Special relativity (SR) (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies". It generalizes Galileo's principle of relativity–that all uniform motion is relative, and that there is no absolute and well-defined state of rest (no privileged reference frames)–from mechanics to all the laws of physics, including both the laws of mechanics and of electrodynamics, whatever they may be. Special relativity incorporates the principle that the speed of light is the same for all inertial observers regardless of the state of motion of the source.
This theory has a wide range of consequences which have been experimentally verified, including counter-intuitive ones such as length contraction, time dilation and relativity of simultaneity, contradicting the classical notion that the duration of the time interval between two events is equal for all observers. (On the other hand, it introduces the space-time interval, which is invariant.) Combined with other laws of physics, the two postulates of special relativity predict the equivalence of matter and energy, as expressed in the mass-energy equivalence formula E = mc2, where c is the speed of light in a vacuum. The predictions of special relativity agree well with Newtonian mechanics in their common realm of applicability, specifically in experiments in which all velocities are small compared to the speed of light.
The theory is termed "special" because it applies the principle of relativity only to frames in uniform relative motion.
Special relativity reveals that c is not just the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation (light)—but rather a fundamental feature of the way space and time are unified as spacetime. A consequence of this is that it is impossible for any particle that has mass to be accelerated to the speed of light.
Equations relevant to flat earth theory
Let's define the event to have space-time coordinates in system S and in S'. Then the Lorentz transformation specifies that these coordinates are related in the following way:
where is called the Lorentz factor and "c" is the speed of light in a vacuum.
Prevention of the flat earth's velocity reaching the speed of light
Differential Equation for velocity on earth
"dv/dt = g/γ^3"
Integrating for velocity:
Limit as t -> infinity
As you can see, it is impossible for dark energy to accelerate the earth past the speed of light.
For more information, especially on the derivation and solution of the differential equation I presented, see p. 37 of "Introducing Einstein's Relativity" by Ray D'Inverno, in particular section 3.8. Information taken from postings from Erasmus.