The identification of a non-inertial reference
frame is used to justify the accelerated-frame
solution to the twins paradox.

There is, however, another kind of accelerated
reference frame, that is not equivalent to a
non-inertial reference frame, ie, a freely falling
reference frame. Such a reference frame, in a uniform
gravitational field, is equivalent to an inertial reference frame,
notwithstanding that it is an accelerated frame.

Why is this? Because the accelerating force is
entirely uniformly applied to all the matter and energy
contained in the frame and, therefore, no accelerating
force is detectable within the frame.

Thus, to accelerate and decelerate the travelling twin,
by using such a uniformly applied accelerating force,
which is at least conceptually possible, would mean
that the travelling twin's reference frame would always
be a freely falling frame, and always equivalent to an inertial
frame, despite the acceleration.

In this way, the paradox cannot be disposed of by appealing
to the distinction between an inertial and non-inertial
reference frame.

Alen.

"alen" wrote in message
news:01c36229$bf335760$3c7ea6cb@default...
The identification of a non-inertial reference
frame is used to justify the accelerated-frame
solution to the twins paradox.

There are other ways to explain the paradox. Like the fact that the proper
time between two events depends on the worldline between the two events.
E.g. Take two clocks of identical construction and set them to the same time
when they are at the same place. Let one clock stay at that same place. Le
the other clock leave, travel at great speed, turn around, and come back to
the same place and compare the reading with a clock which was left behind -
The readings will be different. The reason is that each clock followed a
different worldline in spacetime between the same two events (Event 1: Set
clocks at location R. Event 2: Compare clocks at location R)

There is, however, another kind of accelerated
reference frame, that is not equivalent to a
non-inertial reference frame, ie, a freely falling
reference frame. Such a reference frame, in a uniform
gravitational field, is equivalent to an inertial reference frame,
notwithstanding that it is an accelerated frame.

To be precise you should state what the frame is accelerating with respect
to. The free-fall frame is accelerating with respect to the source of
gravity. The frame is not accelerating with respect to an inertial frame
(locally anyway).


Pmb

Hi Alen, Pete et al...

Alen
The identification of a non-inertial reference
frame is used to justify the accelerated-frame
solution to the twins paradox.

There is, however, another kind of accelerated
reference frame, that is not equivalent to a
non-inertial reference frame, ie, a freely falling
reference frame. Such a reference frame, in a uniform
gravitational field, is equivalent to an inertial reference frame,
notwithstanding that it is an accelerated frame.

Why is this? Because the accelerating force is
entirely uniformly applied to all the matter and energy
contained in the frame and, therefore, no accelerating
force is detectable within the frame.

Thus, to accelerate and decelerate the travelling twin,
by using such a uniformly applied accelerating force,
which is at least conceptually possible, would mean
that the travelling twin's reference frame would always
be a freely falling frame, and always equivalent to an inertial
frame, despite the acceleration.

Agree with everything, one twin could swing around a
planet or star (hyperbolic trajectory) to return to a
rendevous without experiencing an acceleration, as
measured by an accelometer.

In this way, the paradox cannot be disposed of by appealing
to the distinction between an inertial and non-inertial
reference frame.

Agreed, a general solution requires General Relativity.

In relativity, either twin may regard themselves at rest,
and so there is no such thing as a travelling twin.
When the twins Coordinate Systems were initially
coincident, they each set there clocks to t=0, and
agree to transmit at a constant 1 Megahertz ,
(1 Megacycle). Each twin counts the cycles received
from the other twin, and compares that to the number
he sent. We know from the Einstein shift that a
transmitter in a strong g-field appears to reduce it's
rate of transmission compared to a transmitter in a
weaker field, so the twin in the g-field will log fewer
cycles sent compared to cycles received.
Obviously, it would take some math to predict
definite values (I could try if you want), but off hand
it a looks like the twin using the g-field would find
he sent fewer cycles than he received when they
rendevous.

What is intriguing about this is how a *g-field* deflected
trajectory and a *quantized* deflected trajectory (as in
the case of providing rocket thrust, to return a twin to
rendevous), could be computed to provide identical
results.

Alen.

From: Pmb )
"alen" wrote in message
news:01c36229$bf335760$3c7ea6cb@default...

Alen
The identification of a non-inertial reference
frame is used to justify the accelerated-frame
solution to the twins paradox.

Pete
There are other ways to explain the paradox. Like the fact that the proper
time between two events depends on the worldline between the two events.
E.g. Take two clocks of identical construction and set them to the same time
when they are at the same place. Let one clock stay at that same place. Le
the other clock leave, travel at great speed, turn around, and come back to
the same place and compare the reading with a clock which was left behind -

But how does one distinguish the relative speed, which one is
travelling at great speed if that's the speed the were
initially in, that a relative motion.

The readings will be different. The reason is that each clock followed a
different worldline in spacetime between the same two events (Event 1: Set
clocks at location R. Event 2: Compare clocks at location R)

There is, however, another kind of accelerated
reference frame, that is not equivalent to a
non-inertial reference frame, ie, a freely falling
reference frame. Such a reference frame, in a uniform
gravitational field, is equivalent to an inertial reference frame,
notwithstanding that it is an accelerated frame.

To be precise you should state what the frame is accelerating with respect
to. The free-fall frame is accelerating with respect to the source of
gravity. The frame is not accelerating with respect to an inertial frame
(locally anyway).

Pete, you're using a preferred FoR, by using the source of gravity
as a reference, this is a no-no in GR. You may fix your FoR to the
free-fall frame and the laws of physics are certainly valid in that
FoR too. I believe Alen is seeking a Generally Covariant description
and validity to the twin paradox.

Pmb

Regards
Ken S. Tucker

(Ken S. Tucker) wrote

Pete
There are other ways to explain the paradox. Like the fact that the proper
time between two events depends on the worldline between the two events.
E.g. Take two clocks of identical construction and set them to the same time
when they are at the same place. Let one clock stay at that same place. Le
the other clock leave, travel at great speed, turn around, and come back to
the same place and compare the reading with a clock which was left behind -

But how does one distinguish the relative speed, which one is
travelling at great speed if that's the speed the were
initially in, that a relative motion.

Look at the worldlines. Which one has a kink in it?

Pete, you're using a preferred FoR, by using the source of gravity
as a reference, this is a no-no in GR.

That's not a no-no in GR. There's nothing wrong with saying "I'm
accelerating with respect to the Earth's surface." or that "I'm
accelerating with respect to an inertial frame of referance." For
example: I'm at rest with respect to my computer. My computer is at
rest with respect to the surface of the Earth. I'm at rest with
respect to the surface of the Earth. I'm accelerating with respect to
a local free-fall frame.

Recall Einstein's comment on this in his publication of GR
-----------------------------------------------------------
Let K be a Galilean system of reference, i.e. a system relatively to
which (at least in the four-dimensional region under consideration) a
mass, sufficiently distant from other masses, is moving with uniform
motion in a straight line. Let K' be a second system of reference
which is moving relatively to K in uniformly accelerated translation.
Then, relatively to K', a mass sufficiently distant from other masses
would have an accelerated motion such that its acceleration and
direction of acceleration are independent of the material composition
and physical state of the mass.

Does this permit an observer at rest relatively to K' to infer that he
is on a "really" accelerated system of reference? The answer is in the
negative; for the above-mentioned relation of freely movable masses to
K' may be interpreted equally well in the following way. The system of
reference K' is unaccelerated, but the space-time territory in
question is under the sway of a gravitational field, which generates
the accelerated motion of the bodies relatively to K'.

-----------------------------------------------------------

Pmb

"alen" wrote in message news:01c36335$ebd92940$277ea6cb@default...
Tom Roberts wrote:

In GR, it is normal to define a freely-falling object as inertial
motion. "Freely-falling" means not subject to any external force; in GR
gravitation is not a force, it is a geometrical aspect of the manifold.


the accelerating force is
entirely uniformly applied to all the matter and energy
contained in the frame and, therefore, no accelerating
force is detectable within the frame.

You can think of it that way. In GR it is more natural to simply note
that objects follow geodesic paths unless there is an external force on
them.

A discussion about this could lead to a long post, so I will try to
summarise instead.

An accelerated object describes a smooth curve in a spacetime
reference frame. Every smooth curve can be seen as the geodesic
of some manifold, and can be associated with a metric tensor,
and this applies to all spacetime trajectories. Therefore
the geodesic picture and the force field picture are always
two sides of the one coin, as it were. In the manifold picture
there is no external force, because the manifold is already
the result achieved by the force and, therefore, the force
is no longer visible, provided all matter and energy in the
reference frame is following the same geodesic. But if you
view the situation from a flat metric (however difficult that
might be?) you will see force and acceleration. Therefore,
to apply a force uniformly to all the matter and energy
in a reference frame, in a flat metric, justifies calling the
frame 'freely falling'.

Applying this to the twins paradox, both the travelling
twin and the earthbound twin would see each other's
reference frames to be equally 'freely falling', because
there is no change or force detectable within either the
earthbound frame or the travelling frame, because the
accelerating force is applied perfectly uniformly to
the 'travelling' frame, and it therefore qualifies as
'freely falling'.

Alen

No, it doesn't qualify as freely falling. Why don't you say what your
conclusion is wrt the age of both twins upon meeting after the
separation? My guess is you think they are the same age and time
dilation is only an apparent effect rather than a real effect?

alen wrote:
An accelerated object describes a smooth curve in a spacetime
reference frame. Every smooth curve can be seen as the geodesic
of some manifold, and can be associated with a metric tensor,
and this applies to all spacetime trajectories.

I doubt that this is true, but cannot prove it offhand for the general
case. But it is easy to prove it is not true for two particles, one of
which is charged and one of which is not, and there are external
electromagnetic fields present.


Therefore
the geodesic picture and the force field picture are always
two sides of the one coin, as it were.

No. Geometry applies to all particles, as there is but a single world in
which they all exist and move. Different particles can have different
external forces impressed upon them.

Remember that there is a unique geodesic through a given point
with a given set of initial conditions. This means that forces
can only be modelled geometrically if every particle that can be
placed at that point has a force proportional to its mass (here
I use the Newtonian approximation, as it's good enough to make
my point). If different particles have a different ratio of
f/m, they will accelerate differently and will not follow
identical trajectories -- therefore they cannot both be
following geodesic paths.

So the only known "force" for which a geometrical interpretation is
possible is gravitation.


[...] But if you
view the situation from a flat metric

In general it is not possible to apply a flat metric to a manifold that
has a curved metric. For instance: the surface of a sphere S^2 is
inconsistent with a flat metric. The topology of the manifold must be
consistent with the metric.


Applying this to the twins paradox, both the travelling
twin and the earthbound twin would see each other's
reference frames to be equally 'freely falling', because
there is no change or force detectable within either the
earthbound frame or the travelling frame, because the
accelerating force is applied perfectly uniformly to
the 'travelling' frame, and it therefore qualifies as
'freely falling'.

I don't get your point. In a curved manifold (or indeed a flat one with
topology different from that of R^n) it is possible to have multiple
geodesics between a given pair of points, and in general they will have
different "lengths" between their intersections (path length in a
Riemannian manifold, elapsed proper time in a Lorentzian manifold).
There is no "twins paradox" in GR, because there is no expectation
whatsoever in a curved manifold that the elapsed proper time will be the
same for two different paths (i.e. there's no "paradox" because
different elapsed proper time is the expected behavior).


Tom Roberts

Bruce Pew wrote:

No, it doesn't qualify as freely falling. Why don't you say what your
conclusion is wrt the age of both twins upon meeting after the
separation? My guess is you think they are the same age and time
dilation is only an apparent effect rather than a real effect?

I have said elsewhere what my conclusion is, and will repeat it here
since you require it.

Yes, I do claim that they are always the same age
when at rest in the same frame, but that each appears to the other to
be younger when in relative motion. But I don't say that this effect is
only apparent. Each is looking at a present manifestation of the
past history of the other, and this manifestation is 'real'.

Alen.

Tom Roberts wrote:

An accelerated object describes a smooth curve in a spacetime
reference frame. Every smooth curve can be seen as the geodesic
of some manifold, and can be associated with a metric tensor,
and this applies to all spacetime trajectories.

I doubt that this is true, but cannot prove it offhand for the general
case. But it is easy to prove it is not true for two particles, one of
which is charged and one of which is not, and there are external
electromagnetic fields present.

The only difference between a charged particle accelerated by a
gravitational field and one accelerated by an electrical field is that
a particle accelerated by gravity via its mass will
respond via the same mass, whereas a particle accelerated via its
charge will still respond via its mass. They will therefore
follow different manifolds, but these are equally valid.

There is no such thing as 'curved space', but only a curved
manifold within space, and space can contain distinct curved
manifolds.

Remember that there is a unique geodesic through a given point
with a given set of initial conditions. This means that forces
can only be modelled geometrically if every particle that can be
placed at that point has a force proportional to its mass (here
I use the Newtonian approximation, as it's good enough to make
my point). If different particles have a different ratio of
f/m, they will accelerate differently and will not follow
identical trajectories -- therefore they cannot both be
following geodesic paths.

So the only known "force" for which a geometrical interpretation is
possible is gravitation.

That is what I dispute. It assumes that space itself is curved by
gravitation, and therefore by gravitation only. In reality, space only
contains curved manifolds, and forces other than gravity can
generate manifolds also. It only means that different particles
will follow different manifolds. The only thing unique about gravity
is that the property generating the force and that generating
the inertial response is the same.

There is no "twins paradox" in GR, because there is no expectation
whatsoever in a curved manifold that the elapsed proper time will be the
same for two different paths

Each twin thinks that the other is the one following the geodesic of
a curved manifold.

Alen

alen wrote:
: Joe Fischer wrote:
: : alen wrote:
: : An accelerated object describes a smooth curve in a spacetime
: : reference frame.
:
: No it doesn't Einstein, in space-time it
: follows a straight line.
: In Minkowski World Line Diagrams it follows
: a smooth curve because the diagram shows the time
: axis and a single pertinent spatial dimension axis
: separately.
:
: OK - the worldline of an accelerated object describes a smooth
: curved 4-d trajectory.
: Alen

I didn't really mean "an accelerated object",
in the way you interpret "accelerated", I apologize.

But you still don't get it, not that it is
all your fault, with the mathematics talk of a
curved manifold in a flat space.

Let me make the statements, and see if
it helps.
Manifolds is math talk, but the concepts
used in General Relativity describe nature so
well that they should be considered valid, and
Newtonian concepts of acceleration are not in
all cases.

Objects which are accelerated by outside
means, except gravity are really accelerated, and
feel accelerated, and an accelerometer attached
will read acceleration.

It is the objects that _appear_ to be
accelerated, but are in freefall or in orbit,
that are not accelerated in the GR sense.
Forget the nonsense about the "field"
acting on all parts equally.

Only then do my original remarks stand.

Joe Fischer

--
3

(Gauge) wrote in message . com...

(Ken S. Tucker) wrote

Pete
There are other ways to explain the paradox. Like the fact that the proper
time between two events depends on the worldline between the two events.
E.g. Take two clocks of identical construction and set them to the same time
when they are at the same place. Let one clock stay at that same place. Le
the other clock leave, travel at great speed, turn around, and come back to
the same place and compare the reading with a clock which was left behind -

But how does one distinguish the relative speed, which one is
travelling at great speed if that's the speed the were
initially in, that a relative motion.

Look at the worldlines. Which one has a kink in it?

Ha, look at these two world lines...
t=0
/\
\ \
/ \
\ \
/ /
\ /
/ /
\/
rendevous.

Are you advocating the kinkiest? Of course not, each twin
may rightfully regard his worldline as straight. There is no
third person reference. The calculation can only involve the
relativity of the twins.

Pete, you're using a preferred FoR, by using the source of gravity
as a reference, this is a no-no in GR.

That's not a no-no in GR. There's nothing wrong with saying "I'm
accelerating with respect to the Earth's surface." or that "I'm
accelerating with respect to an inertial frame of referance." For
example: I'm at rest with respect to my computer. My computer is at
rest with respect to the surface of the Earth. I'm at rest with
respect to the surface of the Earth. I'm accelerating with respect to
a local free-fall frame.

Agreed, that's a CS specialization, and doesn't apply here.

Regards
Ken S. Tucker

Recall Einstein's comment on this in his publication of GR
-----------------------------------------------------------
Let K be a Galilean system of reference, i.e. a system relatively to
which (at least in the four-dimensional region under consideration) a
mass, sufficiently distant from other masses, is moving with uniform
motion in a straight line. Let K' be a second system of reference
which is moving relatively to K in uniformly accelerated translation.
Then, relatively to K', a mass sufficiently distant from other masses
would have an accelerated motion such that its acceleration and
direction of acceleration are independent of the material composition
and physical state of the mass.

Does this permit an observer at rest relatively to K' to infer that he
is on a "really" accelerated system of reference? The answer is in the
negative; for the above-mentioned relation of freely movable masses to
K' may be interpreted equally well in the following way. The system of
reference K' is unaccelerated, but the space-time territory in
question is under the sway of a gravitational field, which generates
the accelerated motion of the bodies relatively to K'.

Pmb