The theory of relativity is traditionally broken into two parts, special and general relativity. Special relativity provides a framework for translating physical events and laws into forms appropriate for any frame of reference. General relativity addresses the problem of accelerated motion and gravity.
Special Theory of Relativity :
A key problem for Newtonian physics was the need for absolute space and time when referring to events or interactions. In particular, the problems of light propagation required an medium, an aether, for the light waves to exist within. The Michelson-Morley experiment showed that there was not absolute space and that inertial frames were relative only to themselves.
Einstein realized that there is a logical contradiction with respect to Newtonian physics and electromagnetism with respect to what a light ray ``looks like'' when the observer is moving at the speed of light. The solution is that only massless photons can move at the speed of light, and that matter must remain below the speed of light regardless of how much acceleration is applied.
The key point to special relativity is that the speed of light (c=299,790 km per sec) is constant in all frames of reference. What this means can be best demonstrated by the following scenario:
In Newtonian mechanics, quantities such as speed and distance may be transformed from one frame of reference to another, provided that the frames are in uniform motion (i.e. not accelerating).
Under special relativity, there is a natural upper limit to velocity, the speed of light. And the speed of light the same in all directions with respect to any frame.
There are two important consequences of the speed of light limit; Lorentz contraction and time dilation. Lorentz contraction states than an object moving near the speed of light appears shorter to an observer at rest.
Similarly, time is compressed in frames of reference that travel at velocities near the speed of light, as compared to rest frames. This effect is called time dilation.
Time dilation leads to the famous Twins ``Paradox'', which is not a paradox but rather a simple fact of special relativity. Since clocks run slower in frames of reference at high velocity, then one can imagine a scenario were twins age at different rates when separated at birth due to a trip to the stars.
The effects of relativity are dramatic, but only when speeds approach the speed of light. At normal, everyday speeds the changes to clocks and rulers are too small to be measured. However, near extreme objects, such as black holes and neutron stars relativity dominates over Newtonian physics.
Special relativity describes changes in size and time through the use of Lorentz transformations. For an event that lasts to seconds in your frame, the same event will appear to last t in a frame that is moving with velocity v such that:
where c is the speed of light.
Spacetime:
Special relativity demonstrated that there is a relationship between spatial coordinates and temporal coordinates. That we can no longer reference spatial points without some reference to when. Einstein introduced a new concept, that there is an inherent connection between geometry of the Universe and its temporal properties. The result is a four dimensional (three of space, one of time) continuum called spacetime which can best be demonstrated through the use of Minkowski diagrams and world lines.
Spacetime makes sense from special relativity since it was shown that spatial coordinates (Lorentz contraction) and temporal coordinates (time dilation) vary between frames of reference.
Einstein also discovered that there is a relationship between mass, gravity and spacetime. Mass distorts spacetime, causing it to curve.
Gravity can be described as motion caused in curved spacetime .
General Relativity :
The second part of relativity is general theory of relativity and lies on two empirical findings that he elevated to the status of basic postulates. The first postulate is the relativity principle: local physics is governed by the theory of special relativity. The second postulate is the equivalence principle: there is no way for an observer to distinguish locally between gravity and acceleration.
The primary result from general relativity is that gravitation is a purely geometric consequence of the properties of spacetime. In this sense, general relativity is a field theory, relating Newton's law of gravity to the field nature of spacetime.
There were two classical test of general relativity, the first was that light should be deflected by passing close to a massive body. The first opportunity occurred during a total eclipse of the Sun in 1919.
Measurements of stellar positions near the darkened solar limb proved Einstein was right. Direct confirmation of gravitational lensing was obtained by the Hubble Space Telescope last year.
The second test is that general relativity predicts a time dilation in a gravitational field, so that, relative to someone outside of the field, clocks (or atomic processes) go slowly. This was confirmed with atomic clocks flying airplanes in the mid-1970's.
The general theory of relativity is constructed so that its results are approximately the same as those of Newton's theories as long as the velocities of all bodies interacting with each other gravitationally are small compared with the speed of light--i.e., as long as the gravitational fields involved are weak. The latter requirement may be stated roughly in terms of the escape velocity. A gravitational field is considered strong if the escape velocity approaches the speed of light, weak if it is much smaller. All gravitational fields encountered in the solar system are weak in this sense.
A low speeds and weak gravitational fields, general and special relativity reduce to Newtonian physics.
Black Holes:
The fact that light is bent by a gravitational field brings up the following thought experiment. Imagine adding mass to a body. As the mass increases, so does the gravitational pull and objects require more energy to reach escape velocity. When the mass is sufficiently high enough that the velocity needed to escape is greater than the speed of light we say that a black hole has been created.
Another way of defining a black hole is that for a given mass, there is a radius where if all the mass is compress within this radius the curvature of spacetime becomes infinite and the object is surrounded by an event horizon. This radius called the Schwarzschild radius and varys with the mass of the object (large mass objects have large Schwarzschild radii, small mass objects have small Schwarzschild radii).
The Schwarzschild radis is easy to determine for an object of mass M. It is simply the radius where a test particle of mass m must move at the speed of light to exceed the gravitational energy of the primary object. So, we equate the kinetic energy and the gravitational potential energy such that:
which can be written as
where G = 6.668x10-11 and c = 3x108 meters per second and mass is in kilograms.
The visual image of a black hole is one of a dark spot in space with no radiation emitted. Any radiation falling on the black hole is not reflected but rather absorbed, and starlight from behind the black hole is lensed. So even though no radiation escapes a black hole, its mass can be detected by the deflection of starlight. In addition to mass, a black hole can have two other properties, electric charge and angular momentum.
Even though a black hole is invisible, it has properties and structure. The boundary surrounding the black hole at the Schwarzschild radius is called the event horizon, events below this limit are not observed. Since the forces of matter can not overcome the force of gravity, all the mass of a black hole compresses to infinity at the very center, called the singularity.
A black hole can come in any size. Stellar mass black holes are thought to form from supernova events, and have radii of 5 km. Galactic black hole in the cores of some galaxies are built up over time by cannibalizing stars. Mini black holes formed in the early Universe (due to tremendous pressures) down to masses of asteroids with radii the sizes of grains of sand.
Note that a black hole is the ultimate entropy sink since all information or objects that enter a black hole never returns. If an observer entered a black hole to look for the missing information, he/she would be unable to communicate their findings outside the event horizon.
Of course if the objects falling into the black hole form an accretion disk, then we can detected the x-rays from the infalling gas. This is our only method of indirectly finding black holes, as companions to other stars.
