Special Relativity
One of the consequences of Maxwell's equations of electro-magnetics is that the speed of a light wave is independent of the reference frame it is observed in. This is in direct violation of the Galilean Transformation rules. In the Galilean system if an observer in frame S', moving with velocity v, sends out a light signal, with velocity c, in the direction of travel then the velocity of the light wave as measured by a stationary observer in frame S, c', is c' = v + c. But that is not what Maxwell's equations say. They say that the velocity of light is c no matter what the frame observed in. So observers in S and S' measure c for the velocity of the same light wave. This is where the breakdown begins. This effect has now been measured in laboratories and has been confirmed the world over. The velocity of light is c (300,000,000 m/s) in every reference frame. This is one of the foundations of the theory of special relativity.
Another foundation of the theory of relativity is a postulate that states that the laws of physics are invariant in all inertial reference frames. This is the postulate that lead to the discovery of a mathematical transformation to replace the Galilean Transformation. The transformation is named the Lorentz Transformation after the mathematician Hendrick Lorentz who developed it. He developed the transformation 2 years before Einstein did but by different means than Einstein. Lorentz was searching for a transformation that would keep Maxwell's equations invariant. The transformation is
Einstein recognized the physical significance of this transformation and thus formulated the special theory. The transformations now show that time is not the immutable dimension we thought it was. Later I will discuss the time travel opportunities present in this theory. However, it is important to note that in the Lorentz transformation there seems to be a mixture of space and time coordinates. It was at this point that people began to realize that time and space were not the separate entities that they appeared to be. They are somehow bound together in a 4-dimensional spacetime continuum.
Before I move on to time in General Relativity it will be useful for me to digress into some mathematical formalism that will be useful in our later discussions of spacetime warps. In relativity it is useful to talk about metrics that describe the geometry of spacetime. Often we wish to find a metric that can describe the behavior of spacetime in a specific region. By metric, I mean an equation that is written by computing the distance element of a spacetime in terms of the coordinate system being used. For example, in a 2-dimensional flat space with Cartesian coordinates, x and y, the the distance, Delta s, can be found by forming a right triangle and applying the Pythagorean theorem
We often write this in as an infinitesimal distance element, ds. So in three dimensional space we have
Shortly after Einstein published his first paper on special relativity, Rudolph Minkowski recognized that the Lorentz transformation described a 4-dimensional spacetime with the metric equation
In this metric there are two different times. First is of course the coordinate time, the time measured by 2 stationary observers to the coordinate system. Second is the proper time, the time measured by a lone observer whom measures their own motion to be zero. The proper time is found by simply dividing the distance in the spacetime by a velocity, the speed of light.
The proper time and the coordinate time are related by the metric equation.
This relation will help us travel in time, but we shall come back to that shortly.