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General Relativity

Brief Introduction

In 1907, while Einstein was sitting in the Patent Office, he suddenly thought to himself: "Falling objects don't feel gravity". He discussed this idea with a fellow scientist, Marcel Grossmann, and in 1916, he published a new way of looking at gravity and the universe.

Problems with Newton's Theory

Isaac Newton pictured gravity as a force acting at a distance. He knew the force of gravity was stronger with increasing mass and he used this to explain why the earth and all the other planets orbited the sun. He concluded that the force of gravity grew weaker by the inverse square law: (Force of Gravity) = 1 (Distance squared) The force of gravity seemed to differ from this equation in powerful gravitational fields and very fast objects. Also, people wondered what would happen if the sun suddenly disappeared. Newton guessed that the earth would instantly fly into space. But according to the Minkowski diagram (mentioned before) the earth would continue orbiting an imaginary sun until it went into the sun's FUTURE cone. Movement of the Perihelion of Mercury: Astronomers noticed that Mercury's perihelion (the closest point to the sun in its orbit) changed slightly with every orbit. Astronomers tried to solve this by predicting a mystery planet they called Vulcan. It was never found. This led more people to think that there was something wrong with Newton's theory.

Euclidean and Non-Euclidean Continuums

Let’s say a surface of a flat table is spread out in front of us. We can get from any point on the table by passing continuously from one point to a neighbouring one without executing jumps. This is a continuum.

Imagine I mark the table like this:

If we choose to measure the distance between any two adjacent points, that distance will be the same for every other pair of adjacent points. The distance between two diagonal points (45) is of the distance between two adjacent points. This holds for every other pair of diagonal points.

The Special Theory of Relativity’s space-time continuum and the continuums used in Minkowski’s diagrams are Euclidean ones. Einstein realised that even the Special Theory of Relativity was limited to inertial (unaccelerated) frames of reference. Einstein’s General Theory of Relativity, in order to include accelerated frames of reference and frames of references in gravitational fields, must use a Non-Euclidean continuum—one with distortions in it. We will look at non-Euclidean continuums in the next section.

The Famous Rubber Sheet*

Many beginner's and non-mathematician's guides to relativity often describe space as a huge rubber sheet with objects with mass distorting it. If the earth were an elephant sitting on a bed sheet on a bed (rubber sheet) the elephant would create a huge valley. Objects on the bed would "fall" toward the elephant, similar to how objects fall toward the earth.

Einstein modified Newton's theory by making the space-time sheet distort with the presence of mass. (Distorting space and time of other objects) Any object coming near the mass of the sheet curves its path toward it). Einstein predicted this would happen to everything--even light. As we get farther and farther away from the mass, gravity would be weaker. Without the presence of mass the space-time sheet is uniform and can be regarded as an Euclidean continuum like a flat grid. This theory seems to create the same effects of Newton's theory. There are a few differences, though. Before Einstein, a scientist thought an object with 500 times the sun's mass would trap in light. This statement would be true. But after Einstein a scientist would also say the object would collapse into infinite density but no volume (called a singularity) or in other words, it forms a black hole (the region of space-time around singularity where gravity is so intense that light can't escape).

Gravitational Waves

When there is a change in the shape of the rubber sheet, the change would cause very slight gravitational waves to form. It's like ripples in a pond of water--when there is a disturbance, ripples are formed. But the gravitational ripples caused by an event are very, very slight. Scientists still have not detected them. Supernovas (the explosion of stars) cause gravitational waves because of the sudden change of shape in the rubber sheet but the gravitational waves emitted are still very weak.

In a binary star system, two stars are orbiting each other. As the stars slowly emit light and other radiation, they lose mass (E=mc2). This gradual loss of mass will create very, very, very, very small gravitational waves. But after about 10,000,000,000,000,000,000,000 years, the stars will get closer and closer in their orbits until they collide.

Black Holes

In 1916, only a few months after the General Theory of Relativity was published, Karl Schwarzchild indirectly predicted black holes. He calculated the size a body had to be squeezed into in order to form a black hole. This radius is now known as the Schwarzchild radius.

Black holes are formed when a star of over 3 solar masses runs out of fuel. The energy emitted from the star can no longer keep gravity from squeezing it smaller. Gravity then takes over and makes the star decrease in volume. The star, meanwhile, blows off its outer layers in an explosion known as a supernova. The core of the star continues getting smaller and denser until its radius is smaller than the Schwarzchild radius. The star "stops" collapsing when it has zero volume but infinite density. This is called a singularity. The region extending from the collapsed star's centre to the end of the Schwarzchild radius is known as the black hole. The edge of the Schwarzchild radius is known as the event horizon. Once within the event horizon, nothing, not even light can escape. One of the first black holes to be suspected was Cygnus X-1, a blue giant star circling a black hole. They suspected a black hole because the star appeared to be circling something invisible. The black hole was pulling the matter on the surface of the star into the black hole. According to GR (General Relativity) black holes make such a deep well in space-time nothing can escape the black hole's gravity outside the event horizon.

Principle of Equivalence

Einstein predicted that the effects of the Special Theory of Relativity would be felt in large gravitational fields, like the ones found in black holes. After considering the "falling man" idea, he came to the conclusion that gravity and acceleration were equivalent. That is, if you were on a rocket, which was uniformly accelerated, you would feel a force pulling you toward the "floor" of the rocket. Also, if you were in freefall (falling in a vacuum without any air) you would feel weightless. You would feel the same effect if you were accelerating in space or just on the ground. He said that gravity and acceleration were equivalent.

If you were in a very powerful gravitational field, you would feel the same effects that you would feel if you were accelerating very quickly. Someone outside would notice that your length is shorter and time seems to be running slower for you.

Special Relativity is Insufficient! — The Twin Paradox

The Special Theory of Relativity was made so relativistic effects were the same for all observers. If a rocket were hurdling by at 99% of the speed of light relative to an observer, the astronauts in the rocket would also see the observer hurdling by at 99% of the speed of light. The observer would see the rocket’s length contract in the direction of its motion and its time run slower. Likewise, the astronauts aboard the rocket would also see the observer’s length contract and time dilate. There is a problem that arose in the Special Theory of Relativity.

Let’s say there are two twins, Robert and William. One of them, Robert, goes on a high-speed journey (very nearly the speed of light) while William stays on Earth. After a while, Robert returns. According to William, Robert is gone for 50 years. But according to Robert, he is gone for only 2 years. Robert could have claimed he was at rest and William was zooming away at high speed—relative to Robert, the earth would have been rapidly receding from him. If that was true, Robert should have also observed William’s length contracting and time dilating. But this is not true. The events are not as symmetrical as we think. If we include the Principle of Equivalence, Robert would feel a "gravitational force" as he accelerates away. William would not feel this. Thanks to the Principle of Equivalence in the General Theory of Relativity, the Twin Paradox is solved.

Powerful Gravitational Fields (including black holes)

At the event horizon of a black hole, the escape velocity (the speed needed to escape gravity's pull) is equal to the speed of light. Beneath the event horizon the escape velocity is greater than the speed of light. As a consequence of the special theory of relativity, nothing can escape. The region between the event horizon and singularity is called the black hole. According to the General Theory of Relativity, the path of a beam of light would appear to curve near gravitational fields. This was first proven in the solar eclipse of 29 May 1919. On May 29, 1919, astrophysicist Arthur Stanley Eddington (who was also quite interested in Einstein's work) and his team went to St. Helena to photograph the brilliant solar eclipse. After examining the photographs, they noticed that two stars that would normally have been hidden behind the sun actually appeared to them. Because gravity and acceleration are equivalent, this effect would also seem to appear when accelerating very fast.

When stars are accelerating away from us, the light emitted is shifted toward the red end of the spectrum, in an effect known as red-shift or the Doppler Effect. The Doppler Effect also works with sound. That's why a police car's siren, when rapidly approaching, sounds differently than when the siren is rapidly receding from us. Because of the of the Equivalence Principle, gravitational fields will also create this effect. The light from a neutron star (the remnants of a star not massive enough to form black holes) appears to be more "reddish" in colour. Some people call this effect the "Einstein" Shift, but it is identical to the Red Shift.

 

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