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Have been really struggling for some considerable time to understand the universal limitation of speed i.e. it is not possible to travel faster than the speed of light. In particular - what is the limiting factor? Is it something physical that can be understood (e.g. is there an equivalence to terminal velocity of a falling body, which can easily be understood by a layman of balancing the wind resistance against the force of gravity).

Firstly, in trying to understand this I have learned that speed is only a relativistic quantity, everything can be considered to be at rest or in motion in makes no difference? So the limitation is only between two (or more) bodies. But if there was nothing in existence other than two independent particles separated by some arbitrary distance (let's say 10 light years for example) how can one influence the other? By which I mean the (relative) speed between the two is limited, so they must somehow influence each other? And if so how do they influence each other? Or is it the space-time vacuum between them that marks the relative distance & speed? What is the limiting mechanism and the reference??

Asked another way, in trying to understand this problem I have learned from various sources that the universe is expanding at an incredible rate...at more than half light speed in one direction, and more than half light speed in the opposite direction. I have learned that this is "allowed" since the two opposite ends are outside of each others sphere of influence...no information can ever be communicated between, neither can have cause or influence over each other (or are even aware of each others existence, and could perhaps be considered to be independent realities), yet it does seem possible that something could exist in between the two to observe that, relative to each other they are travelling faster than light...

So if we imagined just three particles in space, A moving "left" of B and C moving "right" of B (and saying B is "stationary" for the sake of argument). Let's say that initially A & C are within a sphere of influence (i.e. close enough to each other) so that they are limited in not being able to travel more than 0.5c away from each other. Let's say they are going 0.49c relative to B (in opposite directions). Over time they get further and further away...Suddenly, they reach the event horizon (the A-C event horizon) and they both increase speed to 0.9c relative to B. Particle B sees that A and C are now suddenly moving away from each other at 1.8c, almost as if they've hit a turbo switch. They have suddenly been allowed to increase speed by the universe.

When they fall off each others causality event horizon, what is it that allows them to increase speed? What is the limiting factor just before they reach the horizon? And finally, in going over that horizon would they experience any acceleration?

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    $\begingroup$ I did a quick count and found at least eleven questions which is far too many for one post. This isn't a good first post for this reason. Have you taken some time at the help center for some tips on writing good questions for this format? $\endgroup$ – Alfred Centauri Jun 23 '18 at 0:15
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    $\begingroup$ as far as we know, the speed of light is a fundamental limit of the geometry of spacetime. No theory so far derives the geometry from some more fundamental concepts $\endgroup$ – lurscher Jun 23 '18 at 0:32
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    $\begingroup$ "then happy to delete it and live in ignorance" - an alternative would be to delete it and then spend a modest amount of time searching this site's many questions regarding the light speed limit. Many of the regulars here that volunteer their time to answer questions have posted high quality answers to such questions and maybe, just maybe, after perusing these, you won't feel the need to 'live in ignorance'. And then, if you still have a question, reference these other answers and identify what, in particular, isn't clear to you. $\endgroup$ – Alfred Centauri Jun 23 '18 at 0:35
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    $\begingroup$ For a start, try this one: If all motion is relative, how does light have a finite speed? $\endgroup$ – Alfred Centauri Jun 23 '18 at 0:39
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    $\begingroup$ Possible duplicate of If all motion is relative, how does light have a finite speed? $\endgroup$ – sammy gerbil Jun 23 '18 at 2:06
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Either the structure of space-time has no speed limit (unbounded relative velocities would then be possible), or the structure of space-time has a maximum speed (limitation on the maximum relative speed possible).

The first case corresponds to Newtonian space & time. We know that this is incorrect for fast moving objects, and evidence has been accumulating since before Einstein's work of 1905.

The second case, where there is a speed limit, is explained by Einstein's Special Relativity. Here the maximum speed is c, which also happens to be the speed that light travels.

To consider the cosmological case of the expanding universe, one needs Einstein's General Relativity. Here we have a dynamic space-time, which responds to local density of matter and energy. From observation it is known that the cosmos is expanding, and the further away (in time and space), the faster it is moving away from the observer, such as us. There are two elements to this motion: one is ordinary motion, the other is due to the expansion (stretching) of space.

The stretching of space is not limited by the speed of light, and thus there are elements of the cosmos which used to be observable, but which now have attained speeds, due to the expansion of space, such that they can no longer be seen.

For details, see How Can the Universe Expand Faster Than the Speed of Light?

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The limiting factor is the speed of time. In relativity, time and space are related (hence "relativity"). The faster you move in space (relative to me), the slower your time moves (as I see it). As you approach the speed of light, I see your time slowing down to zero. Nothing is smaller (in the absolute value) than zero. Therefore no faster speed exists in relativity (in the hyperbolic geometry of spacetime) than the speed of light.

Speed is distance divided by time. If time and space were independent, naturally, any speed would be possible (as in the Galilean or Newtonian spacetime). However, in special relativity (Minkowski spacetime), time and space relate to each other. So when you divide distance by time, but time is relative to space, it so works out in this (counterintuitive) geometry, that the result of your division is never larger than one (as in one light second per second).

So the speed of light is the maximum speed, not because of a technical limitation or limiting factor. It is not the case where a higher speed conceptually exists, but nothing can achieve it. The actual case is that a faster speed simply does not exist in the (Minkowski) geometry of spacetime. You cannot move faster than light for a similar reason as why you cannot move to the North of the North Pole. In the geometry of the globe, no place exists to the North of the North Pole. Similarly, in the geometry of spacetime, no speed exists faster than one (the speed of light in natural units).

Speed geometrically is represented by a rotation of coordinates, say, $x$ and $t$. Well, these coordinates in our spacetime are very dissimilar and behave quite differently. One is space, the other is time. Mathematically this is described by the hyperbolic geometry of Minkowski. In this geometry, the speed also is represented by a rotation of coordinates, except the space and time coordinates rotate in different directions, one clockwise, the other counterclockwise. Accordingly, you cannot rotate them more than 45 degrees (visually, the hyperbolic angle is calculated differently) before they meet. And at 45 degrees the speed is one. So the maximum possible speed in this type of geometry (in which we live) is one lightsecond per second. There simply is no faster speed. Even if you magically could achieve any speed, no faster speed exists in our spacetime, so you'd still be limited to the speed of light.

Now let's move onto two objects (A and B) flying apart from you (C) in the opposite directions each near the speed of light relative to you. Is their relative speed a double speed of light? No. "Relative" means the speed of an object relative to another object. So, if you figure the speed of A relative to B, then, according to the relativistic speed addition formula, this relative speed is still slower than light. Wait, but don't you see that these two objects are flying apart from you with a double speed of light? Yes, but it is not their relative speed (as just explained). There is no such thing as "the speed of A and B relative to C". "Relative" means "A relative to B" or "A relative to C" or "B relative to C" where all three of these speeds are slower than light. So, when you see two objects flying apart from you with a double speed of light, it's OK. No laws are broken. No relative speed is faster than the speed of light. These objects would not disappear from you or from each other behind any horizon (in a flat non-expanding space of special relativity).

Finally, moving on to the expansion of space faster than light. One flaw in your reasoning is when you say that two objects separated by a distance are limited in their relative speed by the speed of light. This would be true only in the Minkowski (flat and non-expanding) spacetime. However, our spacetime is flat only locally (asymptotically). It is not flat globally or even at a relatively small distance in gravity where spacetime is curved by a heavy mass (e.g. of the Earth). And when spacetime is curved, the geometry is no longer Minkowski, and speed is no longer limited by the speed of light. However, in a small region (locally), the geometry is still Minkowski and the speed of light limitation is still there. Things far away from you can conceptually move faster than light relative to you, by nothing can fly near you faster than the speed of light.

For example, consider the frame of the rotating Earth. In this frame you simply sit and watch the stars rotating around the sky. Alpha Centauri is 4.4 light years away making a 27-light-year circle around you in just 24 hours. This is ten thousand times faster than the speed of light. Another example, the universe expands. The further the galaxy is from us, the faster it moves away. Can it move faster than the speed of light? Sure, because it is not a local speed. The speed of light limitation is only local - nothing can fly faster than the speed of light (relative to you) near you. However, far far away, there are different rules. The universe expands. Black holes slow down time to a halt. Spacetime is curved. Frames are not inertial. So sure, under certain conditions, things that are not near you can move faster than the speed of light relative to you. This is OK, no laws are broken. The law of the maximum speed is only local. Globally the universe is not limited by any speed.

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The limiting factor is the speed of light itself. Nothing can travel faster than this speed.

This is both an empirical fact deduced from the Michaelson-Morley experiment and a theoretical fact deduced from Maxwells theory of electromagnetism.

In fact, as particles begin to approach light speed their mass begins to increase so it progressively takes more and more energy to increase their speed; this is why particles that do travel at this speed are massless.

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