JesseM said:
But you could equally well say that he could prove the previous measurements (made on a ruler/clock system at rest relative to the Earth) were "fictional" by an analogous method.
phyti said:
Hi JesseM;
Those were measurements in his own frame, no dilation, no contraction.
What does that matter? If we were talking about an alien planet which moved path Earth at 0.8c inertially, and which had
always been moving at that speed relative to Earth rather than accelerating away from Earth like the ship, would that somehow change your answer about whose measurements are "fictional"?
phyti said:
time for spacecan-earth event: s=0 e=0 a=3.2
time for spacecan-alpha event: s=3 e=1.8 a=5.0
Yes, these are the simultaneous clock-readings in the frame where the ship is at rest and Earth and Alpha Centauri are moving at 0.8c.
phyti said:
Here you show the spacecan time for the trip is 3 yr, the same as when it moves at .8c!
Of course, all frames must agree that if the relative velocity between the ship and Earth+Alpha Centauri is 0.8c, and the distance between Earth and Alpha Centauri is 4 ly in their own rest frame, then the ship's clock will have advanced 3 years between the moment it passes Earth and the moment it passes Alpha Centauri. That's just how relativity works, all frames must always make the same predictions about local physical events like what a clock reads at the moment it passes right next to some landmark. The point is that there is no real basis for calling one frame's measurements "real" and the other's "fictional".
phyti said:
If you synchronize the clocks in the e-a frame, they use the 'rest' distance of 4 lyr.
What are you talking about? They don't use "distance" at all, they just set off a flash at the midpoint, and zero their clocks at the moment the light from the flash reaches each one.
phyti said:
The time difference for Earth and alpha-c is 2sbg=5.33 yr., with s= spatial separation (2), b=v/c (.8) and g=gamma factor (1/.6) This includes time dilation.
If you're talking about the time difference between clocks at Alpha Centauri and Earth as seen in the frame where they are moving at 0.8c, it's 3.2 years.
JesseM said:
Alpha Centauri clock now reads 0.6 * 5.333... = 3.2 years
phyti said:
Here because you used the contracted distance (2.4, a result of time dilation), you have shortened the time twice.
The contracted distance is not "a result of time dilation", they are separate phenomena. If the spaceship-observer looks at an enormous ruler which is at rest with respect to himself, and there are clocks at the 0 light-year mark and the 2.4 light-year mark which are synchronized in the ruler rest frame, then the reading on the 0 light-year mark clock as Earth passes it will be the same as the reading on the 2.4 light-year mark as Alpha Centauri passes it; thus at the "same moment" (in this frame) that Earth is at the location of one clock, Alpha Centauri is at the location of the other clock, 2.4 light years away.
phyti said:
With the traveler on earth, all three agree on a 4 lyr distance.
The synchronization of earth-alpha clocks is irrelevant because they are in the same frame.
Yes, but if they are synchronized in their own frame, then when viewed in another frame they will be out-of-sync. Do you disagree?
phyti said:
With s not moving:
time for spacecan-earth event: s=0 e=0 a=u
time for spacecan-alpha event: s=5 e=3 a=u+3
With s moving:
time for spacecan-earth event: s=0 e=0 a=0
time for spacecan-alpha event: s=3 e=5 a=5
You seem to be completely confused about the relativity of simultaneity. If Earth and Alpha Centauri clocks are synchronized in their own rest frame, then when s is not moving relative to them, of course they will be synchronized in the current rest frame of s (since this is the same as their own rest frame), so if e reads 0 then a should read 0 as well, not "u". Then when s is moving relative to them, Earth and Alpha Centauri's clocks will be out-of-sync in the current rest frame of s, so if e reads 0 then a reads 3.2.
phyti said:
If you don't already use them, the Minkowski space-time diagrams are very helpful in maintaining clarity for exercises like these.
Minkowski diagrams simply illustrate the coordinates assigned to events by different frames which are related to one another by the Lorentz transformation. Let's call frame A the rest frame of Earth and Alpha Centauri, and the coordinates of that frame x and t, while frame A' is the frame where they are moving at 0.8c and the ship is at rest (after it has accelerated), and the coordinates of that frame x' and t'. In the A frame, say that Earth is always at position x = 0 light years, Alpha Centauri is always at position x = 4 light years. Their clocks match the coordinate time of this frame, so that at t=0 years the Earth clock reads 0 and the Alpha Centauri clock reads 0, at t=5 years the Earth clock reads 5 and the Alpha Centauri clock read 5, etc.
Now, let's look at the event of the Earth clock reading 0, and also look at the event of the Alpha Centauri clock reading 3.2. Event #1 has coordinates x=0, t=0, while event #2 has coordinates x=4, t=3.2. Now we can use the Lorentz transformation to find the coordinates x' and t' of both events in the A' frame. The Lorentz transformation equations are:
x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)
where gamma = 1/sqrt(1 - v^2/c^2).
With v=0.8c, gamma = 1.6666...
So for event #1 we have:
x' = 1.666... * (0 - 0.8*0) = 0
t' = 1.666... * (0 - 0.8*0) = 0
And for event #2 we have:
x' = 1.666... * (4 - 0.8*3.2) = 2.4
t' = 1.666... * (3.2 - 0.8*4) = 0
So, you can see that in this frame, both events--the Earth clock reading 0, and the Alpha Centauri clock reading 3.2--happen simultaneously at time t'=0. You can also see that in this frame, event 1 happens at x'=0 while event 2 happens at x'=2.4, showing that they are 2.4 light years apart in this frame.
phyti said:
Einstein was intelligent enough to know that merely moving did not affect distant clocks or spatial intervals physically, i.e. alter physical processes elsewhere. He felt confident enough to state this as postulate one of the SR theory.
Er, where did he say that? The first postulate of SR is that the laws of physics work the same in every frame.
phyti said:
I call it a 'black box' theory because you put in a set of values and a different set comes out, without explaining the details of how it works. The main source of confusion is in my opinion, the distinction between the physical event and the perception of the event. A simple observation, you see a star. You don't see the star 'now' as a coincident/synchronous event, you see it as it was, e.g., 1 million yrs ago. That's the time difference between the event and its perception.
When I
see an event has nothing to do with what coordinate I assign it in my reference frame. If I look through my telescope and see a star exploding at the same time I drop a quarter in the ground, and the star is 1 million light-years away in my frame, then this means that in my frame the event of the star exploding has a t-coordinate 1 million years less than the t-coordinate of my dropping the quarter on the ground. The Lorentz transformation deals with the time-coordinates assigned to events in this way, not with the times that I
see events.
