One of the most popular questions asked about relativity concerns what would happen as an object approaches the speed of light. If it were moving fast enough, would it become a black hole? The same question could be asked another way: are you a black hole? The answer to the second question is obviously no. So is the answer to the first question. Confused? 

The basic argument is as an object approaches the speed of light from our perspective, three things happen. It’s time appears to slow down (time dilation), it appears to get shorter along its direction of travel (length contraction), and its mass appears to increase (relativistic mass).

These are all consequences of relativity and the fact that light has the same speed in all frames of reference. Relativity also says that a large enough mass in a small enough volume becomes a black hole. So it seems reasonable that an object near the speed of light should become a black hole.

If that were true, you must be a black hole.

Here’s why: the Universe is expanding. It’s not just that galaxies are racing away from us, space itself is expanding. This means that the more distant an object is from us, the faster it is moving away from us. There are distant galaxies moving away from us at nearly the speed of light. That means we are moving away from them at nearly light speed, so we should appear to be a black hole to them.

But the thing about black holes is that you either are one or aren’t one. A black hole’s event horizon must exist in all frames of reference. So if you aren’t a black hole in your frame of reference (and you’re not) then you’re not a black hole in any frame of reference. So a fast moving object does not become a black hole.

So what gives? It turns out the original argument is flawed. It’s true that an object’s relativistic mass does increase with speed. This means they are increasingly difficult to push, making it impossible for them to reach the speed of light. But everything is relative. From our perspective we have to keep pushing harder. From the object’s perspective our pushes keep getting smaller. As far as the object is concerned it’s mass hasn’t changed. So naturally it doesn’t become a black hole.

Just as you aren’t a black hole because you’re speeding away from distant galaxies. An object speeding away from us doesn’t become a black hole.

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Black Holes are where God divided by zero, so the saying goes. As short sayings go, that’s not a bad description of a black hole’s singularity, and it gives one a good idea why singularities are so problematic in physics.

In my last post I wrote about the cosmic censorship conjecture, and how it might be violated in hypothetical 5-dimensional black holes. I didn’t delve too deeply into the conjecture itself because there are actually multiple versions of the conjecture. The weak cosmic censorship conjecture is basically as I stated, that a singularity must be enclosed by an event horizon. It’s a bit more subtle than that, since an event horizon isn’t a local object in space, but rather a global structure in spacetime. So the formal definition is that a singularity can’t be seen by an observer sitting far away from a black hole (what is called null infinity). If you were close to a black hole you might catch a glimpse of a singularity, but the structure of spacetime is such that you couldn’t tell someone far from a black hole what you saw.

The upshot of all this is that anybody reasonably distant from a black hole could never see a singularity or interact with it. Since it can’t effect things outside a black hole, we don’t have to care about how it would affect things. As long as Pandora’s box stays closed, there’s no need to worry. As I mentioned in the earlier post, there are theoretical examples in general relativity where a singularity can be seen by a distant observer, but none of them seem likely to occur in a real situation.

The strong cosmic censorship conjecture takes a different approach. This is where the “dividing by zero” idea comes in. If you were to divide a number by zero, you might think the answer is infinity. After all, zero can go into a number like 1 an infinite number of times. But the actual answer is undefined. The formal reason has to do with the subtleties of mathematics, but for our purposes suppose we first divided by a very small number. For example, if we divide 1 by 0.01, the answer would be 100. If we divided 1 by 0.0001 we get 10,000. If we kept dividing by an ever smaller number, our answer would get bigger and bigger. This seems to say that 1/0 is infinity, but suppose instead we divided 1 by -0.01. In that case the answer would be -100. Dividing by -0.0001 we get -10,000. That would make 1/0 negative infinity. Starting with a small negative number or a small positive number gets us to the same 1/0, so is the answer positive or negative? The ratio 1/0 is meaningless without knowing how we approached zero.

A singularity is similar to this in that it is indeterministic. If all you have is a singularity, you have no idea how it became a singularity. Likewise if a singularity were to interact with other objects in the universe, the outcome would be unpredictable. So the strong cosmic censorship conjecture proposes that general relativity must be deterministic. As a result, singularities must be excluded from interaction with the rest of the universe.

It turns out there are solutions to Einstein’s field equations that satisfy the weak conjecture but not the strong conjecture, and vice versa. By itself general relativity is not bound by either censorship conjecture. But general relativity is also perfectly fine with warp drive and time travel as well, and they don’t seem to be physically possible for reasons beyond relativity. So it’s likely that some physical process prevents these kinds of weird singularities from occurring.

Then again, we’ve been wrong before.

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A fast radio burst (FRB) is a short burst of intense radio energy originating from outside our galaxy. We aren’t sure what causes FRBs, though the likely candidate is a white dwarf or neutron star falling into a black hole. They only last a few milliseconds, which makes them a challenge to study, but their brief duration may also allow us to test the limits of general relativity.

If relativity is wrong, then different wavelengths from an FRB should arrive at different times. Credit: Purple Mountain Observatory, Chinese Academy of Sciences

The foundational idea of general relativity is known as the principle of equivalence. On a basic level it states that two objects of different masses should fall at the same rate under the influence of gravity. The principle is necessary to equate the apparent force of gravity with a curvature of spacetime. So far all tests of the equivalence principle have confirmed it to the limits of observation, but there’s an interesting catch. Since relativity also states that there is a connection between mass and energy, the equivalence principle should also hold for two objects of different energy. Specifically, two beams of light with different wavelengths (and therefore different energies) should be affected by gravity in the same way.

We know that the path of light is changed by the curvature of space (an effect known as gravitational lensing), but the curvature also affects the travel time of light from its source to us (known as the Shapiro time delay). According to relativity, the amount of curvature and the time delay shouldn’t depend upon the wavelength of light. This means we can in principle use FRBs to test this idea.

Since FRBs only last milliseconds, they provide a sharp pulse of light at a range of frequencies. If relativity is correct, then the pulse we observe won’t be affected by gravity. If the equivalence principle is wrong, then shorter wavelengths of radio waves from the burst could arrive at a different time than longer wavelengths. We already see different wavelengths arrive at different times due to the interaction between the radio waves and the interstellar plasma in our galaxy, but we know from other observations how much that shift should be. The key is to test whether there is an additional shift not accounted for by standard physics.

Relativity is an extremely well-tested scientific theory, so I wouldn’t count on FRBs showing an energy-based effect, but it’s great that we could have yet another way to test our model. It’s a win-win, since we’ll either confirm our theory yet again, or we’ll discover something new to explore.

Paper: Y. F. Huang & J. J. Geng. Collision between Neutron Stars and Asteroids as a Mechanism for Fast Radio Bursts. arXiv:1512.06519 [astro-ph.HE] arxiv.org/abs/1512.06519

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While general relativity is an extremely well tested theory, we’re always looking for more precise ways to test it further. Either relativity will pass the test yet again or we’ll find evidence of something beyond relativity. But tests of general relativity are hard to come by, particularly for the most subtle effect of the theory, gravitational waves.

The best tests we have of gravitational waves is through binary pulsars. The first indirect evidence came from the Hulse-Taylor system, which consists of a pulsar orbiting a neutron star. Most of the systems we can use to test GR contain a pulsar orbiting another object such as a neutron star or white dwarf. But a system known as J0737-3039 consists of two pulsars. One pulses at a rate of 23 milliseconds, while the other at about 2.8 seconds. Their orbital period is only about 2.4 hours, so they are quite close to each other.

The regular radio pulses of a pulsar allow us to determine their motion with great accuracy, and since we can measure the motion of both pulsars we can test relativity with greater precision. While Hulse-Taylor system confirms relativity to within 0.2% of the theory, J0737-3039 allows for confirmation to within 0.04%. So far, general relativity still wins.

Paper: Michael Kramer. Experimental tests of general relativity in binary systems. 28th Texas Symposium on Relativistic Astrophysics, (2015)

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One of the predictions of special relativity is that the speed of light in a vacuum is a universal constant. This prediction has held up so well that we now use the speed of light to define part of the metric system. The first verification of special relativity is typically seen as the Michelson-Morley experiment, which demonstrated there wasn’t a luminiferous aether. But this experiment was actually done before Einstein proposed relativity, and so it wasn’t technically a prediction. It took two other experiments to completely verify Einstein’s model.

The Michelson-Morley experiment focused on determining the speed of the Earth through the aether. It wasn’t designed as a test of special relativity, and so it only tested that the speed of light was the same with different orientations. No matter which way you orient your device, the travel time back and forth along your experiment is the same. That’s certainly a prediction of relativity, but the theory goes further to claim that light speed is the same even if you’re moving at different speeds.

It took two other experiments to fully pin down the veracity of relativity. One, known as the Ives-Stilwell experiment looked at the time dilation effects of the model. In order for the speed of light to be the same in every reference frame, the clock of an experiment moving relative to you must appear to tick more slowly than that of an experiment sitting next to you. This effect is known as time dilation, and is one of the stranger aspects of relativity.

The Ives-Stilwell experiment looks at the light emitted or absorbed by fast moving particles and compares them with the transverse Doppler effect. If an object speeds past you from left to right, when it is directly in front of you would you see any Doppler shift of its light? Since the relative motion along your line of sight at that moment is zero, you might think there would be no shift. But since the object is speeding past you, its time should be dilated. As a result there should be a Doppler shift. The experiment confirmed the Doppler shift just as relativity predicts.

But relativity also predicts that space and time are connected, so a time dilation must also create a change of apparent length (known as length contraction). In other words not only must the clock of a moving experiment appear slower, then length of the experiment must appear shorter. Ives-Stilwell confirmed the first part, but not the second. To do that took a different test known as the Kennedy-Thorndike experiment.

Schematic of the Kennedy-Thorndike experiment.

The Kennedy-Thorndike experiment is similar to the Michelson-Morley. A beam of light is split to travel along two different paths. The separate beams of light are then recombined to create an interference pattern. The main difference is that the path length of the two beams is radically different. Since (according to Michelson-Morley) the speed of light is independent of orientation, the travel time of each path is different. Since Ives-Stilwell verified time dilation, as the apparatus moves with Earth, the amount of time dilation along one path is different from the other. This would produce a shift in the resulting interference pattern unless the lengths of the two paths also contract as relativity predicts.

The Kennedy-Thorndike experiment found no apparent shift in the interference pattern. Combined with the results of Michelson-Morley and Ives-Stilwell, this confirms that the speed of light is constant, and time dilation and length contraction both occur in agreement with special relativity.

And that’s why relativity is the strangest theory we know is true.

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When you think of the strangest scientific theory ever, you probably think of quantum mechanics. With its particle-wave duality and collapsing wavefunctions, it can certainly be considered a strange theory. But I would argue that even more bizarre is the theory of relativity. In quantum theory objects behave strangely in space and time, but in relativity the very nature of space and time is questioned. Despite all of its consequences such as curved space, time dilation and interchange of mass and energy, we know relativity to be true. Not just “kind of” true, but provably true to a high degree of precision.

Michelson and Morley’s 1887 interferometer.
Credit: Case Western Reserve Archive

The foundational experiment for both special and general relativity is the verification that the speed of light in a vacuum is the same for all frames of reference. This was first demonstrated by the famous Michelson-Morley experiment in 1887. The idea that the speed of light is constant is deeply counterintuitive. It would be as if you are riding a skateboard at 5 mph, and you toss a ball at 15 mph, and someone standing on the sidewalk sees the ball move not at 20 mph, but 15 mph just as you do. Michelson and Morley weren’t trying to prove light speed is constant, they were actually trying to determine how the speed of light changes as the Earth orbits the Sun. Instead of observing a changing speed of light, they found no variation to the limit of their experiment, which was about 1 part in 40,000. Because of this, the Michelson-Morley experiment is sometimes called the most successful failed experiment ever.

Since then, the result has been confirmed numerous times. With the advent of lasers, the experiment could be performed to extremely high accuracies. The results were so precise that in 1983 the speed of light became the standard by which the meter is defined. The most recent tests of the constancy of light speed find that it varies by no more than a few parts per quintillion.

Since the speed of light is constant for all frames of reference, there should be a difference between the way time is observed in different reference frames. We can see how this works if we imagine a clock made with light. Take two mirrors and place one above the other and facing each other, then bounce a pulse of light between them. We can measure time by counting the number of times the light bounces off a mirror. Each bounce is like the tick or tock of a mechanical clock. If you could watch the pulse of light, you would see it move up and down between the mirrors at the speed of light. Up and down at a constant rate. Now suppose you took your clock on a fast moving train. Standing in the aisle of the train, you would see the light pulse move up and down at the same rate as before. Up and down at the speed of light.

Credit: John D. Norton

But an observer watching the train pass by would see something slightly different. They would also see the pulse move at the speed of light, but from their perspective the light can’t move straight up and down because it must also be moving along with you. They would see the pulse move diagonally up then diagonally down, which is a slightly longer distance between each bounce. That means it would take the light longer to travel from bounce to bounce. So from their viewpoint the ticks and tocks of your clock are slower than the ticks and tocks as you see them. Your clock appears to be running slow because of your motion relative to them.

This effect is known as time dilation, and it can be demonstrated by what is known as the Ives–Stilwell experiment. This experiment looks at the light emitted or absorbed by fast moving particles and compares them with the transverse Doppler effect. If an object speeds past you from left to right, when it is directly in front of you would you see any Doppler shift of its light? Since the relative motion along your line of sight at that moment is zero, you might think there would be no shift. But since the object is speeding past you, its time should be dilated. As a result there should be a Doppler shift. The Ives-Stilwell experiment is thus a direct test of time dilation. It was first performed in 1938, and has been improved upon over the years. Just this week, results on a new version of the experiment has confirmed time dilation to 2 parts in a billion.

Other experiments have looked at time dilation by comparing the time of two clocks in motion relative to each other. These experiments assume light obeys special relativity, so they are model dependent rather than the direct Ives-Stilwell experiments. However when comparing the relative time dilation of two atomic clocks, we find that they agree with the predictions of relativity to within one part in 10¹⁶. Special relativity makes our observations of space and time relative, but it is astoundingly accurate.

Of course relativity can be broadened to include the effects of gravity. This general theory of relativity begins with the observation that any two objects will fall in a gravitational field in the same way. It was first noted by Galileo who (according to legend) dropped lead balls of different size from the tower of Pisa. Galileo actually tested the idea by rolling balls down an inclined plane, and could find no observable difference.

Robert DuBois of Missouri Science & Tech, with an Eotvos apparatus

In 1908 Loránd Eötvös made the first precision test of the equivalence principle using a torsion balance, and found that it was true to within one part in a billion. Over the last century this result has been improved to the point that we now know the equivalence principle is true to within 1 part in 10¹⁴.
Using the equivalence principle, Einstein developed a gravitational theory even more strange than special relativity. In his theory the fabric of space and time can be bent and twisted by the presence and motion of masses. One of the first tests of the gravitational curvature of space was the deflection of starlight during a solar eclipse, first observed by Eddington in 1919. Eddington’s results supported Einstein’s model, but not very strongly. Given the radical approach of general relativity, Eddington’s results were initially disputed by some. But subsequent observations confirmed Einstein’s predictions.

A distant galaxy (blue) is gravitationally lensed by a closer galaxy (red). Credit: ESA/Hubble & NASA

The most recent tests of light bending agree with general relativity to within 0.3%. That doesn’t seem very strong, but it’s very difficult to measure accurately. At Earth’s distance, the light of a star seen at a direction 90 degrees away from the Sun is deflected by 4 milliarcseconds. That’s roughly equivalent to the width of a human hair seen from 3 kilometers away.

Our astronomical instruments have gotten so precise that even this small deflection must be taken into account. Satellites such as Hipparcos (and soon Gaia) make very precise observations of the positions of millions of stars. As the satellites orbit the Sun, measured stars are sometimes aligned more closely or more distant from the Sun. As a result, the gravitational curvature of the Sun can shift the observed position of the stars by a measurable amount. In fact, measurements of Gaia will be so precise that it could provide a more stringent confirmation of light bending.

Perhaps the weirdest prediction of relativity is that rotating masses twist space around them. This effect is known as frame dragging, and it is most dramatic around black holes. But even the Earth’s rotation twists space ever so slightly. In 2011 a spacecraft known as Gravity Probe B successfully observed this effect due to the Earth.

Gravity Probe B tests general relativity. Credit: Gravity Probe B Team, Stanford, NASA

The frame dragging effect of Earth is so small that it’s astounding we can perceive it at all. To observe these effects, the team had to create quartz spheres so precise that their surface varied no more than 40 atoms from a mathematically perfect sphere. They were then covered with a thin layer of niobium so they could be suspended within an electric field. Their rotation created a small magnetic field, which was measured by superconducting quantum interference devices. Of course all of this is packed into a probe and shot into space for an 18 month mission. Over the duration of the experiment, the rotation of the spheres had to be measured with milliarcsecond precision. Despite the challenges, Gravity Probe B confirmed the Earth’s gravitational curvature of space to within 1% of predictions, and confirmed frame dragging to within 19%.

Decay of a binary pulsar’s orbital period over time.

Despite all these successes, there is one confirmation of relativity which still eludes us: direct observation of gravitational waves. According to general relativity, when large masses such as neutron stars or black holes orbit each other, they should create ripples in spacetime. These ripples should radiate outward as gravitational waves. Because these waves carry energy away from the system, the orbits of these massive objects should decay. We’ve seen this kind of orbital decay in binary pulsars, so we’re pretty certain these gravitational waves exist. But to truly be certain we’d like to detect them directly.

In principle we should be able to observe their effect as an oscillation in the separation distance of two mirrors. But even if the mirrors were separated by 4 kilometers, the variation in their distance would only be about 10^-18 meters, or about a thousand times smaller than the width of a proton. We haven’t confirmed gravitational waves yet, but with projects such as LIGO, and the proposed space-based LISA, we will likely succeed.

Of course, these experiments are only possible using lasers, and the knowledge of the invariant speed of light. Perhaps it is fitting that we use the first triumph of relativity to strive for its last great confirmation.

Note: This post originally appeared at Starts With A Bang!

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One of the consequences of general relativity is that light can be deflected by nearby masses. Mass curves space, and this curvature causes light to bend slightly. It was first observed during a total eclipse in 1919. The effect is extremely small unless the light passes close to a large mass, so gravitational lensing (as it is typically known) is usually only noticed with objects such as lensed galaxies, or specific tests of general relativity. But even though the effect is small as you get further from a mass, it isn’t zero. As our astronomical measurements become more precise, the effects of gravity are starting to become something we can’t ignore.

The deflection of light by the Sun’s gravity isn’t a binary thing. It is largest when light grazes the surface of the Sun, but gets gradually smaller with distance. Even with starlight well clear of the Sun, there is a small deflection. At Earth’s distance, the light of a star seen at a direction 90 degrees away from the Sun is deflected by 4 milliarcseconds. That’s an extraordinarily tiny shift, and usually one you can ignore. But when we launched the Hipparcos satellite in 1989, it was capable of measuring the positions of stars to about 3 milliarcseconds. Over its lifetime it measured about 3.5 million relative positions of stars. Since they were taken at different angular separations from the Sun, these measurements have varying gravitational deflections. The effect can be seen in the data, and can even be used to confirm Einstein’s prediction to within about 0.3%.

When the Gaia spacecraft comes online, the effect will be even more clear. Gaia will measure the positions of about a billion stars, and has an accuracy of about 24 microarcseconds. At that point the affects of gravity aren’t just a minor statistical fluctuation, but are in fact a major effect that must be corrected. One upside of this is that the Gaia data can be used to test light bending in general relativity to a greater precision than we have so far.

No one expects that Gaia will fail to confirm general relativity, but it does demonstrate just how precise modern astronomy has gotten. An effect that could barely be detected under the best conditions a century ago has now become a major factor in the apparent wandering of stars in our data.

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It’s a well known law of physics that the speed of light (in a vacuum) is always the same, regardless of your frame of reference (essentially your vantage point). But this isn’t entirely true. It actually depends on how you define “speed”.

There are different ways to define speed, which is easy to see if you think about a car. One way to determine the speed of a car would be to look at the speedometer. If it says you are travelling 50 miles per hour (80 kilometers per hour), then that is the car’s speed at that moment in time. We sometimes call this the instantaneous speed. Another way to determine the car’s speed is to time how long it takes the car to travel a certain distance. So if we observe the car travel 50 miles in one hour, then we would say its speed if 50 mph. This is known as an average speed.

So what does this have to do with the speed of light? After all, if the speed of light is constant, doesn’t that mean the instantaneous speed and average speed will always be the same? They would be if light always travelled in a straight line (in flat space), but we know that space is actually curved. Gravity is the result of the curvature of space and time around a mass, so near a large mass like the Sun space is not flat. This means the average speed isn’t necessarily the same as the instantaneous speed.

We can see this with the car example again. Suppose two towns were 100 miles apart, and connected by a perfectly straight road. Then suppose we travel in a car with cruise control set at exactly 50 mph. The instantaneous velocity of the car is constant, so we would expect the travel time between the cities to be exactly 2 hours. But halfway on our journey, there is a bit of construction, so we must take a slight detour. Our speed remains 50 mph, but because of the detour our path is not perfectly straight. So instead of 2 hours, it takes us a bit longer, say 2 hours and 4 minutes. Thus the average speed is 48.4 mph.

We see this same effect with light. The mass of the Sun warps space near it, therefore light passing near the Sun has a slight detour. That means light from a planet on the other side of the solar system from earth reaches us a tiny bit later than we would otherwise expect. The first measurement of this time delay was in the late 1960s by Irwin Shapiro. Radio signals were bounced off Venus from Earth when the two planets were almost on opposite sides of the sun. The measured delay of the signals’ round trip was about 200 microseconds, just as predicted by general relativity. This effect is now known as the Shapiro time delay, and it means the average speed of light (as determined by the travel time) is slightly slower than the (always constant) instantaneous speed.

This effect has been used to determine the mass of certain neutron stars. In 2010, as described in a paper from Nature, a binary system consisting of a neutron star and a white dwarf was observed. The neutron star is a pulsar, meaning that it sweeps out a beam of intense energy as it rotates (kind of like a lighthouse). We see this as repeating radio pulses.

The two stars orbit each other in such a way that the white dwarf almost passes in front of the pulsar each orbit. This means the radio pulses pass close to the white dwarf before reaching us, and experiences a Shapiro time delay. Since the amount of the delay depends on the amount of spatial warping, which depends on the mass of the white dwarf, the team could use the delay to determine the mass of the white dwarf. Knowing that, they could determine the mass of the neutron star.

What they found was that this particular neutron star (PSR J1614-2230) is the most massive neutron star currently known, with a mass of about twice the Sun. It is so massive that we aren’t entirely sure how it exists.

In our everyday lives, a flight delay can be a bit of an annoyance. For light, a flight delay is a useful astronomical tool.

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If you toss a ball into the air, it will slow down as it rises.  The Earth’s gravity pulls on the ball as it moves upward, causing it to slow down until it comes to a momentary stop at its highest point. Then it will begin to move downward, speeding up as it does.  Suppose, then, that you were to shine a flashlight upward. What would happen? You might argue that gravity would pull on the photons, causing them to slow down, but we know that light has a constant speed, and can’t slow down. You might argue that since photons are massless gravity doesn’t affect them, but we know that the Earth’s mass, like any other mass, can cause light to change directions. So neither of these can be the answer. The real answer is pretty interesting, and it turns out to be one of the tests of Einstein’s theory of relativity.

To understand what happens to our beam of light, we need to look at gravity in a slightly different way. Normally we think of gravity as a force, but it can also be described in terms of energy. If you take a mass and drop it, it speeds up as it falls, thus gaining kinetic energy.  When you toss a ball in the air, it slows down, thus losing kinetic energy. But energy is conserved, so energy gained or lost by the ball has to come from or go somewhere. This energy is typically called gravitational potential energy, since gravity has the potential to cause the ball to move. So rather than thinking of gravity as a force, we can see it as having a gravitational potential that can give or take away energy from the ball.

This brings us to the case of light. When we shine the flashlight upward, Newtonian gravity would say that the light is unaffected, since light is massless, but under general relativity light is affected by gravity, so as the light travels upward it must lose energy. But how is that possible if it can’t slow down? It turns out that the energy of light doesn’t depend upon its speed, but upon its wavelength. Red light with long wavelengths has less energy than blue light with short wavelengths. So as the light travels upward, its wavelength is stretched, and the light becomes more red. This is known as gravitational redshifting. For those of you wondering what this has to do with curved space, in general relativity the radial distance close to the Earth is shorter than the radial distance farther away.  As the light moves upward its wavelength is stretched by the curvature of space due to gravity, which again means it is redshifted.

This effect is typically quite small, and it wasn’t confirmed until 1959, when Robert Pound and Glen A. Rebka performed an experiment with radioactive iron (Fe 57). This isotope emits gamma rays at a very specific wavelength. To detect these gamma rays they used a detector made of the same element. Since the detector can only catch the gamma rays at the same wavelength, any redshift of the gamma rays would make detection unlikely. Pound and Rebka showed that if you oscillate the detector at a particular frequency it would detect the emitted gamma rays. This is because the oscillation of the detector meant it saw the gamma rays redshifted due to relative motion. In this way they were able to determine how much the gamma rays were redshifted (or blueshifted) by its upward or downward motion. Their results agreed with Einstein’s predictions.

The Pound-Rebka experiment for gravitational redshift is now considered one of the three principal tests of general relativity. The other two are the precession of Mercury’s orbit and the deflection of light by gravity. As strange as general relativity may seem, it really is how the universe works.

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More general relativity today. This time a bit on how to calculate the perihelion advance of Mercury in general relativity. When you derive the central force equation for relativistic gravity you find there is an extra term not seen in Newton’s gravity. The extra term is small, but enough to make Mercury’s orbit (any orbit really, but we typically use Mercury as an example) deviate slightly from an ellipse. Since the deviation is small, you can make some broad approximations, get an approximate solution for Mercury’s orbit, then determine the perihelion advance for one orbit.

That’s all fine and good if all you want is an approximate solution, but what if you really want to grind through things and find an exact answer. It turns out you can’t get a general analytic solution for the GR case, but you can solve it computationally. Do this in Mathematica and you can get an interpolating function. Okay, so get your interpolating function and figure the shift for one orbit. Problem solved, right?

Well, not so fast. Just doing it for one orbit is not very accurate, so its better to calculate the solution over several hundred orbits. But then you run into a subtlety of relativistic orbits. It turns out the perihelion advance is only part of what is going on. There are actually two types of non-Newtonian behaviors. One is a secular deviation (the perihelion advance), and the other is a periodic deviation.

You can see both of them in the first figure above. I’ve plotted the deviation between Newton’s orbit and Einstein’s, and what you can see is there is a kind of periodic motion with increasing amplitude. If we just take an average of this difference, everything washes out. What we want to find is how the amplitude increases over time. So we have to do a fit to the average amplitude increase. The result is seen in the second figure, which gives the steady advance of the perihelion.

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A while back I wrote about how general relativity predicts gravitational waves. While we haven’t yet observed gravity waves directly, we know they exist. That’s because gravitational waves carry energy away from their source, just as light waves carry light energy.

When two stars orbit each other, they produce gravitational waves. The gravity waves in turn take away some of the energy from the binary system. This means that the two stars gradually move closer together, an effect known as inspiralling. As the two stars inspiral, their orbital period gets shorter (because their orbits are getting smaller). Eventually they become so close that they merge, usually creating a supernova or black hole.

For regular binary stars this effect is so small that we can’t observe it. However in 1974 two astronomers (Hulse and Taylor) discovered an interesting pulsar. You may remember that pulsars are rapidly rotating neutron stars that happen to radiate radio pulses in our direction. The pulse rate of pulsars are typically very, very regular. Hulse and Taylor noticed that this particular pulsar’s rate would speed up slightly then slow down slightly at a regular rate. They showed that this variation was due to the motion of the pulsar as it orbited a star. They were able to determine the orbital motion of the pulsar very precisely, calculating its orbital period to within a fraction of a second. As they observed their pulsar over the years, they noticed its orbital period was gradually getting shorter. The pulsar was inspiralling, and would eventually merge with its companion star in about 300 million years.

In the figure above, I’ve plotted the measured orbital periods of pulsar with the theoretical period shortening due to gravity waves. As you can see, the data lines up almost exactly. The system is losing energy just as general relativity predicts. Hulse and Taylor had demonstrated that gravity waves exist. Which is why they were awarded the Nobel prize in physics in 1993.

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One aspect of general relativity that always amazes me is the level of precision needed to distinguish it from Newtonian gravity. Take, for example, the advance of Mercury’s perihelion. When you count in the gravitational tugs from the sun and all the planets, Newton predicts Mercury’s perihelion will advance about 531.65 arcseconds per century. When we measure the orbit of Mercury, we find its perihelion actually advances 574.10 arcseconds per century. This means Newton’s prediction is off by about 42.45 arcseconds per century. I say “about” because there is an uncertainty in our observations of about 0.65. General relativity predicts an “extra” perihelion advance of 42.98, which agrees exactly with experimental observation.

The difference between Newton’s model and Einstein’s amounts to 28 millionths of a degree each orbital revolution. Put another way, Mercury makes one orbit every 87.969 days, but it reaches its perihelion about a half second later than Newton says it should. The difference between Newton and Einstein is less than a human heartbeat in time.

The most amazing thing about all this? This deviation from Newton was first accurately measured by Urbain Le Verrier in 1859.

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