Interstellar space propulsion...if a spaceship were to get beyond our Sun's gravitational pull and since there is no atmosphere/wind/friction in space...does that mean, if an engine was constantly thrusting, that the spaceship would continue to accelerate to the speed of light, as long as the engine was propulsing forward? In other words, does speed constantly increase in a vacuum? or are the two ideas (Space/Vacuum) separate ideas?
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Assuming that our perceptions of Relativity hold up in interstellar space...
The ship would continue to accelerate as long as fuel held out.
Speed would continue its asymptotic clime towards C...
until drag was sufficient to end that climb. Drag would also increase due to interstellar medium collisions as speed increased. But note also: a sufficiently high thrust will not be exceeded by drag.
Note that as speed increases, and drag increases, so does radiation. The interstellar medium results in Alpha & Beta radiation increases, and high energy photon releases from same producing X-ray and even gamma radiation.
Note that several science fiction works have relativity have a cutoff point. In a few more, Relativity is ignored entirely; the most egregious offender is E.E. "Doc" Smith's Lensmen series. John Ringo's Looking Glass series has heliopause boundary changes to the nature of physics, including relativity
You asked three questions...
1.) ... does that mean, if an engine was constantly thrusting, that the spaceship would continue to accelerate to the speed of light, as long as the engine was propulsing forward?
No. For reasons ghoppe gave.
2.) ... In other words, does speed constantly increase in a vacuum?
Yes, speed would increase continuously, for the example you gave, asymtotically approaching (but never reaching) the speed of light.
3.) ... or are the two ideas (Space/Vacuum) seperate ideas?
No. A vacuum is an excellent and practical description of what most space is.
The answer is no, according to Einstein's Theory of Relativity.
Also, there is wind/friction in space. There is no absolute vacuum, there is an interstellar medium. Indeed, relativistic rockets would have to account for the medium, and perhaps even use it as a source of fuel, see: Bussard Ramjet.