Neptune

[Luigi Fortunati]

Neptune is about 30 astronomical units (au) from Earth.

If I look at the planet Neptune from Earth, I am not looking at a
stationary object.

In my reference, Neptune makes a full 360-degree turn in 24 hours.

I was curious to check at what speed Neptune is moving with respect to
my terrestrial frame of reference from which I am observing it and I
have discovered that, incredibly, Neptune (with respect to me) is
moving faster than the speed of light and exactly at 327,000 km per
second.

Obviously I am wrong in my calculations which are these.

The space traveled by Neptune is a circumference that has its center in
the Earth and a radius of 30 au.

Thus, the circumference traveled (in my reference) by Neptune in the 24
hours is 2*pi*r*30=2*3.14*30=188.4 au long.

In one hour Neptune travels 188.4/24=7.85 au.

In one minute Neptune travels 7.85/60=0.13 au.

In a second Neptune travels 0.13/60=0.00218 au.

Since an au corresponds to approximately 150,000,000 km, Neptune
travels 0.00218*150,000.00=327,000 km per second, with respect to me.

Can you tell me where is the conceptual or calculation error?

[[Mod. note -- I see no conceptual or calculation error here.

As a quick sanity check, https://en.wikipedia.org/wiki/Neptune
gives the radius of Neptune's orbit about the sun as 4.5e9 km,
so the apparent speed in the rotating-with-the-Earth reference frame
is 2*pi*4.5e9 km / (24*3600 s) = 330,000 km/s.

So, relative to your rotating-with-the-Earth reference frame, Neptune
is moving a bit faster than the speed of light.

There's no contradiction with relativity here -- this is just a
(non-inertial) *coordinate* speed; there's no *physical object* in
whose inertial reference frame another *physical object* is moving
faster than the speed of light.

In fact, we can carry your argument much farther: instead of considering
a reference frame attached to the Earth (rotating about once per 24 hours),
[Actually, you probably want the siderial rotation period,
once per 23 hours 56 minutes.]
let's consider a reference frame attached to the rotor of an ultracentifuge
(sitting in a lab on the Earth's surface) rotating at 60,000 rpm = 1000 Hz.
Relative to that (rapidly-rotating) reference frame, an object (stationary
on the Earth's surface) about 50 km away would be moving slightly faster
than the speed of light.

Nature doesn't know about coordinates (which are solely a human construct),
so (as a famous relativist once said in a slightly different context)
coordinates can change "at the speed of thought".
-- jt]]

[Luigi Fortunati]

Post by Luigi Fortunati
Neptune is about 30 astronomical units (au) from Earth.
If I look at the planet Neptune from Earth, I am not looking at a stationary object.
In my reference, Neptune makes a full 360-degree turn in 24 hours.
I was curious to check at what speed Neptune is moving with respect to my terrestrial frame of reference from which I am observing it and I have discovered that, incredibly, Neptune (with respect to me) is moving faster than the speed of light and exactly at 327,000 km per second.
Obviously I am wrong in my calculations which are these.
The space traveled by Neptune is a circumference that has its center in the Earth and a radius of 30 au.
Thus, the circumference traveled (in my reference) by Neptune in the 24 hours is 2*pi*r*30=2*3.14*30=188.4 au long.
In one hour Neptune travels 188.4/24=7.85 au.
In one minute Neptune travels 7.85/60=0.13 au.
In a second Neptune travels 0.13/60=0.00218 au.
Since an au corresponds to approximately 150,000,000 km, Neptune travels 0.00218*150,000.00=327,000 km per second, with respect to me.
Can you tell me where is the conceptual or calculation error?
[[Mod. note -- I see no conceptual or calculation error here.

Okay.

In my animation
https://www.geogebra.org/m/sn4eav7h
I have highlighted on the left the point of view of the reference of the
fixed stars where Neptune is (almost) stationary and the Earth rotates
on itself by 360° in 24 hours.

And on the right, the point of view of the terrestrial reference where
the Earth stands still and Neptune rotates 360° around the Earth in 24
hours.

Speed ​​is relative and, therefore, from our point of view, we are
observing a body moving at relativistic speed (with respect to us) and
it is a condition that affects all planets and all stars eternally
moving at different speeds.

Some speeds (indeed most of them) are not such that they can be
considered negligible compared to the speed of light.

So, my question is this: Why with our telescopes do we always (and only)
see perfectly spherical celestial bodies and have we never seen one
contracted in the direction of motion like the one at the top right of
my animation?

[Phillip Helbig (undress to reply)]

Post by Luigi Fortunati
So, my question is this: Why with our telescopes do we always (and only)
see perfectly spherical celestial bodies and have we never seen one
contracted in the direction of motion like the one at the top right of
my animation?

It is a misconception that spheres look contracted when moving at
relativistic speeds:

A. Lampa, _Z. f. Physik_, 27, 138, 1924.
J. Terrell, _Phys. Rev._, 116, 1041, 1959.
R. Penrose, _Proc. Camb. Phil. Soc._, 55, 137, 1959.

I can turn around in a second but the relative motion of the Moon, much
faster than the speed of light, doesn't correspond to the notion of
relative motion normally discussed in SR.

[Luigi Fortunati]

Phillip Helbigundress to reply martedì 15/11/2022 alle ore 13:17:21 ha

Post by Phillip Helbig (undress to reply)

Post by Luigi Fortunati
So, my question is this: Why with our telescopes do we always (and only)
see perfectly spherical celestial bodies and have we never seen one
contracted in the direction of motion like the one at the top right of
my animation?

It is a misconception that spheres look contracted when moving at
A. Lampa, _Z. f. Physik_, 27, 138, 1924.
J. Terrell, _Phys. Rev._, 116, 1041, 1959.
R. Penrose, _Proc. Camb. Phil. Soc._, 55, 137, 1959.
I can turn around in a second but the relative motion of the Moon, much
faster than the speed of light, doesn't correspond to the notion of
relative motion normally discussed in SR.

In my animation
https://www.geogebra.org/m/pxcxznqz
I added a light clock on Neptune, where the photon (in the reference of
the inhabitant of neptune) moves vertically (up and down) along the red
line.

Instead, for the terrestrial observer, the same photon follows a zigzag
path.

Thus, Neptune's space *must* be contracted (in the direction of motion)
for the terrestrial observer.

This is the correct notion of relative motion normally discussed in SR.

[Jonathan Thornburg [remove -color to reply]]

Post by Phillip Helbig (undress to reply)
It is a misconception that spheres look contracted when moving at
A. Lampa, _Z. f. Physik_, 27, 138, 1924.
J. Terrell, _Phys. Rev._, 116, 1041, 1959.
R. Penrose, _Proc. Camb. Phil. Soc._, 55, 137, 1959.
I can turn around in a second but the relative motion of the Moon, much
faster than the speed of light, doesn't correspond to the notion of
relative motion normally discussed in SR.

Luigi Fortunati <***@gmail.com> wrote:
[[question about an apparent paradox involving special relativity
and a rotating reference frame]]

I think the underlying cause of Luigi's apparent paradox may be that
special relativity implicitly assues that the geometry of space is
Euclidean... but the geometry of a rotating reference frame is non-Euclidean.
(The non-Euclidean nature of rotating reference frames results in things
like the Sagnac effect, the Ehrenfest paradox, etc.)

There are interesting and relevant discussions in
https://en.wikipedia.org/wiki/Sagnac_effect
https://en.wikipedia.org/wiki/Ehrenfest_paradox
https://en.wikipedia.org/wiki/Born_coordinates

[Jonathan Thornburg [remove -color to reply]]

Post by Jonathan Thornburg [remove -color to reply]
[[question about an apparent paradox involving special relativity
and a rotating reference frame]]
I think the underlying cause of Luigi's apparent paradox may be that
special relativity implicitly assues that the geometry of space is
Euclidean... but the geometry of a rotating reference frame is non-Euclidean.
(The non-Euclidean nature of rotating reference frames results in things
like the Sagnac effect, the Ehrenfest paradox, etc.)
There are interesting and relevant discussions in
https://en.wikipedia.org/wiki/Sagnac_effect
https://en.wikipedia.org/wiki/Ehrenfest_paradox
https://en.wikipedia.org/wiki/Born_coordinates

Two other excellent discussions which directly address the complexities
of rotating reference frames in relativity are physics FAQ entries:
https://math.ucr.edu/home/baez/physics/Relativity/SR/rotatingCoordinates.html
https://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

ciao,

[Richard Livingston]

Post by Jonathan Thornburg [remove -color to reply]

Post by Phillip Helbig (undress to reply)
It is a misconception that spheres look contracted when moving at
A. Lampa, _Z. f. Physik_, 27, 138, 1924.
J. Terrell, _Phys. Rev._, 116, 1041, 1959.
R. Penrose, _Proc. Camb. Phil. Soc._, 55, 137, 1959.
I can turn around in a second but the relative motion of the Moon, much
faster than the speed of light, doesn't correspond to the notion of
relative motion normally discussed in SR.

[[question about an apparent paradox involving special relativity
and a rotating reference frame]]
I think the underlying cause of Luigi's apparent paradox may be that
special relativity implicitly assues that the geometry of space is
Euclidean... but the geometry of a rotating reference frame is non-Euclidean.
(The non-Euclidean nature of rotating reference frames results in things
like the Sagnac effect, the Ehrenfest paradox, etc.)
There are interesting and relevant discussions in
https://en.wikipedia.org/wiki/Sagnac_effect
https://en.wikipedia.org/wiki/Ehrenfest_paradox
https://en.wikipedia.org/wiki/Born_coordinates
--
currently on the west coast of Canada
"!07/11 PDP a ni deppart m'I !pleH" -- slashdot.org page footer, 2022-10-16
"eHpl !'I mrtpaep dnia P PD1 /107" -- slightly more plausible message
given PDP-11 little-endian byte order

The misunderstanding is that the rotating coordinate frame is not an
inertial reference frame. SR only applies in inertial frames.
Velocities in a rotating frame are not real and you can't use SR with
these coordinates.

Rich L.

[Jonathan Thornburg [remove -color to reply]]

Post by Jonathan Thornburg [remove -color to reply]
[[question about an apparent paradox involving special relativity
and a rotating reference frame]]
I think the underlying cause of Luigi's apparent paradox may be that
special relativity implicitly assues that the geometry of space is
Euclidean... but the geometry of a rotating reference frame is non-Euclidean.
There are interesting and relevant discussions in
[[references]]

On further thought, I think the questions Luigi raised don't actually
involve the rotating-coordinate issues discussed in those references.
Instead, Luigi's questions are "just" about what we see if we observe
something (Neptune) in a reference frame which is moving *faster*
than the speed of light.

As Phillip Helbig noted, it's easy to see observationally that the
answer is "nothing special" -- if you spin your body around at an
angular frequency of faster than about 1 revolution per 8 seconds,
your body reference frame will have the Moon moving faster than the
speed of light, and empirically the Moon looks pretty ordinary
when you do this.

[Tom Roberts]

[...] the rotating coordinate frame is not an inertial reference
frame.

True.

A minor point: in SR all possible frames are inertial, because "frame"
implies the coordinate axes are mutually orthogonal, and that only
happens for Minkowski coordinates at rest in an inertial frame. Rotating
and otherwise-accelerated coordinates do not have mutually orthogonal
coordinate axes.

SR only applies in inertial frames.

False. SR applies in any coordinates if the physical situation is within
its domain of applicability. That domain is restricted to flat manifolds
with the topology of R^4, which means that gravitation is absent (or at
least negligible).

Note, however, that standard presentations of SR give equations only in
inertial coordinates (within its domain). To determine what equations
apply in rotating or otherwise-accelerated coordinates, one starts with
the usual equations in inertial coordinates and applies the appropriate
coordinate transform to the desired coordinates.

Velocities in a rotating frame are not real and you can't use SR
with these coordinates.

That is merely repeating the above mistake.

Tom Roberts