According to Mach's principle and the principle of Equivalence it is
as valid to regard the Earth as stationary with the stars orbiting it
once every 24 hours, as it is to regard the Earth as rotating on its
axis every 24 hours.

If we take the reference frame of a stationary Earth with the stars
orbiting it every 24 hours, given that every star is at least one
light-year distant, if it orbits the Earth in 24 hours then you would
think that it must be travelling faster than the speed of light which
is supposed to be impossible.

Do the stars in this refernce frame travel faster than the speed of
light? If not how do they get round in 24 hours?
 

Note: I am aware that not all Physicists accept the Principal of
Equivalence  but there is a large body of respectable opinion which
does. What I want to understand is how those who accept the Principle
of Equivalence would answer this question.

Hello again, antell

   So you know now that to ask about whole universe rotating or not,
   you would have to consider a cosmological model,  which is an
complex
   issue requiring tensor calculus.
   That Issue is active, albait not mainstream, as disscussed in this
review
   by a retired physics professor here:
   http://universe.gsfc.nasa.gov/docs/roadmap/Cosmo-22.pdf

   The mainstream cosmological model, big-bang and expansion model,
   as you know, has a preffered frame of reference, model in which the
   stars are not rotating, but just moving away from each other.
   You have cleverly reformulated your question and now you talk about
the
   stars few light-year away, so we can consider a single galaxy,
   which (like earth or a solar system) is an inertial system,
   meaning system freely falling through  space, as described here
   http://www.fnal.gov/pub/inquiring/questions/centripetal.html

   The concept of intertial system is simple, fundamental and people,
   serious pople, are still discussing it:
   http://www.lns.cornell.edu/spr/2001-10/msg0036114.html

   It implies that there is a preffered system, which can be attached
   to your galaxy, and  you can determine that this system is rotating
with
   respect to the other galaxies, in accordance with Mach's Principle:
   http://www.bun.kyoto-u.ac.jp/~suchii/mach.pr.html

  But you do not care about that,
  you are somewhere inside your galaxy
  in a spaceship (it can be spaceship Earth),
  and you can neglect the effect of the other galaxies
  since the rotation of a galaxy is very very slow
  (typically millions of years per revolution).
  You are just looking at the nearby stars.
  You see they rotate and you know that laws of physics are covariant,
  formulated so that you can pick any frame of reference
  and you pick one attached to your spaceship.
  You call the period of rotation you see a day.

  You ask how come that the stars you see far away, are orbiting you
  faster then speed of light, completing whole circle in  one day.

  Search term : rotation general relativity

  does bring some papers, often related to the rotating black holes
and
  associated Kerr metrics or Kerr space.

  This one calculates the  metric tensor g(a,b) in your frame:
   http://hypatia.ss.uci.edu/lps/home/fac-staff/faculty/malament/papers/RotationNoGo.pdf

   This one says that your question is ambiguous  but still  procceds
to
   calculate very complex tensor fields which will arise  in your
frame.
   philsci-archive.pitt.edu/documents/disk0/00/00/01/17/
PITT-PHIL-SCI00000117-00/RelOrbitalRotation.pdf


    But  you said that your eyes glaze over
    when you see tensor equations which are
    longer than three line sand  you just  want
    to know "how come".

    OK.:
You have to stop thinking in classical terms, such as
    'in one day this happened'. There is no univeral time.

    Your clock  is the relative rotation you see , 
    (you see the same star overhead again
 and YOU call that a day.

  It is measuring your time. The remote stars are
    at the bottom of very deep gravitational well. You know that it
    is true, since if you drop somethings, it will fly away from you;
    It falls 'down' to the stars. (in the inertial system that would
be
called centrifugal force, but we know it's just gravity, right).
    Since the stars are in the gravitational well, their clock runs
very slow.
    They have lot of time to complete the circle at sub-luminal spped.

    Search terms: gravitational red shift

    will bring many papers on this, including experimental check:
    http://phyun5.ucr.edu/~wudka/Physics7/Notes_www/node99.html

    In your case the graviational filed is a bit strange, since you
insist
    on using non-inertial frame (one attached to your ship, instead
    of one the  attached to your galaxy) but numbers will agree: the
far out
    stars will have even slower clocks.

    In conclusion, I would like to recommend a book. It is on this
list
     http://www.ucolick.org/~burke/class/gr/books.html

     and it is available at:
     http://www.amazon.com/exec/obidos/ASIN/0486655822/

     All tensor expressions in it are shorter then 3 lines (well,
     almost all) and it can truly be called: fun with tensors.

     Hope this help - and please do provide feedback - so that we
     know if this was useful.

     hedgie.

I think this is a very good answer which has made me realise that I
need to ask another question!

The Earth turns, the stars don't move as quickly as they appear to,.

Hi,

This is my first time using google answers, and I can't figure out how
to answer, so i'll try this.  If it just shows up as a comment, oh
well.

This Principle of Equivalence is a bit new to me, but I'll take your
word for it that that a "stationary earth" interpretation is one of
its consequences.

In essence, the difference between the two interpretations is really
just a transformation of space-time coordinates.  F(x,y,z,t) =
(x',y',z',t').

Fine.  In the frame where the earth is stationary, the stars can be
considered to be moving around the earth at horrendous velocities.

However, the laws of physics in this frame are very different. 
Einstein's theory of relativity (one of the consequences of which is
the predictions that stars cannot move faster than the speed of light)
is one of many theories that do not apply.  Other examples:

1) Gravity.  In the typical inertial frame, the earth is rotating, and
a geostationary satellite orbits the earth once every 24 hours.  This
orbit is made circular due to the effects of gravity.  In the
'stationary earth frame', the satellite is also stationary, just
hanging there in space.

2) Newton's laws of motion.  The classic "F=ma".  In the "stationary
earth frame", lots of things are accelerating, but with no force
applied to them.  Stars, the sun, other planets, etc.

3) The coriolus (sp?) effect.  In the classic inertial frame, the
rotation of the earth gives rise to an effect that causes hurricanes
to rotate the way they do.  In the "stationary earth frame",
hurricanes are just weird.

So, while it is true that it is possible to do a relatively
straightforward transform of space-time coordinates between the two
frames, the laws of physics are different.  The "inertial frame" is
generally preferred, because the laws of physics are easier to
formulate and make much more sense.

Ken Kelley (Ph.D. in physics)

Your first statement is incorrect. The principal of equivalence states
that there is no local, experimental test to distinguish between a
uniformly accelerating frame of reference and a uniform gravitational
field. A rotating frame of reference is not uniformly accelerating,
and so is not globally equivalent to any gravitational field. We have
various experimental tests available to see that we're in a rotating
frame of reference -- balls curve due to Coriolis forces, distant
stars move in circles every day, etc. As noted by another commentator,
the laws of physics are different in a rotating frame.

You might say, "Yes, but any small part of the rotating reference
frame is locally (in space and time) uniformly accelerating, and
therefore, by the principal of equivalence, obeys standard laws of
physics with a gravitational field". And indeed, all the experimental
tests to realize that it's part of a globally rotating frame require
looking outside the region where it is locally uniform. Eg., by using
a telescope to look at distant stars.

See http://einstein.stanford.edu/STEP/information/data/gravityhist2.html
for more on equivalence, including a mention of Mach's role.

Hello, Antell. I think I have an answer for you...

The short answer to your question is "no". In the frame reference of
the earth, the stars will be moving slower than the speed of light.
This seemingly counter-intuitive answer has to do with the way gravity
(and acceleration) affects space itself in general relativity.

In general relativity, gravity curves space: In a gravitational field,
light goes in curved lines. And in relativity, the path of light rays
DEFINES space itself.

Since being in an accelerating (or rotating) reference frame is
equivalent to being in a gravitational field, the previous statement
fact also holds for accelerating systems.

In other words, when you accept the highly unusual reference frame of
a rotating earth, the geometry of the universe becomes unusual as
well. Under these conditions, there is no gurantee a star going full
circle around the earth really travelled the distance it seemed to
have travelled.

Without any complex math, one can prove the stars are moving below the
speed of light by the following thought experiment:

Let us examine two light rays emitted by one of the stars: One light
ray which is emitted in the same direction the star seems to be
rotating, and another light ray in the opposite direction.

What is the speed of the two light rays, relative to the earth? We
know it must be 186,000 miles per second, because light ALWAYS travels
at this speed in ANY reference frame (rotating or not)! In other
words, their velocities are -c and +c.

Now, let us wait 24 hours. By now, the star and the two light rays
have completed full circle around the earth. One light ray is 24
light-hours ahead of the star, while the second light ray is 24
light-hours behind it. So although it might seem absurd, we've proven
that the star's velocity relative to the earth is between -c and +c,
which is less than the speed of light!

This result might seem absurd at first glance, but if you follow the
logic closely you will see that it is clearly valid.

I hope this helps you. If you still want further clarification, don't
hesitate to ask.

Later,

Odyssey.

I'm grateful for your answer, odyssey2001-ga, because it is the first
answer I have had which shows understanding of the question, the other
answers simply missing the point!

I don't find your example of the two light rays emitted from one of
the stars takes us very far however. This may be because I've missed
something but this example seems to raise as many questions as it
answers. For example, we can't detect the light ray emitted by the
star orbiting the earth. Granted we can postulate its existance but
this doesn't seem to be going much further than simply saying "of
course nothing goes faster than the speed of light, therefore the
light ray must make it round before the star therefore the star must
be travelling slower than the speed of light". It is possible to
accept this logic but still not understand how it can be given the
distances involved.

Your statement that:

"Since being in an accelerating (or rotating) reference frame is
equivalent to being in a gravitational field, the previous statement
fact also holds for accelerating systems.
 
In other words, when you accept the highly unusual reference frame of
a rotating earth, the geometry of the universe becomes unusual as
well. Under these conditions, there is no gurantee a star going full
circle around the earth really travelled the distance it seemed to
have travelled."

makes a lot of sense but I'm trying to understanding it in a little
more detail. In ordinary geometry the circumference of a circle is 2
pi times the radius. Now I know that according to GR, in the case of a
rotating frame the circumference is a bit less that 2 pi but can it
ever be less than the radius? More to the point is it possible for the
radius to be say 1 light year whilst the circumfrence is only one
light day? It may be that the answer to this is "Yes" but is is so
counter-intuitive that I need if possible something in the way of an
explanation.

I hope you don't mind that no one has dared to research your question
yet.  This is one of the most difficult (and interesting) questions I
have seen posed on Google Answers.  While these comments may not be
what you had in mind, I hope you find them thought provoking, or at
least amusing.  In any event, at least they are free.

To continue the thought experiment began by others here, some
questions to ponder.
If stars are orbiting the Earth at velocities greater than the speed
of light, would they be visible to the human eye?

If stars vary in their distance from the Earth, wouldn't each star
need to orbit at a different faster-than-light speed in order to
maintain it's relative position?  If so, would we anticipate the
appearance of a star to vary depending upon the speed it travels. 
Again, would it's movement at speeds faster than the speed of light
make it invisible to us?

Is Moby correct?... "Nothing Can Stop Us Now, We Are All Made of
Stars" <grin>
Perhaps the "black space" in outer space is comprised of stars that we
are unable to see because they are orbiting the Earth at speeds
greater than the speed of light.  On the other hand, we would expect
these stars to exert gravity, whether or not their "faster than the
speed of light" light is visible or invisible to us.

Peter Kosso's book on philosphy and physics might help, entitled
"Appearance and Reality"
http://www.oup-usa.org/toc/tc_0195115155.html

An interesting overview of the debated theories featuring Dicke's
experiments can be found here:
http://www.aspsky.org/mercury/mercury/9404/dicke.html

And finally, another thought that won't answer your question:
Imagine if there is no "time" as such.  Imagine if all of time has
already "happened."  Everything that seemed to happen before is
actually happening "now"... events that we perceive to happen in the
future actually happened "now".  What if what we observe as time is
simply the rate at which our senses and mind are able to process
events, all of which have already occured?  This could explain Deja
Vu.  This (and a six-pack) could explain fortune tellers.  This could
explain why observed time seems to "slow down" for the individual
observer at various times, such as an athlete making a last second
play, or a driver seeing the world in slow motion during a car
accident.

OK, now I return to surfing between the Sci-Fi channel and MTV.  Good
luck in your search for this answer.

Glad I could be of assistance, Antell.

As I've thought, my reply would require further clarification. So here
I'll try to clarify the points you've found problematic:

You've mentioned the fact that the light in my thought experiment
would not been seen on earth. While true, this could easily be
remedied by placing mirrors at the two relevant points on both sides
of the star. Or to be a bit more realistic, placing two small planets
which partially reflect the light to earth.

Now we could see the light on earth. Moreover, we would be able to
know exactly where the light came: from the observed position of the
two planets. The star itself would be clearly between these two
points, thus proving it was going below the speed of light.

As for your second question: "Now I know that according to GR, in the
case of a
rotating frame the circumference is a bit less that 2 pi but can it
ever be less than the radius?"

The answer is clearly yes. There is nothing special about the point
where the circumference and the radius are equal. The ratio between
the two can be any positive number, depending on the geometry of space
inside the circle. And in the case of the rotating earth, it really
gets as low as my answer imply it does.

To further understand why rotating reference frame give such strange
results, let us think for moment what a "rotating reference frame"
means:

Rotationary motion is accelerated motion - this is evident by the
centrifugal force felt by people standing on a rotating object.
Furthermore, since the centrifugal force always pulls OUTWARD, we know
the direction of acceleration is always INWARDS - towards the center
of the rotating object.

But because the object is rotating, the direction of "inwards" (at a
specific point) also rotates. Thus in a rotating system the direction
of acceleration CHANGES as the system rotates.

Since acceleration is equivalent to gravity, and gravity is equivalent
to space curvature, we reach the following conclusion:

Given a rotating reference frame, the curvature of space itself is
rotating! As the geometry of space itself is rotating, it "drags" the
stars with it.

Now you might ask: But if the geometry of space is so "crazy", why
don't we feel it?

The answer is, we DO feel it. The centrifugal force which makes
gravity a little weaker at the equator can be seen as a result of this
"crazy geometry". The same holds for the Coriolis forces - a unique
phanomena existing only in rotating frames of reference.  And
of-course, we see the outside universe whizzing around us at seemingly
impossible speeds...

I hope this serves as the explanation you were looking for.

Later,

Odyssey

My understanding of the principle of equivalence is that it does not
hold for rotating reference frames. In fact I believe that it is alway
possible to state absolutely whether or not you are rotating which is
completely different from the case of uniform motion and uniform
acceleration.

The standard thought example is a bucket of water in an otherwise
empty
universe. If the bucket is rotating then the surface of the water will
become
curved. Taken literally Mach's principle would imply that the water's
surface
would remain flat. 

So I am willing to go out on a limb and state that I do not believe in
Mach's principle. So I know this isn't an answer to your question but
I think
your assumption that the two frames of reference are equilivant is
wrong.

Neil

Thanks Odyssey.

I studied mathematics and physics to "A" level standard. If you are
not familiar with school qualifications in England, as a rough guide
this means that the equations and concepts you meet in Special
Relativity hold no fears for me but when it comes to Tensors and the
equations you meet in General Relativity my eyes start to glaze over!

Nevertheless I would like to be able to understand General Relativity
to the extent that my background allows me to (I have not studied
mathematics and physics beyond A level: I studied law at university).

Can you recommend a book or a website which is aimed at my level.
Essentially I'm looking for explainations pitched at about the level
at which you so kindly answered by question.

I'm totally flummoxed by Odyssey's argument concerning the two
hypothetical light rays. He says:

"By now, the star and the two light rays have completed full circle
around the earth. One light ray is 24 light-hours ahead of the star,
while the second light ray is 24 light-hours behind it."

But how do we know that the first light ray is ahead of the star at
all? Perhaps the star has kept pace with it for the whole 24 hours or
even overtaken it. And since the question was whether
faster-than-light speed is possible at all under this paradigm, it
seems to me a tad arbitrary to invoke the speed of light in the
argument like this. Aren't you just hiding the conclusion in the
premise?

I'm no physicist - just a humble logician - so maybe I'm missing
something here!

A question for Hedgie:

Your answer, in general, makes sense. Especially the "gravity well"
part.

But how did you reach the conclusion that the gravity well is so deep?

If the stars' clocks are really as slow as your explanation imply,
wouldn't we expect to see a huge gravitation redshift? And if the
"reverse" gravity well was as deep as you state it is, how come the
feeble gravity of the earth is more than enough to cancel it's
effects?

I agree that the "well" would be wide - many light-years wide. But I
cannot see why one would say it is "deep". Are you sure this is
correct?

Odyssey

A question for Odyssey and/or Hedgie:

The simple way I am thinking about the question of the validity of
using the earth as a reference frame is this:

If we are considering constant (non-accelerating) motion then no frame
of reference is "better" than any other. All are equally valid and
there is nothing special about measuring speed relative to the Earth
because (to Physicists) there is nothing special about Earth.

If we are considering accelerating motion in a straight line
("non-rotating") then again, no frame of reference is "better" than
any other.

If we are considering rotational motion then the laws of Physics
"work" whether we imagine the Earth stationary with the universe
rotating around it, or whether we imagine the Earth spinning on its
axis. However the Earth spinning is a "preferred" way of looking at it
because in the reference frame in which the Earth is spinning, every
Galaxy is moving away from our own in a straight line ("non-rotating")
which is much simpler mathematically than a rotating universe.

Have I got this about right?

Antell, you got it right. :-)