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Imagine 1 million objects travelling in space, at a constant speed, along an imaginary line. They don't deviate from that line for the sake of this argument.

Now, object 1 has a velocity of $v_{1}=1$ km/s relative to a point in space.

Object 2 has a velocity of $v_{2,1}=1$ km/s relative to object 1, which means $v_{2}=2$ km/s when calculating the velocity related to the point in space we considered with object 1.

Object 3 has a velocity of $v_{3,2}=1$ km/s relative to object 2, which means $v_{3}=3$ km/s when calculating the velocity related to the point in space we considered with object 1.

Then I would assume that object 1 million as a velocity of $v_{1M}=1$ million km/s when calculating the velocity related to the point in space we considered with object 1. However, this goes against the principle that nothing can exceed the speed of light.

So:

1.) Where is the error in this reasoning?

2.) How can we talk about the existence of a maximum velocity when $velocity$ is actually vector-based measurement which changes with the reference we consider?

Qmechanic
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Dayman75
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    You aren't using the relativistic addition of velocities. See e.g. http://physics.stackexchange.com/q/107820/. You should also not be thinking about the speed of light as a speed limit. The universe doesn't have a "speed limit". One can go from here to Andromeda in an hour ship time with sufficient energy. "The speed of light is constant in every inertial system" is the correct phrase and it means something totally different than "speed limit". – CuriousOne Apr 11 '16 at 23:01
  • But if "infinite" energy is necessary to move an object at the speed of light, how could you get to Andromeda in an hour? Thank you for your comment :D – Dayman75 Apr 11 '16 at 23:06
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    The distance between here and Andromeda at the speed of light is zero, i.e. it wouldn't take any time to get there. Shrinking the distance to one light-hour wouldn't take infinite energy and it wouldn't require going at the speed of light in either of the rest systems. Is it realistic to get there with a ship made from baryonic matter at that speed? No. Is it physically forbidden? No. A smart engineer wouldn't do it, though. A smart interstellar spacecraft engineer will make his "ship" out of light. :-) – CuriousOne Apr 11 '16 at 23:09
  • But for us that we are traveling to Andromeda, the journey wouldn't be subjectively instantaneous or would it? I understand that for someone i.e. on Earth it would feel instantaneous, but what about the one making the trip to Andromeda? :) – Dayman75 Apr 11 '16 at 23:13
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    If you could make a ship like that it would feel like an hour for the passengers. The acceleration would be rather unpleasant (and deadly), but other than that it would be a short trip. For everyone on Earth (and Andromeda) the ship would be traveling near the speed of light, i.e. it would take millions of years. What relativity does is not to limit fast travel, it limits the return to ones own time. Traveling x lightyears means one can not return earlier than after 2x years, but that's not a speed limit. It's a choice between one's own future and the future of a remote part of the universe. – CuriousOne Apr 11 '16 at 23:18
  • Oh okay :) So it's the other way around for what I was thinking. Thank you sir, for answering so quickly and succinctly :) I actually understood the error in my reasoning and learned a few more things :) – Dayman75 Apr 11 '16 at 23:20

1 Answers1

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As per the comments, I wasn't taking into account the relativistic addition of velocities, which is becomes relevant when designing scenarios with such high velocities.

So for a observer in the point specified in my argument, the fastest objects (object #1 million, object #999.999, ...) would appear to have velocities close to light speed, but they would never reach it.

For more information check out the Website http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel.html

Also, this question might be a duplicate of others. If you have questions and feel this answer isn't enough, consider checking the others out: Relativistic addition of velocities of spaceships

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Dayman75
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