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A Tale of Two Clocks: A Thought Experiment
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Titus Piezas III
science forum Guru Wannabe


Joined: 10 Mar 2005
Posts: 102

PostPosted: Thu Jul 06, 2006 4:57 am    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

Ben Rudiak-Gould wrote:
Quote:
titus_piezas@yahoo.com wrote:
1. For the sake of argument, suppose there is this gigantic clock, call
it clock A, at point A which is a billion light-years from Earth.


(snip)

Thanks for the thorough analysis, Ben. It was just something I thought
of while I had too much time in my hands. ;-)

(Normally, I write stuff about number theory. See "Ramanujan and
3^3+4^3+5^3 = 6^3" at
http://www.geocities.com/titus_piezas/ramanujan_page9.html )

-Titus
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Ben Rudiak-Gould
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Joined: 04 May 2005
Posts: 382

PostPosted: Wed Jul 05, 2006 8:39 pm    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

titus_piezas@yahoo.com wrote:
Quote:
1. For the sake of argument, suppose there is this gigantic clock, call
it clock A, at point A which is a billion light-years from Earth.

That's pretty far. Strictly speaking you can't ignore general relativity at
that scale, but I suppose it doesn't matter.

Quote:
Suppose further we have this very powerful telescope that enables us to
see its display and it shows "1:00 pm, Year 0". Exactly a year
later it shows "1:00 pm, Year 1".

What, no month or day? Well, I suppose that doesn't matter either.

Quote:
2. Earth builds a similar gigantic clock, call it clock B, at a point B
just outside the orbit of Pluto and its display shows "1:00 pm, Year
0" though it is not started yet.

3. Now suppose we have this advanced spaceship that can travel just
under the speed of light c. Its top velocity is fc where f is a
fraction very close to 1 such that by using the formulas for
relativistic time dilation, while a billion years pass outside the
travelling ship, only a year would have passed inside it.

Your f is normally called beta. Presumably you want gamma = 10^9, which
corresponds to beta ~ 1 - 5 x 10^-19.

Quote:
4. Two powerful telescopes will be put on board: one pointed towards
clock A and the other at clock B so the crew (one of which, Albert, has
a twin on Earth) can look at the two clocks at the same time.

5. Clock B is started, the ship takes off and maintains velocity fc,
and travels for a billion years towards clock A. Since the ship is not
traveling at 1c, after a billion years it will not cover the full
distance D of a billion light-years but will stop short at a point P =
D(1-f) light-years away from point A. For example, if f = 0.999999999,
then point P is just 1 light-year away.

Actually f = 0.9999999999999999995, and point P will be a few thousand
kilometers away.

Quote:
6. The ship stops.

Ouch.

Quote:
1. From point P, using the onboard telescope, what year will be seen on
the display of clock A?
2. What year will be displayed for clock B?

To answer this you can use the Doppler shift formula, which is

k = sqrt ((1 + beta) / (1 - beta))

In this case k is about 2 x 10^9. Clock A will have been blueshifted by this
factor during the whole trip, and clock B will have been redshifted by the
same factor. We've already established that the trip takes one year of
proper time. So assuming that both clocks were seen to read 0 at the start
of the trip, clock A will read two billion years, and clock B will read one
two-billionth of a year, or about 16 milliseconds.

Quote:
Possible answers:

i) Since clock A has been running for 2 billion and 1 years already,
and since point P is 1 light-year away, the crew will see its display
as "Year 2,000,000,000".

Right.

Quote:
ii) Since clock B has been running for 1 billion years and first light
from it passed by point P only a year ago, then the crew will read its
display as "Year 1".

Wrong, but right general idea. Your physical intuition seems good.

Quote:
3. If the crew in their subjective travel time of one year looked at
clock A the whole time, what would they see? Would the display numbers
have rapidly changed from Year 1 to 2,000,000,000 in that span of time?

Yes. (With such a large blueshift, they wouldn't actually be able to see the
clock, but I suppose that doesn't matter.)

Quote:
Would that imply two billion years sped by to their year? Shouldn't
the dilation be only a billion to one?

The gamma factor is mainly important for the Lorentz transform. To work out
what you see through a telescope, you need the Doppler shift factor instead.

Quote:
4. If they also looked at clock B the whole time, if it only went from
Year 0 to 1, would that imply they observed its display moving at a
normal speed relative to clocks onboard their ship?

No, it will appear to run very slowly the whole time (and be redshifted out
of visibility).

A thorough discussion of the twin paradox can be found here:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

it includes an analysis in terms of telescopes and Doppler shifts.

-- Ben
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Sorcerer1
science forum Guru


Joined: 09 Jun 2006
Posts: 410

PostPosted: Wed Jul 05, 2006 8:32 pm    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

"Paul Cardinale" <pcardinale@volcanomail.com> wrote in message
news:1152124894.784484.37370@l70g2000cwa.googlegroups.com...
| You're confusing seeing with observing.
|
| Paul Cardinale
|
You are confusing physics with sanity.

"Tom Roberts" <tjroberts137@sbcglobal.net> wrote in message
news:P4Hqg.60105$Lm5.3167@newssvr12.news.prodigy.com...

| This is PHYSICS, not math or logic, and "proof" is completely irrelevant.

Humpty Roberts has his hand in the beloved insane FAQs. Do not expect
math or logic.
Androcles.
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Paul Cardinale
science forum Guru Wannabe


Joined: 08 May 2005
Posts: 177

PostPosted: Wed Jul 05, 2006 6:41 pm    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

You're confusing seeing with observing.

Paul Cardinale
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AllYou!
science forum Guru


Joined: 08 May 2005
Posts: 1088

PostPosted: Wed Jul 05, 2006 12:59 pm    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

<titus_piezas@yahoo.com> wrote in message
news:1152089300.082997.316560@l70g2000cwa.googlegroups.com...
Quote:
Hello all,

This might be of some interest:

"A Tale of Two Clocks: A Thought Experiment in Relativity"

I. Introduction

Since this is just a thought experiment, we need the following:

a. two very large digital clocks
b. two very powerful telescopes
c. a very fast spaceship

II. Experiment

1. For the sake of argument, suppose there is this gigantic clock,
call
it clock A, at point A which is a billion light-years from Earth.
Suppose further we have this very powerful telescope that enables us
to
see its display and it shows "1:00 pm, Year 0". Exactly a year
later it shows "1:00 pm, Year 1".

2. Earth builds a similar gigantic clock, call it clock B, at a
point B
just outside the orbit of Pluto and its display shows "1:00 pm, Year
0" though it is not started yet.

3. Now suppose we have this advanced spaceship that can travel just
under the speed of light c. Its top velocity is fc where f is a
fraction very close to 1 such that by using the formulas for
relativistic time dilation, while a billion years pass outside the
travelling ship, only a year would have passed inside it.


Wrong. For the observer in the ship, 1/1,000,000,000 years would pass
outside the ship for every year passed in the ship. For the observer
outside the ship who views the ship as travelling at fc,
1/1,000,000,000 years would pass inside the ship for every year passed
outside the ship.

[snipped the rest which is based upon this fallacy]
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shuba
science forum Guru Wannabe


Joined: 08 May 2005
Posts: 160

PostPosted: Wed Jul 05, 2006 12:02 pm    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

Titus Piezas wrote:

Quote:
A good discussion can be found in the Wikipedia article
http://en.wikipedia.org/wiki/Twin_paradox

That article deliberately confuses the isssue with historical
baggage and philosophical fluff.


---Tim Shuba---
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Sue...
science forum Guru


Joined: 08 May 2005
Posts: 2684

PostPosted: Wed Jul 05, 2006 8:58 am    Post subject: Re: A Tale of Two Clocks: A Thought Experiment Reply with quote

titus_piezas@yahoo.com wrote:
<< Hello all, >>
Quote:

....and a tho't experiment.



Abstract
Einstein addressed the twin paradox in special relativity
in a relatively unknown, unusual and rarely cited paper
written in 1918, in the form of a dialogue between a
critic and a relativist. Contrary to most textbook versions
of the resolution, Einstein admitted that the special
relativistic time dilation was symmetric for the twins,
and he had to invoke, asymmetrically, the general relativistic
gravitational time dilation during the brief periods
of acceleration to justify the asymmetrical aging.
Notably, Einstein did not use any argument related to
simultaneity or Doppler shift in his analysis. I discuss
Einstein's resolution and several conceptual issues
that arise. It is concluded that Einstein's resolution using
gravitational time dilation suffers from logical and
physical flaws, and gives incorrect answers in a general
setting. The counter examples imply the need to reconsider
many issues related to the comparison of transported
clocks. The failure of the accepted views and
resolutions is traced to the fact that the special relativity
principle formulated originally for physics in empty
space is not valid in the matter-filled universe.

C. S. Unnikrishnan
Gravitation Group,
Tata Institute of Fundamental Research,
Homi Bhabha Road, Mumbai 400 005, India
http://www.iisc.ernet.in/currsci/dec252005/2009.pdf
-----

Sue...
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Titus Piezas III
science forum Guru Wannabe


Joined: 10 Mar 2005
Posts: 102

PostPosted: Wed Jul 05, 2006 8:48 am    Post subject: A Tale of Two Clocks: A Thought Experiment Reply with quote

Hello all,

This might be of some interest:

"A Tale of Two Clocks: A Thought Experiment in Relativity"

I. Introduction

Since this is just a thought experiment, we need the following:

a. two very large digital clocks
b. two very powerful telescopes
c. a very fast spaceship

II. Experiment

1. For the sake of argument, suppose there is this gigantic clock, call
it clock A, at point A which is a billion light-years from Earth.
Suppose further we have this very powerful telescope that enables us to
see its display and it shows "1:00 pm, Year 0". Exactly a year
later it shows "1:00 pm, Year 1".

2. Earth builds a similar gigantic clock, call it clock B, at a point B
just outside the orbit of Pluto and its display shows "1:00 pm, Year
0" though it is not started yet.

3. Now suppose we have this advanced spaceship that can travel just
under the speed of light c. Its top velocity is fc where f is a
fraction very close to 1 such that by using the formulas for
relativistic time dilation, while a billion years pass outside the
travelling ship, only a year would have passed inside it.

4. Two powerful telescopes will be put on board: one pointed towards
clock A and the other at clock B so the crew (one of which, Albert, has
a twin on Earth) can look at the two clocks at the same time.

5. Clock B is started, the ship takes off and maintains velocity fc,
and travels for a billion years towards clock A. Since the ship is not
traveling at 1c, after a billion years it will not cover the full
distance D of a billion light-years but will stop short at a point P =
D(1-f) light-years away from point A. For example, if f = 0.999999999,
then point P is just 1 light-year away.

6. The ship stops.

III. Questions

1. From point P, using the onboard telescope, what year will be seen on
the display of clock A?
2. What year will be displayed for clock B?

Possible answers:

i) Since clock A has been running for 2 billion and 1 years already,
and since point P is 1 light-year away, the crew will see its display
as "Year 2,000,000,000".
ii) Since clock B has been running for 1 billion years and first light
from it passed by point P only a year ago, then the crew will read its
display as "Year 1".

Further questions:

3. If the crew in their subjective travel time of one year looked at
clock A the whole time, what would they see? Would the display numbers
have rapidly changed from Year 1 to 2,000,000,000 in that span of time?
Would that imply two billion years sped by to their year? Shouldn't
the dilation be only a billion to one?
4. If they also looked at clock B the whole time, if it only went from
Year 0 to 1, would that imply they observed its display moving at a
normal speed relative to clocks onboard their ship?

IV. Conclusion

First, a disclaimer. Regarding the value for f given here, it
is just an example chosen for simplicity and is not meant to be
understood as the exact value that will give a time dilation of a
billion to one (the exact formula involves the Lorentz factor).
Second, while this was just a thought experiment, there are in
fact natural objects that can serve as a "clock A", namely quasars,
most of them more than a billion light-years away from us.
Third, question 3 gives a twist to the so-called "twin
paradox". A good discussion can be found in the Wikipedia article
http://en.wikipedia.org/wiki/Twin_paradox. Since time has slowed down
in the moving ship, one might naturally assume the crew will find the
external world to be correspondingly "sped up". But the special
theory of relativity gives a counter-intuitive answer to this. Albert
and his twin on Earth would see the image of each other, via a
relativistic Doppler shift, as aging at a certain rate times his own.
So whose time has slowed down?
The resolution to this paradox is when Albert turns around
(hence, an acceleration which changes frames) and returns. But
supposing he does not return? Supposing he travels for a billion years
(relative to his twin on Earth) at the near light-speed fc. Thus,

5) What would the crew see as the year on the display of clock A a few
milliseconds before they stop at point P? What would it be when they
exactly stop?

Of course, answers/comments are welcome.


Titus Piezas III
July 5, 2006
tpiezasIII(at)gmail.com (Pls remove antispam III.)
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