The equation looks simpler than it is (okay, just kidding). The big R represents the properties of space (‘Ricci curvature tensor’), the big T corresponds to the properties of matter (‘stress-energy tensor,’ containing mass and energy) and g is the ‘metric tensor’ (which provides the definition of distances and angles). Tensors are mathematical functions that can be represented as matrices (in the case of the field equation), i.e. rows and columns of variables or numbers. Lambda is the cosmological constant (more on that later), G the gravitational constant. The tensors represented in the field equation are practically symmetrical, so that there are not 4x4=16, but just ten independent equations.
What does this equivalence system reveal? It sums up the general theory of relativity wonderfully. Mass (in the stress-energy tensor) tells the space (in the curvature tensor) how to bend. At the same time, the space determines how the masses move over its curvature. The gravitational effect is only an apparent force. Objects only move along certain paths, purely based on the geometry of space. However, concrete statements can only be made if we solve the system of equations.
Before we go into that, first a few words about the peculiarities of space in Riemannian geometry. Curvature is not just curvature. First of all, we have to clear up a preconception about parallels. The fact that these never intersect, as we learned in school, only applies to flat, Euclidean geometry. In general, one can call two geodesics parallel if they are parallel at one point (measurable because their angles of intersection with another geodesic are equal at this point).
The Earth’s lines of longitude, for example, fulfill this condition because they all intersect the equator (also a geodesic) at right angles, so here they are parallel. Nevertheless, they also intersect with each other, namely at the North and South Poles. The Earth's surface is positively curved. The classic counterexample is a horse saddle (mathematical term: hyperboloid). Here the geodesics in the middle of the saddle are parallel. But then they move apart in both directions, their distance increases and they never meet. Such a curvature is called negative. Finally, the curvature may be zero as on a cylinder surface or in Euclidean geometry.
The type of curvature has interesting consequences for the sum of the interior angles of a triangle. On a flat surface, this is always 180 degrees. On a positive curvature, it’s more than 180 degrees, and on a negative curvature it’s less. The different types of curvature can therefore be recognized not only by the behavior of the geodesics, but also by the shape of a triangle.
Both a cylinder and a flat piece of paper have a curvature of zero. But one crucial factor differentiates them. An ant can follow a geodesic line and end up back at the starting point (it walks around the cylinder). But on a flat sheet of paper it would never come back to the starting point. That’s because a cylinder—unlike a flat piece of paper—also has an external curvature (we’ll call the other curvature ‘inner curvature’). The presence of external curvature is generally evident from the fact that the vertical lines on a geodesic are not parallel.
To grasp the nature of spacetime, we just have to accept one more fact, that the curvature isn’t constant (like it is on a sphere or a cylinder). The relationship between curvature, mass/energy, and four-dimensional location establishes the field equations.
When viewed from up close, spacetime appears flat to any observer. Or have you ever noticed when looking out the window that the lawn outside your house is curved? Therefore, Newton's equations apply in these dimensions as a very good approximation.
Evidence in outer space
So we need to look a bit further afield if we want to find evidence of the general theory of relativity. Back in the day, it was easy to analyze a hitherto unsatisfactorily explained phenomenon with the help of GTR. There is a unique feature in the movement of the planet Mercury around the sun, and astronomers have been aware of it for some time. The point on its orbit that is closest to the sun changes in a way that can’t be calculated from the gravitation of the sun and the other planets using Newton's laws. In 1919, scientists had new evidence fall into their laps. Observing a solar eclipse off the coast of Africa, Arthur Eddington was able to show that space curvature, caused by the mass of the sun, changes the position of the constellations in the immediate vicinity of the sun—so much so that the GTR presents itself as the only explanation.
However, the early evidence was not yet complete enough to rule out other theories. That is why Einstein received his Nobel Prize in 1921 not for the GTR, but for his explanation of the photoelectrical effect. The lack of tests was also because there were initially no alternatives available to differentiate the general theory of relativity using experimental measurements.
This didn’t change until the 1960s—for one example, with the Brans-Dicke theory. In this theory, additional factors that influenced space curvature were entered into the field equations. A level of certainty could be achieved using modern measurement methods for clarification.
The highest level of precision is achieved by measuring the gravitational lens effect with a radio telescope. The radio waves of distant galaxies are diverted by large masses on their way through space, in such a way that their position in space appears to shift for an Earth-bound observer.
Another consequence of the equivalence principle of the GTR (predicted by Einstein as early as 1911) is gravitational time dilation. It has nothing to do with the time dilation of the theory of special relativity. When an object moves away from a gravitational mass, its clock speeds up. When it moves toward the mass, time goes more slowly (which becomes very interesting when approaching a black hole). This means that, for photons, with their wave and particle properties, the frequency is decreased (fewer oscillations are measured per time unit) or the wavelength is increased. So, the photon reaches the observer with a wavelength shifted in the red direction. This has nothing to do with the Hubble redshift of light from a star (which indicates the expansion of the universe). It is very small, and was only able to be measured in the 1970s using atomic clocks.
Black holes as a solution
Today, the general theory of relativity is one of the best-tested theories in modern physics, alongside quantum physics. But what do their solutions say about the universe?
To answer that, you have to solve the field equations. So far, physicists have only been successful in special cases. If we take an area of space without matter (the outer space around a star, for example), we can set the stress-energy tensor to zero. We get the vacuum field equations (always remember that what looks like one equation is really ten):
In the case of a homogeneous, non-rotating, and uncharged sphere as the only field source (mass) in the center, this results in the Schwarzschild solution, which was calculated in 1916 by the astronomer Karl Schwarzschild. It consists of two components, an exterior and an interior solution. The exterior solution is sufficient, for example, to describe our solar system and to derive the peculiarities of Mercury’s orbit.
In special cases, this solution describes a black hole or a wormhole. This case occurs when the body lies completely within its Schwarzschild radius, calculated thus:
With the mass of the sun, this radius is 2,952 meters, and the Earth would have to shrink to a 9-millimeter radius. What happens next? Inside the object, a singularity emerges that is surrounded by an event horizon. Everything that disappears beyond this horizon is trapped forever in the black hole, because the space is curved in such a way that there is no escape. Space-like geodesics become time-like, and vice versa. It could be interpreted as turning a movement through space into a movement through time. And vice versa.
If a spaceship approached this limit, what happened next would depend on the observer. To an outside observer, the vehicle never reaches the event horizon. The closer it gets, the slower it moves. For the astronauts, on the other hand, a finite amount of time passes.
The singularity inside the black hole, where all physical concepts lose their meaning, could theoretically (mathematically) be linked t
o a ‘white hole.’ The behavior of a white hole is opposite to that of a black hole—it only lets matter out, not in. The link, then, would be a wormhole, also known as an Einstein-Rosen bridge.
Wormholes are popular in science fiction for enabling fast travel through space in spite of the limitations of the speed of light. Unfortunately, Wheeler and Fuller demonstrated in 1962 that wormholes are unstable in the general theory of relativity. Only forms of matter with negative energy density (which would also have an anti-gravitational effect) could prevent the collapse of a wormhole—but no such forms of matter are known or prognosticated in the Standard Model. The wormhole would also have to arise in a manner that prevents the formation of the singularity.
The cosmological constant
Einstein inserted the term ‘lambda’—Greek character Λ—into his field equations as the cosmological constant. He chose its value based on the idea that the universe would become static—corresponding to the prevailing assumption of the times. After Hubble proved the expansion of the universe, Einstein discarded his cosmological constant. It seems he acted prematurely, because today all measurements indicate that it must have a positive value, albeit a very small one.
Its interpretation depends on the model used. However, today it is no longer regarded as a selectable parameter of the theory, but as an expression of the energy density of the vacuum. The thinking behind this is that if, as assumed, the energy density of the vacuum is constant in time despite the expansion of the cosmos, negative pressure must be arising that accelerates the expansion. This would provide an explanation for the dark energy associated with the accelerated expansion that is actually being observed.
The expanding FLRW universe
The Friedmann-Lemaître-Robertson-Walker universe (also known in other combinations of the surnames of the scientists involved, and their abbreviations) is an approximate solution to the Einstein equations that does a good job of describing the standard cosmological model. Where it differs from reality is that the model presupposes the homogeneity of the universe. In fact, the portion of the cosmos we can observe is ‘lumpy.’
What the FLRW universe does well is to say from the outset that the universe has no option other than to expand. What’s more, its expansion must have been tiny at time=0—compliments of the Big Bang. The fate of the universe depends on its external curvature. If this is positive, a new collapse must follow at some point. If its curvature is non-existent or negative, however, the universe will continue to expand forever.
The extent of its curvature is determined by the mean density of matter in the universe. If this is higher than the critical density (10-29 grams per cubic centimeter), the universe must be positively curved and thus spatially and temporally finite. If its density is exactly the same as the critical density, that means we’re living in a flat, static universe, which is nevertheless unstable. If it’s lower, then the universe is infinite and will continue to expand endlessly. The actual density must be somewhere near the critical density, however. Otherwise the universe would either have already collapsed, or would have expanded so fast that no stars or galaxies would have formed. But whether the curvature is positive or negative has so far not been confirmed.
One thing is clear, though—the universe is continuously expanding, at an ever-increasing rate. For example, very distant supernovas are registered on Earth less brightly than could be expected given their distance. This would imply (unless another currently unknown effect is responsible) a positive cosmological constant that has a repulsive effect. The repulsive force increases with distance, which explains the expansion. The physical basis for this expansion is assumed to be the (otherwise mysterious) dark energy that accounts for about 60 percent of the energy of the universe.
Another problem emerges when you measure the rotation of distant galaxies. Scientists frequently observe that the mass of the galaxy is actually not enough to hold all the stars in their orbits. The assumed explanation for this is that a galaxy’s mass includes about one-quarter dark matter, which can’t be observed, but which is still attractive.
Gravitational waves
Although Newton still assumed that the effect of gravity spread out infinitely fast, it has been clear since the theory of relativity that the speed of light is the upper limit. Every system of moving bodies has to generate gravitational waves. Their existence is shown by the field equations of the GTR. But physicists have still not found a universal form for the distribution equations.
The only thing they’re certain about is that the gravitational waves are very weak. The orbit of the Earth around the sun, for example, causes the radiation of gravitational waves of about 300 Watts. So in this case the effect is in no way measurable. But scientists hope for clues from other locations—more intense waves must arise or have arisen from supernovas, orbiting neutron stars, or black holes and, of course, the Big Bang.
You don’t need to construct any exotic measuring instruments to prove their existence. Gravitational waves are temporal changes in the metric tensor. So it’s enough to carry out sufficiently precise length measurements, which should show periodic effects. However, there’s one problem with that—the measuring apparatus must be well-insulated from its environment. The changes in length, which are what is being measured, are roughly a ten-thousandth of the size of a proton. Such measurements have been running for several years in various countries. In many cases, gravitational waves have already been found (for example, at Virgo in Italy and LIGO in the United States.)
The nature of dark energy
Scientists only have vague ideas about the composition of dark energy. The simplest but also the least satisfying answer is to postulate a suitable value for the cosmological constant, and accept it as a property of the universe.
Quantum field theory (QFT) assumes the presence of an energy in empty space— vacuum energy. However, this results in a value for the cosmological constant that is too large by several dimensions. So far no one has been able to explain this difference.
Some scientists consider dark energy to be the effect of a scalar field, which they call quintessence (a scalar field gives a value to every point in space). The fluctuations of such a field mostly spread out at close to the speed of light. Basically, it behaves like a gas—that is, the fluctuations quickly flow from dense regions into less dense ones, resulting in an even distribution. The elementary particles that would belong to such a scalar field would be very, very light (about 10−82 electron masses) and would be unlikely to interact with normal matter except gravitationally.
Evidence of dark energy
Cosmic background radiation
The cosmic microwave background is a distant echo from the beginning of the universe. By analyzing it, we can deduce that the geometry of the universe is approximately flat. But this suggests that the mean mass density of the universe multiplied by its size would give the total mass. However, measurements of the microwave spectrum show that this is not the case. Something is missing. Matter accounts for only 30 percent of the total density.
Exploding stars
A type Ia supernova happens in a binary system in which a white dwarf absorbs mass from its companion until—having gorged itself—it explodes and shines brighter than all other stars in the universe for a short time. These explosions are visible from anywhere in the universe. Their brightness is a known quantity, which means we can determine their distances. Scientists have been able to demonstrate from this that the expansion of the universe is accelerating.
The shape of the universe
The shape of the cosmos can be derived from the general theory of relativity. We know that the universe is more or less flat. But there isn’t enough matter for this diagnosis. Considering E = m*c2, could dark energy provide the missing portion as the intrinsic energy of space?
The expansion of the universe
The rate of the expansion of the universe is determined by the time it takes for the universe to grow by a certain amount. Whatever causes t
his expansion is not diluted by it, even though it’s always creating new space. In order for the power of this energy to remain constant, more of it must be arising along with the new space. This suggests that dark energy is a property of space itself.
Dark energy and the death of the universe
Cosmologists don’t have it easy. We can find out how the universe began by looking out into space and having a little patience. Traces of what happened about 13.8 billion years ago are still evident. But how will the universe end? To answer this question, its development so far must be extrapolated into the distant future, from data that are at best only a snapshot.
Imagine that you’re allowed to watch a two-second video clip from a murder mystery, and from that you are expected to develop a theory about who the murderer is. Even if you know the rules of the genre, this is not an easy task.
Similarly, it should come as no surprise that there are many different possibilities up for discussion regarding the death of the cosmos. Starting with the most exotic (the descriptions are partly by the author and partly from existing literature):
•Big Trick: It’s possible that we’re involved in a colossal prank of nature. What we think of as space in its default state, the vacuum, may in fact be in a higher state of quantum physics. This cannot technically be ruled out. If that’s true, then one day nothing, absolutely nothing, could move the vacuum into a lower-energy state. This would probably be unfavorable for the present state of matter.
•Big Collapse: If our universe is part of a multiverse also containing universes made of antimatter, there could be a fatal collision.
The Death of the Universe: Ghost Kingdom: Hard Science Fiction (Big Rip Book 2) Page 28