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.
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 first, second 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.
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)
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.
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.
Today we move deeper into the reader-requested 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 slowly creeping up as complex numbers start to appear.
The first, second and third webpages in this series provided a self-contained introduction that concluded with a full cartoon of the triplet model, showing how a small modification of the Higgs mechanism of the Standard Model can shift a “W” particle’s mass upward.
Next, we begin a new phase in which the cartoon is gradually replaced with the real thing. In the new fourth webpage, I start laying the groundwork for understanding how the Standard Model works — in particular how the Higgs boson gives mass to the W and Z bosons, and what SU(2) x U(1) is all about — following which it won’t be hard to explain the triplet model.
