I know that nothing can exceed the speed of light, but

[student34]

... can someone explain to me why the following thought experiment does not give the object a speed greater than 3*10^8 m/s.

Imagine that the experiment takes place in a very large vacuum. A very long toy train is moving at 100 m/s on a track from rocket propusion. On the top of this train, there is another smaller train that starts to run on a track on the top of the larger train. The smaller train now has a speed of 100 m/s relative to the larger train and 200 m/s relative to a still observer. Then, a smaller train does the same thing on the second train. There are 3*10^6 + 1 trains that are all on the backs of another doing 100m/s. Exactly what happens when the
300000th train fires up its rocket relative to a still observer? If possible, what happens when the 300001th train's rocket fires?

 

[Vorde]

Because velocities don't add like we think they do. Well, at low speeds them seem to increase through simple addition, but once you get up to relativistic speeds you have to start using the relativistic velocity addition formula $V_{total} = \frac{v_1+v_2}{1+ \frac{v_1 v_2}{c^2}}$

http://en.wikipedia.org/wiki/Velocity-addition_formula

Either reading the page or doing some quick checks yourself will show you that you can never get something moving faster than the speed of light with this formula.

 

[student34]

Because velocities don't add like we think they do. Well, at low speeds them seem to increase through simple addition, but once you get up to relativistic speeds you have to start using the relativistic velocity addition formula $V_{total} = \frac{v_1+v_2}{1+ \frac{v_1 v_2}{c^2}}$

http://en.wikipedia.org/wiki/Velocity-addition_formula

Either reading the page or doing some quick checks yourself will show you that you can never get something moving faster than the speed of light with this formula.

There can't be anything more counterintuitive than this. It is no wonder why rationalism is dead.

 

[Whovian]

There can't be anything more counterintuitive than this. It is no wonder why rationalism is dead.

Welcome to special relativity. Though I'd have to say quantum mechanics is probably more counterintuitive. (But don't worry about figuring out what everything means in QM, "shut up and calculate.")

 

[Vorde]

Rationalism isn't dead, far from it. All relativity (and quantum physics for that matter) does is limit the usefulness of common sense.

The relativistic velocity formula comes very directly and very rationally from a postulate that violates common sense. If you accept that common sense isn't always correct, then its very intuitive.

If you can get a deep understanding of the initial postulates of relativity, then you can start seeing why relativistic physics makes sense, the trouble is the initial hurdle of "huh?"
that is at the beggining of all relativity classes.

 

[Nugatory]

The relativistic velocity formula comes very directly and very rationally from a postulate that violates common sense.

What is and is not "common sense" is something of a matter of opinion, but it is not at all clear that either postulate of relativity, taken alone, violates common sense. The problem is that although we can make a pretty good common-sense argument for both, they appear to be contradictory at first look, and much of the second half of the 19th century was spent trying to reconcile them before Einstein succeeded in 1905.

There's a long thread in the relativity forum recently: https://www.physicsforums.com/showthread.php?t=670253

 

[Nugatory]

There can't be anything more counterintuitive than this.

Oh yes there is...

Relativity just requires you to consider that the intuition you've developed in a lifetime of dealing with speeds much less than that of light might not be applicable at higher speeds. It's somewhat like the adjustment we go through as we learn that, all appearances and common sense notwithstanding, the world is not flat and "up" is a completely different direction in Australia and America.

But quantum mechanics... That's big-league counterintuitive.

 

[student34]

Rationalism isn't dead, far from it. All relativity (and quantum physics for that matter) does is limit the usefulness of common sense.

The relativistic velocity formula comes very directly and very rationally from a postulate that violates common sense. If you accept that common sense isn't always correct, then its very intuitive.

If you can get a deep understanding of the initial postulates of relativity, then you can start seeing why relativistic physics makes sense, the trouble is the initial hurdle of "huh?"
that is at the beggining of all relativity classes.

I think that one would have a really tough time figuring out relativity just by thought alone. We might be using rationalism differently.

 

[Vorde]

What is and is not "common sense" is something of a matter of opinion, but it is not at all clear that either postulate of relativity, taken alone, violates common sense.

Interesting thread, I'll have to read the rest of it. But I think you'd have a hard time arguing that the consistency of the speed of light obeys common sense.

I think that one would have a really tough time figuring out relativity just by thought alone. We might be using rationalism differently.

I think we are a little bit, but my point is that if you take the two postulates for granted, then you can figure out of the rest of special relativity by thought alone.

It's the taking the two postulates for granted that's the hard part.

 

[Dale]

I think that one would have a really tough time figuring out relativity just by thought alone.

I agree. That is why it took a mind of Einstein's caliber to do just that.

 

[danR]

I think that one would have a really tough time figuring out relativity just by thought alone. We might be using rationalism differently.

As one non-expert to another, I can tell you that the matter was extremely difficult to deal with until I decided it lay in the assumption that distance and time were something deeply fundamental, and that a meter stick cloned from some Geneva (or Greenwhich, or wherever it is) invar standard ought always measure the same at Alpha Centauri or observed by me in a rocket ship going past at .99c.

The fact is they don't. I could only conclude that there is a fundamental measure distance/time together, that is invariant in all (non-accelerating) reference frames. It's not a terrible sacrifice of intuition, and great deal of fruit falls out quite naturally from that tree, once I accept it exists.

(Edit: I should hastily add that the d/t thing is not mine. It's an old variant of one of Einstein's postulates. I only decided it was true)

 

[Nugatory]

Interesting thread, I'll have to read the rest of it. But I think you'd have a hard time arguing that the consistency of the speed of light obeys common sense.

As I said, "common sense" is somewhat in the eye of the beholder...

If we apply the first postulate to all the laws of physics, then we cannot leave out Maxwell's laws of electrodynamics. These laws predict electromagnetic radiation propagating at speed c in a vacuum; so I don't find t such a huge leap of common sense and intuition to accept that either different observers will have different electrodynamical laws in violation of the first postulate, or that all observers must get the same value for the speed of light.

(Of course if I had studied Maxwell's electrodynamics in the 1870s instead of the 1970s common sense and intuition would have led me to an ether theory instead. Historical context matters).

 

[Vorde]

As I said, "common sense" is somewhat in the eye of the beholder...

If we apply the first postulate to all the laws of physics, then we cannot leave out Maxwell's laws of electrodynamics. These laws predict electromagnetic radiation propagating at speed c in a vacuum; so I don't find t such a huge leap of common sense and intuition to accept that either different observers will have different electrodynamical laws in violation of the first postulate, or that all observers must get the same value for the speed of light.

(Of course if I had studied Maxwell's electrodynamics in the 1870s instead of the 1970s common sense and intuition would have led me to an ether theory instead. Historical context matters).

I agree that common sense is in the eye of the beholder, but I think there are some things which violate all human common sense.

Your point is good, but I'd just say that it implies just that maxwell's laws also imply things that are contrary to common sense. Do you really think saying that an object moving past you that doesn't seem to slow down if you move in its direction of motion follows common sense?

 

[Nugatory]

Do you really think saying that an object moving past you that doesn't seem to slow down if you move in its direction of motion follows common sense?

Well, it seemed pretty weird to me the first time I came across it... But not as weird as ditching either of Einstein's postulates would have been.

When you say that "the relativistic velocity formula comes very directly and very rationally from a postulate that violates common sense" I agree with you that it follows directly and rationally from a postulate; I just don't think the postulate violates common sense.

 

[Vorde]

The first postulate I'd completely agree with you, the second seems to me to be completely against common sense. However, I think you're right that this seems a matter of individuality.

 

[harrylin]

There can't be anything more counterintuitive than this. [..].

Oh yes: QM's Bell theorem !

In contrast, you can intuitively understand that SR equation by accounting for two relativistic effects in combination with the synchronization convention; it then becomes common sense. However, this implies - if you are a normal human being - that you will need to do some calculation exercises to develop that intuition.

It may be helpful to realize that this "velocity sum" equation technically isn't really an addition but a transformation; the fact that 2+2=4 isn't challenged!

Note also that the two postulates of SR (in their original form) were considered both common sense and rather well established, although they seemed to be incompatible with each other.

 

[ghwellsjr]

The relativistic velocity formula comes very directly and very rationally from a postulate that violates common sense.

The first postulate I'd completely agree with you, the second seems to me to be completely against common sense.

The velocity addition formula does not depend on the second postulate which is a requirement for Einstein's theory of Special Relativity. Even prior to him proposing that second postulate, the velocity addition formula can be derived just from the first postulate which you agreed makes common sense to you and the common belief at the time in an absolute ether rest frame, only in which light propagates at c.

We can illustrate how this works with an example involving three inertial observers who we show as colocated at the origin of a spacetime diagram depicting a single ether rest frame. We refer to the observers by their color. The black observer is stationary in the ether frame. The green observer is traveling to his right at 0.6c and the blue observer is traveling to his left at 0.6c.

Now we are going to see how each observer measures the speeds of each of the other observers using radar signals which will travel at c relative to the ether frame and not relative to each observer. However, because of the Principle of Relativity, we insist that when one observer measures the relative speed another observer, the other observer will get the same answer when measuring the relative speed of the first observer. And for that to happen we must come to the same conclusions that scientists came to prior to Einstein which is that clocks run slower when traveling through the ether. So on our spacetime diagram, we show dots for each observer that mark out one-year intervals of time.

Now a little review about radar signals. An observer aims a radar signal at another observer noting the time on his clock at which he sent it. Eventually that signal hits the other observer and reflects back to the first observer. When the reflected radar signal gets back to the first observer, he notes this second time. Now with these two time measurements, the observer does a little calculation. He adds the two times together and divides by two. This gives him the time at which he will say the "measurement" applies. He also subtracts the two times and divides that by two. This gives him a measurement of the distance (in compatible units where c=1). So he gets a distance applied at a particular time. To calculate the speed of the second observer, he uses that fact that they started out colocated so all he has to do is divide the distance by the applied time.

So let's see how this works first for the black observer measuring the speed of the green observer:

When the black observer's clock reads 1, he aims his radar gun at the green observer and he receives the echo back when his clock reads 4. The radar signals are shown in red and always propagate along 45-degree angles with respect to the ether frame. Black calculates (4+1)/2=2.5 as the time at which the measurement was made and he calculates (4-1)/2=1.5 as the distance that the green observer was away from him at his time of 2.5 and we can see that since he is at rest with respect to the ether, his measurement matches what the ether frame portrays as shown by the yellow line. He can also calculate the speed of the green observer with respect to him as 1.5/2.5=0.6c. Although I don't show it, he can make the same measurement of the blue observer and will get the same results since blue is symmetrically a mirror image of green.

Now let's see how green will measure the speed of black:

We see that the exact same explanation used by black to measure green applies for green measuring black. In fact, since we know that they must get the same answer, that is the reason why scientists concluded that green's clock must be slowed down while he is traveling through the ether. Note that the distance of 1.5 for the yellow line does not correspond to any such distance in the ether frame (but the green observer can't tell that).

Although I don't show it, we could also show that blue would get the same measurement of black's speed since green is symmetrically a mirror image of blue.

Now we'll take a look at how green would measure the speed of blue using exactly the same radar method as before:

Once again, green fires his radar signal at his clock time of 1 but this time, since he is waiting for the reflection from the blue observer, it takes until his clock time of 16 before he gets it. So his calculation of the time the measurement applies is (16+1)/2=8.5 and for the distance is (16-1)/2=7.5. So he concludes that at his time of 8.5, the distance to the blue observer was 7.5 and the speed of the blue observer is 7.5/8.5=0.882c. Note that the length of the yellow line does not correspond to anything in the ether frame. And again because blue and green are mirror images, blue will get the same measurement of green's speed.

But now let's use the relativistic addition formula to see what speed we get when we add 0.6+0.6:

(0.6+0.6)/(1+0.6*0.6) = 1.2/(1+0.36) = 1.2/1.36 = 0.882c

This is the same answer we got using the ether frame and the radar measurement. So you see, it is not necessary to apply Einstein's second postulate to demonstrate the validity or the common sense of the relativistic velocity addition formula.

 

[SinghRP]

I just happen to see the original question by student34 at Post 71. Here is one physicist's intuition.
1. I want you to recall Newton's 2nd law: Force vector = Rate of change of momentum vector. If you keep mass and the direction of velocity vector fixed, then force = (Inertial mass)(acceleration).
2. Only that which has an inertial mass, CAN be accelerated or decelarated.
3. Light (or em waves) do not have inertial mass. That means, light CANNOT be accelerated or decelerated. That means, poor light is stuck with whatever speed Nature assigned it.
4. The same rules apply when adding speeds.
5. A photon's spin is 1 but only two projections: +1 and -1; no zero.
6. There is no such thing as a traveling photon. Em waves travel; photons interact with whatever they can.
I recommend you discuss these with your professor.

 

[OmCheeto]

... 2+2=4 ...

Ha ha!

2+2=3.99999999982224 in special relativity.

$V_{total} = \frac{v_1+v_2}{1+ \frac{v_1 v_2}{c^2}}$

given v1=v2=2 and c=300,000

sorry.



And didn't Einstein use a train in his thought experiment?

What if I were to sit on a train that was traveling at the speed of light? What would light look like? Light has the same speed, no matter what. I would be light! Perhaps I should slow the train down a bit, and see what happens. Now I'm going almost as fast as light, but light is now speeding away from and towards me, at the speed of light. How am I going to reconcile this?

After a bit of googling, it would appear my imagination is pretty far off.


-------------------
Emagination = oM x ²

 

[DrGreg]

2+2=3.99999999982224 in special relativity.

The point harrylin was making was that it's an abuse of terminology to describe that as "addition" or to use the symbol "+" in that way.

In relativity it is still true that 2+2=4, to combine velocities you must use<br /> \frac{2 + 2}{1 + \frac{2 \times 2}{c^2}} = 3.99999999982198<br />(in km/s).

 

[Morgoth]

6. There is no such thing as a traveling photon. Em waves travel; photons interact with whatever they can.

One question on this, although it is not appropriate for a GR or SR topic.
When we have our action, and we add the electromagnetic interaction field (the F_αβ) don't we impy that EM field is everywhere in our spacetime? Doesn't that imply that it does not propagate with a certain speed (c)?

 

[Dale]

The point harrylin was making was that it's an abuse of terminology to describe that as "addition" or to use the symbol "+" in that way.

I believe that the correct term would be "composition of boosts", but "velocity addition" is such a standard abuse of terminology that it is necessary for everyone to understand what is meant (as well as that it is an abuse of terminology).

 

[my_wan]

As one non-expert to another, I can tell you that the matter was extremely difficult to deal with until I decided it lay in the assumption that distance and time were something deeply fundamental, and that a meter stick cloned from some Geneva (or Greenwhich, or wherever it is) invar standard ought always measure the same at Alpha Centauri or observed by me in a rocket ship going past at .99c.

Here is something that occurred to me many years ago, as a teen, and espoused by Edward Arthur Milne. Newtonian physics notwithstanding, IIF we do live in a purely mechanistic Universe I can't personally comprehend how it could be conceived that distance and time have any direct empirical meaning outside the mechanism used to measure it. If the rate of this clock 'mechanism' locally varies wrt a similar mechanism at some distance then that local observer, who is also a mechanism, by definition can't empirically distinguish purely local variations in these clock rates. Under this mechanistic assumption it also follows that space and time are inversely synonymous. Under a purely mechanistic assumption, by what magic could it be assumed that space and time are empirically accessible purely independent absolutes, or empirically "fundamental"? It then follows that any variable that locally covaries with time is itself an empirical constant.

It also seems to me that the principle of relativity is perfectly sensible on the grounds of conservation laws alone. That entails that the addition of two physical quantities is not some random sum, whether the sum is linear or not.

In terms of what is counterintuitive for me quantum mechanics holds ALL the cards.

 

[nitsuj]

I believe that the correct term would be "composition of boosts", but "velocity addition" is such a standard abuse of terminology that it is necessary for everyone to understand what is meant (as well as that it is an abuse of terminology).

 

[1977ub]

When you walk away from me, to me you appear to get smaller. Rationally, from that, I might reason that to you I should appear to be getting larger - but what's that? You say I am also appearing to get smaller to you?! Does that seem counterintuitive? Irrational?

 

[harrylin]

Ha ha!

2+2=3.99999999982224 in special relativity.

$V_{total} = \frac{v_1+v_2}{1+ \frac{v_1 v_2}{c^2}}$

given v1=v2=2 and c=300,000

sorry.

[..]

I stressed that that equation corresponds to a coordinate transformation between different coordinate systems of measurement.
It's similar when you add Singapore dollars to US dollars: 1 SD + 1 USD ≠ 2 dollars. Also in that case you need to transform, either to USD or SD.

 

[OmCheeto]

I stressed that that equation corresponds to a coordinate transformation between different coordinate systems of measurement.
It's similar when you add Singapore dollars to US dollars: 1 SD + 1 USD ≠ 2 dollars. Also in that case you need to transform, either to USD or SD.

hmmm...

Looking at this again, brings up another question in my head.

If you had a train that circled the globe at the equator traveling at 2m/s, with another train riding on top of that train traveling at 2 m/s, a stationary observer would view the top train as traveling at 3.99999999982224 m/s, both coming and going.

Does that sound right?

And with a equitorial circumferance of ≈40,000,000 meters, the lower train would make the trip in 20,000,000 seconds. The upper train would do this in 10,000,000.00044 seconds, or ~4.4 tenth thousandths of a second longer than it should.

Does that sound right also?

 

[OmCheeto]

When you walk away from me, to me you appear to get smaller. Rationally, from that, I might reason that to you I should appear to be getting larger - but what's that? You say I am also appearing to get smaller to you?! Does that seem counterintuitive? Irrational?

This sounds like house size logic.

Bert; "Which is bigger, the inside or the outside of a house"?
Om; "The inside".
Bert; ?
Om; "If you go into a house, it surrounds you. It becomes the universe. It is very large."
..."If you go outside, and walk a little ways, you can squash the house between your fingers. It is very small."
..."Therefore, the inside of the house is larger than the outside".

Also, this appears to have absolutely nothing to do with special and/or general relativity.
I suspect our posts will soon disappear.
Read quickly, as my humor shield is useless against the mighty sword of off-topic-ness.

-------------------
ok to delete

 

[1977ub]

Also, this appears to have absolutely nothing to do with special and/or general relativity.

The issues of rationality / intuitiveness were raised here. I sometimes think of the POV of observers both finding the other to shrink with distance when I see questions about how moving observers can *each* see the other's clock as slowed down.

 

[OmCheeto]

The issues of rationality / intuitiveness were raised here. I sometimes think of the POV of observers both finding the other to shrink with distance when I see questions about how moving observers can *each* see the other's clock as slowed down.

Your original post stuck me non-relativistic. But now you've changed it to a clock. That's different. Now if you'd questioned something originally about the top train becoming shorter, or the lamp on the top of the train emitting a different frequency of light, both coming and going, or why the clock on the top train didn't keep the same time as the bottom train, then that would have been on topic.

 

[1977ub]

Your original post stuck me non-relativistic.

Just referring back to this - what I took to be the major issue of this thread.

There can't be anything more counterintuitive than this. It is no wonder why rationalism is dead.

 

[georgir]

What I want to know is, why did you need 300001 trains to get your point across instead of just 2 each moving at 0.6c like the usual threads we keep getting all the time...

 

[EskWIRED]

I think that one would have a really tough time figuring out relativity just by thought alone.

But isn't that precisely how relativity was figured out? It's not like Einstein took data points and worked backwards.

 

[jalsck]

Would it be correct to say that matter cannot be observed to be traveling at or faster than the speed of light, however, it is theoretically possible to increase the velocity of a spaceship until the velocity is well beyond the speed of light when compared to some initial velocity? Note, I'm not comparing an initial location with the current location and velocity, just the initial velocity with the current velocity.

Thought experiment:
A spaceship is traveling at some undefined velocity that the observers on the ship consider to be at rest. The ship then accelerates and the rate of acceleration is measured by instruments on the ship as well as experienced by the observers (on the ship). The ship stops accelerating and the change in velocity is calculated to be 0.25c. This is repeated until the change in velocity is 2.00c. Absolutely nothing unusual was observed on the ship as it's velocity exceeded the speed of light as measured by the ship's instruments.

 

[Nugatory]

Would it be correct to say that matter cannot be observed to be traveling at or faster than the speed of light, however, it is theoretically possible to increase the velocity of a spaceship until the velocity is well beyond the speed of light when compared to some initial velocity? Note, I'm not comparing an initial location with the current location and velocity, just the initial velocity with the current velocity.

Thought experiment:
A spaceship is traveling at some undefined velocity that the observers on the ship consider to be at rest. The ship then accelerates and the rate of acceleration is measured by instruments on the ship as well as experienced by the observers (on the ship). The ship stops accelerating and the change in velocity is calculated to be 0.25c. This is repeated until the change in velocity is 2.00c. Absolutely nothing unusual was observed on the ship as it's velocity exceeded the speed of light as measured by the ship's instruments.

No. When you speak of the speed of the ship, you have to say what that speed is relative to, and as far as the ship and its instruments are concerned, the ship is always at rest relative to itself. And if we look at the speed of the ship relative to some external object (say, another spaceship that doesn't accelerate) we'll find that each successive increment of acceleration (assuming they're equally long and strong to the crew) will change the relative speed between the two ships by a smaller increment, so that the speed will never reach c.

You might want to google for "relativistic velocity addition" for more details.

 

[1977ub]

But isn't that precisely how relativity was figured out? It's not like Einstein took data points and worked backwards.

Classic Galilean "relativity" existed before Einstein. The Michelson-Morley experiment had heads scratching before Einstein developed relativity.

http://en.wikipedia.org/wiki/Michelson–Morley_experiment#Fallout

Lorentz transformations predate relativity.

http://en.wikipedia.org/wiki/History_of_Lorentz_transformations

"Talent hits a target no one else can hit; Genius hits a target no one else can see." ~ Arthur Schopenhauer

 

[1977ub]

it is theoretically possible to increase the velocity of a spaceship until the velocity is well beyond the speed of light when compared to some initial velocity?

There are different ways to look at how this is impossible. If the traveler uses classical calculations to determine that he can use a certain amount of energy to get to 51% of c relative to the rest frame, and then plans that he'll just do it again from that 51% bringing him above c, when he actually executes that project, people in the RF find his clocks are quite slowed down once he executes his first thrust and he is not even all the way to his projected 51%, and his speed up from the 2nd thrust just doesn't take him beyond c. The more energy he puts out, the more his clock is slowed down, and it just can never happen.

Some explanations refer to how the ship becomes "heavier" and this requires more and more energy as measures from the rest frame.

It turns out that it takes infinite fuel/energy to get up to [what the rest frame calls] c (let alone beyond).

A lot of descriptions of this out there - you might enjoy reading a few.

http://www.edge.org/3rd_culture/hillis/hillis_p3.html

 

[Agerhell]

hmmm...

Looking at this again, brings up another question in my head.

If you had a train that circled the globe at the equator traveling at 2m/s, with another train riding on top of that train traveling at 2 m/s, a stationary observer would view the top train as traveling at 3.99999999982224 m/s, both coming and going.

Does that sound right?

And with a equitorial circumferance of ≈40,000,000 meters, the lower train would make the trip in 20,000,000 seconds. The upper train would do this in 10,000,000.00044 seconds, or ~4.4 tenth thousandths of a second longer than it should.

Does that sound right also?

Well, as the Earth is spinning at a velocity of 40000km/24 hours at the equator eastwards it matters if the train(s) are going east or west. When the trains are going east the rotation of the Earth has the same direction as the trains. Therefore the clocks onboard the trains will tick slower than an earthbound clock and the upper train will move slower than 4m/s as seen from an earthbound observer.

However, if the trains are going west the clocks onboard the trains will tick faster than that of an earthbound observer and the upper train will move faster than 4m/s as seen from the earthbound observer.

 

[jalsck]

Thanks for the replies. I understand that the ship cannot be observed to be traveling at or faster than c. All observes are on the ship and they can measure the acceleration of the ship. Is it true to say that after a period of measured acceleration (measured by the observers in the ship) the observers cannot conclude that their speed has changed?

 

[HallsofIvy]

What speed are you talking about? If there is a non-zero acceleration, then the observers on the ship would determined that some outside objects speed, relative to them, has increased. Of course, the ship's speed, relative to observers on the ship, is always 0.

Of course, if at some time, the speed of an object outside the ship is .9c relative to observers on the ship, and the ship accelerates, yes, the observers will see that speed increase to, say, .99c. Then later, it might increase to .999c, constantly increasing but always slightly less than c.

 

[Nugatory]

Thanks for the replies. I understand that the ship cannot be observed to be traveling at or faster than c. All observes are on the ship and they can measure the acceleration of the ship. Is it true to say that after a period of measured acceleration (measured by the observers in the ship) the observers cannot conclude that their speed has changed?

When you say "their speed has changed", I have to ask you what that speed is relative to.

If all the observers are in the ship and they have no outside reference point, then for all they know they are at rest the whole time and using their rocket engines to hover in a powerful gravitational field.

 

[jalsck]

When you say "their speed has changed", I have to ask you what that speed is relative to.

If all the observers are in the ship and they have no outside reference point, then for all they know they are at rest the whole time and using their rocket engines to hover in a powerful gravitational field.

Instrumentation on the ship would enable the observers to measure any distortions in space/time . For the thought experiment we could say there is only the ship and uniform space/time. I'm not attempting to argue that there is something wrong with relativity. The only observers are on the ship.

 

[jalsck]

I think the cause of some people's confusion is the phrase "nothing can travel faster than the speed of light". We see this statement in the media now and then. I believe it would be better to say "matter cannot be observed to be traveling at or faster than the speed of light".

An unmanned spaceship could potentially travel to a star 100 light years away and return in a matter of weeks (measuring elapsed time on the ship itself). If people were traveling on the spaceship and they could survive 2 G's of acceleration then the round trip would take just over 10 years (as measured by the people on the ship). They would have to accelerate at 2 G's until they were half-way there and then decelerate at 2 G's until they arrived at the star. Same heading home. A little over 202 years would have passed on Earth by the time they were home.

Disclosure: I'm not an expert on the subject. I'm learning... slowly.