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Many of the explanations that I've seen of the Twin Paradox seem to rely on the role of acceleration to explain the differences between the twins. But why can't we simply resolve it just via "length contraction" alone?

If twin A stays on Earth while twin B flies off to some planet located $d$ meters away (that is, $d$ meters away in Earth's frame where both planets are rest) at some relativistic speed $v \approx c$, don't we then already have the requisite "asymmetry" in terms of what both twins see anyway?

Twin A will see Twin B flying away at $v$ a distance of $d$ meters for $d/v$ seconds, while Twin B will see Twin A flying away at $v$ for a distance of $d/\gamma$ for $t/\gamma$ seconds (because length contraction shortens the road length for twin B). Hence, both twins will already "not see the other take the same path" so we would already have an asymmetry from the get-go and therefore the claim in the paradox that "both twins see the same thing" already goes into the trash can.

Since pretty much all the explanations rely on twin B's "acceleration" to explain the paradox, what am I missing here? It seems way too simple to merely say "length contraction = no twin paradox" (even though neither twin will agree on the trip distance anyway) without having to invoke any fancy acceleration and/or change in reference frames, so I'm pretty sure there's something I'm missing here.

**Edit:

In other words -- with time dilation & length contraction in SR, it easily "already" makes sense that different observers can observe time and distance differently. For example, let's say I'm timing my friend who is sprinting on a 100m track from some start to finish line, while both of us are holding a stopwatch and ruler in each hand. His procedure is:

  1. Accelerate instantly to $v \approx c$ from the start line
  2. Then instantly start the stopwatch
  3. Run at constant $v$ from start to finish line
  4. Stop the stopwatch
  5. Decelerate instantly back to rest in my frame

So, from my point of view, he just ran at a speed of $v$ for 100m in 100m/v seconds. But according to him, he also ran at a speed of $v$ (relative to the start/mid/finish line), but instead for $d/\gamma$ meters and $d/(\gamma v)$ seconds from his point of view (he is carrying his own ruler and own stopwatch). So, while we both agree on our relative $v$, we do not agree on the exact distance traveled and the exact duration (only their ratio, i.e v = distance/time). And this makes sense -- speed is a ratio of distance/time, and so just because we agree upon that ratio does not imply that we "also" have to agree on exactly what numerator (distance) and denominator (time) were used.

So, what am I missing here? If we accept that SR allows different observers to measure distances and times differently, there seems to be no contradiction here nor any possible "twin paradox" to speak of. In other words (even if the runner makes a second lap back to the start line again), what is described by the "twin paradox" would actually be my expectation instead, rather than some odd/unexpected result!

ManRow
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1 Answers1

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So in your set up, the inbound and outbound legs of the trip look the same. If I write (per leg):

$$ \tau^A_A = \frac D v $$

for the proper time that $A$ sees $A$ experience, and

$$ \tau_B^A = \tau^A_A / \gamma =\frac 1 {\gamma} \frac D v$$

for the proper time that $A$ sees $B$ experience, then we have:

$$ \tau_B^B = \tau_B^A $$

for the proper time $B$ sees $B$ experience.

No paradox.

The problem appears when we consider $B$ looking at $A$'s proper time, which is dilated relative to $B$'s clock:

$$ \tau_A^B = \tau_B^B/\gamma = \frac 1 {\gamma^2} \frac D v$$

So when we add both legs:

$A$ sees $A$ tick off:

$$ T_A^A = 2\frac D v$$

and $B$ ticks for:

$$ T_B^A = \frac 2 {\gamma}\frac D v = T_B^B $$

in both frames, but $B$ see $A$'s clock only tick off:

$$ T_A^B = \frac 2 {\gamma^2}\frac D v$$

and at the end, $A$ and $B$ share world-lines again so $B$ has missing time:

$$ T_A^A-T_A^B = \frac{2D}v(1 - \frac 1 {\gamma^2}) =\frac{2Dv}{c^2} $$

Where did it go?

JEB
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