Exploring the Truth Behind Time Dilation: Debunking Common Misconceptions

[Mister T]

This means that a traveler observing only the clock "under his nose" as he travels will see (sic) the time in the other frame going more quickly, contrary to the popular argument that clocks should appear to run slower in the alternate frame.

No, he won't. The second sentence in my last post is backwards. When 7.5 hours elapse on the rocket clocks, to the rocket observer 4.5 hours elapse on the Earth clocks. But if the Earth clocks are synchronized with the destination clock in their rest frame, to the rocket observer the clock at the destination will be 8 hours ahead of Earth clocks.

 

[m4r35n357]

No, he won't. The second sentence in my last post is backwards. When 7.5 hours elapse on the rocket clocks, to the rocket observer 4.5 hours elapse on the Earth clocks. But if the Earth clocks are synchronized with the destination clock in their rest frame, to the rocket observer the clock at the destination will be 8 hours ahead of Earth clocks.

Not sure where the 4.5 comes from . . .
I thought we had agreed on the figures (I am referring to post #44); that my spaceship clock advances 7.5 units over the journey, and that the coordinate clock (which I read at both ends) advances 12.5 units.
In which case, the coordinate clock (the one at my coordinates) must run be seen to run more quickly by a factor of $\gamma$, not more slowly.

 

[Mister T]

Not sure where the 4.5 comes from . . .

Since the Earth observer measures the rocket clock to be running slow by a factor of $\gamma=\frac{5}{3}$, that is, the ratio $\frac{12.5}{7.5}$, then the rocket observer must measure Earth clocks as running slow by the same factor $\gamma=\frac{5}{3}=\frac{7.5}{4.5}$.

I thought we had agreed on the figures (I am referring to post #44); that my spaceship clock advances 7.5 units over the journey, and that the coordinate clock (which I read at both ends) advances 12.5 units.

That was in the frame of reference of the Earth observer.

In which case, the coordinate clock (the one at my coordinates) must run be seen to run more quickly by a factor of $\gamma$, not more slowly.

No, because the rocket observer must use two "Earth clocks" to compare to his own rocket clock. One is located on Earth and the other is located at the destination. I call them "Earth clocks" because they are synchronized in Earth's rest frame. The rocket observer will measure the clock located at the destination to be 8 years ahead of the clock located at Earth. When he arrives he will find the clock at the destination reading a time that is 12.5 years more than the clock on Earth read when he left, but that is not because he measures the clocks running faster than his clock, it's (mostly) because the two clocks are not synchronized in his frame.

 

[m4r35n357]

The rocket observer will measure the clock located at the destination to be 8 years ahead of the clock located at Earth.

Now there's a magical seemingly 8 out of nowhere, you really should say how you get to these numbers ;) All my numbers out are in the open in posts #43 and #44. There is no 6 (post #48), 4.5 (#71) or 8 (#73). I see nothing in your arguments that contradicts my assertion at the end of #72.

If I'm wrong I want to know, but by now I'm pretty sure it's right.

 

[Mister T]

The clock on Earth is synchronized with the clock at the destination. Both clocks are at rest relative to each other and separated by a proper distance of 10 light years. To an observer moving from Earth towards the destination at a speed of 0.8c, the clock at the destination will be (0.8)(10)=8 years ahead of the clock on Earth.

 

[PeterDonis]

To an observer moving from Earth towards the destination at a speed of 0.8c, the clock at the destination will be (0.8)(10)=8 years ahead of the clock on Earth.

But this is not a direct "measurement"; it's a calculation. Also, it's a synchronization, not a rate.

 

[Mister T]

Now there's a magical seemingly 8 out of nowhere, you really should say how you get to these numbers ;) All my numbers out are in the open in posts #43 and #44. There is no 6 (post #48), 4.5 (#71) or 8 (#73). I see nothing in your arguments that contradicts my assertion at the end of #72.

If I'm wrong I want to know, but by now I'm pretty sure it's right.

Ok, I've got a few minutes this morning and need a distraction to help avoid the sympathetic panic of finals week ...

First and foremost the 1st Postulate implies that if rocket clocks run slow compared to Earth clocks then Earth clocks run slow compared to rocket clocks.

Second, let's see where all the numbers come from. Recall that we already have the following, based on the statement of the scenario, the definition of $\gamma$, the choice of unprimed coordinates for Earth frame and primed coordinates for rocket frame, and the usual conventions regarding the relationship between the primed and unprimed coordinate systems. $x$ is in light years, $t$ is in years. Note that $c=1$.

$\beta=\frac{4}{5}$
$\gamma=\frac{5}{3}$
$\Delta x=10$
$\Delta t = 12.5$
$\Delta x' = 0$
$\Delta t' =7.5$

So, let's take a look at one of the Lorentz transformation equations: $$\Delta t'=\gamma(\Delta t-\beta \Delta x)$$
Substituting the values, we have
$$(7.5)=\frac{5}{3}(12.5-\frac{4}{5}10)$$
$$(7.5)=\frac{5}{3}(12.5-8.0)$$
$$(7.5)=\frac{5}{3}(4.5)$$

In the second from last line you see the 8.0 you asked about. In the last line you see the 4.5 you asked about.

Let's look at one more Lorentz transformation equation: $$\Delta x=\gamma(\Delta x'+\beta \Delta t')$$
$$(10)=\frac{5}{3}\Big(0+\frac{4}{5}(7.5)\Big)$$
$$(10)=\frac{5}{3}(6)$$
This last line contains the 6 you asked about.

I think I got 'em all.

 

[PeterDonis]

the 1st Postulate implies that if rocket clocks run slow compared to Earth clocks then Earth clocks run slow compared to rocket clocks.

Actually, that's not what the first postulate says. It says the laws of physics are the same in all inertial frames; but that's a statement about physical measurements, not about coordinate conventions, which is what "clocks running slow" refers to. A direct physical measurement would be something like the Doppler shift of light signals traveling between the rocket and Earth; the first postulate then says that the redshift of rocket signals observed on Earth must be the same as the redshift of Earth signals observed on the rocket (assuming that neither Earth nor rocket change their state of motion while the light is traveling).