General Relativity

Theory of General Relativity

• Einstein presented his theory of general relativity in 1915 and it is based on the idea that space and time are linked. We call this the fabric of spacetime. From the name fabric, we infer that spacetime can be bent in any direction. Thus, space can be curved.
• There are a few basic consequences of general relativity (GR):
  • Gravity is a distortion of spacetime - It is not a force that we learned.
  • Time runs slower in a gravitational field.
  • The universe has no boundaries and therefore has no center.
  • Masses emit gravitational waves.
• One of the most important aspects of GR is the EQUIVALENCE PRINCIPLE. Einstein calls this the happiest thought of his life.
• The equivalence principle states "The effects of gravity are exactly equivalent to the effects of acceleration".
• This implies that we cannot tell the difference if we were in a gravitational field or accelerating. The laws of physics, and hence our perception of the world, is the same whether we are sitting here on Earth or accelerating in a rocket through space.
• This realization alone had profound impacts and is the basis for GR.
  
  
• Spacetime is a four dimensional surface. A point has zero dimensions, a line has one, a plane has two, and a cube has three. Adding time into a cube provides the fourth dimension.
  
  
• The curvature of spacetime is difficult to imagine in four dimensions, so we use the surface of the Earth as a representation. If you were looking at a map and wanted to fly from NYC to Seatle, you would draw a straight line on the map. However, the shortest possible route is what we call a great circle, and this would carry you through Canada.
• In other words, the shortest distance between two points is not a straight line. This is because space (the surface of Earth) is curved.
• This leads to the concept of the curvature of spacetime and let's image three different curvatures or geometries.
  • a FLAT geometry is what we are used to: here the circumference of a circle is 2 pi r.
  • a SPHERICAL geometry is much like using the surface of Earth, and here the circumference is smaller than in a flat geometry.
  • a SADDLE-SHAPPED geometry has a circumference that is larger than that in a flat.
  
  
  
  
  
• The shape of spacetime requires a new definition of a "straight line". The way to define a straight line is that trajectory that is followed where object experience weightlessness. Otherwise, if you experience weight, then you must be accelerating (according to the equivalence principle) and, hence, you must be changing your direction through spacetime.
• Since the space station feels weightlessness, then it is traveling in the straightest possible path through spacetime. Similarly the Earth is going straight as it orbits around the Sun. We standing here on the Earth are, however, accelerating.
  
  
• This realization of the curvature of spacetime solve the problem of having an object that is far away exert a force on you (the force of gravity). Instead of a force, gravity is a distortion of spacetime. The rubber sheet analogy helps to demonstrate the bending of space in two dimensions.
• The two ways to change spacetime are to 1) increase the mass and 2) decrease the area in which you keep the mass (i.e., increase the density, or compress the object). Both of these cause the shape of spacetime to become more exaggerated.
• The extreme limit of curved spacetime is a black hole. It has curved spacetime so much, that it has closed in on itself - thus, nothing can escape from it. The point of no return for a black hole is called the EVENT HORIZON.
• Gravitational Time Dilation tells us that time runs slower the closer you are to a gravitational field. Thus, people who live a low elevations on Earth should live longer than people at higher elevations. Thus, for an object falling into a black hole, from the outside we would see their clock get slower and slower as it approached the event horizon and eventually stop.

Observational Evidence for GR

• There are multiple observations now that support GR, but there were two historic observations that convinced everyone. These are the precession of Mercury, and the bending of light by the Sun or GRAVITATIONAL LENSING.
• It had been known for a very long time that the orbit of Mercury actually precesses. It was thought for a long time that Mercury's orbit was off because of an undiscovered planet (some people called this planet Vulcan). It was a big mystery until Einstein explained it, and he calls this discovery the high point of his scientific life.
• Mercury is on a elliptical orbit so sometimes it is closer to the Sun than other times. In straight Newtonian physics, this would not matter and we should be able to measure the orbital time accuractely. But according to GR, the closer to get to a massive object, in this case the Sun, then both space and time are altered. Thus, Mercury was going in and out of different spacetime geometries.
• In 1915, Einstein used GR is predict exactly the proper orbit of Mercury and explained the discrepency.
• Einstein also predicted that light should be bent as it travels next to a massive object. If spacetime is curved, then any particle, including massless ones like photons, will have to obey the shape of space. This was unheard of before because we had assumed that gravity only acts on particles with mass.
• So in 1919, Einstein predicted that if we observe stars close to the Sun during a solar eclipse, those stars should appear in a different place since the light is bent as it travels through the curved spacetime near the Sun. Einstein's calculation was exact, and this observation gave him instant fame.
• This effect is called gravitational lensing (the massive object acts as a lens) and we have now seen many effects from it. The images below show two astronomical examples of it. The first is an Einstein Cross and the second is an Einstein Ring.
  
  
  
  
• Since spacetime can be curved, then sudden changes in a mass can cause ripples through this spacetime. The effect is a gravitational wave. Two massive objects that orbit each other are similar to an eggbeater in spacetime. They stir up the shape of spacetime and send out a ripple. There are a few programs now that are trying to detect these ripples.
  
  
• Due to the curvature of spacetime, there are ways in which we can time travel and travel distant parts in the universe in a short amount of time. Using the analogy of Earth, if we want to travel of NYC to Hong Kong, the fastest way to go would be to travel through the Earth, if we could.
• Similarly in the universe, if we could find a shortcut, we would be able to travel to distant places without going in the straight line (and therefore not violated the fact that nothing can travel faster than the speed of light). We call these shortcuts wormholes.
• Wormholes may actually exists, but we do not have the physics or technology to explore the question yet. However, there is a philosophical problem with there existence since they would allow time travel. If we could travel in time, then we could back in time and as soon as that happens, all sorts of paradoxes arise that you have seen in many movies and science fiction novels. Thus, most scientists end up concluding that it cannot happen.