From Newsgroup: sci.physics.relativity

The easy way out
When scientists began formulating the theory of relativity, a question
arose: what happens to a small time intervalrComeasured relative to a
proper time intervalrCowhen studying an accelerated object (like a rocket)
or particle?
They chose the easy path.
"We can treat this as equivalent to an inertial frame of reference, since
the interval is very short."
A matter of convenience.
But that is not how it works.
Strange as it may seem, the derivative is not dt' =
d(tau)/sqrt(1-v-#/c-#), but rather:

<http://nemoweb.net/jntp?xl6IQneaw_eyTt1atyxEFo9HBTI@jntp/Data.Media:1>

When Vr=(Real speed) is Vo/sqrt(1-Vo-#/c-#)

..according to the formula provided by one of the friendly contributors
to this forum.
And we can see that it is not the same thing at all.
We also see that the way relativistic physicists handle
accelerated relativistic frames of reference is a total disaster.
A disaster whose existence is not easily proven through simple
experiments.
But a disaster nonetheless.
The question is: if Doctor Hachel is right, must the entire theory be reformulated down to its very foundations?
Perhaps, friends, perhaps...

R.H.
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From Newsgroup: sci.physics.relativity

For those who haven't followed the previous discussion, the issue
is actually quite simple.

Hachel uses unusual notation:

Tr = tau = proper time measured by the moving clock
To = t = time in the inertial laboratory frame
Vo = v = ordinary velocity

and his "real velocity"

Vr = v/sqrt(1-v^2/c^2)

is what is usually called proper velocity (celerity).


WHAT SR SAYS
------------

For any object, accelerated or not, SR gives locally

d(tau) = dt*sqrt(1-v^2/c^2).

This is not an approximation obtained by pretending that an
accelerated object is inertial for a short time.

It follows directly from the spacetime interval:

c^2 d(tau)^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.

For an accelerated journey, v changes with time, so one integrates:

tau = integral sqrt(1-v(t)^2/c^2) dt.


WHAT HACHEL CLAIMS
------------------

Hachel proposes instead, during the accelerated motion discussed
here,

d(tau) sqrt(1+Vr^2/(4c^2))
------ = --------------------
dt 1+Vr^2/(2c^2)

while for inertial motion he accepts

d(tau)/dt = 1/sqrt(1+Vr^2/c^2),

which is the usual SR result.

So the real disagreement is very precise:

SR:
the local clock-rate relation depends on instantaneous velocity.

Hachel:
at the same instantaneous velocity, it is different depending
on whether the object is accelerating or inertial.


WHY THIS MATTERS
----------------

Take the example already discussed:

Vr = sqrt(2)c
v = sqrt(2/3)c = 0.8165c.

Just before Bella switches her engine OFF, Hachel's accelerated
formula gives

d(tau)/dt = 0.612372...

Immediately after switching it OFF, the velocity has not changed,
but his inertial formula gives

d(tau)/dt = 0.577350...

a difference of about 6.07%.

Hachel has explicitly confirmed this result.

So this is no longer a philosophical argument about "curves",
"chords", or infinitesimal inertial frames.

It is a different physical prediction.


THE MAIN PROBLEM
----------------

Hachel presents his equation as showing that the standard SR
treatment of acceleration is a "total disaster".

It shows no such thing.

Writing down an alternative equation does not establish that the
standard equation is wrong.

We need an experiment for which:

SR predicts: X
Hachel predicts: Y
experiment measures: Z +/- uncertainty.

If Z agrees with Hachel and excludes SR, then Hachel has found
something important.

If not, all we currently know is that Hachel has proposed a
different clock law.

And that is the whole issue:

two equations,
different predictions,
one experiment.

The clocks, rather than the adjectives, should decide.
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From Newsgroup: sci.physics.relativity

Le 17/09/2026 |a 21:42, Python a |-crit :

For those who haven't followed the previous discussion, the issue
is actually quite simple.

Hachel uses unusual notation:

Tr = tau = proper time measured by the moving clock
To = t = time in the inertial laboratory frame
Vo = v = ordinary velocity

and his "real velocity"

Vr = v/sqrt(1-v^2/c^2)

is what is usually called proper velocity (celerity).

WHAT SR SAYS
------------

For any object, accelerated or not, SR gives locally

d(tau) = dt*sqrt(1-v^2/c^2).

This is not an approximation obtained by pretending that an
accelerated object is inertial for a short time.

It follows directly from the spacetime interval:

c^2 d(tau)^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2.

For an accelerated journey, v changes with time, so one integrates:

tau = integral sqrt(1-v(t)^2/c^2) dt.

WHAT HACHEL CLAIMS
------------------

Hachel proposes instead, during the accelerated motion discussed
here,

d(tau) sqrt(1+Vr^2/(4c^2))
------ = --------------------
dt 1+Vr^2/(2c^2)

while for inertial motion he accepts

d(tau)/dt = 1/sqrt(1+Vr^2/c^2),

which is the usual SR result.

So the real disagreement is very precise:

SR:
the local clock-rate relation depends on instantaneous velocity.

Hachel:
at the same instantaneous velocity, it is different depending
on whether the object is accelerating or inertial.

WHY THIS MATTERS
----------------

Take the example already discussed:

Vr = sqrt(2)c
v = sqrt(2/3)c = 0.8165c.

Just before Bella switches her engine OFF, Hachel's accelerated
formula gives

d(tau)/dt = 0.612372...

Immediately after switching it OFF, the velocity has not changed,
but his inertial formula gives

d(tau)/dt = 0.577350...

a difference of about 6.07%.

Hachel has explicitly confirmed this result.

So this is no longer a philosophical argument about "curves",
"chords", or infinitesimal inertial frames.

It is a different physical prediction.

THE MAIN PROBLEM
----------------

Hachel presents his equation as showing that the standard SR
treatment of acceleration is a "total disaster".

It shows no such thing.

Writing down an alternative equation does not establish that the
standard equation is wrong.

We need an experiment for which:

SR predicts: X
Hachel predicts: Y
experiment measures: Z +/- uncertainty.

If Z agrees with Hachel and excludes SR, then Hachel has found
something important.

If not, all we currently know is that Hachel has proposed a
different clock law.

And that is the whole issue:

two equations,
different predictions,
one experiment.

The clocks, rather than the adjectives, should decide.

Tr|?s bonne analyse.

R.H.
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