Newton

The Impossible Commentary: Is Gravity a Force? Is it an Illusion?

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[This is a tricky one… it’s easy to make confusing statements about Einstein’s theory of gravity (general relativity), and so I am especially hopeful of getting readers’ feedback on this subtle issue, to make sure what follows is 100% clear and correctly stated.]

  • (Quote) On Earth’s surface, we are roughly 4,000 miles from Earth’s center. But if you ascended another 22,000 miles, where you’d find the GOES weather satellites that monitor Earth’s weather patterns, you’d find your weight (but not your mass!) reduced to one-fortieth of what it is on Earth… And if you traveled out into deep space, far from any large object, you’d weigh virtually nothing. Yet all the while, your body’s mass—the difficulty I would face if I tried to speed you up or slow you down—would never change.
  • (Endnote) Confusingly, astronauts orbiting Earth inside nearby space stations appear to float as though weightless. From Newton’s perspective, they are not truly weightless; if they were, they’d coast, leaving the Earth’s vicinity and moving rapidly into deep space.
    Instead, they and their spaceship are pulled by gravity into a common orbit around the Earth. Since they travel on the same path as their container and as the camera which films them, they seem and feel weightless. (This subtle issue is turned on its head in Einstein’s view of gravity.)

Astronauts in a space station seem to float, as though they are weightless. Are they truly weightless? Or are they only apparently weightless?

The same issues arise for people in a freely falling elevator, accelerating downward with ever greater speed. They will feel weightless, too. But are they?

Newton would have said they are apparently weightless, subject to gravity but all falling together along with their vehicle. A naive (but instructive!) reading of Einstein might lead us to say that they are truly weightless… that the gravity that Newton claims is present is a pure illusion, a fictitious force. But a precise Einsteinian would say they are almost but not quite weightless — and the lack of perfect weightlessness is a clue, a smoking gun in fact, that they are indeed subject to gravity.

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The Impossible Commentary: Newton, Gravity, and the Speed of the Moon

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Additional supplementary material for the upcoming book; your comments/corrections are welcome. This entry has to do with how Newton realized that weight and mass aren’t the same thing — that the pull of Earth’s gravity depends on how far you are from the Earth’s center.

  • (Quote) Newton knew right away that if the force of gravity were as powerful out by the Moon as it is at Earth’s surface—if the Moon accelerated toward the Earth at the same rate that your dropped keys do—then motion and gravity would be wildly out of balance [and so the Moon would have fallen and crashed into the Earth.]
  • (Endnote) To avoid disaster, the Moon’s orbital speed would need to be 40 miles per second, leading it to circle Earth twice a day.

Here I’ll explain why this is true, using a little math. (If you already know something about Kepler’s laws of planetary orbits, additional relevant discussion can be found in this post from 2022.)

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

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

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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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From Kepler’s Law to Newton’s Gravity, Yourself — Part 2

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Sometimes, when you’re doing physics, you have to make a wild guess, do a little calculating, and see how things turn out.

In a recent post, you were able to see how Kepler’s law for the planets’ motions (R3=T2 , where R the distance from a planet to the Sun in Earth-Sun distances, and T is the planet’s orbital time in Earth-years), leads to the conclusion that each planet is subject to an acceleration a toward the Sun, by an amount that follows an inverse square law

  • a = (2π)2 / R2

where acceleration is measured in Earth-Sun distances and in Earth-Years.

That is, a planet at the Earth’s distance from the Sun accelerates (2π)2 Earth-distances per Earth-year per Earth-year, which in more familiar units works out (as we saw earlier) to about 6 millimeters per second per second. That’s slow in human terms; a car with that acceleration would take more than an hour to go from stationary to highway speeds.

What about the Moon’s acceleration as it orbits the Earth?  Could it be given by exactly the same formula?  No, because Kepler’s law doesn’t work for the Moon and Earth.  We can see this with just a rough estimate. The time it takes the Moon to orbit the Earth is about a month, so T is roughly 1/12 Earth-years. If Kepler’s law were right, then R=T2/3 would be 1/5 of the Earth-Sun distance. But we convinced ourselves, using the relation between a first-quarter Moon and a half Moon, that the Moon-Earth distance is less than 1/10 othe Earth-Sun distance.  So Kepler’s formula doesn’t work for the Moon around the Earth.

A Guess

But perhaps objects that are orbiting the Earth satisfy a similar law,

  • R3=T2 for Earth-orbiting objects

except that now T should be measured not in years but in Moon-orbits (27.3 days, the period of the Moon’s orbit around the Earth) and R should be measured not in Earth-Sun distances but in Moon-Earth distances?  That was Newton’s guess, in fact.

Newton had a problem though: the only object he knew that orbits the Earth was the Moon.  How could he check if this law was true? We have an advantage, living in an age of artificial satellites, which we can use to check this Kepler-like law for Earth-orbiting objects, just the way Kepler checked it for the Sun-orbiting planets.  But, still there was something else Newton knew that Kepler didn’t. Galileo had determined that all objects for which air resistance is unimportant will accelerate downward at 32 feet (9.8 meters) per second per second (which is to say that, as each second ticks by, an object’s speed will increase by 32 feet [9.8 meters] per second.) So Newton suspected that if he converted the Kepler-like law for the Moon to an acceleration, as we did for the planets last time, he could relate the acceleration of the Moon as it orbits the Earth to the acceleration of ordinary falling objects in daily life.

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From Kepler’s Law to Newton’s Gravity, Yourself — Part 1

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Now that you’ve discovered Kepler’s third law — that T, the orbital time of a planet in Earth years, and R, the radius of the planet’s orbit relative to the Earth-Sun distance, are related by

  • R3=T2

the question naturally arises: where does this wondrous regularity comes from?

We have been assuming that planets travel on near-circular orbits, and we’ll continue with that assumption to see what we can learn from it. So let’s look in more detail at what happens when any object, not just a planet, travels in a circle at a constant speed.

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How Einstein Trumped Newton

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Sometimes I encounter people whose impression is that what Einstein’s 1905 theory of special relativity (the one that said no object’s speed can exceed the speed of light in vacuum, etc.) did in “overthrowing” the ideas of the past was somehow like what the Bolsheviks did to the Czars twelve years later– out with the … Read more

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