A MACHO is passing the solar system at the velocity of .5c.

[OneEye]

Okay, everyone, I know that this is sounds like a set-up question. And that's because it is. But please humor me, because I'm trying to get this straight:

A MACHO is passing the solar system at the velocity of .5c.

Therefore, the MACHO experiences a time dilation. Its time is slowed by a factor of 4:3.

Right?

 

[selfAdjoint]

The MACHO doesn't experience any time dilation in its own frame. It sees the Earth experiencing that dilation, and of course Earth sees the MACHO's time dilated.

 

[OneEye]

The MACHO doesn't experience any time dilation in its own frame. It sees the Earth experiencing that dilation, and of course Earth sees the MACHO's time dilated.

Excellent. This is exactly the point that I was trying to set up. Thank you!

So, then, we have a principle which says that time dilation between any two comoving reference frames is always collateral: Both frames see the other as experiencing time dilation. Correct?

(BTW, got my math wrong. 0.5c produces a dilation factor of about 1000:866, not 4:3. Forgot to sqrt.)

 

[selfAdjoint]

Comoving is the wrong word here. Two frames are comoving if the velocity difference between them is (instantaneously) zero. Like a car driving along beside a train and going the same speed.

Now the two frames in your example, in order for the dilations to be valid, have to be inertial, which essentailly means not accelerated. So with that proviso, what you say is true; the dilations are reflexive. This is shown by the Lorentz transformations, which are matrices. If you use some L.T. A to transform from the Earth frame to the MACHO frame, then to transform in the other direction you use A^{-1}.

So now, on to the punch line. And it better not be about twins!

 

[OneEye]

Selfadjoint - thanks for the terminological clarification. Also, thanks for the substantiation.

No, the punchline is not about twins. It is this: It is well-remarked that a person living in the Rocky Mountains will experience an "extra second" or so of life because of the relativistic effect associated with the greater tangential velocity resulting from the constant rotational velocity of the earth.

Surely, given what we have discussed above, this cannot be an effect of special relativity. Right?

Also, years ago, when I worked on GPS, it was common for the officers to breathlessly remark how that it was necessary to compensate for relativistic effects when calculating one's position and time from the satellite signals. Someone on the board just recently pointed out that this was due to both SR and GR effects, and that the GPS clock actually appeared faster than those on the earth.

So, the punchline is this: Does the fellow on top of Pike's Peak really get that "extra second" relative to the sea-level population of the earth. If so, is that really an SR effect?

 

[russ_watters]

So, the punchline is this: Does the fellow on top of Pike's Peak really get that "extra second" relative to the sea-level population of the earth. If so, is that really an SR effect?

He gets a little due to GR and gives a little back due to SR. The "extra second" (if that's accurate) is the net result.

edit, clarification: a person on Pikes Peak notices no change in lifespan, its just that he thinks he's a second or two older than a guy near sea level.

 

[OneEye]

...because, if we are considering only SR, then the guy on Pike's Peak gets his second from the perspective of a guy in New York, but the guy in New York gets a second from the perspective of the guy on Pike's Peak.

Right?

 

[Janus]

...because, if we are considering only SR, then the guy on Pike's Peak gets his second from the perspective of a guy in New York, but the guy in New York gets a second from the perspective of the guy on Pike's Peak.

Right?

The point here is to remember that both are in acceleratedframes (they are both traveling in circles and circular motion is accelerated motion). SR can deal with accelerated frames but the application is somewhat different than that when dealing with inertial frames.

 

[russ_watters]

...because, if we are considering only SR, then the guy on Pike's Peak gets his second from the perspective of a guy in New York, but the guy in New York gets a second from the perspective of the guy on Pike's Peak.

Right?

It sounds like you're trying to create a twins paradox (sorry, SA). None exists here. Only one gets a second and it isn't just because of SR. If you want to limit it to just SR, then neiter gets a second.