Special Relativity: Time Measurements on Two Systems - Mike's Question

[xcom2112]

I saw this question on special relativity on the internet:
"A spaceship traveling to Alpha Centauri, at 0.95C it takes 4.5 years to get there as measured on Earth.how long does it seems to the spaceship passengers?"

the answer was 1.4 years (Lorentz factor of 3.2).

this answer makes no sense to me.
if the spaceship passengers will measure on their clocks 1.4 years, they will know they are moving,
and this is a contradiction to the basic assumption of inertial systems ( as I understand it...)

I would expect them both to measure 4.5 years in their clocks.
(but the 4.5 years are not the same for them (the units are not the same) due to relativity)

Can anyone please help me to understand where and if I'm getting something wrong?

Thank you,

Mike

 

[Doc Al]

if the spaceship passengers will measure on their clocks 1.4 years, they will know they are moving,
and this is a contradiction to the basic assumption of inertial systems ( as I understand it...)

The spaceship knows it's moving relative to the earth. Or rather, from its view, the Earth is moving relative to the spaceship. (Everyone views themselves as at rest in their own frame.) Why in the world would you think that contradicts anything?

 

[phinds]

if the spaceship passengers will measure on their clocks 1.4 years, they will know they are moving

No, they will not.

and this is a contradiction to the basic assumption of inertial systems ( as I understand it...)

It would be if it were true

I would expect them both to measure 4.5 years in their clocks.

Well, you would expect incorrectly.

(but the 4.5 years are not the same for them (the units are not the same) due to relativity)

Absolutely not true, and likely the heart of your misunderstanding. Everybody everywhere always measures time at one second per second. You are confusing this with the number of seconds that pass for each of them.

Can anyone please help me to understand where and if I'm getting something wrong?

You are confusing time dilation with differential aging.

 

[PeroK]

if the spaceship passengers will measure on their clocks 1.4 years, they will know they are moving,
and this is a contradiction to the basic assumption of inertial systems ( as I understand it...)

There's no local experiment that the passengers can carry out on their ship that will tell them that they are in any sense moving at a definite velocity (relative to space itself).

If the ship started out on Earth, then they do know that they accelerated relative to their original rest frame and that they moved relative to that original frame.

They also know if they turn round and travel back to Earth that they again changed their inertial reference frame.

 

[Ibix]

Can anyone please help me to understand where and if I'm getting something wrong?

In the frame where the ship is stationary and the planets are moving, the distance between the planets is length contracted, so the ship's crew have no problems with their elapsed time being short - the distance was short too.

They also have no problem with the planets' clocks saying they took longer. Although they will see the planet clocks ticking slowly due to time dilation (since the planets are moving from their perspective), they will also see that the planets' clocks were incorrectly zeroed. This is due to an effect called the relativity of simultaneity, which is much more important to understanding relativity than is time dilation, but much less well known. Basically they say that Alpha Centauri's clocks show 4.5 years because they were started long before the ship left Earth.

 

[Herman Trivilino]

if the spaceship passengers will measure on their clocks 1.4 years, they will know they are moving,
and this is a contradiction to the basic assumption of inertial systems ( as I understand it...)

Earth and Alpha Centauri are assumed to be at rest relative to each other, and separated by a distance of 4.2 ly, as measured in a frame of reference where both are at rest. In a frame of reference where the ship is at rest that distance is only $\frac{4.2}{3.2}$ ly.

There is no contradiction because the two frames are equivalent. The passengers on the ship can claim that they're in motion, but people on Earth can make the same claim. Both claims are valid.

 

[Nugatory]

Can anyone please help me to understand where and if I'm getting something wrong?

You are not allowing for the relativity of simultaneity, the most overlooked consequence of the invariant speed of light (hence member @Orodruin's sig). Google for "Einstein train simultaneity" to learn more about it. The essential point is that "at the same time" is not universally true; things that happen "at the same time" according to one observer do not happen "at the same time" according to observers.

In the situation here, we have four relevant events - and remember that an even is something happening at a single point in space at a single time:
1) At a single point on Earth the spaceship leaves its lauchpad, we set an earthbound clock to zero, and we set a clock on on the ship to zero.
2) The clock on Earth reads 4.5 years. Of course 4.5 years pass between event #1 and this event according to someone standing next to clock and watching it tick.
3) The spaceship lands on alpha centauri and at the moment of touchdown the spaceship clock reads 1.4 years.
4) The clock on Earth reads $.45\approx1.4/3.2$ (here 3.2 is the Lorentz factor, and the relevance of this event will become clear in a moment). Of course .45 years pass between event #1 and this event according to someone standing next to clock and watching it tick.

When we use the frame in which the Earth is at rest, we conclude that events #2 and #3 happened at the same time, 4.2 years after the ship left earth. We will say that 4.2 years passed on Earth while only 1.4 years passed on the ship.

However, if we use the frame in which the ship is at rest (so the Earth is behind the ship and moving away at .95c while alpha centauri is in front of the ship and approaching at .95c) we conclude that events #2 and #3 do not happen at the same time. Using this frame, event #3 happens at the same time as event #4 and we will say that 1.4 years passed on the ship while .45 years passed on earth.

One description says that the ship is moving and its clock is running slow, while the other description says that the Earth is moving and its clock is running slow. Both descriptions are equally valid and there is no reason to prefer one over the other.