The classical view is easy to understand using common sense. For a long time, physicists tried to reconcile their observations with it. But this became difficult when scientists looked at the distribution of light. Light, it was assumed, consists of waves that spread out at a constant speed through a carrier medium, called ether. This medium was believed to be completely still, just like the space it filled. All celestial bodies moved in relation to it, even the Earth.
Light is different
That meant it should have been possible to determine the speed of the Earth relative to the ether using a simple experiment—sending two light beams in opposite directions to equally distant receivers. Due to the movement of the Earth through the ether, one receiver would have moved toward the light beam, while the other moved away from it. The result should have been that the light took a different amount of time to reach each receiver, because it had to cover a path of a different length.
In 1887, physicists Albert Michelson and Edward Morley reproduced this experiment with a slight modification. They used two beams of light. One moved in the direction of the Earth’s rotation, the other perpendicular to it. To achieve a measurable effect, the scientists ran the light back and forth numerous times between two mirrors.
Michelson and Morley measured whether there was a difference in speed, using the phenomenon of interference. If two wave crests arrive simultaneously, the deflection increases—if not, it’s weakened. Even if the two beams are only slightly different in length, this should be evident from the light intensity. And yet, to the surprise of the scientists, there was no difference!
Obviously, the speed of light is dependent on the choice of reference system. Many other physicists later repeated the experiment, all of them arriving at the same zero result, even with varying error margins. It should be noted that the concept of ether hadn’t been wholly dismissed at this point.
In fact, next followed the thesis that mass-affected bodies carried ether around with them. Another idea was that the object of the experiment moving with the Earth’s rotation underwent a shortening, so that the difference was no longer measurable. Other experiments showed that both of these supplementary theories had various weaknesses.
The same conclusion is reached—i.e., that all is relative—when you solve Maxwell’s equations of electrodynamics. This also leads to a constant, reference system-independent speed of light.
Based on this fact, which at the time was still unproven, Albert Einstein formulated his theory of special relativity (TSR) in 1905, in an essay entitled On the Electrodynamics of Moving Bodies. Einstein made some very daring predictions:
•Space and time depend on the state of motion. Moving objects appear shortened in the direction of movement (length contraction), and moving clocks run slower (time dilation).
•Mass and energy are equivalent.
•The speed of light in a vacuum is absolute and can’t be exceeded.
In his essay, Einstein also provided concrete formulas that made it possible for his claims to be evaluated. The famous formula E=mc2 for the energy content of an object was actually:
E2 = m2*c4 + p2*c2
That’s how it was first formulated by Max Planck in 1906. E is energy, m is mass, c is light speed, and p is momentum (the product of mass and speed). For a stationary object with v=0, (v is velocity), the well-known Einstein formula is obtained by extracting the square root on both sides.
Light particles (photons), on the other hand, have no resting mass (m). The last part of the formula: E=p*c applies to these. What is the significance of momentum here? It results from the fact that the energy of a light wave also has to be calculated using E=h*f, where h is Planck's constant of quantum physics, and f is the frequency of light.
Paul Dirac derived the concept of antimatter from Einstein's formula in 1928. You may know—or be able to recall—from your mathematics classes that a square root in an equation actually produces two solutions, so in addition to E=mc2 there is also E=-mc2. The first anti-particle predicted by Dirac, the anti-electron (positron), was discovered in cosmic radiation four years later.
Farewell to simultaneity
The constancy of the speed of light has consequences that are no friend to common sense. Let’s take another look at the situation on the train. Instead of walking to the restaurant car yourself, you point a laser at the end of the car, in the direction of travel. You use another laser to point toward the back of the train, where there is a wall at the same distance.
From your point of view, the light arrives at the same time in both cases. But that’s obvious, because for you the train system is self-contained. An observer from outside, then, should actually see the light spreading out at the speed of light plus the speed of the train—but this is not the case. The speed of light is dependent on the choice of reference system. First of all, this leads to the collapse of the concept of simultaneity. Imagine again that you’re sitting on the train, in the middle of a car. At the front and back, the ‘Toilet Occupied’ lights turn red, because two people have simultaneously gone inside and locked the cubicles.
While you’re sitting there comfortably, another passenger moves past you in the direction of one of the toilets. The man is not resting, like you. He’s moving at a particular speed. But because the speed of light must also be constant for him, he first sees the light he’s walking toward, and then sees the other light turn on. Because the speed of light is so fast, this effect is barely measurable, but it has been proven experimentally. A process that happened simultaneously for you lost its simultaneity for the moving observer, even though he was in precisely the same place as you.
Slower clocks, shorter spaceships
Depending on the system, the constancy of the speed of light also leads to the two interesting phenomena of time dilation and length contraction. According to these, time passes more slowly the faster you are moving, at least from the point of view of the stationary observer. An astronaut moving at almost the speed of light may only be traveling for a few months from his point of view, and yet when he returns to Earth, all his relatives may be dead. For example, on a round trip to a planet 28 light-years from Earth, under gravitational acceleration, just under 14 years would pass for the astronaut, while over 60 years would have passed on Earth.
Time dilation can be calculated with this formula:
T0’ = T / γ
where T0’ is the time on the moving clock, T is the time on the stationary clock, and γ is the Lorentz factor. This results from speed and light speed, as follows:
It’s clear that at low speeds γ approaches 1, so that the clocks run synchronously again.
The Lorentz factor can be easily explained using the example of the light clock (image below). This is an imaginary clock, which always changes by one unit of time when a transmitted light flash is mirrored back to it again. A time unit is thus 2*d/c seconds long (d = distance between mirror and transmitter). Now, if the light clock is moving, from the point of view of an outside observer, the light has a longer distance to travel. So the clock runs more slowly, because the speed of light is constant. But anyone traveling with the clock won’t notice this. The length of the additional path can be calculated using the Pythagorean theorem, and the result is the formula above.
Not only does the poor astronaut have to live with the time conversion, but length measurements are also not the same as they were on Earth. If he could objectively measure it, he would find his spaceship shrinking in the direction of flight, according to the formula:
L = L0 / γ
This effect is called length contraction. Once again, γ is the Lorentz factor (see above). In any case, the astronaut wouldn’t notice it because his onboard measuring tape would also shrink, by the same factor.
For a stationary observer, there is also another exciting effect. He wouldn’t observe a shrinking object, but rather its apparent rotation. This is because the sides of the object that are actually not in view (red in the image, left is the statio
nary object, right the moving object) become visible because the object is moving itself out of the way and therefore is no longer an obstacle. The faster the object moves, the stronger the impression that it’s rotating.
Even Einstein noted that time dilation seemed to result in a paradox. The abovementioned astronaut would notice on returning that his clock was considerably slow, compared to clocks on Earth. But why? Shouldn’t the systems of the observer on Earth and the astronaut be the same? That would mean that from the point of view of the astronaut, the clocks on Earth should also slow down (the ‘twin paradox’).
In fact this doesn’t happen. This is because, although time dilation is symmetrical, the astronaut has changed the direction of his movement. When he arrived on the distant planet, he could still claim that it was not his clock but our clocks that had changed speed. But with the return journey he changes the reference system, thereby destroying the symmetry himself.
Another interesting thought experiment is the tank paradox. A 10-meter-long tank (military vehicle) is moving toward a 10-meter-wide trench. If it’s moving fast enough, the trench will contract to a narrower width from the tank’s point of view. From the point of view of the trench, however, the tank shrinks, and the trench remains 10 meters wide.
Does the tank fall into the trench, or does it easily drive over it? The problem here is that, from the point of view of the theory of relativity, there are no rigid objects, since effects in the object propagate maximally with the speed of light. The front end of the tank—and from the tank’s point of view there is enough space for it—falls immediately into the ditch. But now it is inside its frame of reference and is small enough anyway. This happens step by step with the other parts of the tank. So the tank ends up in the trench, no matter what angle you observe it from.
Edmund Dewan and Michael Beran thought up an equally exciting experiment in 1959. We have two spaceships, and a thin rope that will break easily is stretched from the back of one ship to the front of the other. Both spaceships undergo identical acceleration. Does the rope break? From the point of view of the astronauts on both ships, the rope appears not to be moving, so its length doesn’t change. And because the spaceships are traveling equally fast, the distance between them remains constant. So the rope doesn’t break.
However, if you put yourself in the position of an outside observer, the rope is subject to length contraction. After all, it’s moving. But because the distance between the ships remains constant (identical acceleration), the rope must break. What is true here? The rope always breaks. From the crew's point of view, there can be no simultaneity. The captain of the spaceship in front accelerates earlier, from the point of view of the crew (not from that of the observer), than the one behind. This breaks the rope.
Numerous experiments have confirmed Time dilation. For example, it can be demonstrated that the short-lived muons in cosmic radiation reach Earth only because they age more slowly due to their high velocity. The twin paradox can be reproduced by comparing a stationary and a moving atomic clock.
All that’s lacking here is the mass-energy equivalence. It has very practical effects. For example, no spaceship can be accelerated to the speed of light. The problem is that, with increasing speed, the kinetic energy of the spaceship increases—and therefore so does its mass. The faster it goes, the greater the effort required for additional acceleration. The accelerating mass then becomes infinite when it approaches the speed of light. But you don’t need to climb into a spaceship to encounter mass-energy equivalence. For example, your vehicle weighs 300 milligrams more with a charged battery than with an empty battery. The conservation of mass in chemical reactions is, strictly speaking, only approximate because binding energy also plays a role.
By the way, Einstein’s famous formula had nothing to do with the atomic bomb. The energy released in nuclear fission is about one percent of the maximum.
The equivalence principle
There are two factors that the theory of special relativity doesn’t take into account; accelerated motion, and gravitation. Imagine the following situation. You are traveling through space as an astronaut. You forgot to strap yourself in when you went to sleep in zero gravity. In the middle of the night you fall painfully to the floor. What happened? Did the autopilot accelerate the spaceship? Or did a heavy object pass by your spaceship? Without looking out the window you can’t determine whether you fell to the floor due to acceleration (your inertial mass) or gravitation (your gravitational mass).
The inertial mass (as it appears in Newton’s laws of motion) and the gravitational mass (from Newton’s law of gravity) are obviously closely related, being results of the same phenomenon—this is called the equivalence principle. As it turns out—and Albert Einstein also recognized this—the two ‘omissions’ of the theory of special relativity result in the same problem. Gravity is evidently no ordinary force like, for example, the electromagnetic force. Rather, it’s a property of space itself.
Einstein then extended Newton's law of inertia with the general statement, “As long as there is no force acting on it, a body moves in spacetime along the shortest distance between two points.” In everyday life, at low speeds, this results in Newton's law. Because, when dealing with short distances, the shortest distance between two points is approximately a straight line.
The curvature of space
However, this doesn’t apply universally. You may have noticed this already during a long flight. When the screen on the back of the seat in front of you shows the flight path, you see a curve—even though the pilot would undoubtedly have chosen the most direct route. This is because the Earth is a sphere. The shortest distance from A to B would only be a straight line if the plane could fly through the interior of the Earth. But it’s restricted to flying over the surface. And the geometric shape of the Earth’s surface is spherical. Here, the shortest distance between two locations is always a great circle, that is, a circle whose middle point is the center of the Earth, and whose circumference runs above both locations. So the pilot flies in a curve, called a ‘geodesic line’ in the jargon of aviation.
Spacetime isn’t flat either, as was assumed for a long time. It possesses a geometry determined by the masses moving around in it. The way a mass can influence its geometry is demonstrated by your pet or significant other. Their mass leaves an unmistakable imprint in a soft cushion. An ant walking across the cushion would have to select a different route depending on whether the mass had just influenced its geometry or not. (However, this analogy is a bit limp because, in the case of cat or partner, the outline of the mass, not the quantity, determines the geometry.)
But what, exactly, does the geometry of the cosmos look like? I’ve already mentioned the term ‘spacetime,’ which consists of the four dimensions of space—length, height, and width, plus the fourth, time. Now, time is obviously unlike the other dimensions, in that an object can only move forward through it.
To illustrate this, we’ll compose a four-dimensional coordinates system, a so-called spacetime diagram. As this is impossible on a flat medium such as paper or a screen, we’ll flatten space into a plane (coordinates x and y) and take time as the direction z. Each point on this diagram is a ‘position’ in spacetime, which we’ll call an ‘event,’ due to the time component and the distinctness of a position in normal space.
If we now enter all the possible paths of light particles in the z direction, this results in a cone (future light cone). On it lie all events that a photon can reach from its origin (observer). If we mirror the cone into the past, we get all the events from which a proton can reach the observer (past light cone). The three-dimensional present lies at the level of zero, which can also be described as a hypersurface.
In the spacetime diagram, there are now three types of shortest distance, all associated with different speeds. Time-like geodesics are circles on the surface of the cone. Ordinary particles, which are not exposed to any force, move along them. Null geodesics run from their origin alon
g the cone’s surface as straight lines. They are reserved for particles that move at the speed of light—that is, light—and are therefore called light-like geodesics. Distances outside of the light cone would be space-like geodesics. These have a speed higher than c—which is why they are physically meaningless (assuming that particles faster than light, such as tachyons, don’t exist... which, it currently seems to be true).
Unfortunately, the light cone is still not sufficient to describe the geometry of the universe. We need a method for calculating distances, because the distances in spacetime correspond to the classical equations of motion. Such a method is called a metric. You will be familiar with metrics in everyday life. Trains use a metric to convert distance into ticket price. Despite the same basic structure (trains, tracks...), the metric is quite location-dependent. In India, people pay much less for a train journey than in Germany. The metric of the train is (presumably) a simple conversion factor. However, it only needs to describe one-dimensional space (the only measurement here is the distance). If you want to describe the curvature of a surface (such as a spherical or cylindrical surface), you need two pieces of data for each point on the surface, the curvature in the north-south direction and the curvature in the east-west direction.
Four-dimensional spacetime unfortunately makes this significantly more difficult. It requires ten independent pieces of data to describe the curvature at any given position. Another obstacle that emerges is that time is a directional dimension. Riemannian geometry presented itself to physicists as a solution that also allows for negative distances.
Einstein’s field equations
These are also the essential ingredients for the field equations of the general theory of relativity. Einstein was the first, in 1915, to publish them in the formula that is still considered correct to this day:
The Death of the Universe: Ghost Kingdom: Hard Science Fiction (Big Rip Book 2) Page 27
