throopw@rtp47.UUCP (Wayne Throop) (06/01/85)
> I've heard the twin paradox, and until recently I thought I had it > straight. I thought of something recently, though. When one twin > takes off, leaving the other here, why does the one in space age > more slowly? Why can't you use a reference frame travelling with > him and say that the earth is travelling at a great velocity? Am > I missing something? (Obviously I am.) > > Rob St. Amant You may have have heard the twin paradox before, but apparently you just realized why it is a paradox. After all, if one twin simply ages faster than the other, where is the paradox? The paradox is that *each* twin ages faster than the other (depending on your point of reference). To be more specific, imagine two folks of the same age, each traveling at some large fraction of C with respect to the other. Each will find that the other fellow seems to be aging slower. One way to resolve the paradox is to note that, unless one or the other fellow accelerates, they will never meet again, and hence the paradox can never be realized. On the other hand, if they *do* accelerate, that takes it out of the realm of special relativity, and *general* relativity accounts for the situation. (I have heard this hand-wave from several science popularizers, including either Arthur Clarke or Issac Asimov, I can't remember which). As I understand it, the situation *does* key on the acceleration, but the resolution of the paradox doesn't require general relativity. The key to the thing is what is considered "simultaneous" by each person in the situation. Let's tag the people here left-traveling and right-traveling (where "traveling" is simply in relation to the other person). The set of events in space-time considered "simultaneous" by the left-traveling person is *not* the same set of points considered "simultaneous" by the right-traveling person. For example, the left-traveling person considers the event "I've aged 1 year", simultaneous with event "the right-traveler has aged half-a-year". Also, the right-traveling person considers the event "I've aged 1 year", simultaneous with the event "the left-traveler has aged half-a-year". However, these two "facts" don't conflict, since they are derived from different reference frames. Now then, assume that the right-traveler accelerates after having traveled 1 year, and aquires the same velocity as the left-traveler. This acceleration changes the set of events that the right-traveler considers simultaneous. The right-traveler now considers the event "I've aged one year" to be simultaneous with "the left-traveler has aged two years". One way to look at it is that the right-traveler sees the left traveler age one-and-a-half years during the acceleration, and the paradox evaporates in a puff of exhaust gasses. :-) I hope this clarifies things rather than making them worse. It is a little easier to understand with diagrams, but I can't get my space-time charts to look good using simple character graphics. Sigh. -- Wayne Throop at Data General, RTP, NC <the-known-world>!mcnc!rti-sel!rtp47!throopw
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (06/05/85)
Throop's analysis of the twins paradox is essentially accurate. Accelerations turn out to have nothing to do with it, other than allowing one to determine that his state of motion has undergone a change. A carefully drawn space-time diagram, including the propagation of regularly-spaced light "ticks" from each twin, will show what is happening using ordinary special relativity. (Send the traveling twin from point A, where the stationary twin lives, and back.)
dgary@ecsvax.UUCP (D Gary Grady) (06/06/85)
Sometimes simplifying a notion leads to a misunderstanding (how's that
for a generic opening line?). For instance, we've found that telling
people that a floppy disk is like a phonograph record causes no end of
confusion. When you pull the disk out of the drive without stopping
the program, they are as astonished as they would be if ZZTop kept
caterwauling even after you removed the record from the turntable.
In relativity people are often told that "all motion is relative" and
they wonder then why we can't view the traveling twin as being at rest
and the other twin as taking the round-trip journey. In fact, asking
that question shows good physics intuition; they are looking for
symmetries.
The problem is that NOT all motion is relative. Inertial reference
frames are equally valid, which says something quite different. The
twin at rest is at rest in one inertial reference frame the whole time.
The traveling twin has to use her rocket motor to change directions, so
that twin is NOT at rest in an inertial frame for the whole time.
That's the non-symmetry in the Twin Paradox.
APS used to sell an excellent book of reprints on Special Relativity
that contained papers on the Twin Paradox (including the famous Dingle
argument in Nature) and other fascinating things. Of interest to
science fiction buffs is one on how far one could go accelerating
continuously at 1g. Turns out you could get to another galaxy in a
little over two decades of ship time, or get anywhere in the universe in
(I think) thirty-some years. Of course, no such journey could be
round-trip in any real sense of the word...
--
D Gary Grady
Duke U Comp Center, Durham, NC 27706
(919) 684-3695
USENET: {seismo,decvax,ihnp4,akgua,etc.}!mcnc!ecsvax!dgaryvallath@ucbcad.UUCP (Vallath Nandakumar) (06/11/85)
> > I've heard the twin paradox ... > > The paradox is that *each* twin > ages faster than the other (depending on your point of reference). >... I have a followup question. Excuse the imprecise language. If the universe is closed (I mean space is of the type where all straight lines meet), wouldn't the twins be able to meet without either of them having undergone acceleration? Who would then be older? Or does all this need the addtional math of general relativity? Or isn't our universe closed anyway? Or should I just go home and sleep it off? Vallath Nandakumar ucbvax!ucbesvax!vallath, esvax.vallath@berkeley.arpa
chrisa@azure.UUCP (Chris Andersen) (06/12/85)
> > The problem is that NOT all motion is relative. Inertial reference > frames are equally valid, which says something quite different. The > twin at rest is at rest in one inertial reference frame the whole time. > The traveling twin has to use her rocket motor to change directions, so > that twin is NOT at rest in an inertial frame for the whole time. > That's the non-symmetry in the Twin Paradox. > I wonder, can the twin who remains behind *really* be considered to be in an inertial reference frame? After all, he is subject to the gravity of the earth. Doesn't that make his reference frame non-inertial? > -- > D Gary Grady > Duke U Comp Center, Durham, NC 27706 > (919) 684-3695 > USENET: {seismo,decvax,ihnp4,akgua,etc.}!mcnc!ecsvax!dgary Chris Andersen USENET: tekronix!azure!chrisa
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (06/16/85)
> If the universe is closed (I mean space is of the type where > all straight lines meet), wouldn't the twins be able to meet > without either of them having undergone acceleration? In some cosmological models, they might meet. However, the original twin paradox assumes that special relativity can be applied. For cosmological problems one has to use heavier artillery.
gwyn@brl-tgr.ARPA (Doug Gwyn <gwyn>) (06/18/85)
> I wonder, can the twin who remains behind *really* be considered > to be in an inertial reference frame? After all, he is subject to the gravity > of the earth. Doesn't that make his reference frame non-inertial? The Earth has nothing to do with it. Let the "home" twin remain free- floating in space or something like that. The twin paradox is entirely special-relativistic.
ptb@ukc.UUCP (P.T.Breuer) (06/25/85)
In article <28@ucbcad.UUCP> vallath@ucbcad.UUCP (Vallath Nandakumar) writes: >> > I've heard the twin paradox ... >> >> The paradox is that *each* twin >> ages faster than the other (depending on your point of reference). >>... > >I have a followup question. Excuse the imprecise language. > >If the universe is closed (I mean space is of the type where >all straight lines meet), wouldn't the twins be able to meet >without either of them having undergone acceleration? >Who would then be older? >Or does all this need the addtional math of general >relativity? Or isn't our universe closed anyway? Or should I >just go home and sleep it off? > >Vallath Nandakumar >ucbvax!ucbesvax!vallath, esvax.vallath@berkeley.arpa This had me worried for a while about a year ago. General relativity isn't relevant since you can put the twins in a hypothetical universe which is flat but closed (say a cylinder with cooordinates (x,y,z,t) identified with (x+1,y,z,t)). If one twin sits at (0,0,0,0) and the other goes past him with speed v in the x-direction, they will certainly meet again without either having accelerated. In fact there's no paradox. If you work out the elapsed times for each twin (which is very easy once you refuse to be worried by the rather weird-looking coordinates the second twin uses if she thinks of herself as at rest), you'll see they're the same.