The challenge is part of an experiment with Gravity Probe A, in 1976.

Assume that the Probe is stationary with respect to the center of Earth,
suspended at a height of 10,000 Km and emitting toward Ground Station
a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
H-maser clock available, with stability better than 10E-16 in short term.

At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
of any kind exists, and the effect of atmospheric layers are neglected.

Display a formula that contains a gravitational time dilation, when the received
signal is compared to an exact clone H-Maser at the reception side.

Have in mind that the gravitational potential is changing continuously as the
Probe goes down, and that classic Doppler effect has to be subtracted.

This is, more or less, what Vessot&Co studied for four years.

HINT: The equations are non-linear, due to the constant change in the value
of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.

Challenges:

1) Obtain a formula for the velocity v(t).
2) Obtain a formula for the height h(t).
3) Obtain a formula for classic Doppler effect D(t)

4) Obtain a formula for gravitational time dilation Δf(t)/fo.

You can focus only in (1), (3) and (4), dismissing (2).

Help relativity by proving that you understand what this problem means.

On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> The challenge is part of an experiment with Gravity Probe A, in 1976.
>
> Assume that the Probe is stationary with respect to the center of Earth,
> suspended at a height of 10,000 Km and emitting toward Ground Station
> a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> H-maser clock available, with stability better than 10E-16 in short term.
>
> At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> of any kind exists, and the effect of atmospheric layers are neglected.
>
> Display a formula that contains a gravitational time dilation, when the received
> signal is compared to an exact clone H-Maser at the reception side.
>
> Have in mind that the gravitational potential is changing continuously as the
> Probe goes down, and that classic Doppler effect has to be subtracted.
>
> This is, more or less, what Vessot&Co studied for four years.
>
> HINT: The equations are non-linear, due to the constant change in the value
> of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
>
> Challenges:
>
> 1) Obtain a formula for the velocity v(t).
> 2) Obtain a formula for the height h(t).
> 3) Obtain a formula for classic Doppler effect D(t)
>
> 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
>
> You can focus only in (1), (3) and (4), dismissing (2).
>
> Help relativity by proving that you understand what this problem means.

I forgot to add the radius of Earth, Re=6.378136550E+06 meters.

You can start with the formula:

d²h/dt² = - GM[Re + h(t)]² , with h(0) =ho = 10E+07 meters AND v(0) = 0

On Thursday, August 25, 2022 at 11:16:53 AM UTC-7, Richard Hertz wrote:
>
> HINT: The equations are non-linear, due to the constant change in the value
> of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
>
> Challenges:
>
> 1) Obtain a formula for the velocity v(t).
> 2) Obtain a formula for the height h(t).
> 3) Obtain a formula for classic Doppler effect D(t)
>
> 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
>
> You can focus only in (1), (3) and (4), dismissing (2).
>
The above are the idiotic steps from your crackpot attempt at deriving time dilation from your botched equations of motion (there is no time dilation in Newtonian mechanics since time is absolute, dumbestfuck!). The above idiocies will never produce anything coherent, they are the ramblings of the demented crank Richard Hertz. The correct derivation is based on the Schwarzschild solution and can be found in the 1976 Vessot paper.

On Thursday, August 25, 2022 at 11:29:29 AM UTC-7, Richard Hertz wrote:
> On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > The challenge is part of an experiment with Gravity Probe A, in 1976.
> >
> > Assume that the Probe is stationary with respect to the center of Earth,
> > suspended at a height of 10,000 Km and emitting toward Ground Station
> > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > H-maser clock available, with stability better than 10E-16 in short term.
> >
> > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > of any kind exists, and the effect of atmospheric layers are neglected.
> >
> > Display a formula that contains a gravitational time dilation, when the received
> > signal is compared to an exact clone H-Maser at the reception side.
> >
> > Have in mind that the gravitational potential is changing continuously as the
> > Probe goes down, and that classic Doppler effect has to be subtracted.
> >
> > This is, more or less, what Vessot&Co studied for four years.
> >
> > HINT: The equations are non-linear, due to the constant change in the value
> > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> >
> > Challenges:
> >
> > 1) Obtain a formula for the velocity v(t).
> > 2) Obtain a formula for the height h(t).
> > 3) Obtain a formula for classic Doppler effect D(t)
> >
> > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> >
> > You can focus only in (1), (3) and (4), dismissing (2).
> >
> > Help relativity by proving that you understand what this problem means.
> I forgot to add the radius of Earth, Re=6.378136550E+06 meters.
>
> You can start with the formula:
>
> d²h/dt² = - GM[Re + h(t)]² , with h(0) =ho = 10E+07 meters AND v(0) = 0

Dumbestfuck,

The above is the equation of motion of a particle of mas "m" falling radially in a gravitational field. It is an improvement over the imbecilities you posted in the other thread where I ripped you a fresh one but it will not get you any closer to deriving time dilation. The above is the wrong starting point but you are claerly too ignorant to figure that out. Hence, you get yet another fresh one ripped.

On Thursday, August 25, 2022 at 3:37:14 PM UTC-3, Dono. wrote:
> On Thursday, August 25, 2022 at 11:16:53 AM UTC-7, Richard Hertz wrote:

<snip>

> > Challenges:
> >
> > 1) Obtain a formula for the velocity v(t).
> > 2) Obtain a formula for the height h(t).
> > 3) Obtain a formula for classic Doppler effect D(t)
> >
> > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> >
> > You can focus only in (1), (3) and (4), dismissing (2).
> >
> The above are the idiotic steps from your crackpot attempt at deriving time dilation from your botched equations of motion (there is no time dilation in Newtonian mechanics since time is absolute, dumbestfuck!). The above idiocies will never produce anything coherent, they are the ramblings of the demented crank Richard Hertz. The correct derivation is based on the Schwarzschild solution and can be found in the 1976 Vessot paper.

For the imbecile Dono:

I clearly wrote that this challenge is based on a SIMPLIFICATION of the 1976 Gravity Probe A.

NONE of these formulae ARE AVAILABLE on the Internet. Vessot's paper only contain a formula FOR AVERAGE VALUES, just ONE
STATIC FORMULA!

The challenge is TO DERIVE FORMULAE that ARE VALID FOR ANY VALUE OF h (height). It means that the formulae HAVE TO BE
CAPABLE of expressing the value of variables at ANY INSTANT between t=0 and t=3200 sec (aprox. duration of the fall).

On Thursday, August 25, 2022 at 1:55:20 PM UTC-7, crank Richard Hertz ate some extra shit:
> On Thursday, August 25, 2022 at 3:37:14 PM UTC-3, Dono. wrote:
> > On Thursday, August 25, 2022 at 11:16:53 AM UTC-7, Richard Hertz wrote:
> <snip>
> > > Challenges:
> > >
> > > 1) Obtain a formula for the velocity v(t).
> > > 2) Obtain a formula for the height h(t).
> > > 3) Obtain a formula for classic Doppler effect D(t)
> > >
> > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > >
> > > You can focus only in (1), (3) and (4), dismissing (2).
> > >
> > The above are the idiotic steps from your crackpot attempt at deriving time dilation from your botched equations of motion (there is no time dilation in Newtonian mechanics since time is absolute, dumbestfuck!). The above idiocies will never produce anything coherent, they are the ramblings of the demented crank Richard Hertz. The correct derivation is based on the Schwarzschild solution and can be found in the 1976 Vessot paper.
> I clearly wrote that this challenge is based on a SIMPLIFICATION of the 1976 Gravity Probe A.

Yours is not a "simplification", it is an utter imbecility, as already explained 3 times. Enjoy your freshly ripped assholes.

On Thursday, August 25, 2022 at 6:05:08 PM UTC-3, Dono. wrote:

[snip]

> > > > Challenges:
> > > >
> > > > 1) Obtain a formula for the velocity v(t).
> > > > 2) Obtain a formula for the height h(t).
> > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > >
> > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > > >
> > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > >
> > > The above are the idiotic steps from your crackpot attempt at deriving time dilation from your botched equations of motion (there is no time dilation in Newtonian mechanics since time is absolute, dumbestfuck!). The above idiocies will never produce anything coherent, they are the ramblings of the demented crank Richard Hertz. The correct derivation is based on the Schwarzschild solution and can be found in the 1976 Vessot paper.

> > I clearly wrote that this challenge is based on a SIMPLIFICATION of the 1976 Gravity Probe A.

> Yours is not a "simplification", it is an utter imbecility, as already explained 3 times. Enjoy your freshly ripped assholes.

Are you admitting that you can't derive any relativistic solution even IF you use your entire arsenal of relativistic math from GR?

After reading how did you use Mathematica in your previous posts, it's no wonder for me that YOU can't even make the first steps.

Go back to college, retarded.

And, BTW, you seem fixated with your homo claims. I'm straight, so try in your usual gay bar.

On Thursday, August 25, 2022 at 3:08:14 PM UTC-7, Richard Hertz wrote:
> On Thursday, August 25, 2022 at 6:05:08 PM UTC-3, Dono. wrote:
>
> [snip]
>
> > > > > Challenges:
> > > > >
> > > > > 1) Obtain a formula for the velocity v(t).
> > > > > 2) Obtain a formula for the height h(t).
> > > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > > >
> > > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo..
> > > > >
> > > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > > >
> > > > The above are the idiotic steps from your crackpot attempt at deriving time dilation from your botched equations of motion (there is no time dilation in Newtonian mechanics since time is absolute, dumbestfuck!). The above idiocies will never produce anything coherent, they are the ramblings of the demented crank Richard Hertz. The correct derivation is based on the Schwarzschild solution and can be found in the 1976 Vessot paper.
>
> > > I clearly wrote that this challenge is based on a SIMPLIFICATION of the 1976 Gravity Probe A.
>
> > Yours is not a "simplification", it is an utter imbecility, as already explained 3 times. Enjoy your freshly ripped assholes.
> Are you admitting that you can't derive any relativistic solution even IF you use your entire arsenal of relativistic math from GR?
>

I sure can. From the Vessot experiment assumptions using the Schwarzschild solution is a simple exercise routinely solved by people who learned GR.
Not from the idiotic assumptions of the relativity denying crank Richard Hertz.

On Thursday, August 25, 2022 at 8:05:21 PM UTC-3, Dono. wrote:
> On Thursday, August 25, 2022 at 3:08:14 PM UTC-7, Richard Hertz wrote:

[snip]

> > Are you admitting that you can't derive any relativistic solution even IF you use your entire arsenal of relativistic math from GR?
> >
> I sure can. From the Vessot experiment assumptions using the Schwarzschild solution is a simple exercise routinely solved by people who learned GR.
> Not from the idiotic assumptions of the relativity denying crank Richard Hertz.

I don't assume anything, idiot. If you don't like my suggestion about using Newton's gravitational law for acceleration (GM/Re²)
because in GR "acceleration" is a BAD newtonian concept connected to FORCES, then use:

d²xᵝ/dt² = - Γᵝᵤᵥ dxᵘ/dt dxᵛ/dt

where your STUPID curvature of spacetime is given by the connection Γᵝᵤᵥ.

Such IDIOTIC EQUATION ALLEGEDLY TELLS that an object in free fall follows a geodesic in a curved spacetime. So, it will be
accelerating (its velocity will change with time) in most coordinate systems. These concepts are extracted from a pure breed
relativistic imbecile with an PhD.

But don't dare to use G (the newtonian gravitational constant) in your idiotic derivation, because G (is the heart of GR) was derived
from Newton's Law of Gravitation USING FORCES (Cavendish and followers).

So, invent your OWN RELATIVISTIC gravitational constant, without using FORCES and Newton.

That's why I call GR an absolute metaphysical pile of crap, starting with the OBVIOUS and OVERRATED Equivalence Principle.

I'd laugh for days reading HOW do you manage to use Schwarzschild's metric to solve the problem (with or without using G). Because
it makes NO SENSE in real physics, and relativistic RATS (like you) have to use Newton IN DISGUISE, scrambling words and messy
tensor notation to HIDE Newton and be trendy at the same time.

So, imbecile, I just ask ONE SOLUTION:

Can you develop the equation of v(t) for those modest 10,000 Km above ground using GR?

Call it free fall, call it inertial motion following a geodesic, call it whatever you want.

Just put A SINGLE FUCKING EQUATION for v(t) using your ARSENAL of mathematical tools from relativity.

On Thursday, August 25, 2022 at 4:42:35 PM UTC-7, crank Richard Hertz frothed at the mouth while lying:
> On Thursday, August 25, 2022 at 8:05:21 PM UTC-3, Dono. wrote:
> > On Thursday, August 25, 2022 at 3:08:14 PM UTC-7, Richard Hertz wrote:
> [snip]
> > > Are you admitting that you can't derive any relativistic solution even IF you use your entire arsenal of relativistic math from GR?
> > >
> > I sure can. From the Vessot experiment assumptions using the Schwarzschild solution is a simple exercise routinely solved by people who learned GR..
> > Not from the idiotic assumptions of the relativity denying crank Richard Hertz.
> I don't assume anything,

Sure you do, these imbecilities, right in your first post:

1) Obtain a formula for the velocity v(t).
2) Obtain a formula for the height h(t).
3) Obtain a formula for classic Doppler effect D(t)
4) Obtain a formula for gravitational time dilation Δf(t)/fo.

Dementia is settling in strongly, you forget all the imbecilities that you just posted.

On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> The challenge is part of an experiment with Gravity Probe A, in 1976.
>
> Assume that the Probe is stationary with respect to the center of Earth,
> suspended at a height of 10,000 Km and emitting toward Ground Station
> a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> H-maser clock available, with stability better than 10E-16 in short term.
>
> At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> of any kind exists, and the effect of atmospheric layers are neglected.
>
> Display a formula that contains a gravitational time dilation, when the received
> signal is compared to an exact clone H-Maser at the reception side.
>
> Have in mind that the gravitational potential is changing continuously as the
> Probe goes down, and that classic Doppler effect has to be subtracted.
>
> This is, more or less, what Vessot&Co studied for four years.
>
> HINT: The equations are non-linear, due to the constant change in the value
> of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
>
> Challenges:
>
> 1) Obtain a formula for the velocity v(t).
> 2) Obtain a formula for the height h(t).
> 3) Obtain a formula for classic Doppler effect D(t)
>
> 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
>
> You can focus only in (1), (3) and (4), dismissing (2).
>
> Help relativity by proving that you understand what this problem means.

After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
to mention challenge (4).

What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).
Permanent changes of coordinate systems, simplification of variables making them constants in local environment, trends
to show that the metric reduced to Minkowski's one, etc, etc, etc.

What's more intriguing, and show the sheer ignorance of relativists with the USE of Schwarzschild's metric, was the constant
ask for help in some serious physics forums, once a developer got stuck with his work and asked hints from others to continue.

Also, it was funny to read how works of physicists with PhD were DESTROYED by others, accusing authors of CHEATING or just
obtain FALSE DERIVATIONS, because they didn't realize that they made MISTAKES and yet continue the work.

If this is the scenario under which GR the simplest solution is used, 100 years after, I pity those who have to make a living working
with this crappy theory. I bet that NONE of them will get a job offer at NASA, ESA, Roscosmos and alike.

Also, this simple research allowed me TO CERTIFY the degree of confusion, ignorance and schizophrenia that reigns within the
"RELATIVISTIC COMMUNITY".

Many papers are from people working with the Perimeter Institute for Theoretical Physics, but NOT EVEN THERE, relativists can
agree about how many coordinate changes they require to use, how to define the boundaries of such elemental problem (infinitesimal,
X distance, infinite) OR when they have to STOP the free fall before the particle approaches to the Schwarzschild's radius. Even
funnier was to read the proposals for the expressions of TIME (proper, relative rest, co-moving, etc.).

I invite doing the same research, and find out by yourselves HOW FUCKED UP is relativistic physics. In particular, if something is NOT
ORBITING something else. Hilarious and sad, at the same time.

Just google: "Schwarzschild metrics applied to a free fall".

On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > The challenge is part of an experiment with Gravity Probe A, in 1976.
> >
> > Assume that the Probe is stationary with respect to the center of Earth,
> > suspended at a height of 10,000 Km and emitting toward Ground Station
> > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > H-maser clock available, with stability better than 10E-16 in short term.
> >
> > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > of any kind exists, and the effect of atmospheric layers are neglected.
> >
> > Display a formula that contains a gravitational time dilation, when the received
> > signal is compared to an exact clone H-Maser at the reception side.
> >
> > Have in mind that the gravitational potential is changing continuously as the
> > Probe goes down, and that classic Doppler effect has to be subtracted.
> >
> > This is, more or less, what Vessot&Co studied for four years.
> >
> > HINT: The equations are non-linear, due to the constant change in the value
> > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> >
> > Challenges:
> >
> > 1) Obtain a formula for the velocity v(t).
> > 2) Obtain a formula for the height h(t).
> > 3) Obtain a formula for classic Doppler effect D(t)
> >
> > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> >
> > You can focus only in (1), (3) and (4), dismissing (2).
> >
> > Help relativity by proving that you understand what this problem means.
> After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> to mention challenge (4).
>

That is because your "challenges" are the work of an ignorant imbecile, kapo.

> What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).

Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.

On Friday, 26 August 2022 at 16:11:08 UTC+2, Dono. wrote:

> Once again, Newtonian mechanics does not admit time dilation, imbecile.

Neither the reality does. It only exists
in your delusional gedankenwelt.

On Friday, August 26, 2022 at 11:11:08 AM UTC-3, Dono. wrote:
> On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> > On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > > The challenge is part of an experiment with Gravity Probe A, in 1976.
> > >
> > > Assume that the Probe is stationary with respect to the center of Earth,
> > > suspended at a height of 10,000 Km and emitting toward Ground Station
> > > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > > H-maser clock available, with stability better than 10E-16 in short term.
> > >
> > > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > > of any kind exists, and the effect of atmospheric layers are neglected.
> > >
> > > Display a formula that contains a gravitational time dilation, when the received
> > > signal is compared to an exact clone H-Maser at the reception side.
> > >
> > > Have in mind that the gravitational potential is changing continuously as the
> > > Probe goes down, and that classic Doppler effect has to be subtracted..
> > >
> > > This is, more or less, what Vessot&Co studied for four years.
> > >
> > > HINT: The equations are non-linear, due to the constant change in the value
> > > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> > >
> > > Challenges:
> > >
> > > 1) Obtain a formula for the velocity v(t).
> > > 2) Obtain a formula for the height h(t).
> > > 3) Obtain a formula for classic Doppler effect D(t)
> > >
> > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > >
> > > You can focus only in (1), (3) and (4), dismissing (2).
> > >
> > > Help relativity by proving that you understand what this problem means.
> > After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> > the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> > to mention challenge (4).
> >
> That is because your "challenges" are the work of an ignorant imbecile, kapo.
> > What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).

> Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.

For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer. Just with this, the
simplest challenge, relativity CRASHES.

There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
Earth, starting from a distant point where such particle is at rest at time ZERO.

Relativists start to fuck things up just with understanding time ZERO. Things get worse when working with what GROUND means.

Relativity is a pseudo-science, with NO PRACTICAL VALUE. It only has generated CHAOS, wasting the working life of thousands.

On Friday, August 26, 2022 at 9:44:39 AM UTC-7, Richard Hertz wrote:
> On Friday, August 26, 2022 at 11:11:08 AM UTC-3, Dono. wrote:
> > On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> > > On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > > > The challenge is part of an experiment with Gravity Probe A, in 1976.
> > > >
> > > > Assume that the Probe is stationary with respect to the center of Earth,
> > > > suspended at a height of 10,000 Km and emitting toward Ground Station
> > > > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > > > H-maser clock available, with stability better than 10E-16 in short term.
> > > >
> > > > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > > > of any kind exists, and the effect of atmospheric layers are neglected.
> > > >
> > > > Display a formula that contains a gravitational time dilation, when the received
> > > > signal is compared to an exact clone H-Maser at the reception side.
> > > >
> > > > Have in mind that the gravitational potential is changing continuously as the
> > > > Probe goes down, and that classic Doppler effect has to be subtracted.
> > > >
> > > > This is, more or less, what Vessot&Co studied for four years.
> > > >
> > > > HINT: The equations are non-linear, due to the constant change in the value
> > > > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> > > >
> > > > Challenges:
> > > >
> > > > 1) Obtain a formula for the velocity v(t).
> > > > 2) Obtain a formula for the height h(t).
> > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > >
> > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > > >
> > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > >
> > > > Help relativity by proving that you understand what this problem means.
> > > After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> > > the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> > > to mention challenge (4).
> > >
> > That is because your "challenges" are the work of an ignorant imbecile, kapo.
> > > What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).
>
> > Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.
> For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer.

Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.

>

.....
> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> Earth, starting from a distant point where such particle is at rest at time ZERO.

Repeating the same lie over and over doesn't make it true, ignorant kapo.

On Friday, August 26, 2022 at 4:20:14 PM UTC-3, Dono. wrote:

<snip>

> Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.

WE?, really WE?. Nobody sides with you, disgusting POS. I know three or four people that post here, who invested their working
lifetime teaching or working with relativity and are highly knowledgeable on that matter, that would prefer to renounce to relativity
before siding with you, romanian gypsy. That's the respect that you collected after 16 years spilling your poison post after post.

> > There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> > Earth, starting from a distant point where such particle is at rest at time ZERO.

> Repeating the same lie over and over doesn't make it true, ignorant kapo.

This is not A LIE. It's the essence of what I collected after doing rather extensive research on this particular topic all over the web.

And regarding the presumption of me seeking EDUCATION here, that's a stupid assertion of yours, based on ignorance and prejudge.

1) I'm not looking for education from others here. I'm a good self-learner since I was a child (60 years and counting). I ventured into
the most complex theories in electronics, mathematics, physics, etc., JUST FOR THE PLEASURE OF DOING IT. Others sought me
for help on particular topics, which I gave without asking returns.

2) The reason that moved me to open this thread was purely and exclusively for entertainment. I thought that it COULD BE INTERESTING
for some people (like Paul) to answer the questions, even with approximations. But, obviously, my challenge didn't prosper.

3) I recognize that my replies are, most of the time, far from BEING POLITE. It's a flaw of my character but, IN NO WAY I posted a bait
to mock at people answering. I'm just too harsh when I reply to some comments, but IN GOOD FAITH.

And this differentiates me from you. I'm mostly naive, and I'm willing to concede if I'm convinced by logical, intelligent replies.

You are a disgusting, bitter and resented person that DREAM ABOUT KNOWING RELATIVITY, which IS ONLY your HOBBY for the
last 20 years, when you were kicked out of a career as an EE. Then, you started your study of relativistic dynamics (at Lithuania).

Conclusion: I post here with no intentions to DECEIVE OR LIE. You post here, always on attack mode, using the CHEAPEST AND LOWEST
tools to insult, downplay others and post LIES without proofs (just quoting some sources, eventually).

The minimized challenge remains: Free fall equations under Schwarzschild, compared with derivations under Newton. Forget
about time dilation, and focus on this elementary proposal. Obtain v(t) expression. That's all.

On 8/26/2022 4:27 PM, Richard Hertz wrote:
> On Friday, August 26, 2022 at 4:20:14 PM UTC-3, Dono. wrote:
>
> <snip>
>
>> Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.
>
> WE?, really WE?. Nobody sides with you, disgusting POS.

I always look forward to Dono's posts where he rips you a new one. Dono
can use "we" (Dono and myself at minimum) even if I'm here more to laugh
at you getting mocked than mock you directly. The only purpose of your
miserable life is to entertain us through your stupidity.

>>> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
>>> Earth, starting from a distant point where such particle is at rest at time ZERO.
>
>> Repeating the same lie over and over doesn't make it true, ignorant kapo.
>
> This is not A LIE. It's the essence of what I collected after doing rather extensive research on this particular topic all over the web.

Then your "extensive research" has failed, hasn't it.
>
> And regarding the presumption of me seeking EDUCATION here, that's a stupid assertion of yours, based on ignorance and prejudge.

The old saying is "Ignorance can be cured through education, but
stupidity is forever." You just demonstrated that you aren't ignorant,
but Stupid.

[snip further examples of stupidity]

> Conclusion: I post here with no intentions to DECEIVE OR LIE.

No, you post here because you are mentally ill, with OCD directed
against Einstein, and SR/GR by extension, and your OCD forces you to post.

On Friday, August 26, 2022 at 11:38:11 PM UTC-3, Volney wrote:
> On 8/26/2022 4:27 PM, Richard Hertz wrote:
> > On Friday, August 26, 2022 at 4:20:14 PM UTC-3, Dono. wrote:
> >
> > <snip>
> >
> >> Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.
> >
> > WE?, really WE?. Nobody sides with you, disgusting POS.
> I always look forward to Dono's posts where he rips you a new one. Dono
> can use "we" (Dono and myself at minimum) even if I'm here more to laugh
> at you getting mocked than mock you directly. The only purpose of your
> miserable life is to entertain us through your stupidity.
> >>> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> >>> Earth, starting from a distant point where such particle is at rest at time ZERO.
> >
> >> Repeating the same lie over and over doesn't make it true, ignorant kapo.
> >
> > This is not A LIE. It's the essence of what I collected after doing rather extensive research on this particular topic all over the web.
> Then your "extensive research" has failed, hasn't it.
> >
> > And regarding the presumption of me seeking EDUCATION here, that's a stupid assertion of yours, based on ignorance and prejudge.
> The old saying is "Ignorance can be cured through education, but
> stupidity is forever." You just demonstrated that you aren't ignorant,
> but Stupid.
>
> [snip further examples of stupidity]
> > Conclusion: I post here with no intentions to DECEIVE OR LIE.
> No, you post here because you are mentally ill, with OCD directed
> against Einstein, and SR/GR by extension, and your OCD forces you to post.

Volney (Moroney) has spoken.

Trying to be the sidekick of someone, physicist wannabe?

First with Bodkin (gone), and now you got closer to Dono?. Well, no wonder, being that both are failed EE.

And as being an EE imply to study and produce REAL THINGS, having value for this world, it doesn't surprise me that both
chose to quit EE and deal with RELATIVITY, because such line of twisted thoughts make both unaccountable for every
stupid post related to that pseudo-science.

It's hard to be a working EE and keep the job. Instead, parroting about relativity is innocuous and sterile. Enjoy the thin shell of your life.

On Friday, August 26, 2022 at 10:12:01 PM UTC-7, Richard Hertz wrote:
> On Friday, August 26, 2022 at 11:38:11 PM UTC-3, Volney wrote:
> > On 8/26/2022 4:27 PM, Richard Hertz wrote:
> > > On Friday, August 26, 2022 at 4:20:14 PM UTC-3, Dono. wrote:
> > >
> > > <snip>
> > >
> > >> Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.
> > >
> > > WE?, really WE?. Nobody sides with you, disgusting POS.
> > I always look forward to Dono's posts where he rips you a new one. Dono
> > can use "we" (Dono and myself at minimum) even if I'm here more to laugh
> > at you getting mocked than mock you directly. The only purpose of your
> > miserable life is to entertain us through your stupidity.
> > >>> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> > >>> Earth, starting from a distant point where such particle is at rest at time ZERO.
> > >
> > >> Repeating the same lie over and over doesn't make it true, ignorant kapo.
> > >
> > > This is not A LIE. It's the essence of what I collected after doing rather extensive research on this particular topic all over the web.
> > Then your "extensive research" has failed, hasn't it.
> > >
> > > And regarding the presumption of me seeking EDUCATION here, that's a stupid assertion of yours, based on ignorance and prejudge.
> > The old saying is "Ignorance can be cured through education, but
> > stupidity is forever." You just demonstrated that you aren't ignorant,
> > but Stupid.
> >
> > [snip further examples of stupidity]
> > > Conclusion: I post here with no intentions to DECEIVE OR LIE.
> > No, you post here because you are mentally ill, with OCD directed
> > against Einstein, and SR/GR by extension, and your OCD forces you to post.
> Volney (Moroney) has spoken.
>
> Trying to be the sidekick of someone, physicist wannabe?
>
> First with Bodkin (gone), and now you got closer to Dono?. Well, no wonder, being that both are failed EE.
>
> And as being an EE imply to study and produce REAL THINGS, having value for this world, it doesn't surprise me that both
> chose to quit EE and deal with RELATIVITY, because such line of twisted thoughts make both unaccountable for every
> stupid post related to that pseudo-science.
>
> It's hard to be a working EE and keep the job. Instead, parroting about relativity is innocuous and sterile. Enjoy the thin shell of your life.
Thanks for the link to this thread, Richard. I note with great satisfaction that, except for some chattering from Dono in the branches above, your challenge has gone completely unanswered by the professional relativists in this forum. Good work.

On Friday, February 10, 2023 at 10:57:57 PM UTC-3, patdolan wrote:
> On Friday, August 26, 2022 at 10:12:01 PM UTC-7, Richard Hertz wrote:
> > On Friday, August 26, 2022 at 11:38:11 PM UTC-3, Volney wrote:
> > > On 8/26/2022 4:27 PM, Richard Hertz wrote:
> > > > On Friday, August 26, 2022 at 4:20:14 PM UTC-3, Dono. wrote:
> > > >
> > > > <snip>
> > > >
> > > >> Remember, we are not here to educate you (you are uneducatable anyways, being a hardened crank), we are here to mock you.
> > > >
> > > > WE?, really WE?. Nobody sides with you, disgusting POS.
> > > I always look forward to Dono's posts where he rips you a new one. Dono
> > > can use "we" (Dono and myself at minimum) even if I'm here more to laugh
> > > at you getting mocked than mock you directly. The only purpose of your
> > > miserable life is to entertain us through your stupidity.
> > > >>> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> > > >>> Earth, starting from a distant point where such particle is at rest at time ZERO.
> > > >
> > > >> Repeating the same lie over and over doesn't make it true, ignorant kapo.
> > > >
> > > > This is not A LIE. It's the essence of what I collected after doing rather extensive research on this particular topic all over the web.
> > > Then your "extensive research" has failed, hasn't it.
> > > >
> > > > And regarding the presumption of me seeking EDUCATION here, that's a stupid assertion of yours, based on ignorance and prejudge.
> > > The old saying is "Ignorance can be cured through education, but
> > > stupidity is forever." You just demonstrated that you aren't ignorant,
> > > but Stupid.
> > >
> > > [snip further examples of stupidity]
> > > > Conclusion: I post here with no intentions to DECEIVE OR LIE.
> > > No, you post here because you are mentally ill, with OCD directed
> > > against Einstein, and SR/GR by extension, and your OCD forces you to post.
> > Volney (Moroney) has spoken.
> >
> > Trying to be the sidekick of someone, physicist wannabe?
> >
> > First with Bodkin (gone), and now you got closer to Dono?. Well, no wonder, being that both are failed EE.
> >
> > And as being an EE imply to study and produce REAL THINGS, having value for this world, it doesn't surprise me that both
> > chose to quit EE and deal with RELATIVITY, because such line of twisted thoughts make both unaccountable for every
> > stupid post related to that pseudo-science.
> >
> > It's hard to be a working EE and keep the job. Instead, parroting about relativity is innocuous and sterile. Enjoy the thin shell of your life.
> Thanks for the link to this thread, Richard. I note with great satisfaction that, except for some chattering from Dono in the branches above, your challenge has gone completely unanswered by the professional relativists in this forum. Good work.

Read this, Pat. GTD is Gravitational Time Dilation, and GRS is Gravitational Red Shift. It's related to P&R, Vessot and other cretins
trying to prove "Einstein's right".

Fallacies of gravitational time dilation and gravitational red shift. Einstein created them.
https://groups.google.com/g/sci.physics.relativity/c/nW6iJfAi76o/m/QAtkw6pFAgAJ

BTW, I solved the challenge for the Gravity Probe A MATHEMATICALLY [h(t), v(t), Doppler(t)], but I can't find the thread.

You see, I do this for fun. I'm not bookmarking my threads to pull them later or to use them for a blog or something.

In the end, I DON'T GIVE A SHIT ABOUT PAPERS ON RELATIVISTIC PSEUDO-PHYSICS. I only care about REAL PHYSICS.

And I JUDGE WHAT IS REAL AND WHAT IS A FAIRY TALE. Don't need consensus, nor publicity. Humble, curious lone wolf.

On Friday, February 10, 2023 at 6:32:42 PM UTC-8, Richard Hertz wrote:
> Humble, curious lone wolf.

You are anything but. You are an idiot, Dick. Arrogant as well. A toxic combination.

On Friday, August 26, 2022 at 6:44:39 PM UTC+2, Richard Hertz wrote:
> On Friday, August 26, 2022 at 11:11:08 AM UTC-3, Dono. wrote:
> > On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> > > On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > > > The challenge is part of an experiment with Gravity Probe A, in 1976.
> > > >
> > > > Assume that the Probe is stationary with respect to the center of Earth,
> > > > suspended at a height of 10,000 Km and emitting toward Ground Station
> > > > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > > > H-maser clock available, with stability better than 10E-16 in short term.
> > > >
> > > > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > > > of any kind exists, and the effect of atmospheric layers are neglected.
> > > >
> > > > Display a formula that contains a gravitational time dilation, when the received
> > > > signal is compared to an exact clone H-Maser at the reception side.
> > > >
> > > > Have in mind that the gravitational potential is changing continuously as the
> > > > Probe goes down, and that classic Doppler effect has to be subtracted.
> > > >
> > > > This is, more or less, what Vessot&Co studied for four years.
> > > >
> > > > HINT: The equations are non-linear, due to the constant change in the value
> > > > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> > > >
> > > > Challenges:
> > > >
> > > > 1) Obtain a formula for the velocity v(t).
> > > > 2) Obtain a formula for the height h(t).
> > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > >
> > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > > >
> > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > >
> > > > Help relativity by proving that you understand what this problem means.
> > > After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> > > the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> > > to mention challenge (4).
> > >
> > That is because your "challenges" are the work of an ignorant imbecile, kapo.
> > > What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).
>
> > Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.
> For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer. Just with this, the
> simplest challenge, relativity CRASHES.
>
> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> Earth, starting from a distant point where such particle is at rest at time ZERO.

Of course there is. It's more complicated than the Newtonian version but
the Newtonian version has been falsified by experiment and the GR version
not yet. I can derive it for you if you'd like to see it.

In general, the behaviour of paths of free fall is very well understood even in
Kerr geometry.

> Relativists start to fuck things up just with understanding time ZERO. Things get worse when working with what GROUND means.
>
> Relativity is a pseudo-science, with NO PRACTICAL VALUE. It only has generated CHAOS, wasting the working life of thousands.

Lies like this one don't become true by repetition.

--
Jan

On Saturday, February 11, 2023 at 1:53:14 PM UTC+1, JanPB wrote:
> On Friday, August 26, 2022 at 6:44:39 PM UTC+2, Richard Hertz wrote:
> > On Friday, August 26, 2022 at 11:11:08 AM UTC-3, Dono. wrote:
> > > On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> > > > On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > > > > The challenge is part of an experiment with Gravity Probe A, in 1976.
> > > > >
> > > > > Assume that the Probe is stationary with respect to the center of Earth,
> > > > > suspended at a height of 10,000 Km and emitting toward Ground Station
> > > > > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > > > > H-maser clock available, with stability better than 10E-16 in short term.
> > > > >
> > > > > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > > > > of any kind exists, and the effect of atmospheric layers are neglected.
> > > > >
> > > > > Display a formula that contains a gravitational time dilation, when the received
> > > > > signal is compared to an exact clone H-Maser at the reception side.
> > > > >
> > > > > Have in mind that the gravitational potential is changing continuously as the
> > > > > Probe goes down, and that classic Doppler effect has to be subtracted.
> > > > >
> > > > > This is, more or less, what Vessot&Co studied for four years.
> > > > >
> > > > > HINT: The equations are non-linear, due to the constant change in the value
> > > > > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> > > > >
> > > > > Challenges:
> > > > >
> > > > > 1) Obtain a formula for the velocity v(t).
> > > > > 2) Obtain a formula for the height h(t).
> > > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > > >
> > > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo..
> > > > >
> > > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > > >
> > > > > Help relativity by proving that you understand what this problem means.
> > > > After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> > > > the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> > > > to mention challenge (4).
> > > >
> > > That is because your "challenges" are the work of an ignorant imbecile, kapo.
> > > > What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).
> >
> > > Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.
> > For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer. Just with this, the
> > simplest challenge, relativity CRASHES.
> >
> > There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> > Earth, starting from a distant point where such particle is at rest at time ZERO.
> Of course there is. It's more complicated than the Newtonian version but
> the Newtonian version has been falsified by experiment and the GR version
> not yet.

I'm now in Lausanne but night is young so let's do it. A particle falling in
radially from far away with initial velocity zero (a particle "dropped").

A central mass field is represented in GR by the unique spherically
symmetric metric in its vacuum portion. The usual (most common)
version of the relevant spacetime metric is the Schwarzschild one.
I'll use this form at first, although the Gullstrand-Painlevé form yields
simpler formulas in the end.

So let's do the Schwarzschild form first:

ds^2 = -(1 - 2M/r) dt^2 + dr^2/(1 - 2M/r) + r^2 dtheta^2 + r^2 sin^2(theta) dphi^2

NB: using the units of G = c =1 throughout. M = the central mass.

It's easy to see that any path of free fall in a spherically symmetric
geometry must be planar (as in Newton) for the reasons of symmetry.
Without loss of generality, we can assume the path sits inside the
plane:

theta = pi/2

This means a general problem of free-falling (geodesic) motion in GR is
really a 3D problem in coordinates (t, r, phi), since theta = pi/2 implies
that:

dtheta = 0 and sin(theta) = 1

....so we have the following 3D metric as far as paths of free fall are
concerned:

ds^2 = -(1 - 2M/r) dt^2 + dr^2/(1 - 2M/r) + r^2 dphi^2

In general, one would at this point write the geodesic equation and
solve it, except:

(1) this equation is of 2nd order and a mess (try writing it down!),
(2) a better method exists which involves only order 1 equations.

This reduction in order is possible because due to the symmetry we
have 3 constants of the motion available: q, E, L (explained in a moment).

Let's denote the particle's (still unknown) path by gamma. So its
components are:

gamma = (t, r, phi)

....and let's denote its (affine, aka. geodesic) parameter by s.
Also, let's denote the vectors tangent to gamma by adding
a prime: gamma'.

Finally, let's denote the inner product defined by the metric ds^2
above by angle brackets <, >. And let the coordinate lines directions
in the directions t, r, phi be d/dt, d/dr, d/dphi, respectively.

OK, with all this notation out of the way, we have the following inner
products are constant along any free-falling particle's path:

q = <gamma', gamma'>
L = <gamma', d/dphi>
E = -<gamma', d/dt> [the minus sign is conventional to make E > 0]

Writing out the RHS of the equation for q and plugging in the relevant
parts of the RHS for L and E one obtains THE standard equation for
the r-component of the geodesic motion is the Schwarzschild metric,
of order 1 only:

(dr/ds)^2 = E^2 + (q - L^2/r^2)*(1 - 2M/r) (*)

The constant q is negative for material particles (and zero for light
rays) and can be taken to be equal to -1 as the exact value only
affects the curve's spacetime reparametrisation (its mass) which
does not affect its motion.

Also, for reasons of symmetry again, the radially infalling particle's
phi coordinate is constant, so that: L = 0. The particle's path is thus
only a path with only (t, r) coordinates being non-trivial.

Finally, for a faraway particle dropped from rest far away (where
the ds^2 is arbitrarily close to being Minkowskian) we have gamma' = d/dt,
and so:

E = -<gamma', d/dt> = -<d/dt, d/dt> = -g_tt = -(-(1-2M/r->infinity)) = -(-1) = 1

So after all this preparation, we have:

q = -1
L = 0
E = 1

with the initial condition for s_0 = -infinity:

t(s_0) = -infinity
r(s_0) = +infinity

So now the equation if motion (*) simplifies to:

(dr/ds)^2 = 1 + (-1 - 0)*(1 - 2M/r)

i.e.,

(dr/ds)^2 = 2M/r

or, given the fact the particle is INfalling:

dr/ds = -sqrt(2M/r)

Although we could easily solve this equation by separating the
variables, what we really want is the observed motion of the
particle, not its exact geodesic parametrisation. IOW, what we'd
like is r dependence on t, not s. So let's get the t-coordinate first.
Recall the equation for E:

E = -<gamma', d/dt> = -dt/ds * g_tt = dt/ds * (1 - 2M/r)

But E = 1, so we have:

dt/ds = 1/(1 - 2M/r)

Hence we have the r-vs-t dependence:

dr/dt = (dr/ds)/(dt/ds) = -sqrt(2M/r) * (1 - 2M/r)

This can be integrated (use a decent online integrator), I won't
bother copying it in ASCII. A simpler formula can be written
down easily using the Gullstrand-Painlevé form of the Schwarzschild
geometry:

ds^2 = -(1 - 2M/r) dt^2 + 2*sqrt(2M/r) dt dr + dr^2 + r^2 dO^2

....where dO^2 is the usual spherical element as before:

dO^2 = dtheta^2 + sin^2(theta) dphi^2

It turns out exactly the same equation (*) comes out in this
form, so most of the stuff above gets repeated in this formulation,
including:

E = 1
dr/ds = -sqrt(2M/r)

....and the relevant Gullstrand-Painlevé metric components:

g_tt = -(1 - 2M/r)
g_rt = sqrt(2M/r)

....we get from the generic E = -<gamma', d/dt> :

1 = -dt/ds * g_tt - dr/ds * g_rt

i.e.:

1 = dt/ds * (1 - 2M/r) + 2M/r

i.e.:

dt/ds = 1

This is a nice property of the Gullstrand-Painlevé metric: a particle
radially free-falling from far away with initial zero velocity has its
proper time equal to the coordinate t-time!

On Wednesday, February 15, 2023 at 10:33:00 AM UTC-8, JanPB wrote:
> On Saturday, February 11, 2023 at 1:53:14 PM UTC+1, JanPB wrote:
> > On Friday, August 26, 2022 at 6:44:39 PM UTC+2, Richard Hertz wrote:
> > > On Friday, August 26, 2022 at 11:11:08 AM UTC-3, Dono. wrote:
> > > > On Friday, August 26, 2022 at 12:46:33 AM UTC-7, Richard Hertz wrote:
> > > > > On Thursday, August 25, 2022 at 3:16:53 PM UTC-3, Richard Hertz wrote:
> > > > > > The challenge is part of an experiment with Gravity Probe A, in 1976.
> > > > > >
> > > > > > Assume that the Probe is stationary with respect to the center of Earth,
> > > > > > suspended at a height of 10,000 Km and emitting toward Ground Station
> > > > > > a pure fo=10.00000 Ghz signal with enough power. It's stabilized by the best
> > > > > > H-maser clock available, with stability better than 10E-16 in short term.
> > > > > >
> > > > > > At t=0 and ho=10,000 Km, it starts a free fall towards Earth. No interference
> > > > > > of any kind exists, and the effect of atmospheric layers are neglected.
> > > > > >
> > > > > > Display a formula that contains a gravitational time dilation, when the received
> > > > > > signal is compared to an exact clone H-Maser at the reception side.
> > > > > >
> > > > > > Have in mind that the gravitational potential is changing continuously as the
> > > > > > Probe goes down, and that classic Doppler effect has to be subtracted.
> > > > > >
> > > > > > This is, more or less, what Vessot&Co studied for four years.
> > > > > >
> > > > > > HINT: The equations are non-linear, due to the constant change in the value
> > > > > > of g(t), as the Probe falls. Assume, for simplicity, that Earth is not rotating.
> > > > > >
> > > > > > Challenges:
> > > > > >
> > > > > > 1) Obtain a formula for the velocity v(t).
> > > > > > 2) Obtain a formula for the height h(t).
> > > > > > 3) Obtain a formula for classic Doppler effect D(t)
> > > > > >
> > > > > > 4) Obtain a formula for gravitational time dilation Δf(t)/fo.
> > > > > >
> > > > > > You can focus only in (1), (3) and (4), dismissing (2).
> > > > > >
> > > > > > Help relativity by proving that you understand what this problem means.
> > > > > After a thorough research on papers and posts from the last 15 years, which used Schwarzschild's metric for
> > > > > the simple problem that I posted here, I affirm that none of them were capable to solve the challenge (1). Not
> > > > > to mention challenge (4).
> > > > >
> > > > That is because your "challenges" are the work of an ignorant imbecile, kapo.
> > > > > What I observed was a myriad of approaches, looking to simplify the derivation to get a result (which is easy under Newton).
> > >
> > > > Once again, Newtonian mechanics does not admit time dilation, imbecile. Whatever mushrooms you are eating , they give you some serious hallucinations.
> > > For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer. Just with this, the
> > > simplest challenge, relativity CRASHES.
> > >
> > > There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> > > Earth, starting from a distant point where such particle is at rest at time ZERO.
> > Of course there is. It's more complicated than the Newtonian version but
> > the Newtonian version has been falsified by experiment and the GR version
> > not yet.
> I'm now in Lausanne but night is young so let's do it. A particle falling in
> radially from far away with initial velocity zero (a particle "dropped").
>
> A central mass field is represented in GR by the unique spherically
> symmetric metric in its vacuum portion. The usual (most common)
> version of the relevant spacetime metric is the Schwarzschild one.
> I'll use this form at first, although the Gullstrand-Painlevé form yields
> simpler formulas in the end.
>
> So let's do the Schwarzschild form first:
>
> ds^2 = -(1 - 2M/r) dt^2 + dr^2/(1 - 2M/r) + r^2 dtheta^2 + r^2 sin^2(theta) dphi^2
>
> NB: using the units of G = c =1 throughout. M = the central mass.
>
> It's easy to see that any path of free fall in a spherically symmetric
> geometry must be planar (as in Newton) for the reasons of symmetry.
> Without loss of generality, we can assume the path sits inside the
> plane:
>
> theta = pi/2
>
> This means a general problem of free-falling (geodesic) motion in GR is
> really a 3D problem in coordinates (t, r, phi), since theta = pi/2 implies
> that:
>
> dtheta = 0 and sin(theta) = 1
>
> ...so we have the following 3D metric as far as paths of free fall are
> concerned:
>
> ds^2 = -(1 - 2M/r) dt^2 + dr^2/(1 - 2M/r) + r^2 dphi^2
>
> In general, one would at this point write the geodesic equation and
> solve it, except:
>
> (1) this equation is of 2nd order and a mess (try writing it down!),
> (2) a better method exists which involves only order 1 equations.
>
> This reduction in order is possible because due to the symmetry we
> have 3 constants of the motion available: q, E, L (explained in a moment)..
>
> Let's denote the particle's (still unknown) path by gamma. So its
> components are:
>
> gamma = (t, r, phi)
>
> ...and let's denote its (affine, aka. geodesic) parameter by s.
> Also, let's denote the vectors tangent to gamma by adding
> a prime: gamma'.
>
> Finally, let's denote the inner product defined by the metric ds^2
> above by angle brackets <, >. And let the coordinate lines directions
> in the directions t, r, phi be d/dt, d/dr, d/dphi, respectively.
>
> OK, with all this notation out of the way, we have the following inner
> products are constant along any free-falling particle's path:
>
> q = <gamma', gamma'>
> L = <gamma', d/dphi>
> E = -<gamma', d/dt> [the minus sign is conventional to make E > 0]
>
> Writing out the RHS of the equation for q and plugging in the relevant
> parts of the RHS for L and E one obtains THE standard equation for
> the r-component of the geodesic motion is the Schwarzschild metric,
> of order 1 only:
>
> (dr/ds)^2 = E^2 + (q - L^2/r^2)*(1 - 2M/r) (*)
>
> The constant q is negative for material particles (and zero for light
> rays) and can be taken to be equal to -1 as the exact value only
> affects the curve's spacetime reparametrisation (its mass) which
> does not affect its motion.
>
> Also, for reasons of symmetry again, the radially infalling particle's
> phi coordinate is constant, so that: L = 0. The particle's path is thus
> only a path with only (t, r) coordinates being non-trivial.
>
> Finally, for a faraway particle dropped from rest far away (where
> the ds^2 is arbitrarily close to being Minkowskian) we have gamma' = d/dt,
> and so:
>
> E = -<gamma', d/dt> = -<d/dt, d/dt> = -g_tt = -(-(1-2M/r->infinity)) = -(-1) = 1
>
> So after all this preparation, we have:
>
> q = -1
> L = 0
> E = 1
>
> with the initial condition for s_0 = -infinity:
>
> t(s_0) = -infinity
> r(s_0) = +infinity
>
> So now the equation if motion (*) simplifies to:
>
> (dr/ds)^2 = 1 + (-1 - 0)*(1 - 2M/r)
>
> i.e.,
>
> (dr/ds)^2 = 2M/r
>
> or, given the fact the particle is INfalling:
>
> dr/ds = -sqrt(2M/r)
>
> Although we could easily solve this equation by separating the
> variables, what we really want is the observed motion of the
> particle, not its exact geodesic parametrisation. IOW, what we'd
> like is r dependence on t, not s. So let's get the t-coordinate first.
> Recall the equation for E:
>
> E = -<gamma', d/dt> = -dt/ds * g_tt = dt/ds * (1 - 2M/r)
>
> But E = 1, so we have:
>
> dt/ds = 1/(1 - 2M/r)
>
> Hence we have the r-vs-t dependence:
>
> dr/dt = (dr/ds)/(dt/ds) = -sqrt(2M/r) * (1 - 2M/r)
>
> This can be integrated (use a decent online integrator), I won't
> bother copying it in ASCII. A simpler formula can be written
> down easily using the Gullstrand-Painlevé form of the Schwarzschild
> geometry:
>
> ds^2 = -(1 - 2M/r) dt^2 + 2*sqrt(2M/r) dt dr + dr^2 + r^2 dO^2
>
> ...where dO^2 is the usual spherical element as before:
>
> dO^2 = dtheta^2 + sin^2(theta) dphi^2
>
> It turns out exactly the same equation (*) comes out in this
> form, so most of the stuff above gets repeated in this formulation,
> including:
>
> E = 1
> dr/ds = -sqrt(2M/r)
>
> ...and the relevant Gullstrand-Painlevé metric components:
>
> g_tt = -(1 - 2M/r)
> g_rt = sqrt(2M/r)
>
> ...we get from the generic E = -<gamma', d/dt> :
>
> 1 = -dt/ds * g_tt - dr/ds * g_rt
>
> i.e.:
>
> 1 = dt/ds * (1 - 2M/r) + 2M/r
>
> i.e.:
>
> dt/ds = 1
>
> This is a nice property of the Gullstrand-Painlevé metric: a particle
> radially free-falling from far away with initial zero velocity has its
> proper time equal to the coordinate t-time!
>
> This simplifies the final r-vs-t dependence:
>
> dr/dt = (dr/ds)/(dt/ds) = -sqrt(2M/r)
>
> ...which is easy to integrate:
>
> t = -(2/3) * sqrt(1/2M) * r^(3/2) + C
>
> This agrees with our boundary assumption that at t = -infinity we
> are at r = +infinity.
>
> The integration constant C is determined by fixing the time of the
> horizon crossing, say:
>
> r = 2M when t = 0
>
> This forces:
>
> C = 4M/3 [exercise]
>
> So we finally have the infalling particle path:
>
> r = ( (3/2)*sqrt(2M)*(t - 4M/3) )^(2/3)
>
> BTW, the particle's proper time between the horizon and the singularity
> is thus:
>
> delta tau = t(r = 0) - t(r = 2M) = 4M/3 - 0 = 4M/3.
>
> --
> Jan

On Friday, August 26, 2022 at 1:44:39 PM UTC-3, Richard Hertz wrote:

Sadly, I have to repeat this post on this thread, from six months ago:

> For me, the challenge was reduced to a simple question: Write the solutions for a free fall that modern physics offer. Just with this, the
> simplest challenge, relativity CRASHES.
>
> There is not a single solution for the simplest problem involving a particle falling, along the radial path toward the center of mass of
> Earth, starting from a distant point where such particle is at rest at time ZERO.
>
> Relativists start to fuck things up just with understanding time ZERO. Things get worse when working with what GROUND means.
>
> Relativity is a pseudo-science, with NO PRACTICAL VALUE. It only has generated CHAOS, wasting the working life of thousands.

Sad, sad. And relativists INSIST resorting TO THE ONLY TOOL AVAILABLE: SCHWARZSCHILD.

And it doesn't work, but relativists are in love with their preferred hand to "write" MATHTURBATION.