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Consider two events A and C. They have same value of x' and time interval between them is $\Delta \tau$. This is the proper time interval between the events. The time interval between the events in the ct-x frame is $\Delta t$.

$ \Delta \tau = t^\prime_c -t^\prime_a \\ \Delta t = t_c -t_b$

Using the idea of invariant separation, I can write

$ c^2 (t_b - t_c)^2 - (x_b -x_c)^2 = c^2 (t^\prime_b - t^\prime_c)^2 - (x^\prime_b -x^\prime_c)^2 \\ c^2 (t_b - t_c)^2 - (x_b -x_c)^2 = c^2 (t^\prime_a - t^\prime_c)^2 - (x^\prime_b -x^\prime_c)^2 \\ c^2 (\Delta t)^2 = c^2 (\Delta \tau)^2 - (\Delta x^\prime)^2 \\ $

Then, If I use Lorentz transformation for $(\Delta x^\prime)^2$, we get $ \Delta t = \frac{\Delta \tau}{\gamma}$

What am I doing wrong here? Why am I getting time contraction instead of time dilation?

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  • $\begingroup$ The $\Delta x'$ should be zero. In the statement you said that the $x'$ coordinats are the same. $\endgroup$
    – MBN
    Feb 7, 2016 at 12:32
  • $\begingroup$ x' are same for A and C, not for B and C. $\endgroup$ Feb 7, 2016 at 13:20
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    $\begingroup$ Why are you comparing $\Delta \tau$, the (primed) time interval between $a$ and $c$, with $\Delta t$, the time interval between $b$ and $c$? Just pick two events. $\endgroup$
    – knzhou
    Feb 7, 2016 at 23:21
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    $\begingroup$ It's not important at all, but just so you know, the "official" symbol you're looking for is not "\triangle" but "\Delta". Doesn't matter for this post (the difference between $\triangle$ and $\Delta$ - and we all know what you mean), but if you ever use LaTeX to type up a paper to be published, a PI might care. $\endgroup$
    – D. W.
    Feb 7, 2016 at 23:23
  • $\begingroup$ For starters, your expression for $\Delta t$ seems wrong. If this is the time interval between $A$ and $C$ in the unprimed frame, it should be $t_c-t_a$. But you've got $t_c-t_b$. Is this just a typo? $\endgroup$
    – WillO
    Feb 8, 2016 at 0:16

2 Answers 2

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When comparing times in the context of time dilation, you are comparing the time interval between two events. The two events must be the same two events in each reference frame.

So lets focus on the time interval between events $A$ and $C$. By invariance of the spacetime interval: $$-c^2 \Delta t^2 + \Delta x^2 = -c^2 \Delta {t^\prime}^2 + \Delta {x^\prime}^2. $$

In the primed reference frame $A$ and $C$ are colocated, so $\Delta x^\prime=0$, and we identify $\Delta t^\prime=\Delta\tau$ as the proper time.

$$-c^2 \Delta t^2 + \Delta x^2 = -c^2 \Delta\tau^2$$ A little algebra gets us: $$\Delta t^2 \left[ 1 - \left(\frac{\Delta x}{c \Delta t}\right)^2 \right] = \Delta\tau^2 \implies \Delta t = \gamma\Delta\tau.$$

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So, just to clarify your approach:

You take events A and C occurring at the same location $x'_a = x'_c$ but different times $t'_a$ and $t'_c$ in the primed frame. You also take event B occurring at the same time as A in the primed frame, $t'_b = t'_a$, but at the same location as C in the unprimed frame, $x_b = x_c$. For events B and C you then apply the invariance of the space-time interval $$ c^2 (t_b - t_c)^2 - (x_b -x_c)^2 = c^2 (t^\prime_b - t^\prime_c)^2 - (x^\prime_b -x^\prime_c)^2 $$ and in view of $x_b = x_c$, $t'_c = t'_a$, obtain $$ c^2 (t_b - t_c)^2 = c^2 (t^\prime_a - t^\prime_c)^2 - (x^\prime_b -x^\prime_c)^2\\ c^2 (\Delta t)^2 = c^2 (\Delta \tau)^2 - (\Delta x^\prime)^2 \\ $$ Lastly, the Lorentz transform of $\Delta x' = x^\prime_b -x^\prime_c$ gives $\Delta x' = \gamma(x_b - v t_b) - \gamma(x_c - v t_c) \equiv \gamma v \Delta t$, and you conclude, correctly, that $c\Delta t = c\Delta \tau/\gamma$.

Your "problem" is that regardless of the substitution $t'_c = t'_a$ your final relation still gives $(t_b - t_c) = \gamma(t'_b - t'_c)$. Let's tally up what we have:

If we use events B and C in the unprimed frame, but A and C in the primed frame, we find that

"The time interval between events B and C occurring at the same location in the unprimed frame appears time dilated wrt the time interval between event C and an event A occurring in the primed frame at the same location as C but at the same time as B".

The last piece of information, "occurring in the primed frame at the same time as B", is the crucial one: we can replace event A with any other event, at any location, as long as it "occurs in the primed frame at the same time as B".

Otherwise, if we dispense with event A and simply refer to events B and C only, we just find that

"The time interval $(t_b - t_c)$ between two events B and C occurring at the same location in the unprimed frame appears time dilated in the primed frame".

Typical time dilation. Check!

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