Changing Course: Introducing General Relativity (String Theory)

General relativity was Einstein’s theory of gravity, published in 1915, which extended special relativity to take into account non-inertial frames of reference — areas that are accelerating with respect to each other. General relativity takes the form of field equations, describing the curvature of space-time and the distribution of matter throughout space-time. The effects of matter and space-time on each other are what we perceive as gravity.

Gravity as acceleration

Einstein immediately realized that his theory of special relativity worked only when an object moved in a straight line at a constant speed. What about when one of the spaceships accelerated or traveled in a curve?
Einstein came to realize the principle that would prove crucial to developing his general theory of relativity. He called it the principle of equivalence, and it states that an accelerated system is completely physically equivalent to a system inside a gravitational field.
As Einstein later related the discovery, he was sitting in a chair thinking about the problem when he realized that if someone fell from the roof of a house, he wouldn’t feel his own weight. This suddenly gave him an understanding of the equivalence principle.
As with most of Einstein’s major insights, he introduced the idea as a thought experiment. If a group of scientists were in an accelerating spaceship and performed a series of experiments, they would get exactly the same results as if sitting still on a planet whose gravity provided that same acceleration, as shown in Figure 6-4.
Einstein’s brilliance was that after he realized an idea applied to reality, he applied it uniformly to every physics situation he could think of.
For example, if a beam of light entered an accelerating spaceship, then the beam would appear to curve slightly, as in the left picture of Figure 6-5. The beam is trying to go straight, but the ship is accelerating, so the path, as viewed inside the ship, would be a curve.
By the principle of equivalence, this meant that gravity should also bend light, as shown in the right picture of Figure 6-5. When Einstein first realized this in 1907, he had no way to calculate the effect, other than to predict that it would probably be very small. Ultimately, though, this exact effect would be the one used to give general relativity its strongest support.
(Left) Scientists performing experiments in an accelerating spaceship.
Figure 6-4:
(Left) Scientists performing experiments in an accelerating spaceship.
(Right) The scientists get the same results after landing on a planet.
Both acceleration and gravity bend a beam of light.
Figure 6-5:
Both acceleration and gravity bend a beam of light.


Gravity as geometry

The theory of the space-time continuum already existed, but under general relativity Einstein was able to describe gravity as the bending of space-time geometry. Einstein defined a set of field equations, which represented the way that gravity behaved in response to matter in space-time. These field equations could be used to represent the geometry of space-time that was at the heart of the theory of general relativity.
As Einstein developed his general theory of relativity, he had to refine Minkowski’s notion of the space-time continuum into a more precise mathematical framework (see the earlier “Building the space-time continuum” section for more on this concept). He also introduced another principle, the principle of covariance. This principle states that the laws of physics must take the same form in all coordinate systems.
In other words, all space-time coordinates are treated the same by the laws of physics — in the form of Einstein’s field equations. This is similar to the relativity principle, which states that the laws of physics are the same for all observers moving at constant speeds. In fact, after general relativity was developed, it was clear that the principles of special relativity were a special case.
Einstein’s basic principle was that no matter where you are — Toledo, Mount Everest, Jupiter, or the Andromeda galaxy — the same laws apply. This time, though, the laws were the field equations, and your motion could very definitely impact what solutions came out of the field equations.
Applying the principle of covariance meant that the space-time coordinates in a gravitational field had to work exactly the same way as the space-time coordinates on a spaceship that was accelerating. If you’re accelerating through empty space (where the space-time field is flat, as in the left picture of Figure 6-6), the geometry of space-time would appear to curve. This meant that if there’s an object with mass generating a gravitational field, it had to curve the space-time field as well (as shown in the right picture of Figure 6-6).
Without matter, space-time is flat (left), but it curves when matter is present
Figure 6-6:
Without matter, space-time is flat (left), but it curves when matter is present
(right).
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In other words, Einstein had succeeded in explaining the Newtonian mystery of where gravity came from! Gravity resulted from massive objects bending space-time geometry itself.
Because space-time curved, the objects moving through space would follow the “straightest” path along the curve, which explains the motion of the planets. They follow a curved path around the sun because the sun bends space-time around it.
Again, you can think of this by analogy. If you’re flying by plane on Earth, you follow a path that curves around the Earth. In fact, if you take a flat map and draw a straight line between the start and end points of a trip, that would not be the shortest path to follow. The shortest path is actually the one formed by a “great circle” that you’d get if you cut the Earth directly in half, with both points along the outside of the cut. Traveling from New York City to northern Australia involves flying up along southern Canada and Alaska — nowhere close to a straight line on the flat maps we’re used to.
Similarly, the planets in the solar system follow the shortest paths — those that require the least amount of energy — and that results in the motion we observe.

Testing general relativity

For most purposes, the theory of general relativity matched the predictions of Newton’s gravity, and it also incorporated special relativity — it was a rela-tivistic theory of gravity. But no matter how impressive a theory is, it still has to be confirmed by experiment before the physics community fully embraces it. Today, scientists have seen extensive evidence of general relativity.
One stunning modern example of applying relativity is the global positioning system (GPS). The GPS satellite system sends carefully synchronized beams around the planet. This is what allows military and commercial devices to know their location to within a few meters or better. But the entire system is based upon the synchronization of these satellites that had to be programmed with corrections to take into account the curvature of space-time near Earth. Without these corrections, minor timing errors would accumulate day after day, causing the system to completely break down.
Of course, such equipment wasn’t available to Einstein when he published his theory in 1915, so the theory had to gain support in other ways.
One solution that Einstein immediately arrived at was to explain an anomaly in the orbit of Mercury. For years, it had been known that Newtonian gravity wasn’t quite matching up with astronomers’ observations of Mercury’s path around the sun. By taking into account the effects of relativity’s curved
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space-time, Einstein’s solution precisely matched the path observed by astronomers.
Still, this wasn’t quite enough to win over all the critics, because another theory of gravity had its own appeal.

Pulled in another direction: Einstein’s competition for a theory of gravity

A couple of years before Einstein completed his theory of general relativity, the Finnish physicist Gunnar Nordstrom introduced his metric theory of gravity that also combined gravity with special relativity. He went further, taking James Clerk Maxwell’s electromagnetic theory and applying an extra space dimension, which meant that the electromagnetic force was also included in the theory. It was simpler and more comprehensive than Einstein’s general relativity, but ultimately wrong (in a way that most physicists then and today see as fairly obvious). But this was the first attempt to use an extra dimension in a unification theory, so it’s worth investigating a bit.
Einstein himself was supportive of Nordstrom’s work to incorporate special relativity with gravity. In a 1913 speech on the state of unifying the two, he said that only his work and that of Nordstrom met the necessary criteria. In 1914, though, Nordstrom introduced a mathematical trick that increased the stakes of unification. He took Maxwell’s electromagnetic equations and formulated them in four space dimensions, instead of the usual three that Einstein had used. The resulting equations included the equation describing the force of gravity!
Including the dimension of time, this made Nordstrom’s theory a 5-dimensional space-time theory of gravity. He treated our universe as a 4-dimensional projection of a 5-dimensional space-time. (This is kind of similar to how your shadow on a wall is a 2-dimensional projection of your 3-dimensional body.) By adding an extra dimension to an established physical theory, Nordstrom unified electromagnetics and gravity! This provides an early example of a principle from string theory — that the addition of extra dimensions can provide a mathematical means for unifying and simplifying physical laws.
When Einstein published his complete theory of general relativity in 1915, Nordstrom jumped ship on his own theory because Einstein could explain Mercury’s orbit while his own theory could not.
Nordstrom’s theory had a lot going for it, though, because it was much simpler than Einstein’s theory of gravity. In 1917, a year after Nordstrom himself had given up on it, some physicists considered his metric theory a valid alternative to general relativity. Nothing noteworthy came out of these scientists’ efforts, though, so clearly they had backed the wrong theory.
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The eclipse that confirmed Einstein’s life Work

One major difference between Einstein’s and Nordstrom’s theories was that they made different predictions about light’s behavior. Under Nordstrom’s theory, light always traveled in a straight line. According to general relativity, a beam of light would curve within a gravitational field.
In fact, as early as the late 1700s, physicists had predicted that light would curve under Newtonian gravity. Einstein’s equations showed that these earlier predictions were off by a factor of 2.
The deflection of light predicted by Einstein is due to the curvature of space-time around the sun. Because the sun is so massive that it causes space-time to curve, a beam of light that travels near the sun will travel along a curved path — the “shortest” path along the curved space-time, as shown in Figure 6-7.
Light from distant stars follows the shortest path along curved space-time, according to Einstein's theory of general relativity.
Figure 6-7:
Light from distant stars follows the shortest path along curved space-time, according to Einstein’s theory of general relativity.
In 1911, Einstein had done enough work on general relativity to predict how much the light should curve in this situation, which should be visible to astronomers during an eclipse.
Astronomers on an expedition to Russia in 1914 attempted to observe the deflection of light by the sun, but the team ran into one little snag: World War I. Arrested as prisoners of war and released a few weeks later, the astronomers missed the eclipse that would have tested Einstein’s theory of gravity.
This turned out to be great news for Einstein, because his 1911 calculations contained an error! Had the astronomers been able to view the eclipse in 1914, the negative results might have caused Einstein to give up his work on general relativity.
When he published his complete theory of general relativity in 1915, he’d corrected the problem, making a slightly modified prediction for how the light would be deflected. In 1919, another expedition set out, this time to the west African island of Principe. The expedition leader was British astronomer Arthur Eddington, a strong supporter of Einstein.
Despite hardships on the expedition, Eddington returned to England with the pictures he needed, and his calculations showed that the deflection of light precisely matched Einstein’s predictions. General relativity had made a prediction that matched observation.
Albert Einstein had successfully created a theory that explained the gravitational forces of the universe and had done so by applying a handful of basic principles. To the degree possible, the work had been confirmed, and most of the physics world agreed with it.
Almost overnight, Einstein’s name became world famous. In 1921, Einstein traveled through the United States to a media circus that probably wasn’t matched until the Beatlemania of the 1960s.

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