Recently, a reader raised a couple of central questions about speed and relativity. Since the answers are crucial to an understanding of Einstein’s relativity in particular and of the cosmos in general, I thought I’d bring them to your attention, in case you’ve had similar questions.
The Questions
I understand that the vacuum speed of light [“c“] is constant throughout the Universe, and I’m familiar with the math that shows that the energy required to accelerate a particle becomes infinite as the speed approaches c. But what physical effect enforces this behavior? If a proton, for example, gets ejected in a supernova explosion, how does it “know” that it’s getting close to c and can’t go any faster?
And as a corollary to this question, what is the reference frame for measuring these relativistic velocities? For example, when a particle beam at CERN is said to be moving at 99.99% the speed of light, is that speed relative to the infrastructure at CERN? Or does it somehow account for the velocity components that arise from the rotation of the Earth, the orbital motion of the Earth around the Sun, the galactic motion of the Sun in the Milky Way, and so on?…
[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.
In my last post I raised a question about the pros and cons of common sense. I left it as a wide-open question, as I was curious to see how readers would react.
Many aspects of common sense affect how we relate to other people, and it’s clear they have considerable value. But the intuitions we have for nature, though sometimes useful, are mostly wrong. These conceptual errors pose obstacles for students who are learning science for the first time.
It’s also interesting that once these students learn first chemistry and then Newtonian-era physics, they gain new intuitions for the natural world, a sort of classical-physics common sense. Much of this newfound common sense also turns out to be wrong: it badly misrepresents how the cosmos really works. This is a difficulty not only for students but also for many adults. If you’ve read about or even taken a class in basic astronomy or physics, it can then be challenging to make sense of twentieth-century physics, where Newtonian intuition can fail badly.
Once we clear away the hype (see the previous posts 1, 2, 3, 4), and realize that no one is doing anything as potentially dangerous as making real wormholes (ones you could actually fall into) in a lab, or studying how to send dogs across the galaxy, we are left with a question. Why bother to do wormhole research at all?
The answer is that it has nothing to do with actually making wormholes… at least, not in the sense of science fiction portals that you and I could use to travel from here to some faraway place across the universe. It has to do with potentially gaining new insight into the quantum physics of gravity, space and time.
How Do We Study Black Holes, and Why?
Why do scientists do research on black holes? There are at least two very different reasons.
Large black holes can be observed in nature. These black holes, which astronomers and gravitational wave experimenters study, are well-described by non-quantum physics — “classical” physics, where the future is (in principle) truly predictable from the past.
Small black holes are a window into quantum gravity — the unknown quantum physics of spacetime, where space itself is governed by the uncertainty principle, meaning that the very shape of spacetime can’t be precisely specified. This is relevant for black holes far too small for us to discover using astronomy, yet far too difficult for us to produce experimentally. They are important because they pose conceptual problems and puzzles for quantum gravity. Theoretical physicists think about black holes, and study their math, in hopes of uncovering quantum gravity’s secrets.
To gain more insight into their workings, scientists also simulate black holes on computers, and study analogues to black holes in laboratories.
Why Wormholes?
In contrast to black holes, there may be no wormholes worthy of the name anywhere in our universe. Though recent research clearly shows that there’s no principle that forbids wormholes from existing, it also shows it’s unlikely that large wormholes can be produced or can endure in our universe. While black holes are a generic outcome of the collapse of a huge star, wormholes are relatively delicate, and difficult to create and maintain.
But wormholes may be even more interesting than black holes for the problems of quantum gravity. This was only appreciated, slowly at first, over the past 10 years.
It’s hard to define the quantum state of a black hole. [In quantum physics, objects don’t just have locations and motions; roughly speaking, they have “states”, in which they have a combination of many locations and motions all at once.] The basic obstacle is entropy, a measure of missing information. The air in your room has entropy, because although you may know its temperature and pressure, you do not know where every atom of air is; that’s missing information. It turns out that a black hole has entropy too, which means that our usual description of a black hole is intrinsically missing some crucial information. That prevents us from knowing precisely what its state is.
But surprisingly, in some circumstances the quantum state of a wormhole can be sharply defined — in which case its entropy is zero. (Such a wormhole is not missing any information. But if you take either half of this wormhole and ignore the other half, you find a black hole. That black hole has entropy precisely because you’re ignoring all the information included in the other half of the wormhole!) To obtain and understand such a wormhole involves giving it two apparently different but actually interchangeable descriptions, one in terms of space-time and gravity, where the wormhole’s geometric shape is clear, and one in terms of what one might call a gravity-independent auxiliary quantum system, in which its quantum state is precisely defined.
The Power of Duality: A Rosetta Stone for Quantum Gravity
One physical object, two quantum descriptions — one with gravity, one without; the first with more space dimensions than the latter. It’s like being able to read the same text in two completely different languages. It’s an example of what physicists often call “a duality.” (I’ve gone into more detail about this in recent posts here and here.)
This is the message of what goes by the mantra of “ER=EPR”, referring to two famous and apparently unrelated papers from 1935 by Einstein and Rosen, with the second having Podolsky as a co-author. ER=EPR asserts that two apparently different things,
a tangible bridge across curved, extra-dimensional space between two regions, and
a less tangible bridge, established with quantum entanglement between objects in the same two regions, without any use of gravity,
are literally the same thing.
An conceptual illustration of the proposal that two perfectly entangled quantum systems (EPR) are equivalent to a wormhole connecting the locations of those systems (ER), and represent two languages for describing exactly the same thing. The wormhole is empty; the slabs shown merely indicate how distances are shrinking as one proceeds to the wormhole’s midpoint. Not shown is that the wormhole changes shape over time; for the situation in this picture, this wormhole is “non-traversable”, because there’s insufficient time to cross from one side of the bridge to the other before it shrinks down to nothing.
Discovering that spacetime is related to quantum entanglement, and that ER and EPR involve the same issues, is somewhat like discovering that two poorly understood and partially readable texts in completely different languages are actually two translations of exactly the same document. It’s a Rosetta stone. Parts of the document can easily be read in one language, other parts in the second language; and putting them together, we find we can read more and more.
Similarly, the math of a wormhole (ER) looks completely different from the math of two quantum-entangled non-gravitational systems (EPR). But in particular cases, Juan Maldacena and Lenny Susskind argued, they are two languages describing the same object. We can combine these two partial views of this single object to learn more and more about it.
Moreover, because we’re using math, not text, we can go a step further. Even in regimes where we cannot “read the document” in either language, we can use computers to explore. Scientists can try to simulate the math of the entangled auxiliary quantum systems on a computer, ideally a quantum computer so that it keeps track of all quantum effects, to learn more about the wormhole’s behavior in regimes where we have no idea how it works — in regions where the quantum uncertainty principle affects space and time.
Even more remarkable would be to actually make — not merely simulate — this entangled pair of auxiliary quantum systems. Then we would be closer to making a wormhole, with laws of nature different from ours and with its own gravity, that connects on to our world. But that’s a long ways off, and not the story for the present.
From ER=EPR to Traversable Wormholes
A further breakthrough, beyond the original ER=EPR idea, came with the work of Gao, Jafferis and Wall (see also here and here) in which it was demonstrated for the first time that “traversable wormholes” — ones that can truly serve as bridges across which objects can be transported — do make physical sense. Astonishingly, they are related by duality to an important and exciting research area in quantum information, called “quantum teleportation.” That’s the process by which, using two entangled quantum systems, quantum information can be brought to one of the systems, destroyed in that system, and recreated in that second system some distance away. Again, don’t expect anyone to be teleporting your dog, but simple information and ultra-microscopic objects might be transportable.
Be warned though; the teleportation only works if additional non-quantum information is traded between the two systems. In the wormhole language, that means you can only get through the wormhole if information is also passed outside the wormhole from the departure region to the arrival region. This makes it impossible to go someplace that you haven’t already been sending messages to, and to use any such wormhole as a shortcut — i.e., to get to your destination faster than could a near-light-speed spacecraft traveling outside the wormhole. Not only do portals to ultra-distant places remain science fiction, they now seem even more likely to stay that way.
Still, with these caveats, there’s still something amazing here: we can now imagine using the Rosetta stone of duality to simulate a traversable wormhole, and learn how it works in quantum gravity. That would be fantastic!
The Dream of Simulating Quantum Gravity
This is a dream, yet to be fulfilled. Computers are nowhere near being able to handle the questions we’d like to answer about the gravity we live with in our “four-dimensional space-time” (our familiar three space dimensions, plus one more for time). But by simplifying the problem in several steps (see the last figure of this post), we can at least hope to answer some early questions in a much simpler sort of wormhole in a simpler sort of gravity. This is what I’d prefer to call an artificially-simulated cartoon wormhole— rather than a “baby” wormhole, because unlike a baby, it isn’t a small version of an adult, nor has it any hope of growing into one. It’s more like a stick figure. It’s in two-dimensional space-time — one space and one time. That’s a big simplification — there’s nothing like normal gravity there! [Worse, we don’t have an exact duality in that case; the auxiliary quantum system we need isn’t really the same “text” as the wormhole. These are two systems, not one, with a limited but useful overlap.]
But cartoons aren’t to be mocked. Don’t underestimate them; cartoons are a powerful tool for educators everywhere, and subversive political cartoons have helped take down governments. For decades, famous physicists — Schwinger, ‘t Hooft, Gross and Neveu, Kogut and Susskind, and many more — have studied cartoon versions of real physics, especially ones in which our four space-time dimensions are replaced with just two. They’ve often learned interesting lessons from doing so, sometimes even profound ones.
[Note: Stick figure physics also can be a very good description of real stick-figure systems, for example a one-dimensional chain of atoms inside a material.]
I hasten to caution you that this technique does not always work. Not all of the lessons learned from stick-figure physics turn out to apply to the corresponding real-world problem. But this method has had enough success that we should take cartoon studies seriously.
This is why exploration of one-dimensional wormholes, and of some sort of auxiliary quantum problem to which they might be approximately related, may be worthwhile. And this is why it’s important to learn to simulate these auxiliary quantum systems on quantum computers, as was done in the paper that generated all the hype, based on proposals made in this paper and this one. Even if we can’t hope soon to understand how three-dimensional quantum space emerges from quantum entanglement, we can perhaps hope to learn more about one-dimensional quantum space, using quantum computer simulation. Maybe what we learn there would already teach us a deep and universal truth about quantum gravity, or at least suggest new ways to think about its subtleties.
The experiment done in the recent paper is a baby step in this direction. Others have attempted something along similar lines, but this is the first experiment that seems to have focused on the truly wormhole-like regime, and found some evidence for what was expected already of wormholes (from direct calculation and from classical computers…I’ll write about those details in a future post.) That seems like a real step forward. But let’s keep things in perspective. No new knowledge was created by this experiment; its achievements were technical and technological. It’s not a conceptual breakthrough. (I’m not alone in this view; Lenny Susskind, Dan Harlow and Scott Aaronson all expressed the same opinion in the New York Times and elsewhere.)
But nevertheless, this experiment represents a little arrow that points to a possible big future… not a future of a new Elon Musk, building wormholes for fun and profit, but one of a future Einstein, comprehending the quantum nature of spacetime itself.
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”.
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.
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.
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.
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.
