people like Ethan Siegel and Sabine Hossenfelder. I think Dr. Hartsfield would have done better to leave the argument to them. 

An Army Made of Straw

Dr. Hartsfield’s article sets up one straw person after another. 

  • The “100 billion” cost is just the first.  (No one is going to propose, much less build, a machine that costs 100 billion in today’s dollars.)  
  • It refers to “string theorists” as though they form the core of high-energy theoretical physics; you’d think that everyone who does theoretical particle physics is a slavish, mindless believer in the string theory god and its demigod assistant, supersymmetry.  (Many theoretical particle physicists don’t work on either one, and very few ever do string theory. Among those who do some supersymmetry research, it’s often just one in a wide variety of topics that they study. Supersymmetry zealots do exist, but they aren’t as central to the field as some would like you to believe.)
  • It makes loud but tired claims, such as “A giant particle collider cannot truly test supersymmetry, which can evolve to fit nearly anything.”  (Is this supposed to be shocking? It’s obvious to any expert. The same is true of dark matter, the origin of neutrino masses, and a whole host of other topics. Its not unusual for an idea to come with a parameter which can be made extremely small. Such an idea can be discovered, or made obsolete by other discoveries, but excluding it may take centuries. In fact this is pretty typical; so deal with it!)
  • “$100 billion could fund (quite literally) 100,000 smaller physics experiments.”  (Aside from the fact that this plays sleight-of-hand, mixing future dollars with present dollars, the argument is crude. When the Superconducting Supercollider was cancelled, did the money that was saved flow into thousands of physics experiments, or other scientific experiments?  No.  Congress sent it all over the place.)  
  • And then it concludes with my favorite, a true laugher: “The only good argument for the [machine] might be employment for smart people. And for string theorists.”  (Honestly, employment for string theorists!?!  What bu… rubbish. It might have been a good idea to do some research into how funding actually works in the field, before saying something so patently silly.)

Meanwhile, the article never once mentions the particle physics experimentalists and accelerator physicists.  Remember them?  The ones who actually build and run these machines, and actually discover things?  The ones without whom the whole enterprise is all just math?

Although they mostly don’t appear in the article, there are strong arguments both for and against building such a machine; see below.  Keep in mind, though, that any decision is still years off, and we may have quite a different perspective by the time we get to that point, depending on whether discoveries are made at the LHC or at other experimental facilities.  No one actually needs to be making this decision at the moment, so I’m not sure why Dr. Hartsfield feels it’s so crucial to take an indefensible position now.

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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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5th Webpage on the Triplet Model is Up

Advanced particle physics today:

Another page completed on the explanation of the “triplet model,”  (a classic and simple variation on the Standard Model of particle physics, in which the W boson mass can be raised slightly relative to Standard Model predictions without affecting other current experiments.) The math required is still pre-university level, though complex numbers are now becoming important.

The firstsecond and third webpages in this series provided a self-contained introduction that concluded with a full cartoon of the triplet model. On our way to the full SU(2)xU(1) Standard Model, the fourth webpage gave a preliminary explanation of what SU(2) and U(1) are.

Today, the fifth page explains how a U(1)xU(1) Standard Model-like theory would work… and why the photon comes out massless in such a theory. Comments welcome!

Long Live LLPs!

Particle physics news today...

I’ve been spending my mornings this week at the 11th Long-Lived Particle Workshop, a Zoom-based gathering of experts on the subject.  A “long-lived particle” (LLP), in this context, is either

  • a detectable particle that might exist forever, or
  • a particle that, after traveling a macroscopic, measurable distance — something between 0.1 millimeters and 100 meters — decays to detectable particles

Many Standard Model particles are in these classes (e.g. electrons and protons in the first category, charged pions and bottom quarks in the second).

Typical distances traveled by some of the elementary particles and some of the hadrons in the Standard Model; any above 10-4 on the vertical axis count as long-lived particles. Credit: Prof. Brian Shuve

But the focus of the workshop, naturally, is on looking for new ones… especially ones that can be created at current and future particle accelerators like the Large Hadron Collider (LHC).

Back in the late 1990s, when many theorists were thinking about these issues carefully, the designs of the LHC’s detectors — specifically ATLAS, CMS and LHCb — were already mostly set. These detectors can certainly observe LLPs, but many design choices in both hardware and software initially made searching for signs of LLPs very challenging. In particular, the trigger systems and the techniques used to interpret and store the data were significant obstructions, and those of us interested in the subject had to constantly deal with awkward work-arounds. (Here’s an example of one of the challenges... an older article, so it leaves out many recent developments, but the ideas are still relevant.)

Additionally, this type of physics was widely seen as exotic and unmotivated at the beginning of the LHC run, so only a small handful of specialists focused on these phenomena in the first few years (2010-2014ish).  As a result, searches for LLPs were woefully limited at first, and the possibility of missing a new phenomenon remained high.

More recently, though, this has changed. Perhaps this is because of an increased appreciation that LLPs are a common prediction in theories of dark matter (as well as other contexts).  The number of new searches, new techniques, and entirely new proposed experiments has ballooned, as has the number of people participating. Many of the LLP-related problems with the LHC detectors have been solved or mitigated. This makes this year’s workshop, in my opinion, the most exciting one so far.  All sorts of possibilities that aficionados could only dream of fifteen years ago are becoming a reality. I’ll try to find time to explore just a few of them in future posts.

  But before we get to that, there’s an interesting excess in one of the latest measurements… more on that next time.

Just a few of the unusual signatures that can arise from long-lived particles; (Credit: Prof. Heather Russell)

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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