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I'm having a heck of a time understand how to account for time dilation and space contraction in the following problem:

A mothership traveling at 0.620 c toward the Earth launches a landing craft. The landing craft travels in the same direction with a speed of 0.790 c relative to the mothership ( as measured by aliens on the mothership ). As measured on the Earth, the spaceship is 0.220 ly from the Earth when the landing craft is launched.

(a) What speed do the Earth-based observers measure for the approaching landing craft?

(b) What is the distance to the Earth at the moment of the landing craft's launch as measured by the observers on the mothership?

(c) What travel time is required for the landing craft to reach the Earth as measured by observers on the mothership?

(d) What travel time is required for the landing craft to reach the Earth as measured by observers on the Earth?

(e) What travel time is required for the landing craft to reach the Earth as measured by observers on the landing craft?

I tried approaching the problem by making a few tables to keep track of what data I know and the reference frames in which it was taken, with reference frames labeled down the first column and the names of each object observed listed across the top of the other columns:

$ \begin{array} {|r|c|c|c|} \hline Distances\ (\ ly\ ) & Earth & Landing\ Craft & Mothership \\ \hline Mothership & (b) & 0 & 0 \\ \hline Landing\ Craft & \ & 0 & 0 \\ \hline Earth & 0 & 0.220 & 0.220 \\ \hline \end{array} $

$ \begin{array} {|r|c|c|c|} \hline Records\ Proper\ Distance\ to\ & Earth & Landing\ Craft & Mothership \\ \hline Mothership & & \checkmark & \checkmark \\ \hline Landing\ Craft & & \checkmark & \checkmark \\ \hline Earth & \checkmark \\ \hline \end{array} $

$ \begin{array} {|r|c|c|} \hline Travel\ Time\ (\ yr\ ) & Landing\ Craft\ \to\ Earth & Records\ Proper\ Time \\ \hline Mothership & (c) & \ \\ \hline Landing Craft & (e) & \ \\ \hline Earth & (d) \\ \hline \end{array} $

$ \begin{array} {|r|c|c|c|c|} \hline Velocities\ (\ c\ ) & Earth & Landing\ Craft & Mothership \\ \hline Mothership & -0.620 & 0.790 & 0 \\ \hline Landing Craft & \ & 0 & \ \\ \hline Earth & 0 & (a) & \ \\ \hline \end{array} $

As you can see from the tables, question (b) is asking for the distance from the mothership to the earth, as measured by observers on the mothership. So, I've labeled the relevant cell in each table with the corresponding question.

The original question provides a velocity for the mothership, but doesn't say from which reference frame that observation was made. We will need two observations of velocity in the Lorentz transform for (a); so, I assume that the given mothership velocity of 0.620 c is in fact an observation from the mothership that the earth is approaching the mothership at 0.620 c. Furthermore, I make observations in this reference frame positive when they record motion away from the mothership, and negative when they record motion toward the mothership.

Also, the distances of the origin of each reference to itself, as well as the velocities of each reference frame relative to itself are all zero. So, I filled those cells with zeroes.

Given the Lorentz Velocity Transformation, finding (a) is simple enough:

$ \LARGE{ (a) = u_x^{'} = \frac{u_x - v}{1 - \frac{vu_x}{c^2}} = 0.946\ c } $

But, answering (b) seems a bit harder.

Assuming the volume of the landing craft is significantly smaller than a cubic lightyear ( which seems like a very safe assumption), and given that the mothership and landing craft are very close together at the moment of separation ( much closer than one lightyear ), observations on the mothership of the distance from the mothership to the landing craft ( at least at the moment of separation ) must record the proper distance.

Similarly, observations on the landing craft of the distance from the landing craft to the spot where it had been attached to the mothership should record the proper distance at the moment of separation. ( Nevermind that this is a magical landing craft that will have a velocity of 0.790 c the very next instant with zero acceleration... )

So, the mothership and landing craft both observe proper distance from each other at the moment of separation. Also, because they are so close to each other at the moment of separation, I assume they both observe the earth to be the same ( distorted ) distance away from themselves. ( The landing craft would observe the earth to be closer to itself than would the mothership, however, if we take into account the landing craft's miraculous acceleration. )

But, both the landing craft and mothership are moving very fast relative to the speed of light. And both observe themselves to be very far away from earth. So, they don't record proper distances when they make these observations.

Furthermore, the book in which this question is found defines proper time as:

the time interval between two events, measured by an observer who sees the events occur at the same point in space.

Observers in Earth's reference frame do not see the two ships separate at the same place they see the ships arrive at earth. In fact, there are no two events observed in this question that take place in the same reference frame, bar the landing craft observing that the mothership leaves from its origin and the earth arrives at its origin.

Is this a correct application of the book's definition of proper time? How am I to approach (b) conceptually?

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    $\begingroup$ Re "There are no two events that take place in the same reference frame" --- events do not "take place in reference frames". Events take place, period. Any (inertial) reference frame encompasses all events (in special relativity). $\endgroup$ – WillO Apr 23 '16 at 2:47
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For part (b) just ignore acceleration issues and assume that the mothership and landing craft share the same position at the moment of separation. You are on the right track with the idea of using proper time, but not quite the way you describe.

As a preamble, the problem with the Earth-mothership distance is that in the Earth frame it is determined by means of two simultaneous events, say $E_0:(x=0, t = 0)$ and $M_0 : (x =-D, t = 0)$ at the Earth and mothership locations respectively (at the moment of separation as seen from Earth), but these events are no longer simultaneous in the mothership frame and therefore cannot provide the corresponding distance from the mothership's point of view. Moreover, if we attempt to rectify the situation by defining an event $E_1$ such that $M_0$ and $E_1$ are simultaneous in the mothership frame, we find that we don't quite know how to specify the distance to $E_1$, since this is exactly what needs to be determined.

One way out of this confusion is to consider Earth and mothership proper times from the moment of landing craft separation to the moment when the mothership itself reaches Earth. In the Earth frame, the rendezvous time follows both from mothership's observed distance and velocity, and from time dilation as applied to mothership's proper time, estimated again from distance and velocity but in the mothership's frame. This gives easily the desired relationship.

A follow-up question: How is all this compatible with the time dilation symmetry between the Earth and mothership frames? The catch is again that what Earth observes as events simultaneous with the landing craft separation at $M_0$ is not what the mothership observes as events simultaneous with the same separation $M_0$. It can be verified (additional exercise?) that at the moment of separation the mothership observes Earth clocks showing an advanced time, hence the mothership sees Earth undergoing time dilation too.

Another important point: there is no reason to assume that the landing craft and the mothership observations at the moment of separation coincide, to the contrary.

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