How about a quick recap of this...
Just out of interest, I'm re-iterating the above, emphasizing what each observer observes and making clear the distinction between what is directly observed
(the raw sensory data) and what might be calculated
from those observations, from the point of view of different inertial reference frames. In this way, I hope it demonstrates how the interpretation of raw observations with respect to different reference frames works, and the precise sense in which each observer "sees" the other's clock running slowly.
All numbers rounded to 2 decimal places.
Let person A stay stationary in an inertial reference frame.
Let person B travel from A, at a speed of 0.866c relative to A, to a point that is 8.66 light-years away as measured by A. We'll call that point C.
Assume no nearby large masses.
The Lorentz factor is:
1 / sqrt(1 - v2
For a speed of 0.866c that is 2.
So, while in transit, person B measures (WRT his reference frame) the distance traveled to be half of what person A measures it to be: 4.33 light-years. And person A measures (WRT his reference frame) person B to be length contracted - squashed in the direction of travel - by a factor of 2.
What A and B Directly Observe
The relativistic Doppler Effect describes the relative tick rates that are directly observed, and is given by:
sqrt( (1 + v/c) / (1 - v/c) )
With v/c = 0.866, this results in a value of 3.73.
If they both zero their clocks when B sets off, then according to B's clock, when B experiences the event of reaching C, he sees that his own clock reads 5 years. That's how long the journey takes him, as measured against his reference frame travelling at 0.866c. 4.33/0.866 = 5. When he gets there, if he momentarily stops and looks back at A, he sees
A's clock reading (10-8.66) = 1.34 years. If he wants to, he may calculate
from this, and from his knowledge of the distance to point C and the speed of light that:
1. measured against his (B's) outward moving
reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 2.5 yrs.
2. measured against his (B's) momentarily stopped
reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 10 yrs.
3. measured against his (B's) inward moving
reference frame, the "Clock A tick event" which is simultaneous with the "B reaching C" event is 17.5 yrs.
But what B actually sees (from a distance) is A's clock reading 1.34 years. And, as Halc says, the A-ticks that he actually sees don't suddenly jump forward by 15 years. But B's judgement as to which A-ticks are simultaneous with which B-ticks does, due to the frame change. This can be seen clearly in this Minkowski diagram that illustrates the lines of simulteneity WRT the two reference frames in this particular case, in the reference frame of A:
It can be seen from this type of diagram that the discontinuity at the point of inertial reference frame switch causes B to calculate
this jump based on
both his observations and the definition of simultaneity.
Likewise, when A sees
B reaching C he sees his own clock reading (10+8.66) = 18.66 years, and sees B's clock reading 5 years. So, if he wants to, he may calculate
from this, and from his knowledge of the distance to point C, measured against his (A's) single reference frame, that the "Clock A tick event" which is simultaneous with the "B reaching C" event is 10 years.
What A sees (from a distance) is B's clock reading 5 years.
Person B then, as measured by his reference frame, takes another 5 years to return from C to A. So at the end of the journey, A and B can both see that B's clock reads 10 years and A's clock reads 20 years.
So, in summary, this is a table of the raw observations of A and B of their own and each other's clocks:
Code: Select all
A as seen by A B as seen by B A as seen by B B as seen by A
Start 0 0 0 0
Observed mid-point 18.66 5 1.34 5
End 20 10 20 10
So you can see from this that, just in terms of their raw uninterpreted senses, they each see the other's clock running 3.73 times as slow during the first leg and they each see the other's clock running 3.73 times as fast during the second leg, in agreement with the relativistic Doppler effect calculation. And at the end they will both agree that B has aged half as much as A.
What A and B Calculate from their Observations about Simultaneity
As I said, above is the situation simply as observed
by the two people. If A and B decide to do some calculations based on that raw sensory data then the results will depend on the reference frame that they use to do those calculations against. This is what Halc was talking about. This is the sense in which they both "see" each other's clocks running slow on both legs of the journey, and it's where the concept of simultaneity comes in.
If B wants to work out which of A's "clock tick events" he ought to regard as being simultaneous with the "B arriving at C" event, he has to consider what he actually sees (the raw sensory data) and then account for the distance from which he sees it and the speed of light. The way that he does this depends on the reference frame against which he does it. But, at C, he switches between two different inertial reference frames.
WRT the first reference frame (that he is stationary WRT during leg 1) he will judge that the "A's clock tick event" that is simultaneous with the "B arrives at C" event is 2.5 years. Which he will judge to be simultaneous with the "B's clock tick event" that is 5 years.
WRT the second reference frame (that he is stationary WRT during leg 2) he will judge that the "A's clock tick event" that is simultaneous with the "B arrives at C" event is 17.5 years. Which he will judge to be simultaneous with the "B's clock tick event" that is 5 years.
On the other hand, B remains in the same inertial reference frame throughout. So he will make just one judgement. He will judge that the "B's clock tick event" that is simultaneous with the "B arrives at C" event is 5 years and that it is simultaneous with the "A's clock tick event" which is 10 years.
Here's a new table to show all this. In this table the columns show the above calculations for A (in his one reference frame) and B (in his two reference frames). The two numbers in each column show the A and B clock events that each observer calculates to be simultaneous.
Code: Select all
A B1 B2
Start 0,0 0,0 0,0
Mid-point 10,5 2.5,5 17.5,5
End 20,10 20,10 20,10
So you can see from this that both A and B calculate
, based on their direct observations, that the other is travelling 2 times more slowly than themselves on both legs of the journey. And the final asymmetry is a result of B's reference frame change. This reference frame change also highlights that the question of whether two events can be regarded as simultaneous depends on the movements of the observer.
You can also see from this that another way of looking at it is that the act of B turning around (accelerating) causes (in B's judgment) A's clock to go forward 15 years. This is where the alternative treatment of this situation, invoking General Relativity and the Equivalence Principle (between gravity and acceleration), comes in. We can either treat this as an idealized instantaneous (zero time, infinite acceleration) transition between two inertial reference frames (SR) or as a transition between two inertial reference frames via a non-inertial reference frame of constant finite acceleration for finite time (GR).
The defining feature of a "special case", as opposed to the more general case, in physics, is that in the special case simplifying assumptions - idealizations - are made. In this instance, the special case is Special Relativity (hence the name). And the idealized simplifying assumption is that the acceleration during the turnaround can be allowed to go arbitrarily high so that the time taken to turn around (the time spent in a non-inertial frame) can be allowed to get arbitrarily close to zero. Hence the non-inertial frame is removed.
This is why, when we consider the problem using Special Relativity, we get the sudden, discontinuous jump in the Minkowski diagram, from 2.5 to 17.5 years. It reflects the discontinuity of the sudden change in inertial reference frame. If we allow ourselves to use General Relativity then we can replace that sharp discontinuity in the diagram with a curve. The rate of curvature would show the size of the acceleration. And the jump from 2.5 to 17.5 years would not be all in one go.