Special relativity all comes from Einstein’s two postulates:
- The laws of physics are the same in all inertial reference frames (keeping this around from Galilean relativity).
- Light travels at the same speed in all reference frames.
Because all of the math and conclusions about various relativistic phenomena are drawn from these, this is the two sentence answer sought by the OP.
msimon’s original Wikipedia quote basically sums up the theoretical implications of superluminal speeds and why it is generally dismissed as impossible. For more blood and gore, read on.
To Al, because light travels at the same speed (in vacuum) in all reference frames, no matter how fast you are going, light will always emanate from a source you are carrying in all directions at the speed of light. To a stationary observer you are passing, the same is true. This is related to the fact that light has no medium (there is no ether), so an source’s motion cannot be measured relative to anything meaningful. This was observed in the Michelson-Morley experiment and was a driving force behind Einstein’s insight.
To Ether, a Lorentz transform will spit out the answer. Let’s establish two events in space time (unprimed coordinates are in frame XY, primed coordinates in frame X’Y’):
Event 1 with both particles at the origin, x1 = x1’ = ct1 = ct1’ = 0.
Event 2 with particle A after time T in frame XY, x2 = c/2T, ct2 = cT, x2’ = ?, ct2’ = ?
Frame X’Y’ is moving with velocity c/2 to the left so B = -0.5 (B = v/c), y = 1/sqrt(1-B^2) = sqrt(4/3) (Beta and gamma are usually the letters used here but B and y will suffice).
The Lorentz transform (which one can derive from the mathematical and physical properties of Einstein’s postulates) is:
x’ = yx - yB * ct
ct’ = -yB * x + y * ct
So plug in above and x2’ = ycT (1/2-B) = ycT, ct2’ = ycT * (-B/2 + 1) = 5/4ycT.
Particle A’s velocity in frame X’Y’, then, is (x2’-x1’)/(t2’-t1’) = 4/5 c.
NOT c as one would expect from Galilean relativity. One could also apply the canned velocity transform (based on this) and get the same result.
The increasing mass concept is something I feel is not quite accurate. It exists to reconcile conceptually whats going on with Newtonian physics. Really, the momentum that is conserved is with respect to a proper velocity (because no one can agree on a common clock anymore), and this screws up all our nice F = ma tools (but F is still dp/dt !). As you go faster and faster, space-time is distorted such that the extra acceleration is accounting for more and more energy. Calculating the Energy of a relativistic particle (again from Lorentz transforms) gives K= (y-1)mc^2, where m is always just the rest mass, and the quickly diverging behavior is due to y as opposed to m getting bigger. E = mc^2 is just a statement about the rest mass of particles, since when v = 0, y = 1, so K = 0. The total energy is E = ymc^2.
Well that was fun.