Archive for the 'General Relativity' category

A muddled article on Relativity in the Oberlin alumni magazine

My wife graduated from from Oberlin college in 1992, and as such she gets the Oberlin alumni magazine. The summer 2012 issue includes a one-page article entitled "The Entirety of Relativity", which I find to be a very unfortunate presentation of Relativity. (As a pedantic point, it's only talking about Special Relativity (SR), and doesn't address General Relativity (GR) at all, but that really is a pedantic point. When a physicist says "Relativity", she likely means GR (especially given that SR is a subset of GR, so nothing is lost), but when presented publicly we often use "Relativity" as a shorthand for SR.)

The basic problem with the article is that it presents the theory as if its nature were the way that SR has been taught to students for a long time. The article starts with three things that are correct as far as they go: moving clocks run slow, a moving rod is short, and moving clocks aren't synchronized. Where the article loses me, however, is on point number 4, "That's All There Is To It.".The brief text after this says that the first three points are the basis of relativity, and the rest of the article claims that all of SR is a consequence of these three points. This is at the very least a perverse way of describing the theory.

A lot of texts at both the high school and college level present Relativity by first presenting these three points. You're given formulae for each of these consequences; parts of them resemble each other, but they're each presented as if they were a fundamental formula that couldn't be derived from anything else, for you to memorize (or, in a more modern way of thinking about it, look up) and use. However, this is a back-assward way of presenting SR, and I would argue that stating that the rest of SR is a consequence of these three observations is not just back-assward, but in fact wrong.

In fact, these three points are themselves consequences of the theory of Relativity. The formulae for them can be derived from more fundamental considerations. They're no more fundamental than all of the various kinematic formulae you memorize or look up (such as da2) when you do a non-calculus Newtonian mechanics class; those kinematic formulae themselves are just results of the definition of velocity and acceleration as, respectively, the rate of change of position and the rate of change of velocity, together with Calculus. Those definitions are the fundamental thing, not all the various kinematic equations you learn to use if you take a non-Calculus physics class. I could start with da2, take a couple of derivatives, and say, "hey, acceleration is the rate-of-change of the rate-of-change of position, and that's a consequence of this kinematic equation". That would be back-assward and indeed wrong, and it's just as wrong to say that everything else in Relativity is a consequence of moving clocks running slow, separated moving clocks not being synchronized, and moving rods being short.

Special Relativity itself starts with just two very simple postulates— "simple" in the sense of "not complex", not in the sense of "easy to understand". Those postulates are:

  • The laws of physics are the same for every freely-falling observer
  • The speed of light is one of those laws of physics; every freely-falling observer will measure the speed of light in a vacuum to be 2.998×108 meters per second.

Everything else in SR— including moving clocks running slowly, separated moving clocks not being synchronized, and moving rods being short, as well as other things (such as the Doppler shift, focusing of light emitted by a moving object in the direction of motion, an apparent rotation of a moving object) are consequences of these two postulates.

I should note that both of these postulates do require more explanation to be truly precise. For the first postulate, you have to carefully define "freely-falling observer". You get it basically right if there are no net external forces other than gravity acting on that observer. (However, if you allow gravity to be around, things can get a little subtly complicated. It doesn't ruin the postulates, but you have to be careful in treating the consequences.) For the second postulate, in fact it's not the speed of light that's absolute, it's the speed of any object that both carries energy and is massless. Light just happens to be the thing that we think about the most that works like this, and thus we call the cosmic speed limit "the speed of light", even though we really ought to call it "the speed of spacetime" (at least in the context of Relativity).

One of the most interesting consequences of these two postulates it that you have to change the way you think about time. Most of us live our lives with a Galilean/Newtonian view of time: it's an absolute, that advances at the same rate and is the same for everybody. However, you can't maintain that idea and have the speed of the same bit of light be measured at the same rate by everybody regardless of how they're moving. Galileo and Newton would say that the latter is wrong; Einstein's postulate, from which all of Relativity springs, was that in fact it's this speed of massless objects that is absolute, and as such we just have to give up on the idea of absolute time. Some of the consequences of this are that separated moving clocks aren't synchronized and moving clocks run slow... as well as other things.

I'm fond of the way that Thomas Moore's Six Ideas That Shaped Physics presents Special Relativity. (This is the book series that I currently use when teaching introductory calculus-based physics.) His Book R of the series is written for college-level physics who have had Calculus (and indeed have had some Calculus-based Newtonian physics). It presents SR not in the old-fashioned and unfortunate pedagogical way that the Oberlin article does— by starting with the consequences such as time dilation and with their formulae, and only later getting to the fundamental structure of spacetime implied by Einsteins postulates— but rather by starting with the fundamental structure of spacetime implied by Einstein's postulates, and then developing the consequences out of that

Yes, it's easier to just learn the formulae and do calculations about time dilation and so forth, and presents fewer difficult abstract conceptual challenges to students coming across this for the first time. However, if you learn it this way, you're given a warped perspective of what the theory of Special Relativity really is. My beef with this Oberlin alumni article is that it presents Relativity as if the theory itself is based and structured in the way that it has often been taught.

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Neutrinos don't travel faster than light

Last September, those who pay attention to discoveries in physics were rocked by an experimental result that suggested that neutrinos had been observed moving faster than the speed of light. Neutrinos produced at CERN in Switzerland (not at the LHC, but at another accelerator there) were detected at Grand Sasso in Italy by the OPERA experiment. The result only showed them moving a couple hundredths of a percent faster than light. That doesn't sound like much, but Special Relativity, one of the pillars of modern physics, indicates that the speed of light is an absolute limit, and that it takes an infinite amount of energy for a massive particle to reach that limit. As such, if a particle is going even a little bit faster— and the reported uncertainties on the reported result were much less than the difference between the measured speed and the speed of light— it would be an important result, indicating that in at least some regimes, Special Relativity breaks down.

Most physicists were dubious of the result. Special Relativity (SR) has withstood a lot of experimental tests. As such, any result that indicates that it isn't absolutely true is going to be subject to a lot of scrutiny, and is going to require robust reproduction before we really take it seriously. Also, the press reports and popular perception of the result missed one of the most important points. "Was Einstein wrong?" was the question asked. There were statements or implications that SR would have to be thrown out. The thing is, even if we'd seen a case where SR was violated, all the other experiments that show that it works still stand. It would only mean that SR isn't an absolute theory, that it's an approximation that works frequently but not always. We have many other theories like that. Newton's theory of gravity is a great theory that we use for a lot of things, but we know that there are situations where it isn't quite right (black holes, gravitational lensing of light, the precise orbit of Mercury, the precision required by the GPS), and we have to go to another theory (Einstein's General Relativity).

As I said, though, most of us suspected that the results from the OPERA experiment indicating neutrinos moved faster than light weren't correct. There was no scientific fraud involved, nor did anybody at OPERA do anything wrong from a moral or professional point of view. What they did was exemplary science. And, the rest of us, in doubting their results and wanting confirmation, also behaved as scientists are supposed to. We suspected that there would be some sort of systematic error that explained the apparent detection of faster-than-light neutrinos. It wouldn't be easy to find, because if it were, the scientists at OPERA would have already found it and would never have reported their result. Indeed, a few weeks ago there were reports that a loose cable, as well as possible shortcomings in corrections to GPS timings, may have been the source of the measured too-short travel time for the Neutrinos from CERN in Switzerland to OPERA in Italy.

However, even if the systematic error were never identified, for scientists as a whole to take the result seriously, there would need to be an independent verification by other scientists using separate techniques and separate equipment. A couple of days ago, a report from the ICARUS team indicated a failure to reproduce the OPERA results. This team was looking at the same source of neutrinos at CERN as the OPERA team, but it was a second team using a different detector. This is exactly what we needed. If they had come up with a measurement consistent with OPERA's measurement (and if the OPERA team hadn't identified the likely culprit systematic), then the scientific community would have to start taking faster-than-light neutrinos seriously. (It would be nice to reproduce the result also with a different source, in case there was some systematic involved with the timing on that side that would affect both results, but this would already be enough that we'd have to sit up and pay attention.) However, the ICARUS team reports timings that are entirely consistent with neutrinos moving at the speed of light. (In fact, they aren't moving at the speed of light. However, because their mass is so low, at the energies of these neutrinos they are moving a tiny, tiny fraction slower than the speed of light, a difference small enough that the experiment wouldn't be able to detect it.)

The story isn't completely over. Because SR is one of the foundations of modern physics, it's always worth testing its fundamental postulates. We'll keep pushing at it and looking for cracks in the theory, trying to see if it breaks down in places where we hadn't yet looked. However, at the moment, there's no indication that the theory breaks down anywhere. The brief suggestion that neutrinos might somehow be able to violate SR (which would have lead to all sorts of potentially cool consequences) has passed, as most of us suspected it would.

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The Minimum Size of the Whole Universe

The Observable Universe

When we talk about our Universe, we make a distinction between "the Universe" and "the Observable Universe". The latter includes only what we can see. By "can see", I don't mean what we have the technology to detect. Rather, I mean all objects out there from which light has had time to reach us given the age of the Universe, the speed of light, and the history and future of the expansion of the Universe. The age of the Unvierse is 13.8 billion years. Because the speed of light is finite, we can't see anything that is so far away that light would have taken longer than that from us to reach us. This isn't a technological limitation; this is a limitation on whether or not there is light, even in principle, for us to see given as much technological prowess as you could want.

Indeed, as we look towards the edge of the Observable Universe, we're looking back in time. If light took us 13.7 billion years to reach us, then we're seeing the Universe out there as it was 13.7 billion years ago, not as it is now.

The Universe as a whole, however, is probably infinite. This is easy enough to say, but it's a rather difficult concept to wrap your brain around when you really start thinking about it. One solution is not to think too hard about it. If you find yourself asking questions like "if it's already infinite, how can it expand?", you're not thinking properly about infinity. Infinity is a concept, not a number.

However, the Universe doesn't have to be infinite. According to General Relativity, there are other possibilities. I'm going to lump those possibilities into two categories, but only really talk about the latter.

Interesting Topologies

It's possible that our Universe has an interesting topology. Topology is different from geometry. Geometry includes things like lengths of lines, radii of curvature, sums of angles inside polygons, and so forth. Topology talks about how different parts of space are connected.

As an example, consider the classic video game asteroids:


This game takes place in a (very small) two-dimensional universe. The geometry of the Asteroids Universe is Euclidean— that is, parallel lines will never cross, the ratio of the circumference to the diameter of a circle is π, the sum of the three interior angles of a triangle is 180°, and so forth. However, if you ever played this game, you know that if you go off of the left of the screen, you come back on the right side of the screen. Likewise, if you go off of the top of the screen, you come back on the top. This universe is unbounded; you never hit a boundary, or an edge. However, it is also finite. Its topology is toroidal; it has the same topology as the surface doughnut, although it does not have the same geometry of a doughnut. (The surface of a doughnut has curvature.)

It's possible that our Universe is similar. It may have a flat geometry, but a topology that means that if you kept going in one direction, you came back where you started. If it does have this topology, it's on spatial scales larger than the Observable Universe. Otherwise, we would have seen the signature of this topology (i.e., the fact that parts of space are effectively repeats of each other if you keep going far enough in one direction) on the Cosmic Microwave Background.

For the rest of this post, I shall assume that the Universe does not have any interesting topologies like this. Either it's just infinite space, or it's finite space that is the 3d equivalent of the surface of a sphere.

Possible Geometries of the Universe

The geometry of our Universe doesn't have to be Euclidean. Depending on the total energy density (including the density of regular matter, dark matter, and dark energy), there are three possibilities for the curvature of our Universe.

[Possible Shapes of the Universe
Two-dimensional visualizations of the possible shape of the Universe. Our Universe would be the three-dimensional equivalent of one of these, depending on the total energy density of the Universe.

The parameter Ω is a convenient way of talking about the density of the Universe. There is a critical density, which depends on the current expansion rate of the Universe. That critical density is about 9×10-30 g/cm. That doesn't sound like a lot, but remember that the Universe is mostly empty space! Where we live, on Earth, is an extremely high-density place compared to most of the Universe. The parameter Ω is defined as the ratio of the density of the Universe to the critical density. If Ω=1, then the Universe has a flat geometry. Note that "flat" here doesn't mean "two-dimensional", the way you may used to be talking about flat. Rather, it means that the geometry of space is Euclidean, just like the geometry you probably learned about in high school.

On the other hand, if Ω>1, the Universe has a "closed" geometry. In this case, the geometry of the Universe is the same as the three-dimensional surface of a four-dimensional hypersphere. If that sounds like gobbledygook, think of it as the 3d equivalent to the surface of a sphere. Note that there doesn't need to really be a fourth spatial dimension or a 4d hypersphere out there. It's just that the geometry of the Universe— how parallel lines will behave when extended, how the angles of triangles will add up, what the ratio of the circumference to the diameter of a circle will be— is the same as the geometry of the surface of a sphere. It's possible to describe the mathematics of this geometry entirely using only three spatial dimensions, so there's no need for a higher spatial dimension in which to embed our Universe. However, for purposes of our visualization, it's worth thinking about the surface of a sphere, as that helps us get some idea about what sorts of things would be true in such a universe. The surface of a sphere is a two-dimensional closed universe. Remember, that the universe is the surface. There is no center to this universe, not within the universe— for everything within the universe is on the surface of the sphere, and no point there is any different from any other point.

If Ω<1, the Universe has an "open" geometry. This is harder to visualize. It turns out that you can't embed a slice of an open 3d universe into three dimensions to visualize it, the way you can with a closed universe (in which case you get the surface of a sphere, as described above). However, the closest two-dimensional equivalent would be a saddle or a potato chip (each of which is a hyperboloid or hyperbolic paraboloid). This is an unbounded and infinite universe. It keeps on going forever. However, it's also clearly not flat, and so will have an interesting geomoetry.

The Geometry of Our Universe

You can figure out the geometry of your universe several ways. One way is to create a triangle by drawing three straight lines through space. Then, measure the angle between each of those pairs of lines. If the three interior angles add up to 180°, you're in a flat universe. If they're more than 180°, then you're in a closed universe; if they're less than 180°, you're in an open universe. The problem is the precision needed for this measurement. In order to be able to tell whether or not the angles add up to 180°, you either need to measure them mind-bogglingly precisely, or you need to draw huge triangles, such that the length of one side of the triangle approaches the radius of curvature of your universe. (How close it approaches it depends on how precisely you can measure angles.)

Effectively, we have done this. Measurements of the Cosmic Microwave Background (CMB) give us triangles. One leg of the triangle is given by the characteristic size of fluctuations in the CMB. We know the physical size of those fluctuations. The other legs of the triangle are given by the path of light travelling from either side of one of those fluctuations. By measuring the angle between the light coming from either side of a fluctuation, we can figure out what the geometry of this isosceles triangle is. We did this. The answer: our Universe is flat. However, as with all physical measurements, there is uncertainty on this measurement. The latest form of this measurement tells us that Ω must be between 0.9916 and 1.0133, to 95% confidence (see "Reference" at the end for the source of these numbers). That means that there still is the possibility that our Universe is either infinite (in the case of Ω≤1) or finite (in the case of Ω>1).

The Minimum Size of Our Universe

The Universe is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to the Universe.

With all due apologies to Douglas Adams, let's quantify how big our Universe is.

First, the age of the Universe is 13.8 billion years. That is a long time compared to you and I, but as compared to the age of the Universe, it's just about right. The edge of the observable Universe, right now, is 48 billion light-years away. "Wait!" you may cry. "How can light from something 48 billion light-years away have reached us in a mere 13.8 billion years!" Remember that while that light was working its way towards us, the Universe was expanding. The light, in a sense, had to try to "catch up" with that expansion. This is imperfect language, and indeed if you know Special Relativity you should object to it. However, it does (sort of) make sense in the context of General Relativity.

How does this size compare to the size of the Universe as a whole? If we make the assumption that Ω=1.0133— the highest total energy density consistent with our current data, and thus the smallest closed universe consistent with our data— it's possible to calculate how big the Universe is. The result looks something like the following:

[Observable Universe in the Minimum Total Universe]
Click for a bigger version

In this picture, the surface of the sphere is meant to represent the whole Ω=1.0133 Universe. The parts that are "greyed out" are outside our Observable Universe; the patch at the top that you fully see is the Observable Universe. The radius of curvature of this universe is 120 billion light years. Its circumference is 760 billion light years. That means that the diameter of our Observable Universe is just 1/8 of the full length of a line you'd have to draw through space if you wanted it to connect back go yourself. The volume of the whole Universe is about 100 times the volume of our observable Universe. (If you object to the fact that 83 is not equal to 100, remember that we're not talking Euclidean space here, so your intuition for how radii and volumes of spheres relate doesn't entirely apply.)

Remember, though, that this is the minimum size of the Universe given our current data. Most of us suspect that the Universe is really a whole hell of a lot bigger than that, and indeed may well be infinite.

Size and Fate Are Separate

If you read almost any cosmology book written more than 12 or so years ago, and some written since, you will probably read something about a closed universe being one that recollapses, and an open universe one that expands forever. This is true only if the dark energy density of the universe is zero! Implicitly, those texts assumed that our Universe was matter dominated, and as such the geometry and fate of the Universe were linked. In a universe such as ours, where there is dark energy, the fate and geometry are not so tightly linked. Dark matter and dark energy both affect both the shape of the Universe and its ultimate fate, but they affect it differently. Exactly what will happen to our Universe depends on the details of what dark energy really turns out to be. However, for what most of us consider to be the most likely versions of dark energy, the Universe will keep expanding forever, with clusters of galaxies getting ever more and more separated. This is true whether the geometry of the Universe is flat, open, or closed.


The numbers for the current expansion rate of the Universe (used to derive the critical density) and for the limits on the curvature of the Universe come from the cosmological implications of the WMAP 7-year data as described in Komatsu et al., 2011, ApJS, 192, 18. The image used to wrap the universe sphere is the Hubble Ultra-Deep Field.

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Online Talk Tomorrow (12-03) About FTL Neutrinos

Tomorrow morning, December 3, at 10:00AM pacific time (18:00 UT), I'll be giving the MICA public outreach talk about the faster-than-light neutrino results from CERN and Grand Sasso. The talk will include an overview of the OPERA experiment that has led to the result, a summary of the result, my own headscratching about whether or not it's real, and some notes about what this does (and, more importantly, does not) imply about our confidence in the theory of Relativity.

The talk will be at the MICA Large Ampitheater, and all are welcome. Remember, a Second Life account is free!

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Online Talk, 10AM Pacific Time : SpaceTime Diagrams

As a part of MICA, the Meta-Institute of Computational Astrophysics, I will be giving a public talk tomorrow morning entitled "Understanding Relativity with SpaceTime Diagrams".

Einstein's theory of Relativity completely changed our notions of reality in the early 20th century. Time, it turns out, is not absolute, but rather mixes with space in a particular way that depends on how fast a clock (or other time measuring device) is moving relative to whoever is asking questions about it. Spacetime diagrams are a great tool for understanding Special Relativity. In this talk I'll introduce a few of the startling results of Special Relativity, and show how they can be described using spacetime diagrams. In next week's talk (September 17), we'll use SpaceTime diagrams as a tool to help us describe the geometry near black holes.

The talk will be at the MICA Large Ampitheater in Second Life. Remember that Second Life accounts are free, so register today!

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The Most Elegant Solution In All Of Physics

Feb 27 2007 Published by under General Relativity

A day or two ago, I posted my nomination for the greatest mystery in all of physics: why is it that the "gravitational charge" (i.e. how strongly you couple to the gravitational field) is identically equal to your inertial mass (i.e. how strongly you resist being pushed around by any kind of force)?

Einstein's General Relativity is our modern theory of gravity, and it answers this question in an extremely satisfying and elegant manner. Specifically, gravity is not a force at all; it's the geometry of spacetime. All objects move through spacetime in as straight a line as they can; if they deviate from a straight line, it's simply because of the curvature of spacetime. Objects of different mass are moving through the same spacetime geometry, so they all will move in the same manner.

This, to me, is an amazingly simple and elegant solution to what seems to be a great conundrum. Yes, it's often convenient to talk about gravity as a force, but when we recognize it not as a force but just as the background of what's out there, the conundrum completely goes away. Quantum Mechanics is in many ways a more successful theory than GR, in that it has been much more widely tested, and its tests are more precise. But I find at least the "gravity is the curvature of spacetime" part of GR to be far more elegant and beautiful than quantum mechanics.

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The Greatest Mystery In All Of Physics

Feb 26 2007 Published by under General Relativity

Because this is me, I must start with a lot of disclaimers. First, the title is catchy, but many would disagree with the mystery I've identified. Even I might. So, please try to avoid flaming me for my choice. Second, very shortly I will post "The Most Elegant Solution In All Of Physics," a post that might allow one to argue that what I'm about to identify as the greatest mystery isn't a mystery at all! Groundwork laid, here we go....

In physics, there is this quantity "mass" that we use to describe how much "stuff" there is in a particle. Technically speaking, "mass" is the energy content of an object measured by an observer when that object is at rest with respect to the observer, and when the observer is viewing that object as a closed system from the outside. (In other words, we don't know anything about "internal energy," because the object is just a thing.)

The thing is, there really are two different kinds of mass. First, there is inertial mass, which describes how much an object resists being pushed around by any kind of force. Second, there is gravitational mass, which describes how strongly an object couples to the gravitational field. And, yet, to the best precision we've been able to measure, these two kinds of mass are exactly the same. So much so that in introductory physics classes we just call it "mass", and students may not even realize that there's anything surprising about the fact that the two are the same!

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