Einstein once wrote: “People like us, who believe in physics, know that the distinction between past, present and future is only a stubbornly persistent illusion,” but Stephen believed it was exactly the opposite: time is nothing but the ratio between displacement and speed, as flows from Galileo's law that s = at2.
Even the effect of time dilation is nothing but a manipulation of movement: the individual particles of an ultrarelativistic objects slow down so that time appears to dilate, but, of course, this doesn't mean, as in Robert Anson Heinlein's “Time for The Stars,” that it doesn't happen simultaneously with the rest of the universe. Lorentz' proposition of relativity of simultaneity in 1895 and Einstein's elucidation thereof in 1905 has been misinterpreted: speed doesn't speed up, but slow down time.
The relativistic mass, that is, the mass of a moving object, is calculable with the formula:
mrel = m₀γ = m₀/√(1-v²/c²)
Here, “rel” stands for relativistic, meaning that it is close to the speed of light.
This means that there is an asymptote in v→c, so that relativistic mass becomes infinite at c.
Ek = mv²/2,
So if m becomes infinite, velocity becomes nought. This leads to the contradictory conclusion that velocity is inversely proportional to itself. Actually, the velocity remains constant when v approaches c, but it changes form: the net speed increases, but the speed of every of its particles decreases (relative to each other), so that the sum of their speeds remains equal.
Then what is the equation of relativistic speed? Speed is defined as distance per time, so this depends on what distance and time you mean. As the word says, relativity is relative.
For instance, for a speed inside a relativistic object (such as a spaceship),
vrel = v0/γ2
Where v0 is the rest-speed or inert speed. This sounds oxymoronic, but it's simply the speed in an inertial frame of reference, that is, the speed for which relativity is ignored and the laws of classical mechanics (Newtonian physics) still count without the multiplication of a Lorentz factor. The inert speed may exceed the speed of light, while the relativistic speed may not.
In this formula, the spaceship itself is used as frame of reference. This is so because both distance and time are affected by the Lorentz factor. At relativistic speeds, distance undergoes a Lorentz contraction:
srel = s0/γ
and time undergoes time dilation:
trel = t0γ
Thus,
vrel = s0/t0 γ2 = v/γ2
However, this only counts inside the object, not outside it. Not all length in the universe is contracted, and not all time in the universe is dilated, when something moves at relativistic speed. Because of this, I'll call this the local relativistic speed or internal relativistic speed, that is, the speed observed by someone inside.
Using the good old formula
Ek = mv2/2,
however, we can get a result which applies more broadly. Ironically, Newton's physics may still be used in relativity. They are just as applicable at relativistic speeds when one considers that the relativistic mass is the inert mass multiplied by the Lorentz factor. Velocity itself doesn't directly experience the effect of the Lorentz factor; it does so indirectly because mass and velocity are inversely and proportional (and, of course, the same counts for relativistic mass and velocity).
At relativistic speeds,
Ek = mrel (vrel)2/2
⇔ vrel = √(2Ek/mrel)
⇔ vrel = √(2Ek/m0γ)
For “inert speed,” the only difference is that mass isn't relativistic:
v0 = √(2Ek/m0)
Thus, if we substitute this in the formula of relativistic speed,
vrel = v0/√γ.
This formula applies universally, and not just to the bent space and time of relativistic speeds. Because of this, I'll call it the global relativistic speed or external relativistic speed, that is, the speed observed by someone from outside. When we compare the local relativistic speed with the global relativistic speed, we get that
v int rel / v ext rel = (v0/γ2)/(v0/√γ) = γ-2 · γ-1/2
⇔ v int rel / v ext rel = γ-5/2
⇔ v ext rel = v int rel γ5/2
If v→c, electromagnetic interaction (carried by photons) has twice the speed of light in direction of travel and zero speed in the opposite direction. Note that this is relative to the moving object itself. This follows from the velocity-addition formula: according to relativity, the relativistic sum of two velocities is calculable with the formula:
v1⊕v2 = (v1+v2)/(1+v1v2/c2)
Actually, this is only so if the first and second velocity are aligned, that is, if they have the same (or opposite) orientation. For any speed at all, it's a bit more complicated: the relativistic sum of speed one and speed two equals sum of the quotient of the sum of speed one and speed two and the square root of the difference between one and the quotient of the product of speed one and speed two and the square of the speed of light and the product of the inverse-square of the speed of light, the inverse of the sum of one and the square root of the difference between one and the square of the quotient of speed one and the light speed, and the quotient of the vectorial product of speed one and the vectorial product of speed one and speed two and the difference between one and the quotient of the product of speed one and speed two divided by the square of the speed of light.
