"Do not worry about your difficulties with mathematics, I assure you mine are still greater," - Albert Einstein
Einstein's theories of special and general relativity are well known today, at least by name.
These are the theories that predict length-contraction, time-dilation, space-time warping, and several other bizarre (by earthly standards) effects.
It is important to note, at this juncture, that the predictions of special and general relativity have been well verified by experiment, and are not in dispute by most respected physicists.
Herein shall be discussed a few aspects of special and general relativity as they apply to Starship design.
Special relativity is the simpler of the two theories. It deals only with constant velocity motion (no acceleration, no gravity).
To understand why it is that special relativity was necessary, one must examine the equations for electromagnetic (EM) wave propagation.
EM waves, or EM radiation, are what we perceive as light (this includes visible light, radio waves, x-rays, gamma rays, ultra-violet, and all the other plethora of frequencies on the continuous spectrum).
Faraday's law of induction states that a changing magnetic flux produces an electric field, and Maxwell's law of induction states that a changing electric flux produces a magnetic field.
Given this knowledge, one might immediately jump to the conclusion that if one produces the other, and the other produces the first, then if set in motion, they ought to propagate forever.
Indeed, this is almost precisely what happens. Observe Figure 1. It is a pictorial representation of how the electric and magnetic fields propagate together.
In this figure, the red curve indicates the magnitude of the electric field as it propagates to the right, and the green indicates the magnitude of the magnetic field as it propagates to the right.
When one combines these laws of induction, one can determine the speed at which the wave should propagate in free space (vacuum). Maxwell did this, and arrived at the following speed for an EM wave:
Where Mu-naught is the magnetic permeability of free space, and Epsilon-naught is the electric permeability of free space. Both of these are constants... which means that the speed of light is constant in a vacuum.
Now, if one makes the reasonable assumption that the laws of physics are the same for all observers, then the conclusion naturally follows that the speed of light must be the same for all observers, since it is based entirely on physical, measurable constants.
If you measured the magnetic or electric permeability of free space while traveling at 150 million meters per second (half light-speed), you would measure the same value as when you were not moving.
Since the speed of light is based only on these two constants, then it too must be measured the same at any observer's velocity. This means that if you shown a flashlight ahead of you while traveling at 150 million meters per second relative to your cousin, both you and your cousin would see the light traveling at exactly c.
How can this be? Well, under classical physics, its impossible, which is why Einstein concluded that classical physics was incorrect for high-velocities, and developed the special theory of relativity.
In special relativity, two objects traveling at a high velocity relative to each other will undergo the following effects: Each object will see the other object as being shorter than it was at rest, and each object will see the others clock running slow.
Furthermore, it makes logical sense that one should never be able to reach the speed of light, since light will always appear to be traveling at light speed, relative to you. Physically, this is represented by the fact that a high-speed particle's effective mass will increase as it approaches light speed, such that it would require infinite energy to reach light speed.
This of course, is a tremendous hamper to starship design, as we are doomed to never locally exceed the speed of light.
Einstein's theory of special relativity says nothing about what happens when an object accelerates, or when it is in the presence of gravity (which, as it turns out, is indistinguishable from acceleration). For this Einstein had to develop the much more complicated general theory of relativity.
As with special relativity, general relativity is based on a very simple (and experimentally verified) postulate. That is that the mass of an object that is pulled on by gravity (gravitational mass), and the mass of the object that opposes acceleration (inertial mass) are the same.
That would require that the m in these equations be exactly equivalent: a=F/m (inertia), and F=(GM/r^2)m (gravity). This certainly seems to be a reasonable assumption, and it has been verified by experiment with an error of less than 0.000000000001 (that is 10 to the power of negative 12).
So, what consequences arise from such a simple postulate? Essentially, Einstein theorized that absolutely no experiment or observation could ever distinguish between a reference frame in which their existed a uniform gravitational field, and a reference frame which was accelerating at an equivalent amount.
There are many effects predicted by Einstein's theory of general relativity (none of which have ever been contradicted by experiment). The most popularly known is that of the warping of space-time by gravity. Let us now examine why it is that this warping is necessary. Observe figure 2:
In the constant acceleration frame (the one on the right) a light beam (the yellow line) that is emitted in a direction perpendicular to the direction of acceleration will look (to an observer in the accelerating frame) as if it is bending. This is nothing new, and it is what you'd expect, since the light is not accelerating with you.
In fact, it is impossible for the light to accelerate, since it must have constant velocity (See above). Since its velocity is constant, and you are accelerating, it will appear to bend. The astute reader may, at this juncture, question whether this is a violation of special relativity.
In fact, though, it is not. Remember: Special Relativity only applies in cases where there is no gravity and no acceleration. Indeed, Special Relativity is a "special" case of General Relativity.
So, if light appears to bend in an accelerating frame, and there is absolutely no difference between acceleration and gravity, then that means that light must bend in a uniform gravitational field as well!
Again, this may come as no surprise (after all, a baseball's trajectory bends toward the earth)... However, light has no mass! So, how could gravity affect it? According to Einstein, the space and time through which the light beam travels is actually warped, resulting in the apparently bent path of the light.
The warping of space-time, of course, gives rise to black holes and other interesting phenomena which may (but probably not) be useful for space-travel.