General relativity

Relativity is known as the special theory of relativity, which is compatible with three of the four known forces of nature: electromagnetism, the strong nuclear force, and the weak nuclear force. Gravity, however, can't be shoehorned into the special theory. In order to make gravity work, Einstein had to generalize relativity. The resulting theory is known as the general theory of relativity.

Our universe isn't Euclidean

Euclid proved thousands of years ago that the angles in a triangle add up to 180°. But what does it really mean to “prove” this? Euclid proved it based on certain assumptions (his five postulates), listed in the margin of this page. But how do we know that the postulates are true?

Only by observation can we tell whether any of Euclid's statements are correct characterizations of how space actually behaves in our universe. If we draw a triangle on paper with a ruler and measure its angles with a protractor, we will quickly verify to pretty good precision that the sum is close to 180°. But of course we already knew that space was at least approximately Euclidean. If there had been any gross error in Euclidean geometry, it would have been detected in Euclid's own lifetime. The correspondence principle tells us that if there is going to be any deviation from Euclidean geometry, it must be small under ordinary conditions.

To improve the precision of the experiment, we need to make sure that our ruler is very straight. One way to check would be to sight along it by eye, which amounts to comparing its straightness to that of a ray of light. For that matter, we might as well throw the physical ruler in the trash and construct our triangle out of three laser beams. To avoid effects from the air we should do the experiment in outer space. Doing it in space also has the advantage of allowing us to make the triangle very large; as shown in figure a, the discrepancy from 180° is expected to be proportional to the area of the triangle.

But we already know that light rays are bent by gravity. We expect it based on \(E=mc^2\), which tells us that the energy of a light ray is equivalent to a certain amount of mass, and furthermore it has been verified experimentally by the deflection of starlight by the sun. We therefore know that our universe is noneuclidean, and we gain the further insight that the level of deviation from Euclidean behavior depends on gravity.

Since the noneuclidean effects are bigger when the system being studied is larger, we expect them to be especially important in the study of cosmology, where the distance scales are very large.

One could protest that effects like these are just an imperfection of the light rays as physical models of straight lines. Maybe the noneuclidean effects would go away if we used something better and straighter than a light ray. But we don't know of anything straighter than a light ray. Furthermore, we observe that all measuring devices, not just optical ones, report the same noneuclidean behavior.

Curvature

An example of such a non-optical measurement is the Gravity Probe B satellite, figure d, which was launched into a polar orbit in 2004 and operated until 2010. The probe carried four gyroscopes made of quartz, which were the most perfect spheres ever manufactured, varying from sphericity by no more than about 40 atoms. Each gyroscope floated weightlessly in a vacuum, so that its rotation was perfectly steady. After 5000 orbits, the gyroscopes had reoriented themselves by about 2×10-3° relative to the distant stars. This effect cannot be explained by Newtonian physics, since no torques acted on them. It was, however, exactly as predicted by Einstein's theory of general relativity. It becomes easier to see why such an effect would be expected due to the noneuclidean nature of space if we characterize euclidean geometry as the geometry of a flat plane as opposed to a curved one. On a curved surface like a sphere, figure c, Euclid's fifth postulate fails, and it's not hard to see that we can get triangles for which the sum of the angles is not 180°. By transporting a gyroscope all the way around the edges of such a triangle and back to its starting point, we change its orientation.

The triangle in figure c has angles that add up to more than 180°. This type of curvature is referred to as positive. It is also possible to have negative curvature, as in figure e.

In general relativity, curvature isn't just something caused by gravity. Gravity is curvature, and the curvature involves both space and time, as may become clearer once you get to figure k. Thus the distinction between special and general relativity is that general relativity handles curved spacetime, while special relativity is restricted to the case where spacetime is flat.

Curvature doesn't require higher dimensions

Although we often visualize curvature by imagining embedding a two-dimensional surface in a three-dimensional space, that's just an aid in visualization. There is no evidence for any additional dimensions, nor is it necessary to hypothesize them in order to let spacetime be curved as described in general relativity.

Put yourself in the shoes of a two-dimensional being living in a two-dimensional space. Euclid's postulates all refer to constructions that can be performed using a compass and an unmarked straightedge. If this being can physically verify them all as descriptions of the space she inhabits, then she knows that her space is Euclidean, and that propositions such as the Pythagorean theorem are physically valid in her universe. But the diagram in f/1 illustrating the proof of the Pythagorean theorem in Euclid's Elements (proposition I.47) is equally valid if the page is rolled onto a cylinder, 2, or formed into a wavy corrugated shape, 3. These types of curvature, which can be achieved without tearing or crumpling the surface, are not real to her. They are simply side-effects of visualizing her two-dimensional universe as if it were embedded in a hypothetical third dimension --- which doesn't exist in any sense that is empirically verifiable to her. Of the curved surfaces in figure f, only the sphere, 4, has curvature that she can measure; the diagram can't be plastered onto the sphere without folding or cutting and pasting.

So the observation of curvature doesn't imply the existence of extra dimensions, nor does embedding a space in a higher-dimensional one so that it looks curvy always mean that there will be any curvature detectable from within the lower-dimensional space.

The equivalence principle

Universality of free-fall

Although light rays and gyroscopes seem to agree that space is curved in a gravitational field, it's always conceivable that we could find something else that would disagree. For example, suppose that there is a new and improved ray called the StraightRay\(^T\)EXTu0002TM. The StraightRay is like a light ray, but when we construct a triangle out of StraightRays, we always get the Euclidean result for the sum of the angles. We would then have to throw away general relativity's whole idea of describing gravity in terms of curvature. One good way of making a StraightRay would be if we had a supply of some kind of exotic matter --- call it FloatyStuff\(^T\)EXTu0002TM --- that had the ordinary amount of inertia, but was completely unaffected by gravity. We could then shoot a stream of FloatyStuff particles out of a nozzle at nearly the speed of light and make a StraightRay.

Normally when we release a material object in a gravitational field, it experiences a force mg, and then by Newton's second law its acceleration is a=F/m=mg/m=g. The m's cancel, which is the reason that everything falls with the same acceleration (in the absence of other forces such as air resistance). The universality of this behavior is what allows us to interpret the gravity geometrically in general relativity. For example, the Gravity Probe B gyroscopes were made out of quartz, but if they had been made out of something else, it wouldn't have mattered. But if we had access to some FloatyStuff, the geometrical picture of gravity would fail, because the “m” that described its susceptibility to gravity would be a different “m” than the one describing its inertia.

The question of the existence or nonexistence of such forms of matter turns out to be related to the question of what kinds of motion are relative. Let's say that alien gangsters land in a flying saucer, kidnap you out of your back yard, konk you on the head, and take you away. When you regain consciousness, you're locked up in a sealed cabin in their spaceship. You pull your keychain out of your pocket and release it, and you observe that it accelerates toward the floor with an acceleration that seems quite a bit slower than what you're used to on earth, perhaps a third of a gee. There are two possible explanations for this. One is that the aliens have taken you to some other planet, maybe Mars, where the strength of gravity is a third of what we have on earth. The other is that your keychain didn't really accelerate at all: you're still inside the flying saucer, which is accelerating at a third of a gee, so that it was really the deck that accelerated up and hit the keys.

There is absolutely no way to tell which of these two scenarios is actually the case --- unless you happen to have a chunk of FloatyStuff in your other pocket. If you release the FloatyStuff and it hovers above the deck, then you're on another planet and experiencing genuine gravity; your keychain responded to the gravity, but the FloatyStuff didn't. But if you release the FloatyStuff and see it hit the deck, then the flying saucer is accelerating through outer space.

The nonexistence of FloatyStuff in our universe is called the equivalence principle. If the equivalence principle holds, then an acceleration (such as the acceleration of the flying saucer) is always equivalent to a gravitational field, and no observation can ever tell the difference without reference to something external. (And suppose you did have some external reference point --- how would you know whether it was accelerating?) If the equivalence principle fails, then general relativity's geometrical interpretation of gravity as a form of curvature fails, and we need to find a new theory of gravity.

Gravitational Doppler shifts and time dilation

An interesting application of the equivalence principle is the explanation of gravitational time dilation. Experiments show that a clock at the top of a mountain runs faster than one down at its foot.

To calculate this effect, we make use of the fact that the gravitational field in the area around the mountain is equivalent to an acceleration. Suppose we're in an elevator accelerating upward with acceleration a, and we shoot a ray of light from the floor up toward the ceiling, at height h. The time Δ t it takes the light ray to get to the ceiling is about h/c, and by the time the light ray reaches the ceiling, the elevator has sped up by v=aΔt=ah/c, so we'll see a red-shift in the ray's frequency. Since v is small compared to c; we can just approximate the Doppler shift factor as 1-v/c≈ 1-\(ah/c^2\). By the equivalence principle, we should expect that if a ray of light starts out low down and then rises up through a gravitational field g, its frequency will be Doppler shifted by a factor of 1-\(gh/c^2\). This effect was observed in a famous experiment carried out by Pound and Rebka in 1959. Gamma-rays were emitted at the bottom of a 22.5-meter tower at Harvard and detected at the top with the Doppler shift predicted by general relativity.

In the mountain-valley experiment, the frequency of the clock in the valley therefore appears to be running too slowly by a factor of 1-\(gh/c^2\) when it is compared via radio with the clock at the top of the mountain. We conclude that time runs more slowly when one is lower down in a gravitational field, and the slow-down factor between two points is given by 1-\(gh/c^2\), where h is the difference in height.

We have built up a picture of light rays interacting with gravity. To confirm that this make sense, recall that we have already observed that light has momentum. The equivalence principle says that whatever has inertia must also participate in gravitational interactions. Therefore light waves must have weight, and must lose energy when they rise through a gravitational field.

Local flatness

The noneuclidean nature of spacetime produces effects that grow in proportion to the area of the region being considered. Interpreting such effects as evidence of curvature, we see that this connects naturally to the idea that curvature is undetectable from close up. For example, the curvature of the earth's surface is not normally noticeable to us in everyday life. Locally, the earth's surface is flat, and the same is true for spacetime.

Local flatness turns out to be another way of stating the equivalence principle. In a variation on the alien-abduction story, suppose that you regain consciousness aboard the flying saucer and find yourself weightless. If the equivalence principle holds, then you have no way of determining from local observations, inside the saucer, whether you are actually weightless in deep space, or simply free-falling in apparent weightlessness, like the astronauts aboard the International Space Station. That means that locally, we can always adopt a free-falling frame of reference in which there is no gravitational field at all. If there is no gravity, then special relativity is valid, and we can treat our local region of spacetime as being approximately flat.

In figure k, an apple falls out of a tree. Its path is a “straight” line in spacetime, in the same sense that the equator is a “straight” line on the earth's surface.

Inertial frames

In Newtonian mechanics, we have a distinction between inertial and noninertial frames of reference. An inertial frame according to Newton is one that has a constant velocity vector relative to the stars. But what if the stars themselves are accelerating due to a gravitational force from the rest of the galaxy? We could then take the galaxy's center of mass as defining an inertial frame, but what if something else is acting on the galaxy?

If we had some FloatyStuff, we could resolve the whole question. FloatyStuff isn't affected by gravity, so if we release a sample of it in mid-air, it will continue on a trajectory that defines a perfect Newtonian inertial frame. (We'd better have it on a tether, because otherwise the earth's rotation will carry the earth out from under it.) But if the equivalence principle holds, then Newton's definition of an inertial frame is fundamentally flawed.

There is a different definition of an inertial frame that works better in relativity. A Newtonian inertial frame was defined by an object that isn't subject to any forces, gravitational or otherwise. In general relativity, we instead define an inertial frame using an object that that isn't influenced by anything other than gravity. By this definition, a free-falling rock defines an inertial frame, but this book sitting on your desk does not.

Black holes

The observations described so far showed only small effects from curvature. To get a big effect, we should look at regions of space in which there are strong gravitational fields. The prime example is a black hole. The best studied examples are two objects in our own galaxy: Cygnus X-1, which is believed to be a black hole with about ten times the mass of our sun, and Sagittarius A*, an object near the center of our galaxy with about four million solar masses.

Although a black hole is a relativistic object, we can gain some insight into how it works by applying Newtonian physics. A spherical body of mass M has an escape velocity v=\(\sqrt{2GM/r}\), which is the minimum velocity that we would need to give to a projectile shot from a distance r so that it would never fall back down. If r is small enough, the escape velocity will be greater than c, so that even a ray of light can never escape.

We can now make an educated guess as to what this means without having to study all the mathematics of general relativity. In relativity, c isn't really the speed of light, it's really to be thought of as a restriction on how fast cause and effect can propagate through space. This suggests the correct interpretation, which is that for an object compact enough to be a black hole, there is no way for an event at a distance closer than r to have an effect on an event far away. There is an invisible, spherical boundary with radius r, called the event horizon, and the region within that boundary is cut off from the rest of the universe in terms of cause and effect. If you wanted to explore that region, you could drop into it while wearing a space-suit --- but it would be a one-way trip, because you could never get back out to report on what you had seen.

In the Newtonian description of a black hole, matter could be lifted out of a black hole, m. Would this be possible with a real-world black hole, which is relativistic rather than Newtonian? No, because the bucket is causally separated from the outside universe. No rope would be strong enough for this job.

One misleading aspect of the Newtonian analysis is that it encourages us to imagine that a light ray trying to escape from a black hole will slow down, stop, and then fall back in. This can't be right, because we know that any observer who sees a light ray flying by always measures its speed to be c. This was true in special relativity, and by the equivalence principle we can be assured that the same is true locally in general relativity. Figure n shows what would really happen.

Although the light rays in figure n don't speed up or slow down, they do experience gravitational Doppler shifts. If a light ray is emitted from just above the event horizon, then it will escape to an infinite distance, but it will suffer an extreme Doppler shift toward low frequencies. A distant observer also has the option of interpreting this as a gravitational time dilation that greatly lowers the frequency of the oscillating electric charges that produced the ray. If the point of emission is made closer and closer to the horizon, the frequency and energy measured by a distant observer approach zero, making the ray impossible to observe.

Information paradox

Black holes have some disturbing implications for the kind of universe that in the Age of the Enlightenment was imagined to have been set in motion initially and then left to run forever like clockwork.

Newton's laws have built into them the implicit assumption that omniscience is possible, at least in principle. For example, Newton's definition of an inertial frame of reference leads to an infinite regress. For Newton this isn't a problem, because in principle an omniscient observer can know the location of every mass in the universe. In this conception of the cosmos, there are no theoretical limits on human knowledge, only practical ones; if we could gather sufficiently precise data about the state of the universe at one time, and if we could carry out all the calculations to extrapolate into the future, then we could know everything that would ever happen.

But the existence of event horizons surrounding black holes makes it impossible for any observer to be omniscient; only an observer inside a particular horizon can see what's going on inside that horizon.

Furthermore, a black hole has at its center an infinitely dense point, called a singularity, containing all its mass, and this implies that information can be destroyed and made inaccessible to any observer at all. For example, suppose that astronaut Alice goes on a suicide mission to explore a black hole, free-falling in through the event horizon. She has a certain amount of time to collect data and satisfy her intellectual curiosity, but then she impacts the singularity and is compacted into a mathematical point. Now astronaut Betty decides that she will never be satisfied unless the secrets revealed to Alice are known to her as well --- and besides, she was Alice's best friend, and she wants to know whether Alice had any last words. Betty can jump through the horizon, but she can never know Alice's last words, nor can any other observer who jumps in after Alice does.

This destruction of information is known as the black hole information paradox, and it's referred to as a paradox because quantum physics has built into its DNA the requirement that information is never lost in this sense.

Formation

Around 1960, as black holes and their strange properties began to be better understood and more widely discussed, many physicists who found these issues distressing comforted themselves with the belief that black holes would never really form from realistic initial conditions, such as the collapse of a massive star. Their skepticism was not entirely unreasonable, since it is usually very hard in astronomy to hit a gravitating target, the reason being that conservation of angular momentum tends to make the projectile swing past. For example, if we wanted to drop a space probe into the sun, we would have to extremely precisely stop its sideways orbital motion so that it would drop almost exactly straight in. Once a star started to collapse, the theory went, and became relatively compact, it would be such a small target that further infalling material would be unlikely to hit it, and the process of collapse would halt. According to this point of view, theorists who had calculated the collapse of a star into a black hole had been oversimplifying by assuming a star that was initially perfectly spherical and non-rotating. Remove the unrealistically perfect symmetry of the initial conditions, and a black hole would never actually form.

But Roger Penrose proved in 1964 that this was wrong. In fact, once an object collapses to a certain density, the Penrose singularity theorem guarantees mathematically that it will collapse further until a singularity is formed, and this singularity is surrounded by an event horizon. Since the brightness of an object like Sagittarius A* is far too low to be explained unless it has an event horizon (the interstellar gas flowing into it would glow due to frictional heating), we can be certain that there really is a singularity at its core.

Cosmology

The Big Bang

Discovered by Hubble, the universe is expanding in the aftermath of the Big Bang: when we observe the light from distant galaxies, it is always Doppler-shifted toward the red end of the spectrum, indicating that no matter what direction we look in the sky, everything is rushing away from us. This seems to go against the modern attitude, originated by Copernicus, that we and our planet do not occupy a special place in the universe. Why is everything rushing away from our planet in particular? But general relativity shows that this anti-Copernican conclusion is wrong. General relativity describes space not as a rigidly defined background but as something that can curve and stretch, like a sheet of rubber. We imagine all the galaxies as existing on the surface of such a sheet, which then expands uniformly. The space between the galaxies (but not the galaxies themselves) grows at a steady rate, so that any observer, inhabiting any galaxy, will see every other galaxy as receding. There is therefore no privileged or special location in the universe.

We might think that there would be another kind of special place, which would be the one at which the Big Bang happened. Maybe someone has put a brass plaque there? But general relativity doesn't describe the Big Bang as an explosion that suddenly occurred in a preexisting background of time and space. According to general relativity, space itself came into existence at the Big Bang, and the hot, dense matter of the early universe was uniformly distributed everywhere. The Big Bang happened everywhere at once.

Observations show that the universe is very uniform on large scales, and for ease of calculation, the first physical models of the expanding universe were constructed with perfect uniformity. In these models, the Big Bang was a singularity. This singularity can't even be included as an event in spacetime, so that time itself only exists after the Big Bang. A Big Bang singularity also creates an even more acute version of the black hole information paradox. Whereas matter and information disappear into a black hole singularity, stuff pops out of a Big Bang singularity, and there is no physical principle that could predict what it would be.

As with black holes, there was considerable skepticism about whether the existence of an initial singularity in these models was an artifact of the unrealistically perfect uniformity assumed in the models. Perhaps in the real universe, extrapolation of all the paths of the galaxies backward in time would show them missing each other by millions of light-years. But in 1972 Stephen Hawking proved a variant on the Penrose singularity theorem that applied to Big Bang singularities. By the Hawking singularity theorem, the level of uniformity we see in the present-day universe is more than sufficient to prove that a Big Bang singularity must have existed.

The cosmic censorship hypothesis

It might not be too much of a philosophical jolt to imagine that information was spontaneously created in the Big Bang. Setting up the initial conditions of the entire universe is traditionally the prerogative of God, not the laws of physics. But there is nothing fundamental in general relativity that forbids the existence of other singularities that act like the Big Bang, being information producers rather than information consumers. As John Earman of the University of Pittsburgh puts it, anything could pop out of such a singularity, including green slime or your lost socks. This would eliminate any hope of finding a universal set of laws of physics that would be able to make a prediction given any initial situation.

That would be such a devastating defeat for the enterprise of physics that in 1969 Penrose proposed an alternative, humorously named the “cosmic censorship hypothesis,” which states that every singularity in our universe, other than the Big Bang, is hidden behind an event horizon. Therefore if green slime spontaneously pops out of one, there is limited impact on the predictive ability of physics, since the slime can never have any causal effect on the outside world. A singularity that is not modestly cloaked behind an event horizon is referred to as a naked singularity. Nobody has yet been able to prove the cosmic censorship hypothesis.

The advent of high-precision cosmology

We expect that if there is matter in the universe, it should have gravitational fields, and in the rubber-sheet analogy this should be represented as a curvature of the sheet. Instead of a flat sheet, we can have a spherical balloon, so that cosmological expansion is like inflating it with more and more air. It is also possible to have negative curvature, as in figure e. All three of these are valid, possible cosmologies according to relativity. The positive-curvature type happens if the average density of matter in the universe is above a certain critical level, the negative-curvature one if the density is below that value.

To find out which type of universe we inhabit, we could try to take a survey of the matter in the universe and determine its average density. Historically, it has been very difficult to do this, even to within an order of magnitude. Most of the matter in the universe probably doesn't emit light, making it difficult to detect. Astronomical distance scales are also very poorly calibrated against absolute units such as the SI.

Instead, we measure the universe's curvature, and infer the density of matter from that. It turns out that we can do this by observing the cosmic microwave background (CMB) radiation, which is the light left over from the brightly glowing early universe, which was dense and hot. As the universe has expanded, light waves that were in flight have expanded their wavelengths along with it. This afterglow of the big bang was originally visible light, but after billions of years of expansion it has shifted into the microwave radio part of the electromagnetic spectrum. The CMB is not perfectly uniform, and this turns out to give us a way to measure the universe's curvature. Since the CMB was emitted when the universe was only about 400,000 years old, any vibrations or disturbances in the hot hydrogen and helium gas that filled space in that era would only have had time to travel a certain distance, limited by the speed of sound. We therefore expect that no feature in the CMB should be bigger than a certain known size. In a universe with negative spatial curvature, the sum of the interior angles of a triangle is less than the Euclidean value of 180 degrees. Therefore if we observe a variation in the CMB over some angle, the distance between two points on the sky is actually greater than would have been inferred from Euclidean geometry. The opposite happens if the curvature is positive.

This observation was done by the 1989-1993 COBE probe, and its 2001-2009 successor, the Wilkinson Microwave Anisotropy Probe. The result is that the angular sizes are almost exactly equal to what they should be according to Euclidean geometry. We therefore infer that the universe is very close to having zero average spatial curvature on the cosmological scale, and this tells us that its average density must be within about 0.5% of the critical value. The years since COBE and WMAP mark the advent of an era in which cosmology has gone from being a field of estimates and rough guesses to a high-precision science.

If one is inclined to be skeptical about the seemingly precise answers to the mysteries of the cosmos, there are consistency checks that can be carried out. In the bad old days of low-precision cosmology, estimates of the age of the universe ranged from 10 billion to 20 billion years, and the low end was inconsistent with the age of the oldest star clusters. This was believed to be a problem either for observational cosmology or for the astrophysical models used to estimate the ages of the clusters: “You can't be older than your ma.” Current data have shown that the low estimates of the age were incorrect, so consistency is restored. (The best figure for the age of the universe is currently 13.8±0.1 billion years.)

Dark energy and dark matter

Not everything works out so smoothly, however. One surprise is that the universe's expansion is not currently slowing down, as had been expected due to the gravitational attraction of all the matter in it. Instead, it is currently speeding up. This is attributed to a variable in Einstein's equations, long assumed to be zero, which represents a universal gravitational repulsion of space itself, occurring even when there is no matter present. The current name for this is “dark energy,” although the impressive-sounding term is really just a label for our ignorance about what causes it.

Another surprise comes from attempts to model the formation of the elements during the era shortly after the Big Bang, before the formation of the first stars. The observed relative abundances of hydrogen, helium, and deuterium (\(^2\)H) cannot be reconciled with the density of low-velocity matter inferred from the observational data. If the inferred mass density were entirely due to normal matter (i.e., matter whose mass consisted mostly of protons and neutrons), then nuclear reactions in the dense early universe should have proceeded relatively efficiently, leading to a much higher ratio of helium to hydrogen, and a much lower abundance of deuterium. The conclusion is that most of the matter in the universe must be made of an unknown type of exotic matter, known generically as “dark matter.” We are in the ironic position of knowing that precisely 96% of the universe is something other than atoms, but knowing essentially nothing about what that something is. As of 2011, there are four experiments, all performed at the bottom of mineshafts to eliminate background radiation, that have accumulated data in a search for direct detection of dark matter. Of these, two claim to have detected dark matter, while two claim negative results that appear to contradict the two positive ones.