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I’m linking to some rather excellent lessons on modern physics from the School of Physics – The University of New South Wales, Sydney, Australia.
website – newt.phys.unsw.edu.au/einsteinlight/index.html
|1. GALILEO – Mechanics and Galilean relativity (Multimedia above right, smaller html version here)
|2. MAXWELL – Electricity, magnetism and relativity (Multimedia above right, smaller html version here)
|3. EINSTEIN – The principle of Special Relativity (Multimedia above right, smaller html version here)
|4. TIME DILATION – How relativity implies time dilation and length contraction (Multimediaabove right, smaller html version here)
|5. E = mc2 – How relativistic mechanics leads to E = mc2 (Multimedia above right, smaller html version here)
|6. BEYOND RELATIVITY. (Multimedia version, or smaller html version)Related Links|
By Natalie Wolchover, Senior Writer, Quanta Magazine
October 23, 2017
In 1985, when Carl Sagan was writing the novel Contact, he needed to quickly transport his protagonist Dr. Ellie Arroway from Earth to the star Vega. He had her enter a black hole and exit light-years away, but he didn’t know if this made any sense. The Cornell University astrophysicist and television star consulted his friend Kip Thorne, a black hole expert at the California Institute of Technology (who won a Nobel Prize earlier this month). Thorne knew that Arroway couldn’t get to Vega via a black hole, which is thought to trap and destroy anything that falls in. But it occurred to him that she might make use of another kind of hole consistent with Albert Einstein’s general theory of relativity: a tunnel or “wormhole” connecting distant locations in space-time.
While the simplest theoretical wormholes immediately collapse and disappear before anything can get through, Thorne wondered whether it might be possible for an “infinitely advanced” sci-fi civilization to stabilize a wormhole long enough for something or someone to traverse it.
He figured out that such a civilization could in fact line the throat of a wormhole with “exotic material” that counteracts its tendency to collapse. The material would possess negative energy, which would deflect radiation and repulse space-time apart from itself. Sagan used the trick in Contact, attributing the invention of the exotic material to an earlier, lost civilization to avoid getting into particulars. Meanwhile, those particulars enthralled Thorne, his students and many other physicists, who spent years exploring traversable wormholes and their theoretical implications. They discovered that these wormholes can serve as time machines, invoking time-travel paradoxes — evidence that exotic material is forbidden in nature.
Now, decades later, a new species of traversable wormhole has emerged, free of exotic material and full of potential for helping physicists resolve a baffling paradox about black holes. This paradox is the very problem that plagued the early draft of Contact and led Thorne to contemplate traversable wormholes in the first place; namely, that things that fall into black holes seem to vanish without a trace. This total erasure of information breaks the rules of quantum mechanics, and it so puzzles experts that in recent years, some have argued that black hole interiors don’t really exist — that space and time strangely end at their horizons.
The flurry of findings started last year with a paper that reported the first traversable wormhole that doesn’t require the insertion of exotic material to stay open. Instead, according to Ping Gao and Daniel Jafferis of Harvard University and Aron Wall of Stanford University, the repulsive negative energy in the wormhole’s throat can be generated from the outside by a special quantum connection between the pair of black holes that form the wormhole’s two mouths. When the black holes are connected in the right way, something tossed into one will shimmy along the wormhole and, following certain events in the outside universe, exit the second.
Remarkably, Gao, Jafferis and Wall noticed that their scenario is mathematically equivalent to a process called quantum teleportation, which is key to quantum cryptography and can be demonstrated in laboratory experiments.
John Preskill, a black hole and quantum gravity expert at Caltech, says the new traversable wormhole comes as a surprise, with implications for the black hole information paradox and black hole interiors. “What I really like,” he said, “is that an observer can enter the black hole and then escape to tell about what she saw.” This suggests that black hole interiors really exist, he explained, and that what goes in must come out.
The new wormhole work began in 2013, when Jafferis attended an intriguing talk at the Strings conference in South Korea. The speaker, Juan Maldacena, a professor of physics at the Institute for Advanced Study in Princeton, New Jersey, had recently concluded, based on various hints and arguments, that “ER = EPR.” That is, wormholes between distant points in space-time, the simplest of which are called Einstein-Rosen or “ER” bridges, are equivalent (albeit in some ill-defined way) to entangled quantum particles, also known as Einstein-Podolsky-Rosen or “EPR” pairs. The ER = EPR conjecture, posed by Maldacena and Leonard Susskind of Stanford, was an attempt to solve the modern incarnation of the infamous black hole information paradox by tying space-time geometry, governed by general relativity, to the instantaneous quantum connections between far-apart particles that Einstein called “spooky action at a distance.”
The paradox has loomed since 1974, when the British physicist Stephen Hawking determined that black holes evaporate — slowly giving off heat in the form of particles now known as “Hawking radiation.” Hawking calculated that this heat is completely random; it contains no information about the black hole’s contents. As the black hole blinks out of existence, so does the universe’s record of everything that went inside. This violates a principle called “unitarity,” the backbone of quantum theory, which holds that as particles interact, information about them is never lost, only scrambled, so that if you reversed the arrow of time in the universe’s quantum evolution, you’d see things unscramble into an exact re-creation of the past.
Almost everyone believes in unitarity, which means information must escape black holes — but how? In the last five years, some theorists, most notably Joseph Polchinski of the University of California, Santa Barbara, have argued that black holes are empty shells with no interiors at all — that Ellie Arroway, upon hitting a black hole’s event horizon, would fizzle on a “firewall” and radiate out again.
Many theorists believe in black hole interiors (and gentler transitions across their horizons), but in order to understand them, they must discover the fate of information that falls inside. This is critical to building a working quantum theory of gravity, the long-sought union of the quantum and space-time descriptions of nature that comes into sharpest relief in black hole interiors, where extreme gravity acts on a quantum scale.
The quantum gravity connection is what drew Maldacena, and later Jafferis, to the ER = EPR idea, and to wormholes. The implied relationship between tunnels in space-time and quantum entanglement posed by ER = EPR resonated with a popular recent belief that space is essentially stitched into existence by quantum entanglement. It seemed that wormholes had a role to play in stitching together space-time and in letting black hole information worm its way out of black holes — but how might this work? When Jafferis heard Maldacena talk about his cryptic equation and the evidence for it, he was aware that a standard ER wormhole is unstable and non-traversable. But he wondered what Maldacena’s duality would mean for a traversable wormhole like the ones Thorne and others played around with decades ago. Three years after the South Korea talk, Jafferis and his collaborators Gao and Wall presented their answer. The work extends the ER = EPR idea by equating, not a standard wormhole and a pair of entangled particles, but a traversable wormhole and quantum teleportation: a protocol discovered in 1993 that allows a quantum system to disappear and reappear unscathed somewhere else.
When Maldacena read Gao, Jafferis and Wall’s paper, “I viewed it as a really nice idea, one of these ideas that after someone tells you, it’s obvious,” he said. Maldacena and two collaborators, Douglas Stanford and Zhenbin Yang, immediately began exploring the new wormhole’s ramifications for the black hole information paradox; their paper appeared in April. Susskind and Ying Zhao of Stanford followed this with a paper about wormhole teleportation in July. The wormhole “gives an interesting geometric picture for how teleportation happens,” Maldacena said. “The message actually goes through the wormhole.”
In their paper, “Diving Into Traversable Wormholes,” published in Fortschritte der Physik, Maldacena, Stanford and Yang consider a wormhole of the new kind that connects two black holes: a parent black hole and a daughter one formed from half of the Hawking radiation given off by the parent as it evaporates. The two systems are as entangled as they can be. Here, the fate of the older black hole’s information is clear: It worms its way out of the daughter black hole.
During an interview this month in his tranquil office at the IAS, Maldacena, a reserved Argentinian-American with a track record of influential insights, described his radical musings. On the right side of a chalk-dusty blackboard, Maldacena drew a faint picture of two black holes connected by the new traversable wormhole.
On the left, he sketched a quantum teleportation experiment, performed by the famous fictional experimenters Alice and Bob, who are in possession of entangled quantum particles a and b, respectively.
Say Alice wants to teleport a qubit q to Bob. She prepares a combined state of q and a, measures that combined state (reducing it to a pair of classical bits, 1 or 0), and sends the result of this measurement to Bob. He can then use this as a key for operating on b in a way that re-creates the state q. Voila, a unit of quantum information has teleported from one place to the other.
Maldacena turned to the right side of the blackboard. “You can do operations with a pair of black holes that are morally equivalent to what I discussed [about quantum teleportation]. And in that picture, this message really goes through the wormhole.”
Say Alice throws qubit q into black hole A. She then measures a particle of its Hawking radiation, a, and transmits the result of the measurement through the external universe to Bob, who can use this knowledge to operate on b, a Hawking particle coming out of black hole B. Bob’s operation reconstructs q, which appears to pop out of B, a perfect match for the particle that fell into A. This is why some physicists are excited: Gao, Jafferis and Wall’s wormhole allows information to be recovered from black holes. In their paper, they set up their wormhole in a negatively curved space-time geometry that often serves as a useful, if unrealistic, playground for quantum gravity theorists. However, their wormhole idea seems to extend to the real world as long as two black holes are coupled in the right way: “They have to be causally connected and then the nature of the interaction that we took is the simplest thing you can imagine,” Jafferis explained. If you allow the Hawking radiation from one of the black holes to fall into the other, the two black holes become entangled, and the quantum information that falls into one can exit the other.
The quantum-teleportation format precludes using these traversable wormholes as time machines. Anything that goes through the wormhole has to wait for Alice’s message to travel to Bob in the outside universe before it can exit Bob’s black hole, so the wormhole doesn’t offer any superluminal boost that could be exploited for time travel. It seems traversable wormholes might be permitted in nature as long as they offer no speed advantage. “Traversable wormholes are like getting a bank loan,” Gao, Jafferis and Wall wrote in their paper: “You can only get one if you are rich enough not to need it.”
A Naive Octopus
While traversable wormholes won’t revolutionize space travel, according to Preskill the new wormhole discovery provides “a promising resolution” to the black hole firewall question by suggesting that there is no firewall at black hole horizons. Preskill said the discovery rescues “what we call ‘black hole complementarity,’ which means that the interior and exterior of the black hole are not really two different systems but rather two very different, complementary ways of looking at the same system.” If complementarity holds, as is widely assumed, then in passing across a black hole horizon from one realm to the other, Contact’s Ellie Arroway wouldn’t notice anything strange. This seems more likely if, under certain conditions, she could even slide all the way through a Gao-Jafferis-Wall wormhole.
The wormhole also safeguards unitarity — the principle that information is never lost — at least for the entangled black holes being studied. Whatever falls into one black hole eventually exits the other as Hawking radiation, Preskill said, which “can be thought of as in some sense a very scrambled copy of the black hole interior.”
Taking the findings to their logical conclusion, Preskill thinks it ought to be possible (at least for an infinitely advanced civilization) to influence the interior of one of these black holes by manipulating its radiation. This “sounds crazy,” he wrote in an email, but it “might make sense if we can think of the radiation, which is entangled with the black hole — EPR — as being connected to the black hole interior by wormholes — ER. Then tickling the radiation can send a message which can be read from inside the black hole!” He added, “We still have a ways to go, though, before we can flesh out this picture in more detail.”
Indeed, obstacles remain in the quest to generalize the new wormhole findings to a statement about the fate of all quantum information, or the meaning of ER = EPR.
In Maldacena and Susskind’s paper proposing ER = EPR, they included a sketch that’s become known as the “octopus”: a black hole with tentacle-like wormholes leading to distant Hawking particles that have evaporated out of it.
The authors explained that the sketch illustrates “the entanglement pattern between the black hole and the Hawking radiation. We expect that this entanglement leads to the interior geometry of the black hole.”
But according to Matt Visser, a mathematician and general-relativity expert at Victoria University of Wellington in New Zealand who has studied wormholes since the 1990s, the most literal reading of the octopus picture doesn’t work. The throats of wormholes formed from single Hawking particles would be so thin that qubits could never fit through. “A traversable wormhole throat is ‘transparent’ only to wave packets with size smaller than the throat radius,” Visser explained. “Big wave packets will simply bounce off any small wormhole throat without crossing to the other side.”
Stanford, who co-wrote the recent paper with Maldacena and Yang, acknowledged that this is a problem with the simplest interpretation of the ER = EPR idea, in which each particle of Hawking radiation has its own tentacle-like wormhole.
However, a more speculative interpretation of ER = EPR that he and others have in mind does not suffer from this failing. “The idea is that in order to recover the information from the Hawking radiation using this traversable wormhole,” Stanford said, one has to “gather the Hawking radiation together and act on it in a complicated way.”
This complicated collective measurement reveals information about the particles that fell in; it has the effect, he said, of “creating a large, traversable wormhole out of the small and unhelpful octopus tentacles. The information would then propagate through this large wormhole.” Maldacena added that, simply put, the theory of quantum gravity might have a new, generalized notion of geometry for which ER equals EPR. “We think quantum gravity should obey this principle,” he said. “We view it more as a guide to the theory.”
In his 1994 popular science book, Black Holes and Time Warps, Kip Thorne celebrated the style of reasoning involved in wormhole research. “No type of thought experiment pushes the laws of physics harder than the type triggered by Carl Sagan’s phone call to me,” he wrote; “thought experiments that ask, ‘What things do the laws of physics permit an infinitely advanced civilization to do, and what things do the laws forbid?’”
Where does magnetism come from?
I’ve heard that special relativity makes the concept of magnetic fields irrelevant, replacing them with relativistic effects between charges moving in different velocity frames. Is this true? If so, how does this work?
Luboš Motl, a Czech theoretical physicist, replies:
Special relativity makes the existence of magnetic fields an inevitable consequence of the existence of electric fields. In the inertial system B moving relatively to the inertial system A, purely electric fields from A will look like a combination of electric and magnetic fields in B. According to relativity, both frames are equally fit to describe the phenomena and obey the same laws.
So special relativity removes the independence of the concepts (independence of assumptions about the existence) of electricity and magnetism. If one of the two fields exists, the other field exists, too. They may be unified into an antisymmetric tensor, FμνFμν.
However, what special relativity doesn’t do is question the independence of values of the electric fields and magnetic fields. At each point of spacetime, there are 3 independent components of the electric field E⃗ E→ and three independent components of the magnetic field B⃗ B→: six independent components in total. That’s true for relativistic electrodynamics much like the “pre-relativistic electrodynamics” because it is really the same theory!
Magnets are different objects than electrically charged objects. It was true before relativity and it’s true with relativity, too.
It may be useful to notice that the situation of the electric and magnetic fields (and phenomena) is pretty much symmetrical. Special relativity doesn’t really urge us to consider magnetic fields to be “less fundamental”. Quite on the contrary, its Lorentz symmetry means that the electric and magnetic fields (and phenomena) are equally fundamental. That doesn’t mean that we can’t consider various formalisms and approximations that view magnetic fields – or all electromagnetic fields – as derived concepts, e.g. mere consequences of the motion of charged objects in spacetime. But such formalisms are not forced upon us by relativity.
Although the relationship between special relativity and magnetic fields is often stated as making magnetic fields irrelevant, this is not quite the correct way to say it.
What actually disappears is the need for magnetic attractions and repulsions. That’s because with the proper choice of motion frames a magnetic force can always be explained as a type of electrostatic attraction or repulsion made possible by relativistic effects.
The part that too often is overlooked or misunderstood is that these changes in the interpretation of forces does not eliminate the magnetic fields themselves. One simple way to explain why this must be true is that if it was not, a compass would give different readings depending on which frame you observed it from. So to maintain self-consistency across frames, magnetic fields must remain in place, even when they no longer play a role in the main attractive or repulsive forces between bodies.
One of the best available descriptions of how special relativity transforms the role of magnetic fields can be found in the Feynman Lectures on Physics. In Volume II, Chapter 13, Section 13-6, The relativity of magnetic and electric fields, Feynman describes a nicely simplified example of a wire that has internal electrons moving at velocity v through the wire, and an external electron that also moves at v nearby and parallel to the wire.
Feynman points out that in classical electrodynamics, the electrons moving within the wire and the external electron both generate magnetic fields that cause them to attract. Thus from the view of human observers watching the wire, the forces that attract the external electron towards the wire are entirely magnetic.
However, since the external and internal electrons move in the same direction at the same velocity v, special relativity says that an observer could “ride along” and see both the external and internal electrons as being at rest.
Since charges must be in motion to generate magnetic fields, there can in this case be no magnetic fields associated with the external electron or the internal electrons.
But to keep reality self-consistent, the electron must nonetheless still be attracted towards the wire and move towards it! How is this possible?
This is where special relativity plays a neat parlor trick on us.
The first part of the trick is to realize that there is one other player in all of this:
The wire, which is now moving backwards at a velocity of -v relative to the motionless frame of the electrons.
The second part of the trick is to realize that the wire is positively charged, since it is missing all of those electrons that now look like they are sitting still.
That means that the moving wire creates an electric current composed of positive charges moving in the -v direction.
The third and niftiest part of the trick is where special relativity kicks in.
Recall than in special relativity, when objects move uniformly they undergo a contraction in length along the direction of motion called the Lorentz contraction.
I should emphasize that Lorentz contraction is not some kind of abstract or imaginary effect. It is just as real as the compression you get by squeezing something in a vice grip, even if it is gentler on the object itself.
Now think about that for a moment:
If the object is also charged at some average number of positive charges per centimeter, what happens if you squash the charged object so that it occupies less space along its long length?
Well, just what you think: The positive charges along its length will also be compressed, resulting in a higher density of positive charges per centimeter of wire.
The electrons are not moving from their own perspective, however, so their density within the wire will not be compressed. When it comes to cancelling out charge, this is a problem! The electrons within the wire can no longer fully cancel out the higher density of positive charges of the relativistically compressed wire, leaving the wire with a net positive charge.
The final step in the parlor trick is that since the external electron has a negative charge, it is now attracted electrostatically to the wire and its net positive charge.
So even though the magnetic fields generated by the electrons have disappeared, a new attraction has appeared to take its place!
Now you can go through all of the details of the math and figure out the magnitude of this new electrostatic attraction.
However, this is one of those cases where you can take a conceptual shortcut by realizing that since reality must remain self-consistent – no matter what frame you view if from – the magnitude of this new electrostatic attraction must equal the magnetic attraction as seen earlier from the frame of a motionless wire.
(If you do get different answers, you need to look over your work!)
But what about the other point I made earlier, the one about the magnetic field not disappearing? Didn’t the original magnetic field disappear as soon as one takes the frame view of the electrons?
Well, sure. But don’t forget: Even though the electrons are no longer moving, the positively charged wire is moving and will generate its own magnetic field. Furthermore, since the wire contains the same number of positive charges as electrons in the current, all moving in the opposite (-v) direction, the resulting magnetic field will look very much like the field originally generated by the electrons.
So, just as the method of attraction switches from pure magnetic to pure electrostatic as one moves from the wire frame to the moving electron frame, the cause of the magnetic field also switches from pure electron generated to pure positive-wire generated. Between these two extremes are other frames in which both attraction and the source of the magnetic field become linear mixes of the two extreme cases.
Feynman briefly mentions the magnetic field generated by the moving positive wire, but focuses his discussion mostly on the disappearance of the electron-generated magnetic fields. That’s a bit unfortunate, since it can leave a casual reader with the incorrect impression that the magnetic fieldas a whole disappears.
It does not, since that would violate self-consistency by making a compass (e.g., the magnetic dipole of that external electron) behave differently depending on the frame from which you observe it.
The preservation of the magnetic field as the set of particles generating it changes from frame to frame is in many ways just as remarkable as the change in the nature of the attractive or repulsive forces between objects, and is worth noting more conspicuously.
Finally, all of these examples show that the electromagnetic field really is a single field, one whose overt manifestations can change dramatically depending on the frame from which they are viewed. The effects of such fields, however, are not up for grabs. Those must remain invariant even as the apparent mechanisms change and morph from one form (or one set of particles) to another.
Special Relativity in 14 Easy (Hyper)steps
14. Why there are magnetic fields
By Professor Michael Fowler, University of Virginia
A Magnetic Puzzle…
Suppose we have an infinitely long straight wire, having a charge density of electrons of –λ coulombs per meter, all moving at speed v to the right (recall typical speeds are centimeters per minute) and a neutralizing fixed background of positive charge, also of course λ coulombs per meter. The current in the wire has magnitude I = λv (and actually is flowing to the left, since the moving electrons carry negative charge).
Suppose also that a positive charge q is outside the wire, a distance r from the axis, and this outside charge is moving at the same exact velocity as the electrons in the wire.
So how does this lead to the effect we know as magnetism? Read on . . .
AP Physics Learning Objectives
Essential Knowledge 1.D.3: Properties of space and time cannot always be treated as absolute.
a. Relativistic mass–energy equivalence is a reconceptualization of matter and energy as two manifestations of the same underlying entity, fully interconvertible, thereby rendering invalid the classically separate laws of conservation of mass and conservation of energy. Students will not be expected to know apparent mass or rest mass.
b. Measurements of length and time depend on speed. (Qualitative treatment only.) physics
Learning Objective 1.D.3.1: The student is able to articulate the reasons that classical mechanics must be replaced by special relativity to describe the experimental results and theoretical predictions that show that the properties of space and time are not absolute.
[Students will be expected to recognize situations in which non-relativistic classical physics breaks down and to explain how relativity addresses that breakdown, but students will not be expected to know in which of two reference frames a given series of events corresponds to a greater or lesser time interval, or a greater or lesser spatial distance; they will just need to know that observers in the two reference frames can “disagree” about some time and distance intervals.]