# Albert Einstein’s General Theory of Relativity

Gravity is one of the most mysterious forces in the Universe. Many of the greatest minds have devoted their time to try to explain this strange and yet familiar entity, but have only produced more questions: What causes gravity? In what ways does it reveal itself? Can it be explained in physical terms? In the early 1900s, Albert Einstein took a stab at the problems inherent in gravity and the Universe in general. His models, most importantly the General Theory of Relativity, led to a better understanding of the world around us, and gave way to new and previously inconceivable physical phenomena.

In mid-1905, Einstein published his Special Theory of Relativity, which stated that observers see an event occur differently if they are in different reference frames or moving at different velocities. A result of this theory is that the speed of light must be constant, and that absolute space and time do not exist. This led to such notions as time dilation and length contraction, in which an object moving at a velocity near the speed of light has a noticeably slower clock and a noticeably contracted length in the direction of motion. The problem with this theory is that it assumed the Universe to be devoid of gravity. This did not mean that the theory was wrong, but that it had yet to incorporate an understanding of gravity.

Two years later, with the success of his paper throughout the scientific community, Einstein was asked to write a review of his special theory of relativity, indicating the consequences of these new physical laws, for the well-known Jahrbuch der RadioaktivitÃ?Â¤t und Elektronik. Forced to fill in gaps in his theory, he turned to the one that stood out the most: gravity. While pondering this problem, it occurred to him that a person who is freely falling does not feel his own weight. If, while falling, the person drops rocks from his hand, it would appear, in his small vicinity, that the rocks are floating. Being in a gravitational field is equivalent to being in an accelerated reference frame. Therefore, the localized reference frame of a freely falling object in our Universe is identical to that of an inertial reference frame in a Universe devoid of gravity. This rule, called the equivalence principle, pioneered Einstein’s work on gravity.

The equivalence principle allowed Einstein to solve a paradox that had arisen in his Special Theory of Relativity. Suppose that a person leaves the Earth in a spaceship going near the speed of light, while his twin stays on Earth to watch. Since special relativity states that relativistic effects are the same for all observers, an observer on Earth will see the astronaut’s clock moving slower, while the astronaut will also sees that his Earthly twin’s clock is moving slower. When the astronaut returns back to Earth, only a few years will have lapsed to him, while several decades have gone by on Earth. The astronaut can claim that he was at rest while the Earth moved away from him, and an observer from Earth can claim the astronaut rocketed off from Earth. Who is correct? Since the astronaut has to accelerate to achieve a larger velocity, and has to decelerate when turning around to return home, he feels weight, like a gravitational force. This acceleration is not felt on Earth. Therefore, the astronaut must deny that he has been at rest the whole time, and this paradox, often called the Twin Paradox, is resolved.

Einstein analyzed every aspect of the equivalence principle, including its consequences. One of the most intriguing is that light, which follows the shortest path between two points in spacetime, should bend around a massive object, such as the sun, since spacetime is curved around the object. This can be understood by looking at a person in an elevator. If a stationary laser is placed in the elevator shaft and shot through a stationary car, the light follows a straight-line path across the car. If the car is moving at a constant velocity with respect to the stationary laser, it appears, from inside the car, that the beam of light follows a straight, but slanted, path through the elevator. However, if the car is freely falling towards the ground, which is equivalent to being in an inertial reference frame, the beam of light curves toward the ceiling of the car as it reaches the other side. The actual bending of light in normal circumstances is unnoticeable, since the phenomenon can only be detected in the presence of VERY strong gravitational fields. In 1919, this effect was observed during a solar eclipse when light from a star appeared in a different place in the sky than expected, due to the large mass of the sun.

A strange phenomenon caused by the bending of light around massive objects is gravitational lensing. Imagine that a source of light is far away from the Earth. When viewed from the Earth, the source is faint, because fewer light rays reach us than if the source was nearby. What happens when a massive body, such as a star or galaxy, is placed in between the Earth and the light source? Light rays that are near the massive object will be pulled inward toward it, bent around it, and aimed at or near the Earth. The resulting image from Earth can be a myriad of things. If the rays are aimed at the Earth, the light source will appear brighter than it previously did, since more rays are hitting the Earth. This gravitational focusing can also cause strange images, such as a ring of light around the massive body, or even the shape of a cross, often called the Einstein Cross. Light is also lowered in frequency by the presence of gravitational fields. Since time runs more slowly near the sun, a second on the Earth is shorter than a second on the sun. Thus, the frequency of the light lowers as the light propagates from the sun. This effect is called gravitational redshifting, because the light is shifted toward the red side of the spectrum.

The problem that eventually led Einstein to the formulation of his General Theory of relativity is that of tidal gravitational forces. This occurs throughout the Universe, but is not normally noticed, due to the small size of the Earth. When a person stands on the surface of the Earth, the pull of gravity isn’t the same throughout his body. Gravity is stronger at his feet. He is also squeezed inward from every side, due to the fact that all of his parts are being pulled toward the same point, the center of the Earth. From his point of view, he feels forces pulling his head upward, his feet downward, and his sides inward. (Some may think at this point that he wouldn’t feel a force upward, because gravity only pulls downward. If he only felt a force downward, then he wouldn’t be stretched, only pulled. Since the strength of gravity is stronger at his feet than at his head, his feet will be pulled down more than his head, and he will be stretched. Since stretching occurs, it seems, only from his point of view, that there is a force pulling him upward. This force isn’t real. It is similar to a person driving in a car. When the car is stopped quickly, he is pulled forward. It feels as though there is a gravitational source at the front of the car pulling him toward it. There isn’t really a force there, though.) Even if the person was in free fall, this effect would still occur, but much more greatly. While on the Earth’s surface, there is a normal force caused by the ground pushing his feet upward. This counteracts the vertical stretching of the body. In free fall, however, the normal force is non-existent, and therefore the pull is more intense. This normally imperceptible effect becomes enormous as the mass of the gravitational body increases.

Einstein initially tried explaining tidal forces by using a model of the Universe in which space is flat but time is warped. Gravitational time dilation, which is the slowing of time that is experienced as a person approaches a massive object, was a deceiving concept; Einstein mistakenly thought that a mere warpage of time could explain away everything. The problem is that time is relative to the way two objects are moving with respect to each other. Therefore, if time is warped in all reference frames, space must also be the same. It turned out that space and time could not be considered as separate entities and still exist. They must be combined to form one cohesive entity. The three dimensions of space and one dimension of time form an absolute four-dimensional spacetime for the Universe. Any warping of time or curvature of space is due to a deformation of spacetime itself.

The fabric of spacetime can be thought of as a rubber sheet or trampoline. While nothing is on the trampoline, the surface is flat. Likewise, the absence of matter from the Universe presents a uniform, undistorted spacetime. If a heavy bowling ball is placed in the center, the rubber surface sinks down and curves around the ball. If marbles are rolled near the bowling ball, they follow a curved path around it, due to the warped surface of the trampoline. In the same way, a planet or a beam of light follows a bent path around the sun, due to the deformation of spacetime caused by its mass. The closer the marbles get to the bowling ball, the more their paths are bent. As a planet gets closer to the sun, it also follows a much more angled path. Space could no longer be considered uniform and flat.

Euclidean geometry, it seemed, could not hold true for our gravity-filled Universe. In the real world, parallel lines will intersect, the interior angles of a triangle do not add up to 180 degrees, and circles aren’t perfectly round. Einstein realized that a different, complex type of geometry must be used to explain the inner workings of the Universe. He conferred with his friend Marcel Grossmann, a professor of mathematics. Grossmann found that this type of geometry, which at the time was called tensor calculus, had been formulated and explained in the late 1800s and early 1900s by Bernhard Reimann, Gregorio Ricci, and Tullio Levi-Civita. Though initially hesitant, Einstein and Grossman studied differential geometry, as it is now called, and all of its details. They spent months attempting to explain exactly how matter causes the distortion of spacetime. Unfortunately, it did not seem as though the mathematics were providing Einstein with the picture of the Universe that he believed was correct. A universal principle is what he wanted, one that did not require any special reference frames, as was the case with his Special Theory of Relativity. The pair was forced to publish what they could on the subject, which still involved a Universe in which certain reference frames were necessary.
Einstein, with the help of others, went on to improve the generalized theory of relativity for application to all reference frames. One problem that evaded his theory was the perihelion shift of the planet Mercury. Every century, Mercury’s perihelion, where it is the closest to the sun, progressed closer by 43 arc-seconds more than what astronomers could explain. Scientists tried explaining this by saying that Mercury may have moons, or an undiscovered planet could be nearby. Einstein knew that such minute cosmological details needed to be explained by his theories, and had hoped that he could explain the perihelion shift. For several years he struggled with the problem, getting closer and closer to explaining the shift completely. Finally, on November 18, 1915, the latest form of his principle successfully accounted for this effect completely. Einstein was so thrilled that he could not do any work for three days. He presented the results, and his finalized general relativity theory, which applied to all reference frames, in his paper, The Field Equations of Gravitation, to the Prussian Academy seven days later.

Einstein’s theory of general relativity, though difficult to understand at the time, was a huge success throughout the scientific community. Immediately, minds went to work applying his theories to the real world. In 1916, Karl Schwarzschild solved Einstein’s field equations for massive, dense objects. As a celestial body becomes denser, or more massive, the force of gravity becomes much stronger close to it. It creates a much greater warpage of spacetime. If the body was massive enough, anything that approached it would be pulled in. These massive bodies, called black holes, have such strong gravitational attractions that not even light can escape. No event that occurs inside a black hole can have an affect on the outside world. The boundary at which nothing can overcome the inward pull is the event horizon. At this distance, the escape velocity is greater than that of light. Since the curvature of spacetime around a black hole is so great, the actual distance of the event horizon from the center cannot be calculated. Therefore, the distance that is used is defined as the radius of the event horizon if it were a flat, Euclidean space. Schwarzschild was able to calculate this radius, accordingly called the Schwarzschild radius, using Einstein’s field equations of gravity.

Though completely theoretical in the time of Einstein and Schwarzschild, black hole candidates have since been indirectly discovered, and their formation has been sufficiently explained. When a star that is at least three times more massive than the sun expends all of its fuel, there is no longer energy being created to counteract the star’s strong gravitation. The inner core of the star is pulled inward toward the center, while the outer layer is ejected in an enormous supernova. The core rapidly collapses, decreasing in volume but increasing in density, until it reaches a point, called a singularity, where the volume is zero and the density is infinite. The black hole itself is the volume around the singularity up to the event horizon. Once again, imagine a bowling ball on a trampoline. As the density of the bowling ball increases, while its volume proportionally decreases, it will sink lower and lower into the trampolines surface. At the same time, the amount of curvature of the surface increases near the center, but decreases more rapidly further away. Therefore, a marble placed near the ultra-dense bowling ball will most likely roll into the pit and never come out. At the event horizon of the black hole, time is infinitely dilated, and light that attempts to escape will use all of its energy in an infinitesimal amount of time, causing the redshift of light due to the loss of energy to be infinite.

The effect of a supernova, stellar implosion, or any other sudden change in the curvature of space also theoretically causes unusual ripples in space, called gravitational waves, according to general relativity. This effect is similar to a stone thrown into a pond. When the stone hits the surface of the water, ripples are created that extend outward from the point of contact. Gravitational waves work in much the same way, but are far weaker than light waves, and move at the speed of light. If two massive stars orbit each other rapidly, ripples of spatial curvature propagate from the center. These waves should slightly distort objects that they interact with as they travel through space. Although gravitational waves have yet to be directly detected, they can be inferred from observations made of rapidly orbiting, highly compressed binary systems. According to general relativity, these systems should be losing a lot of energy in the form of gravitational waves, and therefore their orbital period should be diminishing at a certain rate. The observed results match those predicted by general relativity.

Currently, there is much speculation about what might or might not be allowed by general relativity. Certain loopholes in the current perception of the laws of nature allow for the possibility of strange, but still theoretical, phenomena. One of the most debated is the existence of tunnels through four-dimensional space, called wormholes. Consider the surface of the Earth. If you wanted to get from Cleveland to Beijing, you would normally fly a plane over the Earth’s surface. If, however, a hole was created through the center of the Earth, connecting the two cities, and you flew through the hole, the flight time would be shorter. Applying this concept to the problem of getting from Earth to the nearest star, in three-dimensional space, a hole would need to be created into four-dimensional space for the journey to be shorter. Many scientists reject this idea however, pointing out that bypassing large distances in three-dimensional space by traveling a short distance through four-dimensional space would allow for ability of time travel into the past. If this were possible, then a person would be able to go back in time and prevent his parents from meeting. Then he would never be born, and therefore he could have never gone back in time to stop his parents from meeting. This indirectly shows that, if time travel into the past was possible, causality could be violated, and things could happen for no apparent reason. This is why most believe that general relativity cannot allow time travel.

The contributions of Einstein and others have helped explain many unsolved problems in the Universe, and have also given way to more problems and strange consequences. In any case, general relativity has definitely changed the way we look at the world around us. Space and time, it seems, are not what they seem.