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Starfleet Has a Speed Limit, And It's Lower Than You Would Think

Put your hand on a hot stove for a minute, and it seems like an hour. Sit with a pretty girl for an hour, and it seems like a minute. That's relativity.
― Albert Einstein
time dilation graph
Time Dilation -

A little known fact about the Star Trek Prime Universe, mentioned only in the Star Trek TNG Technical Manual,  is that there is a Starfleet enforced speed limit. If you own the technical manual, it's in section 6.2 Relativistic Concerns. This speed limit only limits impulse speed for normal ship operations, not warp speed. Now, that probably seems backwards because warp speed is much much faster... but actually there's a very good reason for it.

Why the impulse speed limit?
Impulse speed is sub-light speed star ship travel. That means when using the impulse engines, star ships are going less than the speed of light while warp speed is faster than light. Anything traveling at or near the speed of light, as you can see in the graph above, experiences extreme time dilation. This obviously includes the Enterprise D. Time dilation can also be noticed when an object is experiencing extreme gravitates. A well known example of this theory is in the move Interstellar when the crew of the Endurance takes a ship down to Miller's planet and experiences huge amounts of time dilation. Due to the gravitational effects of the nearby black hole, for every hour they experience on the planet, everybody else experiences 7 long years.

Relativity was first described by Einstein in his discovery of General Relativity. For the Enterprise,  the relative rate of time that passes for the crew members when moving close to the speed of light is much slower than the rest of the universe, (ie. Starfleet Command on Earth).

Setting an impulse speed limit prevents star ships from going too fast, approaching the speed of light, and experiencing extreme amounts of time discontinuity. If they were to do this all the time, the respective times of the ship and everyone else would be very different. For example, if the Enterprise D went 99.9% of the speed of light with their impulse engines for 1 month, Captain Picard would experience 1 month of time, while almost 2 years would have elapsed on earth! That is clearly not a sustainable way to run the Federation.

What is the Starfleet speed limit?
The Starfleet speed limit is .25c, or 25% the speed of light. Limiting impulse speeds to .25c will limit the effects of the time dilation. Interestingly, "full impulse power" is actually .25c. Star ships try to avoid going any faster than that except in extreme situations when it may be necessary. As you can see in the figure below, time dilation increases exponentially with respect to c. C is about 300,000,000 m/s.

Use this calculator to help understand the effects of relativity. (.999 = 99.9% the speed of light)

Time will move  times faster for the observer. So, a person traveling   of the speed of light will experience time   times slower.

If a person in a spaceship was orbiting the earth at   C  and they continued doing so for 1 month, 1 month would pass for them, but for the people on earth,    days would have gone by! This calculation was done using the equation below.

Time Dilation equation - time dilation calculator
The Lorentz Transformation Equation
c = speed of light
v = object velocity
T0 = variable of time of the observer
T = the time of the moving object - changed by 'v'

Does traveling at warp (much faster than the speed of light) affect time too? 
Interestingly, no. It's a bit counter-intuitive, but when moving with the warp engines, the ship isn't technically moving, as least to the perspective of the space around it. The warp engines form a warp bubble around the ship. The front of the bubble, using matter and anti-matter, continuously contracts while the space behind the warp bubble continuously expands. This motion effectively moves the ship forward along with the space around it. So using the warp engines doesn't break any of Einstein's theories and is theoretically possible to move faster than light with no time dilation issues. 

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