Important warning: This is just an artist’s depiction of a black hole, and what’s coming out of it certainly isn’t Hawking Radiation
Black Holes are awesome. On this, I think we can all agree. But even though they are insatiable eating machines, black holes won’t be around forever. Ultimately — and that ultimate day will be a long day off — black holes will evaporate via a process known as “Hawking Radiation.” You are probably already familiar with the idea of Hawking Radiation. If not, let me give you the simple version of it.
While we normally think of empty space as being, well, empty, in truth there’s a lot more going on there than you might suppose. The vacuum of the universe is a constantly bubbling cauldron of particles and antiparticles, coming into existence and quickly annihilating with one another. This is a consequence of the famous Uncertainty Principle. Sometimes, particles are created sufficiently close to the event horizon of a black hole that one particle falls in, and the other escapes, ultimately observable as radiation.
This leaves lots of unanswered questions, like: How does the Black Hole evaporate when it has particles falling into it? I tried to address this in an old “Ask a Physicist” but there are some who still aren’t satisfied.
Today, I wanted to give you a different way of thinking about — and a handwaving way of deriving — Hawking Radiation, one based on symmetry arguments, and in particular, on Einstein’s Equivalence Principle that I talked about earlier in the week. I’ll even use some terrible MS Paint diagrams to illustrate! At the end, for those interested, I’ll even give you a few equations to show how everything falls into place.
But first, a reminder. In one of its many, many forms, the Equivalence Principle states that:
Any local physical experiment not involving gravity will have te same result if performed in a freely falling inertial frame as if it were performed in the flat spacetime of special relativity.
Consider the International Space Station, currently in orbit about 400 km above the earth’s surface. We think of the ISS as being “in space,” but in a sense it’s very much in earth’s grasp. Gravity 400km up is only about 11% weaker than on the surface of the earth. And yet, if you’ve ever seen footage, the astronauts float around as if they’re weightless.
It’s no trick. In the case of the ISS, the free fall actually takes the form of a nearly circular orbit, but that’s still falling as far as gravity is concerned. The only reason that the ISS needs to be in space is to avoid colliding into mountains and so that air resistance won’t cause it to crash down to earth. The weightlessness is caused by the simple fact that it’s in free fall. A physicist on board should be able to do all sorts of experiments with the speed of light or muons or moving observers and find all of the effects anticipated by Special Relativity. That is the whole point of the Equivalence Principle.
Or very nearly. There’s a little bit of fine print that I feel compelled to mention. The gravitational pull of the earth gets weaker the further you get from the earth. That means that the earth side of the space station feels a slightly higher gravity than the space side, and that consequently, there’s a very, very tiny tidal effect that acts to “stretch” the station. The total tidal force on the entire 450 ton space station is only about a pound. You won’t get points off for ignoring it, and we will.
The Equivalence Principle sounds really trivial, but it’s not. In a few short steps, we’re going to go from Equivalence Principle to Hawking Radiation. And I suspect you haven’t thought of it in this way before.
- Accelerating Charges Emit Radiation.
If you take a charged particle and accelerate it, it will give off radiation. A radio transmitter works by jittering a bunch of electrons at the source and emitting radiation with a particular frequency. On the other hand, stationary particles — or even particles moving at a constant speed and direction — don’t emit any radiation at all.
- Inside the ISS, freely-falling charges will appear to be standing still.
Imagine yourself as an astronaut inside the ISS. You’re in orbit around the earth, and apparently weightless. In order to do an experiment you take an electron and just let it go. If you’re careful, the electron will just appear to float in the middle of the station. After all, it’s orbiting the earth with the exact same trajectory as the rest of the station, so as far as you’re concerned, it’ll just sit still.
See what I said about the ghetto MS Paint Images?Keep just one thing in mind: from your perspective, the doesn’t appear to accelerate. Ergo, no radiation.
- To an outsider, both the ISS and the electron are accelerated.
Suppose I want to go to extreme measures to keep an eye on you. I build a giant 400km pedestal from the surface of the earth all the way up to the orbit of the ISS. Remember, I do not feel weightless. As I already pointed out, I only lose about 11% of my weight at this altitude.
From my perspective, you, the space station, and the electron speed by at nearly 17000mph. But even more importantly, I see all of you accelerated toward the center of the earth. Yes, even though the electrons are moving in a constant speed, because they are constantly changing directions — moving in a circle, in this case — they are absolutely being accelerated, and accelerated electrons will radiate. This is known as “synchrotron radiation,” and is a very useful experimental light source.
Of course, the direction of the acceleration doesn’t matter. All that matters is that I see a charge being accelerated and giving off light.
- The universe is filled with random charge fluctuations.
One of the most surprising predictions of Quantum Field Theory is that the vacuum isn’t completely empty. Particles-antiparticle pairs are created out of nothing all the time. This shouldn’t cause you too much anxiety. I brought it up earlier, and you’ve had time to acclimate. We don’t encounter these particles because they typically get annihilated very, very quickly. To put things in perspective, electrons-positron pairs only last for about $10^{-21}s$, enough time for them to travel, at most, a bit more than the nuclear radius of an atom.
Particle Creation and Annihilation in the Vacuum. This is most definitely not to scale.More to the point, since particles and antiparticles always have the exact opposite charge from one another, vacuum fluctuations won’t cause an excess of either positive or negative charges. There are reasons to be very wary about the vacuum energy density in the universe. For instance, a naive calculation of the total energy density yields something like
times larger than the observed energy density of the universe. This, to my mind, is the worst problem in physics.
On the other hand, we can’t simply assume that there is no vacuum energy density. In 1948, Hendrik Casimir noticed that if you take two neutral metal plates and place them close together, they will attract one another. This is known as the Casimir Effect, and it basically only works if you imagine that there are a swarm of virtual charged particles between the plates — exactly with the properties predicted by our vacuum energy density.
- If you accelerate through a vacuum, you’ll see radiation.
Now imagine yourself in a rocket ship flying through the vacuum. From your perspective, each of those virtual charged particles appears to be accelerating, and if the equivalence principle is right (it is), then those particles will appear to be accelerating according to you. And, as you now know, accelerating particles radiate. This was discovered by a number of people in the 1970′s, including William Unruh for whom the effect is named.
Very much not to scale.I should point out that under normal circumstances, this is a tiny effect. Accelerating at “g,” you’d only see Unruh radiation of about 2 billionths of a degree Kelvin.
- Being suspended outside a black hole is just like acceleration
Now we finally get to the main event. I’m not going to review all of the properties of black holes at this time, but for the moment, you need only consider one: there is an event horizon inside of which nothing can escape, not even light.
Outside the black hole, the gravity is very, very strong, and normally, the tidal forces near a black hole are sufficient to spaghettify you to death. However, supposing the black hole is big enough — at minimum about 10,000 times the mass of a sun — a freely-falling observer won’t even notice that he’s crossed the event horizon.
If you’re an observer dangling outside, you will absolutely notice the gravitational force; you can tell that you’re being accelerated. By the Equivalence Principle, there shouldn’t be any local observational differences between dangling in a gravitational field (or standing on the surface of the earth, for instance), or being accelerated in a rocket ship at the same rate.
You feel as though you’re being accelerated, and as far as Relativity is concerned, you are. Just as you would see Unruh radiation in an accelerating rocket ship, by the Equivalence Principle, you should see the same thing if you’re “really” in a gravitational field. In this case, we call it Hawking Radiation.
The amazing thing is that you don’t need to know much about gravity in order to get the correct result. Students of GR (including me, when I was one) are often troubled by how weirdly everything conspires in Hawking Radiation to get everything correct. Unruh radiation, on the other hand, only requires a bit of electromagnetic knowledge. That plus the Equivalence Principle and blammo! You have Hawking Radiation.
- Some technical notes
I didn’t mark this overall entry as “technical” because while it’s detailed, there haven’t been any equations yet. Still, I imagine a few of you might want to see how this works out in practice.
First, Unruh radiation. If you’re accelerating at a rate,
through the vacuum then you’ll appear to see a temperature:
where
is the Boltzman constant, and
is the reduced Planck’s constant, as usual. Note that the gravitational constant,
doesn’t appear anywhere in here.
Now, consider the gravity near the event horizon of a black hole. This gets a little confusing (and is far beyond the current discussion), since you need to worry about whether we’re talking about the gravitational force as seen using local coordinates near the black hole, or “global coordinates” seen from far away. We’re going to use the far away coordinates, since those are going to tell us about what the black hole gives off to someone far away.
Near the black hole, the acceleration is:
where
is the Schwarzschild radius:
If you plug all of this in, you find that the apparent temperature of a black hole is:
This is exactly the result that you get from doing a detailed GR analysis! Pretty cool, no?
As a side note, notice that the more massive the black hole is, the cooler it will be. If our sun were to turn into a black hole, it would radiate at a temperature of about 60 nano-Kelvin, and more massive black holes would be even cooler. This is insanely cold, about fifty million times cooler than the background temperature of the universe. Because heat flows from hot to cold, the radiation of the universe actually feeds a typical black hole. Only incredibly puny ones — less massive than the moon — are actually shrinking these days. Stellar mass black holes won’t actually start evaporating until the universe gets fifty million times cooler (and thus fifty million times bigger) than it is now. That won’t be for a few hundred billion years or so.
In other words, there’s no chance we could actually use Hawking Radiation to see black holes today. They’re just too cool. We do, however, see hot material falling on to black holes in the form of quasars, but that’s not the same thing as seeing the black hole, itself, and at any rate is a topic for another day.
- Some technical notes
-Dave
I’ve been thinking a lot about Special and General Relativity lately, and in particular, the Equivalence Principle — the main theme for Chapter 7 of my upcoming book on symmetry. In the process, I started playing around with a really simple way of “deriving” the General Relativistic result that time runs slow near massive bodies. I wish I could take the credit. Like so many things, Einstein thought of it first. In this case, he proposed the basic outline of what follows in 1907, but it’s still fun to work through it.
In writing up a draft of this section for the book, I thought it would be fun to adapt it a bit to include equations here on the blog. That’s why I’ve labeled this as “technical,” though the equations are really at the Freshman level. Only 1 simple bit of calculus.
The only other background you need is the Equivalence Principle. The Equivalence Principle, if you’ve never seen it, was Einstein’s central proposition in General Relativity. He phrased it many ways at different times, but a general way of writing it (courtesy of Schutz) goes something like this:
Einstein Equivalence Principle: Any local physical experiment not involving gravity will have te same result if performed in a freely falling inertial frame as if it were performed in the flat spacetime of special relativity.
though an equally good way of thinking about it is that General Relativity can’t distinguish between “real gravity” and “artificial accelerations.”
Comments, as always, are appreciated.
-Dave
Beyond just the esoteric coolness of it all, the Equivalence Principle can serve as the basis for some rather surprising gravitational results. Einstein, himself, came up with a scenario for relating artificial to real gravity as early as 1907.
In this case, we’re going to imagine life on top of a large spinning disk. This is a lot like the 2-dimensional universes that I’ve written about before — you know, the ones we found couldn’t actually support life. We pretty much dismissed it out of hand, but that doesn’t mean that we couldn’t imagine how gravity would work in such a place.
In our universe, there are a bunch of super-intelligent ants slowly crawling around on the surface . The Queen, sitting at the center of the antworld, sits still as far as outsiders are concerned. Her royal court surrounds her in close proximity. To an outsider (you), her courtiers slowly rotate about the queen. To put this in mathematical terms, the rotational speed of an ant is:
where 
The ants don’t know any of this, however. From their perspective, however, they feel a small tug that pulls them outward. As far as they’re concerned, “out” is “down.” This, as you may recall, is known as centrifugal force. It’s the exact same effect that pulls you to the side of a “gravitron” at a carnival. You may also recall that in order to stay moving in a circular path, you need to have just the right amount of acceleration:
The further the ants are from the queen, the faster they are spinning, and the stronger they are tugged outward. From the perspective of the ants, their antworld feels very much like a hill with the queen at the gently curved top, a hill that gets steeper the further you go out. An ant that loses its grip will roll outward – down the hill – at an ever-accelerating rate.
There’s at least one sense in which this analogy isn’t perfect. If you fall down a hill in our world, you’ll simply roll down in a straight line. An ant falling down the hill will start rolling straight down, but will then slowly start rolling around the hill as well. This is the famous “Coriolis Effect.” It’s the same thing that causes cyclones to spin counter-clockwise in the Northern hemisphere, and clockwise in the Southern.
But the Coriolis Effect becomes irrelevant if all of the ants stand still, each glued to a different point in antworld. From their perspectives, there is no dynamics whatsoever. Everyone seems to be fixed compared to everyone else.
We – standing outside the antworld – know better. The queen isn’t moving at all. Nearby ants are moving slowly. Ants further out move faster. The ants out in the hinterlands are moving fastest of all. We know something about the flow of time of moving ants. The faster they move, the slower time will appear to pass compared to the queen. The further out an ant is, the slower she’ll appear to age. As you may recall, the relationship between the clocks for the stationary observer (the queen), and everyone else is:
But we needn’t be quite so fancy. In the limit of 
The latter term tells us about the fractional slowing of the clock. Plugging in our velocity relationship, we get:
But there’s another way of thinking about the same situation – from the perspective of the equivalence principle. Artificial gravity can be generated by rotating a spaceship. And in similar fashion, the rotating antworld creates an “artificial” gravity outward. One of the best ways of describing gravitational force is in terms of the potential. In 1-d (as is the case here), the relationship looks like:
Given that “down” is “out”, and knowing the relationship above, we immediately get:
The potential is zero at the center, and gets more and more negative further out. Since we fall to lower potentials, this makes sense. You may also notice that this means that from the ants’ perspectives:
The ants notice a curious effect. The further you go down, the slower time seems to be running. And the same is true in our universe. Time seems to run more slowly near massive bodies – nearer to the “down” direction – than further away.
I especially like this approach because — unlike how it’s normally done — we don’t need to introduce photon frequencies and the like to make our point. Everything comes out from a natural extension of Special Relativity.
A symmetry approach to time dilation
Howdy, folks.
As you know, I’m in the midst of writing my book on symmetry, and every now and again, I like to try to run various drafts of different snippets of the manuscript by you, the peanut gallery, for comments and suggestions. In particular, I’m working on Chapter 6, tentatively entitled, “Could we Build an Intergalactic Ansible to Communicate Instantly Across Space?” You may recall that I did an io9 column with a similar title, but a very different topic. The io9 column was about quantum entanglement. This chapter is all about Special Relativity, and in particular, how symmetry gives rise to time dilation and the like.
The text below is an early-ish draft of a few sections on how the Pythagorean Theorem leads to the Minkowski description of spacetime, and how that leads to time dilation. It is the most equation-dense part of the book to date. I put in 3 simple equations, all very similar to one another, but given the tone I’m going for, I still worry that that’s too much.
The images are just placeholders. Don’t worry, the final book will have a professional illustrator.
Comments, either below or via email are greatly appreciated.
-Dave
The Pythagorean Theorem
At some point in elementary school, you no doubt came across the Pythagorean Theorem:
It’s a deceptively simple equation. A and B are the lengths of the two short sides of a right triangle, and C is the length of the long side, the hypotenuse.
The Pythagorean Theorem does far more than simply tell you about triangles for their own sake. It tells you how to compute the distances between points. You may remember problems like this from school: Walk 3 miles East and then 4 miles North. Plug through the calculation and you’ll find that you’re 5 miles from where you started. To connect it to the real world, consider a small piece of a Washington D.C. transit map:
Many cities are conveniently designed so that streets run approximately along the cardinal directions of a compass. Washington D.C. is a perfect example: Numbered streets run North-South and lettered streets run East-West. So – to pick an example found by an intensive search through Google Maps – if you wanted to walk from Judiciary Square Station on the corner of 4th and E Street NW to the Chinatown Metro Station on 7th and G, you would start by walking approximately 600 meters due West (along E), and then 250 meters North (along 7th).
Of course, you could also simply take the Red Line subway, and if you plug through the numbers, you’d find that the subway trip is approximately 650 m. It’s just a practical application of the Pythagorean Theorem, albeit with the slightest relabeling of the numbers:
We owe this convention to the work of the 17th century mathematician and philosopher, Rene Descartes. The Cartesian system imagines describing all of the events and objects in the universe on a sort of map. For instance, in the East-West direction, the convention is to label positions as “x.” In the North-South direction, we normally label positions as “y.” I’m going to ignore the possibility of moving vertically in an elevator, but if you were so inclined, you might label that motion as “z.”
I’d be negligent if I didn’t point out that Cartesian system breaks down on the surface of the earth. The earth, after all, is approximately shaped like a sphere, which means that you can’t make a perfect, undistorted flat map that covers the entire thing. That’s fine. That’s why we’re talking about something much smaller, like a few city blocks.
But back to Washington D.C. Suppose you stepped down into the Judiciary Square subway station and some particularly malevolent and efficient urban planner decided to come along and rotate all of the city streets while you were underground. Instead of the numbered streets running North-South, they are turned a few degrees to the right. The streets still make a grid, just a different grid.
A pedestrian in this new version of Washington still wants to walk from Judiciary Square to the Chinatown Metro stop. He still walks down a lettered street and up a numbered one. Each leg of the trip is different than it was before, and yet the subway trip that you take underground is exactly the same as it was before!
We’ve seen rotational symmetry a number of times already from the apparent isotropy of the large-scale universe down to the experimental fact that microscopic interactions really can’t seem to distinguish one direction from another. We’ve even seen that rotational symmetry gives rise immediately to conservation of angular momentum. As a practical matter, this means that the earth will orbit around the sun at a constant rate. Long story short: you may have learned the Pythagorean Theorem as a kid, but it’s anything but child’s play.
What does “distance” mean in space and time?
Space and time are very similar to one another, but not identical. If Einstein’s postulates of Special Relativity are correct – and to date, they have passed every experimental test thrown at them – then we’re going to have to figure out a way of jamming space and time into a single “spacetime.” Einstein himself warned of the danger of trying too hard to think in four-dimensional spacetime:
No man can visualize four dimensions, except mathematically … I think in four dimensions, but only abstractly. The human mind can picture these dimensions no more than it can envisage electricity. Nevertheless, they are no less real than electro-magnetism, the force which controls our universe, within, and by which we have our being.
Let me put this in familiar terms –- or at least familiar if you’ve memorized the entire Star Trek canon. The Vulcan homeworld is approximately 16 light-years from earth and right now (Footnote: In short order, we’ll find out how poorly defined the concept of “right now” really is, but for now, humor me.) Solkar, the great-grandfather of our own Mr. Spock, is a young spaceship pilot. Because we are separated from Solkar in space, but not in time, calculating the distance is easy: 16 light-years.
Let’s add in time. You are now reading these words and ten seconds ago you were reading “Let me put this in familiar terms.” Assuming you are sitting perfectly still, these two events are separated in time by ten seconds, and not separated in space at all.
But what if events are separated in both space and time? If we were to point a ridiculously powerful telescope at Vulcan right now, we wouldn’t see Solkar flying around his spaceship. Instead, we’d see the events on Vulcan unfolding from 16 years ago. This is, of course, because we’re seeing the signals traveling at the speed of light. The events we see are separated from us by 16 light-years of space and 16 years of time. Light signals will always have this one-to-one separation of space and time.
On earth, of course, events are also separated by both space and time. Go to a baseball game and watch and listen to the batter hit a ball. You probably know from experience that you’ll see the hit before you hear it. That’s because the speed of sound is far slower –- by roughly a factor of a million –- than the speed of light. The delay between a batter hitting the ball and you hearing it is perhaps half a second. The distance that the signal travels, on the other hand, is far less -– only about 500 billionths of a light-second.
On earth, generally, separations tend to be much larger in space than time – at least on the human scale. The circumference of the earth is only about an eighth of a light-second, but the time that it takes us to travel that distance is many hours. We may as well be standing still.
See? We can compare totally space and time, but unlike with different dimensions of space, there doesn’t immediately seem to be any equivalent of the Pythagorean Theorem that tells us how to add them.
I’m going to describe things a bit ahistorically, in large part, because some of the most relevant results to our purpose – symmetry – weren’t discovered all at once. Einstein came up with his theory of Special Relativity in 1905, based in large part on the work of James Clerk Maxwell and a number of mathematicians and physicists who laid the groundwork over the decade prior. It wasn’t until 1907 that the German mathematician Hermann Minkowski finally showed how space and time really fit together in a way that would have made Pythagoras proud.
Minkowski realized that in some sense, space and time work in opposite directions. There’s some suggestion, for instance, that Betelgeuse, the bright red star in the constellation Orion, might go supernova any day now (In astronomical terms, “any day now,” means that it might go off in 100,000 years or more.). Betelgeuse is about 600 light-years from earth, which means that even if we saw it blow itself up tomorrow, the actual explosion took place 600 years ago. How far away is the explosion really? We could describe the distance in space (600 light-years) or time (600 years), but combining the two is a bit trickier.
We have a hint, though. If the light from a supernova explosion is just now reaching us then there is an immediacy to it. It is, in a real sense, happening in the here and now, since the speed limit of light prevented us from learning about it earlier. Minkowski created a variant of the Pythagorean theorem where time behaves almost exactly the same as distance, except for a minus sign:
This “interval” is just a fancy way of combining distances in both space and time (Footnote: The mathematically savvy among you may have noticed that for events that are separated more in time than in space, the interval squared turns out to be a negative number, making the interval an imaginary number. Don’t worry about it. This is just a mathematical device to let us know that the two events are “time-like separated,” which simply means that one event could affect the other. If the interval-squared is positive, they’re space-like separated, which means that causality simply can’t come into play. If math makes you want to rock yourself in a corner, please move on. There’s nothing to see here.). It’s also pretty clever. By definition anything that we’re just seeing now –- regardless of how far away it is in space –- has an interval of zero. More importantly for our purposes, the interval between two events is completely independent on your perspective.
We saw with the Pythagorean theorem that it doesn’t matter how you rotate the orientation of your streets, the distance between any two subway stations will always remain the same. The interval is exactly the same. Einstein said that all inertial observers should measure the same speed of light, which means that the interval between any two events should be the same no matter how fast you’re traveling through space.
We’re in deep now. Spend a moment to think about how abstract thinking of symmetries ends up revealing surprising connections. We just found that, from a mathematical perspective, rotating a coordinate axis is the exact same sort of symmetry in space as moving at different speeds does on spacetime. Both of these two transformations leave something invariant. For rotations, the distance between two points stays the same; for different speeds – “boosts” as relativists call them – the interval stays the same. Who could have seen that coming?
If you have trouble picturing this, imagine a Vulcan astronaut flying the earth to Betelgeuse route at a sizeable fraction of the speed of light. As we’ll see, he’ll measure the total distance of the route to be less than 600 light-years. However, he’ll also measure the delay in between Betelgeuse going kablooey and us measuring it as less than 600 years. All combined, he’ll measure the same zero interval as we do no matter how fast he’s going.
How Time Gets Stretched
It’s not enough to simply say that the speed of light is the same for all observers. Clearly time gets messed up, but so far, we haven’t gotten any idea as to how. Suppose Solkar decided to buzz past the earth at half the speed of light. Provided he’s moving in a constant speed and direction, he feels as though he is sitting still. That is, of course, one way of thinking about Einstein’s first postulate of Special Relativity. As Solkar goes about his business – reading the paper, taking a nap, browsing the intergalactic net -– he is, as far as he’s concerned, moving through time and not through space.
On earth, on the other hand, we see him moving through both space and time. If Solkar settles down for what seems to us to be an eight-hour nap, by the time he wakes up, the ship will have traveled four light-hours.
The beauty of Minkowski’s interval is that it’s the same for all observers. From the earth’s perspective, the space and time between the beginning and end of Solkar’s nap partially cancel each other out. On the other hand, as far as Solkar’s concerned, he’s slept for less than 8 hours. As it turns out when you crunch the numbers, he’s actually only gotten about 7 hours of sleep. Relativity can be even more disruptive to your circadian rhythm than daylight’s savings.
Moving clocks run slow. This is not some trick of measurement; it’s a real effect, albeit a very small one, at least under normal circumstances. To put things in perspective, even on the highest speed Japanese bullet trains, time only appears to run slower by less than one part in a trillion.
At only half the speed of light, things only appear a little bit off. The second hands on clocks would appear to tick only about 52 times for every one of our minute, for instance. If, instead, he sped past at 99% the speed of light, things become crazy – clocks appear to be slowed down by a factor of seven! And understand, this isn’t some weird optical illusion or mechanical effect because of the strain of the speed. Everything appears to be slowed down by the same factor. Solkar’s heart – and all of his metabolic processes – would beat slower than normal; his computers would appear sluggish by normal standards; every single device capable of measuring time would appear pear to have slowed down to a crawl. And yet, from Solkar’s perspective, everything appears to be running perfectly normal within the ship.
While we can’t actually build spaceships capable of moving at relativistic speeds, we can measure time dilation here on earth using particles called muons. A muon is almost identical to an electron – but 200 times heavier. As we’ve seen before, heavy particles, whenever possible will decay into lighter ones, and the muon is no exception. After about 2 millionths of a second on average, a muon will decay into an electron and a couple of neutrinos.
Since muons decay so quickly, it’s a wonder that they’re around to be observed at all. Fortunately, the universe is nothing if not dedicated to the task of producing massive particles. When extremely high-energy particles from space – cosmic rays – strike the upper atmosphere, a cascade of particles get created, culminating in the production of muons. This means that the bulk of muons get produced more than ten kilometers above the surface of the earth. This would be no big deal except for their incredibly short lifetime. Even traveling at the speed of light, a muon could only cover about 600 meters in that time. We’d reasonably suppose that virtually no muons should ever reach detectors on the surface of the earth.
And yet, we do detect atmospheric muons all the time. We can even tell that they originate from cosmic rays because we can see a big empty spot – a shadow – where the moon is. In 1941, Bruno Rossi and David Hall, both from the University of Chicago, measured the number of muons coming from sky as measured at the top of a two kilometer high mountain and at ground level. If Galileo were right, and time flowed the same for everybody, then all of the muons should have decayed from top to bottom. Instead, Rossi and Hall found very clearly that the muons’ “internal clock” seemed to be slowed by roughly a factor of five, meaning that they were hauling ass at roughly 98% the speed of light.
But relativity says far crazier things than “moving clocks run slow.” Einstein’s first postulate of special relativity was that you can’t ever tell if you’re the one moving or standing still. It’s easy to imagine things from Rossi and Hall’s perspective. They’re people, after all, and just like in sci-fi movies, we tend to have an anthropocentric view of things.
But if you can manage a little empathy, put yourself in the muons’ position. The muons don’t feel like they are moving at all. Here they are, newly born, and all of a sudden they see the ground – and Rossi and Hall – hurtling toward them at 98% the speed of light. The muons, provided they have the presence of mind to do the experiments, find that Rossi and Hall seem to be living in slow motion, by the same factor of five that we saw before.
Despite having studied relativity for a long time, this still seems crazy to me. If we pass one another in spaceships traveling close to the speed of light, we each measure the other to be aging slowly and we each measure the other’s ship to be shortened in the direction of motion. This simply seems to be logically inconsistent. And yet, it isn’t.
I get mail: Twin Paradox Edition
Greetings, true believers! Apologies for the relative infrequency in posting, but as most of you know, I’ve been working hard on my upcoming book. I’m over the hump, and sent the first half to my editor a couple of weeks ago. I’m now working on Chapter 6, which will tentatively be entitled something like, “Why can’t you build an ansible?” While my old io9 article on the subject dealt with issues related to quantum entanglement, this chapter will talk about the relativity side of things, why simultaneity doesn’t exist in a relativistic universe, where 
In other news, you may have noticed that I got quoted in Natalie Angier’s column on Emmy Noether in the New York Times a couple weeks ago. That was very exciting. Natalie contacted me after seeing my own column/rough draft for the introductory material for Chapter 5.
But enough of that. Today I’d like to tell you about an email I got from a very earnest reader named Eduardo from Brazil. Eduardo wrote to ask me about the “Twin Paradox.” As this entry is labeled “technical” (meaning that there will be a few equations and an assumption that you remember some of your freshman level physics), you probably know the Twin Paradox already, but if not, here’s a short excerpt from the User’s Guide:
There are two twin sisters, Emily and Bonnie, who are both 30 years old. Emily decides to set out for a distant star system, so she gets in her spaceship, and flies out at 99% the speed of light. After a year, she gets a bit bored and lonely, and returns to Earth, again at 99% the speed of light.
But from Bonnie’s perspective, Emily’s clock (and watch, and heartbeat, and everything else) has been running slow. Emily hasn’t been gone for 2 years; she’s been gone for 14! This is true however you look at it. Bonnie will be 44; Emily will be 32. You can even think of traveling close to the speed of light as a sort of time machine – except it only works going forward and not backwards.
…
Here’s where the paradox comes in. When Emily stepped off her ship back on Earth after traveling to Wolf 359 and back, everyone agrees that she’s only aged 2 years in the same time that Bonnie has aged 14. That is totally inconsistent with pretty much everything we just told you, because we immediately know that Emily was the one who “moved” and not Bonnie, and the first rule was that you could never tell who was moving and who was sitting still. So how do we resolve it?
I’ve written a lot about the time dilation of moving ships: here (in which I figure out how long it will seem to take to get to Gliese 581g), here (in which I compute the amount of energy required using a matter-antimatter engine), here (in which I derive the general relations for the flow of time under constant acceleration) and here (where I do a simpler write-up for io9). The image at top is also a movie illustrating the effect for a trip under constant acceleration and deceleration.
The “paradox” part of this will be clear when we read Eduardo’s question. After some very kind words about how awesome I am, he gets right to it:
Please give me an explanation about the Twin Paradox.
…
I refer to the paradox of *why* is the Earth twin that gets older, and not the spaceship twin, if, during the non accelerated stages of the trip, both are in inertial frameworks, by definition equivalent according to SR.
Yes, I know that the spaceship twin accelerates and decelerates, and feels the acceleration and deceleration, four times during the trip, and the Earth twin remains inertial all the time (disregarding Earth movements).
But acceleration is not part of SR.
And in a ‘thought experiment’ we can have arbitrarily high accelerations and decelerations, in order to have the non inertial stages of the trip arbitrarily short, with the trip essentially being performed in constant speed for practically all its duration.
In this case, why only the Earth twin is older? All slower aging happens to the spaceship twin in the extremely short accelerated/decelerated stages? Or the differential aging continues to happen in the inertial stages by some kind of ‘memory’ of the non-inertial stages?
I know the slower aging actually happens to an object that is moving with relation to an observer that is at rest. I know that experiments with decaying mesons, GPS satellites and atomic clocks in airplanes proved that.
But what the explanation for that *using exclusivelly the SR*, *without using* GR and acceleration arguments?
Why the inertial frameworks for the twins are different? Or, if they are equivalent, why only the Earth twin is older, if the non inertial stages of the trip can be of arbitrarily negligible duration? Why is not the spaceship twin older in this case? Or why both are not with the same age when they met each other after the trip?
Why am I asking it?
Because I’m a member of an internet science discussion list in Brazil, with about a thousand members, and we are having now a ‘flamewar’ on the Twin Paradox.
I didn’t ask Eduardo for the link to the board — who wants to get into the middle of that? — but it’s a very good question. Here’s my answer.
In many ways, however, you have hit at the heart of the problem. The Twin Paradox can’t be resolved by understood by Special Relativity because it doesn’t satisfy the postulates of Special Relativity. Einstein laid out his assumptions in his original 1905 paper. As he put it:
- [T]he same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.
- [L]ight is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.
The first postulate is generally understood to mean that all laws, not just electromagnetism, will remain invariant in all inertial reference frames. And what happens if the reference frame is NOT inertial? The theory can’t say a thing. The short answer is that there is no answer “*using exclusivelly the SR*, *without using* GR” as you put it. It simply isn’t a special relativistic question.
You raise a couple of points. You ask, for instance, what happens if the accelerations are nearly instantaneous. To put that in mathematical terms, if the cruising momentum of the spaceship is p:
What happens if the force is extremely high and the time is extremely short?
Instead of thrusters, we could consider what would happen if we did the same sort of acceleration using a potential well. Suppose (for ease of calculation and for general familiarity with the final result) that the final speed of the ship was enough less than the speed of light that we can use non-relativistic approximations. You are free to generalize later, and I’ll assure you it won’t make a difference. What happens then?
Well, if you fall into a potential well 
This is nothing more than the statement that if you decrease your potential energy, you increase your kinetic energy. As I mentioned above, this is in the non-relativistic limit. For a deeper potential well (nearing the event horizon of a black hole, for instance), it’s more complicated, but the overall result is the same.
In a practical sense, the
Since
For 
Check by plugging in 
In other words, you can imagine the trip as though the traveling twin falls into a very deep potential well, hangs out for a while, climbs out when she reaches her destination, falls in again to return, and climbs out again on reaching earth.
As I said, I did this under the assumption that the relativistic effects were relatively mild. You are welcome to try using the full:
Hope that helps and doesn’t just add fire to your flame war.
-Dave
Just a short post today. I have a few quick announcements:
- The Korean translation of the User’s Guide is out. If you happen to be Korean, then what are you waiting for?
- Tomorrow is Emmy Noether’s 130th birthday. Natalie Angiers of the NY Times called me the other day for an interview. Apparently, they’re going to run a small piece about Noether. I’ll post a link when it’s up.
- I’ve finished the first 5 chapters (aka half of the book) of “The Universe in the Rearview Mirror” and will be shipping them off to my editor in the next week. If you’re a test reader, and want to have maximum impact on the final manuscript, now is the time to act!
- On a personal note: I got an official letter from my provost yesterday, promoting me to full professor!
Thanks for your kind indulgence.
-Dave
Some awesome stuff (that I didn’t do)
We are just finishing up the Winter term here at Drexel. This term, I’ve been teaching an advanced undergraduate/graduate course in cosmology. One of my undergraduate students, Mary Chessey, wanted to do an additional honors elective. She developed a series of hysterically funny, pedagogical, and accurate cartoons to illustrate some core concepts in cosmology. Check it out:
How comoving distances work. Credit: Mary Chessey
Dark Matter is not the same as stars and gas. Credit: Mary Chessey
The true origin of cosmological redshifts. Credit: Mary Chessey
Angular size and distance. Credit: Mary Chessey
What is dark matter? Credit: Mary Chessey
If you use these diagrams, be sure to credit Mary. She did a wonderful job.
-Dave
Greetings, physics junkies. This week, I decided to take a break from particle physics, cosmology and relativity, and do some good old-fashioned thermal physics (though cosmology still manages to sneak in there): Why can’t we get down to Absolute Zero? There’s a fair amount about cold temperature records, lame superweapons, and the ultimate heat death of the universe — something for everyone.
If you’re visiting from io9, take a look around, especially on the “physics” tag. And, as always, feel free to send me questions.
-Dave
I’m teaching an advanced undergraduate/graduate course in cosmology this term, and I thought it might be fun to combine the education and outreach sides of my life. I get sent a lot of interesting questions, and answering them have prompted me to learn about subjects that I might not otherwise have thought much about, as well as improving my ability to focus on the mindset of my readers. I figured I’d give my students the same opportunity for a little extra credit.
I picked a few cosmology oriented questions and asked my students to take a swing at them. I’ll post some over the next few days, beginning with this question about the nature of the expanding universe.
Dear Physicist,
Ok, I know about how astronomers have determined the Universe is expanding but there seems still to be a question of why. I thought of a possible reason but surely a non-physicist as myself is not aware of all the possible reasons already rejected or factors already calculated in explaining the observations. Here is a thought that has me plagued. Please tell me what may be wrong or right about it.
The gravitational force between bodies decreases with distance. That much I know. Maybe the separation of mass in the universe decreases the gravitational forces between bodies at a rate faster than the decrease of stored kinetic energy in those bodies? Could it then be that the combined result of force vectors is causing distance between those bodies to accelerate? Also, perhaps as objects move out into space with less matter, solar winds etc, that the resistance of space decreases with distance from the big bang. That may also cause a further acceleration of expansion from reduced drag. Could those two factors explain the observed expansion of the Universe?
My talent in math and physics is not good enough to figure this out but I would guess yours is. I hope you get back to me and end my sleepless nights.
Thanks,
Paul
From Anna:
Hi Paul,
You’ve asked an important question – why is the universe expanding? – a concept that even Einstein himself had trouble believing. In 1929, Edwin Hubble (whose name you may recognize from the Hubble Telescope) famously demonstrated that galaxies appear to be receding from us at a speed proportional to their distance from us. The relationship is speed = H*distance, where H is now called Hubble’s constant. Upon hearing this, you might think that if every galaxy we observe is moving away from us, then doesn’t that imply that we are at the center of the universe? There is a way out of this of course, if the entire universe is expanding – think stretching out a rubber sheet in al directions – then every point in the universe is moving away from every other point. Hubble’s observations have been repeated numerous times, and we now have a fairly accurate estimate of the H value. So there’s no getting around it – the universe is expanding.But the perplexing question is this: given that we know the universe is made of matter (stars, galaxies, gas), and that matter gets attracted together by gravity, why is the universe expanding and not collapsing on itself? This is precisely the reason Einstein did not like the idea of an expanding universe. In the 1920’s, Einstein’s general relativity equations were being put to use in describing the geometry of the universe (no easy task). Einstein worked with the physicist Willem de Sitter and each developed models of the universe that assumed (as any reasonable person might) that the only “stuff” in the universe is matter. However, both of these models yielded a universe that is unstable. Einstein realized that the model could be made stable if an extra term was added in, what he called the “cosmological constant.” The constant allows for a universe that does not collapse in on itself, but rather expands. The idea greatly disturbed him, because he was hoping that this equations would yield a static universe that was basically unchanged throughout time. In 1947 he said: “Since I introduced this term, I had always a bad conscience… I am unable to believe that such an ugly thing should be realized in nature.” However, today the cosmological constant is accepted in the standard theory of cosmology, and has been measured to have a value of about 0.7.
Sparing you too many details of the physics, the cosmological constant – this factor that allows for expansion in the face of gravity – arises from what you may have heard called “vacuum energy”. There is a result from Quantum Mechanics that says even empty space has a small but non zero energy, a very strange discovery indeed! One consequence of this, is that this small amount of energy has “negative pressure”, yet another strange concept. All this means is if pressure is to “push down on something”, then negative pressure simply means to push out or expand. So this vacuum energy, or more properly we call it dark energy, is what works to expand space itself.
I hope this serves as a satisfactory answer to your question. If it does, and you would like more to think about, I’ll tell you that not only is the universe expanding, but it is expanding at an ever faster rate. Three physicists proved this in 1998, and were just this year awarded the Nobel Prize. If you are in the area, one of them will be giving a talk at Drexel called “The Accelerating Universe” on March 1st. This talk is held every year by the physics department at Drexel and is meant for physicists and non physicists alike. More information can be found here.
From Justin:
Dear Paul,
The current explanation that physicists have concerning the expansion of the universe is something we like to call ‘dark energy’. Mathematically, this is similar to the ‘cosmological constant’ in Einstein’s abandoned addition to his general theory of relativity. As to what this stuff physically is, however, we really don’t know.
We do know that the universe is indeed expanding, from the measurement of the Hubble constant at far distances. It has been only relatively recently that this has been ascertained; in 1998 the two teams of Saul Perlmutter, and Brian P. Schmidt and Adam G. Riess measured the most distant supernova distance and determined the expansion as the result (the accomplishment earning them the 2011 Nobel Prize in Physics). The Hubble constant relates the recession velocity of a particular object with its distance from us. Found by observing the redshift of objects (the lengthening of a known signal’s wavelength by expansion), we know that each object we observe is moving faster away from us the farther it is away from us. To think about this in an analogy, you can think of a rubber sheet. If you can imagine certain marked points on this sheet, when one stretches this sheet out in all directions, these points will spread out from each other, without actually moving on the sheet. In our universe, these marks can be star clusters, galaxies, etc. While each individual component may have its own random velocities compared to other components, in general the entire system is expanding.
You are correct in the reduction of the gravitational force according to distance. However, this gravitational attraction is insufficient to explain the total expansion of the universe. If you can imagine the farthest bodies observable, (let’s say a distant galaxy or quasar) then these may be subject to very small gravitational forces towards the center of the universe, and may very well be moving without any real ‘drag force’ towards the farthest regions of space we know of. However, we also know that the entire universe is expanding, not just those objects extremely far away from us. Knowing the speeds that galaxies and stars are capable of obtaining from other areas of astronomy, we find that the expansion speed far exceeds these characteristic velocities in some cases, and that, among other statistical factors, it is space itself that must be expanding.
This is where the concept of dark energy enters into play. As cosmologists, we must account for what kind of ‘stuff’ comprises our universe. Without getting into mathematical detail, we find these components: matter (both the kind we can see and the kind we can’t – the supposed dark matter), relativistic particles (including radiation and neutrinos, etc.), the inherent curvature of space (do we live on a 3-dimensional globe, or flat space, etc.), and the ‘cosmological constant’ that Einstein predicted (dark energy). This last term is another, completely different, variable in our equations of the universe. Without this, we could not produce a model that is consistent with all the things we know about our universe, from the very beginning straight up to the present day. An important point is that this factor does not have the same behavior as the variables for matter, radiation, and space curvature do. That is, it is something completely unrelated to all of these things that we presently know of, and furthermore, it seems as if it is the largest component of the universe by far. Our current model of physics does not predict what this ‘dark energy’ could be, only that either something like it exists, or that some aspect of established physics must be ‘broken’. As to the form of this dark energy, some theories have been proposed, such as a constant energy density pervading all space, yet remaining undetected by us, or of the existence of a new fundamental force (quintessence), but so far these remain solely postulates with no experimental evidence.
And so we are left with a wide-open definition of ‘dark energy’ explaining the accelerating expansion of the universe, but have no direct evidence of what it really is, only ruling out that it cannot be composed of the normal components of the universe that we already know of.
Justin
Finally, from Karsten:
Paul,
Hello! My name is Karsten and I am a first year graduate student in Dr. Goldberg’s cosmology class. I am going to take a stab at your question and attempt to shed some light on the dynamics of the universe! It is my sincere hope that when you finish my email, this problem will no longer plague you but simply bother you to the point of sleeplessness.
From what I read, you have a solid understanding of the Big Crunch theory – the idea that following the Big Bang, the gravitational attraction of matter will inevitably cause everything to stop expanding and then recompress into a tiny ball of universe. Honestly, that was also the theory that I subscribed to before taking Dr. Goldberg’s class. Our universe, it seems, will inevitably end in a Big Chill, where the universe’s expansion accelerates and stars are no longer able to form.
Why is this, you ask? You are correct that the gravitational force does decrease with distance, but never does the gravitational force between two objects ever go to zero. This means that if you have a universe where gravity is the only thing that’s holding it together, then no matter how large it expands, it will slow down and it will crunch back up. It will just take a really, really long time. This suggests to us that something is, in essence, acting against gravity, at least when it comes to the expansion of the universe. Additionally, I can address your theory that the decreased density of the universe as a result of it getting bigger could play a part in its expansion. Intuitively, I would agree with you that there is less stuff for matter to bump into so it would tend to continue its outward expansion…but, to me at least, that would only lengthen the inevitable Big Crunch and would not explain why the universe is expanding at faster and faster rates!
You have probably heard the terms “Dark Energy” and “Dark Matter” being thrown around. Well, Dark Energy is what cosmologists name as the culprit in the case of universal expansion. Why is it called Dark? Well, that seems to be because we can’t directly measure it and don’t know what it is! Dark Energy is a place holder for whatever energy it is that is causing the universe to accelerate, kind of like how the ancients didn’t understand why lightning struck the Earth and came up with the idea of a god hurling lightning bolts at us. A number of scientific experiments have suggested to us the presence of this Dark Energy, here are a few.
First of all, you might have heard of red or blue shifting. It’s a phenomenon that occurs when cosmic bodies that emit light are moving toward or away from us. When they move toward us, the light they emit (when viewed right up next to the thing) looks bluer to us if its moving toward us and redder if it is moving away. By observing many supernovae of the Type IA variety, we were able to see that almost everything around us is moving away from us! From this data we were able to conclude that the universe is expanding and that expansion is accelerating.
So, to get back to your question about what makes the universe behave as it does, I can tell you that there are four things that contribute to the expansion of the universe as we know it. If you imagine the universe at the very beginning like a big explosion, then you’ll have no trouble in believing me when I tell you that radiation plays a part in the expansion of the universe. Matter affects universal expansion as you well know and now you know that Dark Energy is the fourth component. If you were to keep track of which of these four things (radiation, curvature, matter or dark energy) were important and when, then the timeline would go radiation, curvature, matter, and dark energy. In fact, dark energy is now the dominant factor for why the universe does what it does and the gravitational attraction of visible matter will play less and less a role as time goes on.
In closing, I hope I was able to answer your question and not ramble on too much. Feel free to email me back if you’d like to chat some more about the universe. I don’t know much but I find it a fascinating topic of discussion.
Sincerely,
Karsten
As always, I’m happy to get your email, but if you have any followups for the expert panel, feel free to leave them in the comments.
-Dave
Prof. Brian Schmidt
Hey! Do you live in or near Philadelphia? Are you interested in Cosmology? Do you want to hear a Nobel Laureate talk about his prize-winning work? Would you be willing to pay zero dollars for the privilege? Well have I got some good news for you.
On March 1, the Drexel University Department of Physics is hosting the 17th annual Kaczmarczik Lecture. Each year we invite a leading physicist, frequently a Nobel Prize Winner to give a public talk at the high school/nonmajor level.
This year, I’m pleased to announce that we’ll be bringing 2011 Nobel Laureate Brian Schmidt to campus. He’ll be talking about “The Accelerating Universe.” Registration is free, but make sure you sign up online so we know how many people are coming (and if you’re in high school, if your class will be participating in the physics department open house).
This is going to be an especially exciting talk, so I hope to see you all there.
-Dave
Introducing Emmy Noether
Last week, I took a little poll here, on my twitter feed, on facebook, and among my students. I wanted to know if anyone had heard of Emmy Noether or of Noether’s Theorem. I don’t want to put my disappointed face on. After all, I hadn’t heard of her until well into grad school, and couldn’t have written down or derived Noether’s Theorem until well after that.
And that is a real shame, because almost nobody in the 20th century did more to explain how physics – and by extension the universe – ultimately works. Noether was a mathematician of the highest order, and makes her so important to the world of physics is that Noether’s Theorem finally gives us a real insight into why symmetry is so important.
Beyond just her work, I want to say a few words on Emmy Noether herself. In many respects, Noether’s story parallels that of Einstein’s, and the two of them intersected on a number of occasions. She was to born to a Jewish family in Erlangen, Bavaria in the late 19th century. Her father was eminent mathematician at the University of Erlangen, and Noether decided to follow in his footsteps. However, German Universities in 1900 generally did not allow women either to enroll in classes or to sit for examination. In 1898, the faculty senate of Erlangen went so far as to claim that admitting women would, “overthrow all academic order.” Noether essentially had to pursue her entire undergraduate education in mathematics by auditing classes, but was still able to pass the graduation exams at a nearby university in 1903.
In 1904, Noether began her doctoral studies at Erlangen, after the ban on women had finally been lifted. Her thesis advisor was Paul Gordan, a very close collaborator of her father, Max Noether’s. Like many other ostensibly pure mathematicians of the era, Gordan’s work found it’s way into the newly developing quantum, in this case into the so-called “Clebsch-Gordan Coefficients” which are used to describe the spin and angular momentum of electrons.
She completed her Ph.D. in 1908, and had an extremely tough time finding an official academic appointment, despite her obvious gifts. You may recall that Einstein, famously, faced similar troubles, and was exiled to a Swiss patent office until after he became mega-Einstein and discovered relativity, the photoelectric effect, and explained Brownian motion all in the same year. Meanwhile, Noether spent the next eight years as unpaid researcher at the University of Erlangen, occasionally substituting for Max Noether in his lectures.
In both her thesis and during her remaining time in Erlangen, Noether became an expert in mathematical invariants. These are absolutely crucial to our understanding of symmetry, so I should say a few words about them.
Invariants are the counterpoint to symmetries. While a symmetry describes the sort of transformations that you can apply to a system without changing it, an invariant is the thing itself that is unaltered.
To make matters concrete, think about the force of gravitational attraction between two stars. In this case, there are a number of symmetries: It doesn’t matter where you do the calculation, or when, or how the stars are oriented with respect to one another. The Invariant, however, is the strength of the gravitational attraction between the two stars. No matter how you move the system around, the magnitude of the force remains the same.
If you’re starting to get a picture of why Noether might be just the person to understand how symmetry really works in physical laws, you’re not alone.
In 1915, Einstein published his theory of general relativity. There was something incredibly elegant, and deeply symmetric about the theory, but nobody really understood how it all fit together. The eminent mathematicians David Hilbert and Felix Klein (inventor of the Klein Bottle) invited Noether to the University of Gottingen in 1915 was to help in understanding some mysteries introduced by relativity.
As Herman Weyl described the situation:
Hilbert at that time was over head and ear in the general theory of relativity…Emmy was welcome as she was able to help them her invariant-theoretic knowledge. For two of the most significant sides of the general theory of relativity she gave at the time the genuine and universal mathematical formulation.
Under normal circumstances her work would have undoubtedly allowed her to start work as a professor. But just as at Erlangen, biases against her gender interfered. Hilbert was outraged. At a faculty meeting, he argued:
I do not see that the sex of the candidate is an argument against her admission as a Privatdozent (roughly equivalent to Associate Professor in the U.S.). After all, we are a university, not a bathhouse.
Hilbert and Noether bent the rules by listing Hilbert as a course instructor, and then having Noether as the perennial “guest lecturer.” I should note that all of this was without pay. It wasn’t until 1922 that the Prussian Minister for Science, Art and Public Education gave her any sort of official title at all, and even then only a small stipend.
Almost immediately upon her arrival at Gottingen, Noether derived what’s become known as “Noether’s Theorem,” and by 1918 had cleaned it up enough for public consumption. Simply put, her theorem states:
Noether’s Theorem: Every symmetry corresponds to a conserved quantity.
If you are a bit underwhelmed after such a long build-up, you shouldn’t be. For one thing, this is more than just a blithe statement of fact; there’s a lot of mathematics under the hood. Noether’s Theorem gives a prescription for determining the conserved quantity for any system, including quantum fields. It provided the inspiration and simple interpretation for much of Quantum Electrodynamics and Yang-Mills Theories
Conservation laws are the bread and butter of physics. In the early universe, for instance, the positive charges exactly canceled the negative charges, and charge seems to be a conserved quantity, that means that the total electrical charge in the universe must still be zero today.
What Noether proposed sounds quite simple, almost content-free, unless you look at the implications. To give some simple ones:
- Time Invariance -> Conservation of Energy
- Spatial Invariance -> Conservation of Mometnum
- Rotation Invariance -> Conservation of Angular Momentum
But it goes even further than that. The last 50 or 60 years have all been about understanding quantum fields. Noether’s Theorem explains what happens when a system is symmetric upon a change of quantum mechanical phase. Answer: you get conservation of electric charge.
Likewise, her work describes and explains conservation of spin, of “color” (the equivalent of charge in the strong force) and on and on, ultimately providing the mathematical foundation for much of the standard model (despite the scant amount of attention that she gets in wikipedia).
Her story parallels Einstein’s in other, sadder, ways as well. Like Einstein, she fled to the United States in 1933. Einstein settled in Princeton, at the newly built Institute for Advanced Study. Noether went to nearby Bryn Mawr College. Her story has a rather sad ending, however. Only two years after coming to America, Emmy Noether was diagnosed with a cancerous tumor, and in the aftermath of a surgery, she died from infection. She was only 53. In Einstein’s words:
In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.
At some point in the future — if there’s interest — I may write a technical post on Noether’s Theorem and explain how it actually works in classical particle systems. But so much talk is given over to explaining how the study of physics is really the study of symmetry. I thought it would be nice to give a little credit to the work that explains what this symmetry actually means.
-Dave
Buy the book on amazon!
Become a fan on facebook!
Follow us on twitter!
Write to us!