• #### an enigma of proper velocity

From Mark-T@21:1/5 to All on Mon Sep 25 13:50:30 2023
I'm self taught in this subject, and have a question.

A traveler in a closed vehicle can deduce velocity using
an accelerometer. What if he simply burns energy without
limit, and continuously accelerates? His proper speed
ramps up without limit? When does the c limit kick in?

This issue arises in Bell's spaceship paradox, does it not?
The inertial observer sees the ships maintain a constant
separation, while both accelerate. The occupants within
feel a constant proper acceleration, what's the limit on their
proper velocity? What do they see when they look out the window?

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Dono.@21:1/5 to Mark-T on Mon Sep 25 15:21:18 2023
On Monday, September 25, 2023 at 1:50:32 PM UTC-7, Mark-T wrote:
His proper speed
ramps up without limit?

Err, no.

v=at/sqrt(1+{at/c)^2)

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From patdolan@21:1/5 to Mark-T on Mon Sep 25 21:01:05 2023
On Monday, September 25, 2023 at 1:50:32 PM UTC-7, Mark-T wrote:
I'm self taught in this subject, and have a question.

A traveler in a closed vehicle can deduce velocity using
an accelerometer. What if he simply burns energy without
limit, and continuously accelerates? His proper speed
ramps up without limit? When does the c limit kick in?

This issue arises in Bell's spaceship paradox, does it not?
The inertial observer sees the ships maintain a constant
separation, while both accelerate. The occupants within
feel a constant proper acceleration, what's the limit on their
proper velocity? What do they see when they look out the window?

Mark
Mark-T do not listen to Dono. Instead of the enigma of proper velocity, concentrate all your efforts on the enigma of proper and coordinate RELATIVE velocity. That is where all of relativity's dead bodies are buried. Become proficient in that subject
and someday I will take you on a tour of the Lorentz contraction velocity--it will make you forget all about proper velocity.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Mark-T@21:1/5 to Dono. on Tue Sep 26 09:50:52 2023
On September 25, 2023, Dono. wrote:
On September 25, 2023, Mark-T wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

I can't find that equation in Einstein's 1905 paper. Is there
a derivation somewhere?

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Dono.@21:1/5 to Mark-T on Tue Sep 26 10:35:22 2023
On Tuesday, September 26, 2023 at 9:50:55 AM UTC-7, Mark-T wrote:
On September 25, 2023, Dono. wrote:
On September 25, 2023, Mark-T wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)
Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

I can't find that equation in Einstein's 1905 paper. Is there
a derivation somewhere?

It is not in the 1905 paper. Google "hyperbolic motion in SR" or "accelerated motion in SR"
Ignore patdolan , he's an imbecile.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Tom Roberts@21:1/5 to Mark-T on Wed Sep 27 00:00:37 2023
On 9/25/23 3:50 PM, Mark-T wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An object
with constant proper acceleration executes "hyperbolic motion". That is, relative to any inertial frame (this is SR) its trajectory is a
hyperbola, which asymptotically approaches c as time increases without
bound. This, of course, is a highly unrealistic scenario, as it requires
an infinite amount of energy....

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper velocity are identically zero, making them useless. (Some poorly worded and/or crank websites use those terms in nonstandard ways.)

[In astronomy, "proper velocity" has a very different,
and useful, meaning.]

This issue arises in Bell's spaceship paradox, does it not?

No. The same analysis applies.

The inertial observer sees the ships maintain a constant separation,
while both accelerate. The occupants within feel a constant proper acceleration, what's the limit on their proper velocity?

Their proper velocity remains zero. Their velocity relative to their
initial inertial frame is a hyperbola that asymptotically approaches c.

What do they see when they look out the window?

That depends in detail on what is located outside their window.

Tom Roberts

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Richard Hachel@21:1/5 to All on Wed Sep 27 07:40:35 2023
Le 27/09/2023 à 07:00, Tom Roberts a écrit :
On 9/25/23 3:50 PM, Mark-T wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An object
with constant proper acceleration executes "hyperbolic motion". That is, relative to any inertial frame (this is SR) its trajectory is a
hyperbola, which asymptotically approaches c as time increases without
bound. This, of course, is a highly unrealistic scenario, as it requires
an infinite amount of energy....

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper velocity are identically zero, making them useless. (Some poorly worded and/or crank websites use those terms in nonstandard ways.)

[In astronomy, "proper velocity" has a very different,
and useful, meaning.]

This issue arises in Bell's spaceship paradox, does it not?

No. The same analysis applies.

The inertial observer sees the ships maintain a constant separation,
while both accelerate. The occupants within feel a constant proper
acceleration, what's the limit on their proper velocity?

Their proper velocity remains zero. Their velocity relative to their
initial inertial frame is a hyperbola that asymptotically approaches c.

What do they see when they look out the window?

That depends in detail on what is located outside their window.

Tom Roberts

I'm really surprised to see that regulars on this forum still ask
questions like this.
This shows both the extreme incomprehension of the theory by men in
general (including professors who do not know how to explain it or explain
it very poorly); but also, take out your tissues, friends; the extreme arrogance of those who think they know, but know nothing at all, and
believe they are entitled to insult, defame, despise, mock people who know
much more than they do. There is surrealism in this shaky psychological situation.

R.H.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Laurence Clark Crossen@21:1/5 to Mark-T on Wed Sep 27 13:43:20 2023
On Monday, September 25, 2023 at 1:50:32 PM UTC-7, Mark-T wrote:
I'm self taught in this subject, and have a question.

A traveler in a closed vehicle can deduce velocity using
an accelerometer. What if he simply burns energy without
limit, and continuously accelerates? His proper speed
ramps up without limit? When does the c limit kick in?

This issue arises in Bell's spaceship paradox, does it not?
The inertial observer sees the ships maintain a constant
separation, while both accelerate. The occupants within
feel a constant proper acceleration, what's the limit on their
proper velocity? What do they see when they look out the window?

Mark
Their speed continues to increase continuously without limit. There is no cosmic speed limit of c. If you look out the window you won't be hoodwinked by the relativists.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Tom Roberts@21:1/5 to Laurence Clark Crossen on Wed Sep 27 15:57:52 2023
On 9/27/23 3:43 PM, Laurence Clark Crossen wrote:
On Monday, September 25, 2023 at 1:50:32 PM UTC-7, Mark-T wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? [...]
Their speed continues to increase continuously without limit. There
is no cosmic speed limit of c.

In your personal fantasy world, sure, anything can happen.

In the world we inhabit this is wrong. It is also very poorly stated,
and your inability to be precise in wording is a major part of your
confusion. (lack of precision also hides your confusion from yourself.)

Tom Roberts

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Paul Alsing@21:1/5 to Mark-T on Wed Sep 27 16:46:38 2023
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading
as his accelerometer?

Ignore patdolan , he's an imbecile.

how should I proceed?

Reference a university textbook... pretty much any one of them will refute anything Dolan has to say...

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Mark-T@21:1/5 to Dono. on Wed Sep 27 16:43:16 2023
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading
as his accelerometer?

Ignore patdolan , he's an imbecile.

how should I proceed?

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From patdolan@21:1/5 to Paul Alsing on Wed Sep 27 16:54:15 2023
On Wednesday, September 27, 2023 at 4:46:40 PM UTC-7, Paul Alsing wrote:
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks >> out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading as his accelerometer?

Ignore patdolan , he's an imbecile.

how should I proceed?
Reference a university textbook... pretty much any one of them will refute anything Dolan has to say...
Mark-T, mark how they flee in terror before me. You would do well to become my disciple.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Paul Alsing@21:1/5 to patdolan on Wed Sep 27 17:10:20 2023
On Wednesday, September 27, 2023 at 4:54:16 PM UTC-7, patdolan wrote:
On Wednesday, September 27, 2023 at 4:46:40 PM UTC-7, Paul Alsing wrote:
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading as his accelerometer?

Ignore patdolan , he's an imbecile.

Reference a university textbook... pretty much any one of them will refute anything Dolan has to say...

Mark-T, mark how they flee in terror before me. You would do well to become my disciple.

NO ONE flees in terror from you Dolan... I have a mud fence out back that knows more physics than you do... along with my dead dog...

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Gary Harnagel@21:1/5 to Mark-T on Thu Sep 28 05:55:17 2023
On Wednesday, September 27, 2023 at 5:43:17 PM UTC-6, Mark-T wrote:

On September 26, Dono. wrote:

His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading
as his accelerometer?

You haven't specified what you mean by "external observer" -- is he moving along with the ship, is he stationary wrt the ship at t = 0, or what?

Ignore patdolan , he's an imbecile.

how should I proceed?

Mark

“All opinions are not equal. Some are a very great deal more robust, sophisticated and well supported in logic and argument than others.”

“What I cannot create, I do not understand." -- Richard P. Feynman

First, learn principles, then you can create, and then:

"Don't pay attention to 'authorities.' Think for yourself." -- Richard Feynman

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Mark-T@21:1/5 to patdolan on Thu Sep 28 14:28:50 2023
On September 27, 2023, patdolan wrote:
Ignore patdolan , he's an imbecile.

how should I proceed?

Reference a university textbook... pretty much any one of them will refute anything Dolan has to say...

Mark-T, mark how they flee in terror before me. You would do well to become my disciple.

What are you selling?

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Mark-T@21:1/5 to Gary Harnagel on Thu Sep 28 14:26:50 2023
On September 28, 2023, Gary Harnagel wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.

And an external observer sees him accelerate at rate a, the same reading
as his accelerometer?

You haven't specified what you mean by "external observer" -- is he moving along with the ship, is he stationary wrt the ship at t = 0, or what?

The observer is stationary, relative to the ship at t=0.
Does he see the ship accelerate at the same rate as the

Accounting for length contraction, if necessary.

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Mark-T@21:1/5 to Tom Roberts on Thu Sep 28 14:36:09 2023
On September 26, 2023, Tom Roberts wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An object
with constant proper acceleration executes "hyperbolic motion". That is, relative to any inertial frame (this is SR) its trajectory is a
hyperbola, which asymptotically approaches c as time increases without
bound.

I don't get this hyperbolic thing. I can draw a hyperbola on a blank sheet
of paper. I don't know what a hyperbolic velocity is.

The traveler drives along a straight highway. Where's the hyperbola?

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper velocity are identically zero, making them useless.

So if he computes his velocity, as he watches his accelerometer,
it's meaningless?

What do they see when they look out the window?

That depends in detail on what is located outside their window.

He drives straight along a highway, with 1 km markers.

Mark

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Gary Harnagel@21:1/5 to Mark-T on Thu Sep 28 15:53:00 2023
On Thursday, September 28, 2023 at 3:26:52 PM UTC-6, Mark-T wrote:

On September 28, 2023, Gary Harnagel wrote:

And an external observer sees him accelerate at rate a, the same reading as his accelerometer?

You haven't specified what you mean by "external observer" -- is he moving along with the ship, is he stationary wrt the ship at t = 0, or what?

The observer is stationary, relative to the ship at t=0.
Does he see the ship accelerate at the same rate as the

Accounting for length contraction, if necessary.

Mark

No. You can learn about it here:

https://en.wikipedia.org/wiki/Space_travel_under_constant_acceleration

As you can see, elapsed time and distance traveled involve hyperbolic
sine and cosine functions.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Dono.@21:1/5 to Mark-T on Thu Sep 28 16:27:44 2023
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks
out the window?

Yes. Asymptotically.
And an external observer sees him accelerate at rate a,

This is "coordinate " acceleration. dr/dt

as his accelerometer?

This is "proper" acceleration, dr/d\tau. They are not the same. You really need to read a book or take a class.

Ignore patdolan , he's an imbecile.
how should I proceed?

At your own risk. Listening to imbeciles will only confuse you .

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Paul Alsing@21:1/5 to Dono. on Thu Sep 28 16:33:40 2023
On Thursday, September 28, 2023 at 4:27:46 PM UTC-7, Dono. wrote:
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks >> out the window?

Yes. Asymptotically.
And an external observer sees him accelerate at rate a,
This is "coordinate " acceleration. dr/dt
as his accelerometer?
This is "proper" acceleration, dr/d\tau. They are not the same. You really need to read a book or take a class.

Ignore patdolan , he's an imbecile.

how should I proceed?

At your own risk. Listening to imbeciles will only confuse you .

Well, there you go, straight from the horse's own mouth!

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Dono.@21:1/5 to Dono. on Thu Sep 28 16:40:32 2023
On Thursday, September 28, 2023 at 4:27:46 PM UTC-7, Dono. wrote:
On Wednesday, September 27, 2023 at 4:43:17 PM UTC-7, Mark-T wrote:
On September 26, Dono. wrote:
His proper speed ramps up without limit?

Err, no.
v=at/sqrt(1+{at/c)^2)

Then it approaches c. And that's what the traveler sees, when he looks >> out the window?

Yes. Asymptotically.
And an external observer sees him accelerate at rate a,
This is "coordinate " acceleration. d^2r/dt^2
as his accelerometer?
This is "proper" acceleration, d^2r/d\tau^2. They are not the same. You really need to read a book or take a class.
Ignore patdolan , he's an imbecile.
how should I proceed?

At your own risk. Listening to imbeciles will only confuse you .

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Tom Roberts@21:1/5 to Mark-T on Thu Sep 28 22:51:01 2023
On 9/28/23 4:36 PM, Mark-T wrote:
On September 26, 2023, Tom Roberts wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? His proper speed ramps up without
limit? When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An
object with constant proper acceleration executes "hyperbolic
motion". That is, relative to any inertial frame (this is SR) its
trajectory is a hyperbola, which asymptotically approaches c as
time increases without bound.

I don't get this hyperbolic thing. I can draw a hyperbola on a
blank sheet of paper. I don't know what a hyperbolic velocity is.

Given inertial frame S with coordinates (x,t), and a traveler starting
from rest in S at x=0,t=0 with constant proper acceleration a along the
x axis, the velocity u of the traveler relative to S for \tau>=0 is:

u(\tau) = c*tanh(a*\tau/c)

where \tau is the traveler's proper time with \tau=0 when the
acceleration begins. Note that u(0) = 0, and as \tau -> \infinity,
u(\tau) -> c.
The traveler drives along a straight highway. Where's the
hyperbola?

In the velocity relative to S as a function of \tau.

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper
velocity are identically zero, making them useless.

So if he computes his velocity, as he watches his accelerometer,
it's meaningless?

No he just has to be careful about what he is doing, and use the
relativistic equations. tis includes understanding to which inertial
frame his calculation applies.

The observer is stationary, relative to the ship at t=0. Does he see
the ship accelerate at the same rate as the accelerometer reading?

Let the observer be at rest in S (see above). Since a single observer
cannot observe the traveler's acceleration, let me use the inertial
frame S and its (x,t) coordinates to measure it. For the traveler:
at t=0: d^2x/dt^2 = a
for t>0: 0 < d^2x/dt^2 < a
as t->\infinity: d^2x/dt^2 -> 0.

Tom Roberts

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Maciej Wozniak@21:1/5 to Tom Roberts on Thu Sep 28 22:20:24 2023
On Wednesday, 27 September 2023 at 07:00:51 UTC+2, Tom Roberts wrote:
On 9/25/23 3:50 PM, Mark-T wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?
The accelerometer he carries displays his proper acceleration.

No more than real clocks of real GPS display
"proper" time.

Note also that in physics, "proper" means

"matching the Holiest Postulates of our
beloved Giant Guru"

[In astronomy, "proper velocity" has a very different,
and useful, meaning.]

In astronomy and wherever.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Volney@21:1/5 to Tom Roberts on Fri Sep 29 02:21:21 2023
On 9/28/2023 11:51 PM, Tom Roberts wrote:
On 9/28/23 4:36 PM, Mark-T wrote:
On September 26, 2023, Tom Roberts wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An
object with constant proper acceleration executes "hyperbolic
motion". That is, relative to any inertial frame (this is SR) its
trajectory is a hyperbola, which asymptotically approaches c as time
increases without bound.

I don't get this hyperbolic thing.  I can draw a hyperbola on a
blank sheet of paper.  I don't know what a hyperbolic velocity is.

Given inertial frame S with coordinates (x,t), and a traveler starting
from rest in S at x=0,t=0 with constant proper acceleration a along the
x axis, the velocity u of the traveler relative to S for \tau>=0 is:

u(\tau) = c*tanh(a*\tau/c)

where \tau is the traveler's proper time with \tau=0 when the
acceleration begins. Note that u(0) = 0, and as \tau -> \infinity,
u(\tau) -> c.
The traveler drives along a straight highway.   Where's the hyperbola?

In the velocity relative to S as a function of \tau.

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper velocity
are identically zero, making them useless.

So if he computes his velocity, as he watches his accelerometer,
it's meaningless?

No he just has to be careful about what he is doing, and use the
relativistic equations. tis includes understanding to which inertial
frame his calculation applies.

The observer is stationary, relative to the ship at t=0. Does he see
the ship accelerate at the same rate as the accelerometer reading?

Let the observer be at rest in S (see above). Since a single observer
cannot observe the traveler's acceleration, let me use the inertial
frame S and its (x,t) coordinates to measure it. For the traveler:
at t=0:        d^2x/dt^2 = a
for t>0:        0 < d^2x/dt^2 < a
as t->\infinity:    d^2x/dt^2 -> 0.

Thank you, Tom, for your patient explanations, even if it's "pearls
before swine" for 95% of the posters here. A few of us learn from them.

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)
• From Maciej Wozniak@21:1/5 to Volney on Thu Sep 28 23:54:26 2023
On Friday, 29 September 2023 at 08:21:26 UTC+2, Volney wrote:
On 9/28/2023 11:51 PM, Tom Roberts wrote:
On 9/28/23 4:36 PM, Mark-T wrote:
On September 26, 2023, Tom Roberts wrote:
A traveler in a closed vehicle can deduce velocity using an
accelerometer. What if he simply burns energy without limit, and
continuously accelerates? His proper speed ramps up without limit?
When does the c limit kick in?

The accelerometer he carries displays his proper acceleration. An
object with constant proper acceleration executes "hyperbolic
motion". That is, relative to any inertial frame (this is SR) its
trajectory is a hyperbola, which asymptotically approaches c as time
increases without bound.

I don't get this hyperbolic thing. I can draw a hyperbola on a
blank sheet of paper. I don't know what a hyperbolic velocity is.

Given inertial frame S with coordinates (x,t), and a traveler starting
from rest in S at x=0,t=0 with constant proper acceleration a along the
x axis, the velocity u of the traveler relative to S for \tau>=0 is:

u(\tau) = c*tanh(a*\tau/c)

where \tau is the traveler's proper time with \tau=0 when the
acceleration begins. Note that u(0) = 0, and as \tau -> \infinity,
u(\tau) -> c.
The traveler drives along a straight highway. Where's the hyperbola?

In the velocity relative to S as a function of \tau.

Note also that in physics, "proper" means "in the rest frame of the
object in question". So an object's proper speed and proper velocity
are identically zero, making them useless.

So if he computes his velocity, as he watches his accelerometer,
it's meaningless?

No he just has to be careful about what he is doing, and use the relativistic equations. tis includes understanding to which inertial
frame his calculation applies.

The observer is stationary, relative to the ship at t=0. Does he see
the ship accelerate at the same rate as the accelerometer reading?

Let the observer be at rest in S (see above). Since a single observer cannot observe the traveler's acceleration, let me use the inertial
frame S and its (x,t) coordinates to measure it. For the traveler:
at t=0: d^2x/dt^2 = a
for t>0: 0 < d^2x/dt^2 < a
as t->\infinity: d^2x/dt^2 -> 0.

Thank you, Tom, for your patient explanations, even if it's "pearls
before swine" for 95% of the posters here. A few of us learn from them.

He has also explained that when saying that
there is no second apart of ISO second
you're, hmmmm... mistaken. Have you learnt
it, stupid Mike?

--- SoupGate-Win32 v1.05
* Origin: fsxNet Usenet Gateway (21:1/5)