I am going to use a variant of the Twin Paradox from
EinsteinLight with some very slight editing for the sake of clarity:

*Jane and Joe are twins. Jane travels in a straight line at a relativistic speed v to some distant location. She then decelerates and returns. Her twin brother Joe stays at home on Earth. …*

*Joe observes that Jane's on-board clocks (including her biological one), which run at Jane's proper time, run slowly on both outbound and return leg. He therefore concludes that she will be younger than he will be when she returns. On the outward leg, Jane observes Joe's clock to run slowly, and she observes that it ticks slowly on the return run. So will Jane conclude that Joe will have aged less? And if she does, who is correct? According to the proponents of the paradox, there is … symmetry between the two observers, so, just plugging in the equations of relativity, each will predict that the other is younger. This cannot be simultaneously true for both so, if the argument is correct, relativity is wrong.*

The author, Joe Wolfe, goes on to explain that asymmetry resolves the paradox,
an explanation that I do not find to be entirely satisfactory. He uses a flash animated pair of diagrams to support his argument:

**Are the space-time diagrams symmetrical?**Parts of them are. The first three years of the diagrams for Joe's frame and Jane's departing frame are symmetrical: each twin sends three greetings but only receives one. The last year and a half of Joe's frame and Jane's returning frame are also symmetrical: each sends two greetings and receives four. But the diagrams are not symmetrical in between. Why not?

Look
at Jane's diagram. From Jane's point of view,

*immediately*after she has fired her engines, she begins receiving Joe's greetings more frequently. This does not surprise her: she has gone from travelling away from the sender of the greetings and is now travelling towards him.
Jane
observes this change as soon as she turns around, which is for her the midpoint
of her voyage. (She now receives blue shifted messages instead of red shifted
ones. One could apply the same relativistic Doppler factor to the frequency of
arrival of the messages.) Joe, on the other hand, doesn't start to receive
messages at a higher frequency (blue shifted messages) until

*considerably after*the midpoint between Jane's departure and arrival, simply because*the effect of Jane's acceleration and changed reference frame takes a while to get to him*: he doesn't see the high frequency arrival of messages until the arrival of the first message that Jane sends after she turns around.**This is a clear example of where the asymmetry of the twins appears**. The causes of this asymmetry are the fact that Jane reverses direction and Joe does not, and the finite time that light takes to transmit this information to Joe means that Joe doesn't get the news immediately. Jane leaves one inertial frame and joins another, and she has the effect of that change immediately. Joe, on the other hand, doesn't notice the effects of Jane being in a different inertial frame until much later because she is a long way away from him when it happens. The asymmetry is as simple as that.

There
are a few if not hidden, then obscured assumptions, which are perhaps only
obvious when one takes time to search for them.

"(S)ome
distant location" appears sufficiently vague as to avoid creating problems
but an inherent assumption is that this location shares the same frame as Joe.

By
placing Joe on Earth we hide (or obscure) the other assumption, which is that
we also share the same frame as Joe.

"(Jane)
decelerates and returns" is distracting. As the author correctly points
out this is a point of asymmetry. However, a similar scenario (to be shown
shortly) shows that it doesn't matter which frame undergoes deceleration and a
change in direction – that of Jane or the entire universe. It is generally
assumed that the period during which Jane changes direction is insignificant
enough to ignore.

Finally,
"Jane travels in a straight line at a relativistic speed v" begs the
question "relativistic speed v relative to what?" The obscured
assumption is "relative to both Joe and the distant location" (and to
us, the readers). This is a direct consequence of the assumption that Joe and
the distant location share the same frame (and that we also share that frame).

Let
me provide a scenario which is analogous to the scenario described in the twin
paradox.

Joe floats in space (in a protective
space suit) with two clocks (marked as Joe’s).

Jane sits at one end of an
extremely long structure

**which also floats in space, unattached to anything bar Jane and the beacon**with another two clocks (marked as Jane’s). At the other end of the structure is a beacon. According to Jane, the structure has a length of L. Joe knows this.**Observe that Jane represents "the Earth" and the beacon represents "some distant location" in the twin paradox. The assumption that "the Earth" and "some distant location" have a fixed separation is inherent, but unstated, in the twin paradox.**

**Jane and Joe are sufficiently distant from any masses as to be considered to be alone in the universe, with no gravitational field in effect. The gravitation exerted by Jane and her structure on Joe is negligible. (The absence of gravitational effects is another unstated assumption in the twin paradox.)**

Jane and Joe pass each other
twice, at relativistic velocities of v and -v. Joe and the beacon pass each
other twice, also at relativistic velocities of v and -v (Jane and the beacon
are fixed to the same structure and hence share the same frame).

Four
noteworthy events take place:

1. Joe and Jane are
collocated as they pass for the first time. Their clocks begin measuring time
elapsed.

2. Joe and the beacon are
collocated as they pass for the first time. Joe's clocks are paused and the
beacon sends a message to Jane's clocks to pause.

3. Joe and the beacon are
collocated as they pass for the second time. Joe's clocks restart measuring
time elapsed and the beacon sends a message to Jane's clocks to resume
measuring time elapsed.

4. Joe and Jane are
collocated as they pass for the second time. Their clocks stop measuring time
elapsed and Joe and Jane exchange clocks such that they both have one marked
Joe’s and one marked Jane’s. Neither consults the other as they each attempt to
work out what the other's clock will read.

Observe
that I quite specifically do

**not**say who reverses direction. For the purposes of the thought experiment, we can say that both Joe and Jane were anaesthetised while one of them reversed direction, so that neither knows which has changed direction relative to any third observer (such as the reader).**By virtue of the scenario, both clocks are paused while any acceleration takes place and therefore****no****acceleration affects the measured time elapsed.**
There
is an asymmetry in this scenario, but Jane and Joe cannot determine on whose
part that asymmetry lies.

__Jane's calculations__:

Joe
is in motion relative to Jane. Jane calculates that the total time elapsed between
events 1 and 2 and events 3 and 4 will have been 2L/v and her clock confirms
that this is the case. Assuming that the
clock is sophisticated enough, Jane will be able to see that the time elapsed
between events 1 and 2 was (L/v + L/c).
This is because the signal to stop timing would have taken a period of
L/c to reach Jane’s clocks. The time
measured for the return trip, events 3 and 4, was (L/v - L/c), again due to the
time taken for the signal to reach Jane’s clocks (a start signal this time).

Jane
further calculates that because Joe is in motion, his clocks will run slow and
will show a time elapsed of (2L/v) / γ where γ = 1 / √(1-v

^{2}/c^{2}) (see*The Lightness of Fine Tuning (Part 2)*for an explanation of how this value of gamma (γ) is calculated, look up “time dilation” or*here*for confirmation that t’ = t / γ).
Jane
can check Joe’s clock and see:

time elapsed (event 1 – event
2) = (L/v + L/c) / γ;

time elapsed (event 3 –
event 4) = (L/v - L/c) / γ; so

total time elapsed = (L/v +
L/c) / γ + (L/v - L/c) / γ = (2L/v) / γ

Therefore:

Jane’s clock, according to
Jane = 2L/v

Joe’s clock, according to
Jane = (2L/v) / γ

__Joe's calculations__:

Jane
is in motion relative to Joe. Joe therefore calculates that Jane's structure is
foreshortened by a factor of 1/γ. Therefore the time elapsed while the entirety
of the structure passes twice will be (2L/v) / γ. Sure enough, Joe checks his
clock and sees that this is the case.

Working
out what Jane's clock will read is a little more complex. Joe knows that not
only is Jane's structure foreshortened, but that Jane's clocks will also run
slow by a factor of γ.

The
first period elapsed can therefore be calculated as follows (noting that Jane's
relative motion is in the same direction as the message from the beacon to Jane's
clocks):

t1 = γ.(γ.L/v + γ.L/(c-v))

= γ

^{2}.(L/v + L/(c-v))
= γ

^{2}.(L/v.(c^{2}-v^{2})/(c^{2}-v^{2}) + L(c+v)/(c^{2}-v^{2}))
= γ

^{2}.(c^{2}.L/v - Lv + Lc + Lv)/(c^{2}-v^{2})
= γ

^{2}.(c^{2}.L/v + Lc)/(c^{2}-v^{2})
but
since γ

^{2}= 1 - v^{2}/c^{2}= (c^{2}- v^{2})/c^{2},
t1 = (c

^{2}- v^{2})/c^{2}. (c^{2}.L/v + Lc)/(c^{2}-v^{2})
= (c

^{2}.L/v + Lc)/c^{2}
= L/v
+ L/c

The
same process can be used to calculate that the second period elapsed is (L/v - L/c).
The total time elapsed on Jane's clock, as calculated by Joe, will be 2L/v -
precisely the same as calculated by Jane and as shown on the clock labelled as
Jane’s.

Therefore:

Joe’s clock, according to
Joe = (2L/v) / γ

Jane’s clock, according to
Joe = 2L/v

If
both Jane and Joe agree about what the other’s clock should read, and this
agreement is confirmed by measurement, then there is no paradox.

I believe also that
my scenario demonstrates pretty conclusively that neither the acceleration nor the subsequent change to Jane’s inertial frame in Joe Wolfe’s scenario has a direct impact, since (in my scenario) there is no indication as to which
inertial frame changed.------------------------

Please note that Joe Wolfe clearly states that there is no paradox and nothing in this article should be taken as implying that he is a proponent of the Twin Paradox.

cool post. I have heard of the twin paradox but never knew the details in much depth. Now I feel like I have some sense of what it is and why it isn't really a paradox.

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