Resultant time dilation from both gravity and motion

[JesseM]

kev didn't derive any result, kev puts in results by hand.

True, kev didn't derive the relation between local velocity and coordinate velocity, but unless you had definite reason to think the relation he used was incorrect (as opposed to possibly correct but not sufficiently justified in his post), I don't see why you would say "I don't think the expressions put down by kev in that post are correct."

The question came up in the different thread, the one about "Why do all clocks tick at the same rate on the geoid" by Dmitry7. I pointed out repeatedly to you why kev's formulas were not appropiate for answering that thtrad.

What do you mean by "the question"? My post bringing up kev's result was in direct response to espen180's OP on this thread, so it doesn't seem to make any sense to cite some completely different thread in order to back up your claim that "kev's equations did not apply to the OP."

 

[starthaus]

True, kev didn't derive the relation between local velocity and coordinate velocity, but unless you had definite reason to think the relation he used was incorrect (as opposed to possibly correct but not sufficiently justified in his post), I don't see why you would say "I don't think the expressions put down by kev in that post are correct."

What do you mean by "the question"? My post bringing up kev's result was in direct response to espen180's OP on this thread, so it doesn't seem to make any sense to cite some completely different thread in order to back up your claim that "kev's equations did not apply to the OP."

Why don't you read post 29? kev's formulas that you keep citing are wrong, ok?

 

[JesseM]

Because post #8(and all subsequent posts based on it) by kev contains a glaring mistake. I have corrected it in post 25.
Citing kev's posts does nothing but perpretrate mistakes.

In post 25 you say that the mistake was originally pervect's which was just perpetuated by kev, but then earlier in post 13 you said you didn't dispute pervect's results, I guess you changed your mind? It does look like pervect used the wrong formula there.

 

[yuiop]

Because post #8(and all subsequent posts based on it) by kev contains a glaring mistake. I have corrected it in post 25.
Citing kev's posts does nothing but perpretrate mistakes.

The very next line in that post https://www.physicsforums.com/showpost.php?p=2446850&postcount=8 states that I have found and corrected for pervect's rare mistake. I made the edit over a year ago and cleary state that the remaining calculations have been edited to correct for the typo by pervect.

 

[starthaus]

The very next line in that post https://www.physicsforums.com/showpost.php?p=2446850&postcount=8 states that I have found and corrected for pervect's rare mistake. I made the edit over a year ago and cleary state that the remaining calculations have been edited to correct for the typo by pervect.

You are right, you also corrected another error that you made further down in your post.

 

[JesseM]

You are right, you also corrected another error that you made further down in your post. But the derivation in post 8 applies to orbital motion, the equation in this post cited by JesseM). is also for orbital motion and not applicable to this thread.

Why do you think an equation for the time dilation experienced by an orbiting object (an equation which you now agree is correct, I take it?) is "not applicable to this thread"? The OP didn't say anything about the precise state of motion of the object, just that it was in a gravity well and was moving (which would certainly be true for an orbiting object!):

When a frame is moving in relation to an observer in his rest frame at infinity, and the frame is in a gravitational well, is the resultant time dilation simply the sum of the motional and gravitational dilation

 

[starthaus]

Why do you think an equation for the time dilation experienced by an orbiting object (an equation which you now agree is correct, I take it?) is "not applicable to this thread"? The OP didn't say anything about the precise state of motion of the object, just that it was in a gravity well and was moving (which would certainly be true for an orbiting object!):

You are going around in circles. Let's put a stop to this, I gave you the correct expressions , including the derivations for both orbital and radial motion at post 6. My post 6 really belongs in the Dmitry7 thread, whoever split the threads made a mistake.
The reason for all the confusion is that espen180 thread was split from the Dmitry7 thread. The two threads (espen180 and Dmitry7) deal with different situations. The answer I gave you at post 6, stands, the correct answer to Dmitry7 question is not the kev posts you cite but the answer I derived.
Spcifically:

-The correct answer to Dmitry7's question is:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/c\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/c\sqrt{1-r_s/r_2})^2}}

-The answer to espen180 question is :\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{r\omega sin(\theta)/c}{\sqrt{1-r_s/r}})^2}

 

[JesseM]

You are going around in circles. Let's put a stop to this, I gave you the correct expressions , including the derivations for both orbital and radial motion at post 6.

I don't dispute your expressions, but it seems you also do not dispute that kev's expressions are correct in the case of an orbiting object or that they are relevant to the OP by espen180, correct? So will you acknowledge that all your previous argumentative statements saying that kev was wrong and that I was wrong to cite him were made too hastily?

My post 6 really belongs in the Dmitry7 thread, whoever split the threads made a mistake.
The reason for all the confusion is that espen180 thread was split from the Dmitry7 thread. The two threads (espen180 and Dmitry7) deal with different situations. The answer I gave you at post 6, stands, the correct answer to Dmitry7 question is not the kev posts you cite but the answer I derived.

But I was the one who originally brought up kev's derivation, and I brought it up in response to espen180's post, I never claimed that it was supposed to be relevant to Dmitry7's post. Did you misunderstand who I was responding to? My original post on this thread was made 8 minutes before Dmitry7's first post according to the timestamps on the upper left, so even before the thread split my post should have appeared before his and it should have been clear that I was not responding to his question.

 

[starthaus]

I don't dispute your expressions, but it seems you also do not dispute that kev's expressions are correct in the case of an orbiting object or that they are relevant to the OP by espen180, correct?

Yes, but NOT in the context of the original thread as started by Dmitry7. This is where my objections started. With post 6. Do you now understand what my objection is to your citing the inappropriate material for answering Dmitry7's OP?

So will you acknowledge that all your previous argumentative statements saying that kev was wrong and that I was wrong to cite him were made too hastily?

No. See above.

But I was the one who originally brought up kev's derivation, and I brought it up in response to espen180's post, I never claimed that it was supposed to be relevant to Dmitry7's post.

The thread started as one thread, the Dmitry7 thread. Your citation was inappropriate in the context. It is quite possible that when the split was made, the timestamps were messed up as well. Anyways, I have posted clearly what formula goes with what thread.

-The correct answer to Dmitry7's question is:

\frac{d\tau_1}{d\tau_2}=\sqrt{\frac{1-r_s/r_1}{1-r_s/r_2}}\sqrt{\frac{1-(r_1sin\theta_1\omega/c\sqrt{1-r_s/r_1})^2}{1-(r_2sin\theta_2\omega/c\sqrt{1-r_s/r_2})^2}}

-The correct answer to espen180's question is :\frac{d\tau}{dt}=\sqrt{1-\frac{r_s}{r}}\sqrt{1-(\frac{r\omega sin(\theta)}{c \sqrt{1-r_s/r}})^2}

I hope that this clarifies things once and for all.

 

[JesseM]

The thread started as one thread, the Dmitry7 thread.

No it didn't, this seems to be your basic misunderstanding. As I already said, you can look at the timestamps in the upper left of each post to see that my post responding to espen180's post was posted 8 minutes before Dmitry7's very first post. The actual time displayed on your browser may depend on your time zone, but on my browser espen180's OP was from Jun2-10, 02:43 PM, my post #3 responding to him (and citing kev's posts) was from Jun2-10, 03:03 PM, while Dmitry7's first post on the split thread was from Jun2-10, 03:11 PM.

It is quite possible that when the split was made, the timestamps were messed up as well.

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

 

[starthaus]

No it didn't, this seems to be your basic misunderstanding. As I already said, you can look at the timestamps in the upper left of each post to see that my post responding to espen180's post was posted 8 minutes before Dmitry7's very first post. The actual time displayed on your browser may depend on your time zone, but on my browser espen180's OP was from Jun2-10, 02:43 PM, my post #3 responding to him (and citing kev's posts) was from Jun2-10, 03:03 PM, while Dmitry7's first post on the split thread was from Jun2-10, 03:11 PM.

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

espen180 thread was split from Dmitry7 thread. Besides, if you paid attention to the correct formulas, they both need to contain sin(\theta) and \omega is \frac{d\phi}{dt}, not \frac{d\theta}{dt}. The reason for the error is that kev picked up not only a wrong formula from pervect but also a truncated one. It is the \phi coordinate that describes the complete circle, not \theta. See here. So, kev's post 8 is still wrong becuse he started with the wrong metric and used the wrong definitions all along.

 

[espen180]

A more general equation is:

\frac{\text{d}\tau}{\text{d}t}= \sqrt{\frac{1-r_s/r}{1-r_s/r_o}} \sqrt{1- \left (\frac{1-r_s/r_o}{1-r_s/r} \right )^2 \left (\frac{\text{d}r}{c\text{d}t} \right )^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right ) \left(\frac{r \text{d}\theta}{c\text{d}t}\right)^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right) \left(\frac{r \sin \theta \text{d}\phi}{c\text{d}t}\right)^2 }

where r_o is the Schwarzschild radial coordinate of the stationary observer and r is the Schwarzschild radial coordinate of the test particle and dr and dt are understood to be measurements made by the stationary observer at r_o in this particular equation.

For r_o = r the time dilation ratio is:

\frac{\text{d}\tau}{\text{d}t} = \sqrt{1-\frac{v'^2}{c^2}}

in agreement with the generally accepted fact that local measurements made in a gravitational field are Minkowskian.

Let's first combine \text{d}\theta^2+\sin^2\theta \text{d}\phi^2=\text{d}\Omega^2 and so simplify the equation to

\frac{\text{d}\tau}{\text{d}t}= \sqrt{\frac{1-r_s/r}{1-r_s/r_o}} \sqrt{1- \left (\frac{1-r_s/r_o}{1-r_s/r} \right )^2 \left (\frac{\text{d}r}{c\text{d}t} \right )^2 - \left (\frac{1-r_s/r_o}{1-r_s/r} \right ) \left(\frac{r \text{d}\Omega}{c\text{d}t}\right)^2 }

Working backwards to get back to the metric gives me

c^2\left(\frac{\text{d}\tau}{\text{d}t}\right)^2=c^2\frac{1-r_s/r}{1-r_s/r_o}-\left(\frac{1-r_s/r}{1-r_s/r_o}\right)^{-1}\left (\frac{\text{d}r}{\text{d}t} \right )^2-r^2\left (\frac{\text{d}\Omega}{\text{d}t} \right )^2

c^2\text{d}\tau^2=c^2\frac{1-r_s/r}{1-r_s/r_o}\text{d}t^2-\left(\frac{1-r_s/r}{1-r_s/r_o}\right)^{-1} \text{d}r^2-r^2\text{d}\Omega^2

I was hoping that doing this would lead me to an explanation as to where the \frac{1-\frac{r_s}{r}}{1-\frac{r_s}{r_0}} came from, but it seems it did not.

I do observe that in modeling this metric the metric coefficients are found by taking the ratio of the coefficients of the particle wrt an observer at infinity to the coefficients of the observer at r_0 to the same observer at infinity, but could I have an explanation of why that works?

 

[JesseM]

espen180, can you settle this? When you originally wrote the OP, were you starting a new thread at the time or were you just responding to a prior thread that had been started by Dmitry67?

 

[espen180]

Isn't it a little more likely that your memory is playing tricks on you? For myself, I remember pretty clearly that espen180's post was in fact the original post when I responded to it.

espen180 thread was split from Dmitry7 thread.

This thread was not split from Dmitry7's thread. I started a new thread with the OP. I hope this settles that dispute.

Besides, if you paid attention to the correct formulas, they both need to contain sin(\theta) and \omega is \frac{d\phi}{dt}, not \frac{d\theta}{dt}. The reason for the error is that kev picked up not only a wrong formula from pervect but also a truncated one. It is the \phi coordinate that describes the complete circle, not \theta. See here. So, kev's post 8 is still wrong becuse he started with the wrong metric and used the wrong definitions all along.

Why not just contract the angle differentials into \text{d}\theta^2+\sin^2\theta\text{d}\phi^2=\text{d}\Omega^2 and avoid the problem alltogether?

Kev's post #8 is in agreement with all the references I can find on the Schwartzschild metric, and the algebra checks out. What, in your opinion, is the right metric and definitions?

 

[starthaus]

Why not just contract the angle differentials into
\text{d}\theta^2+\sin^2\theta\text{d}\phi^2=\text{d}\Omega^2 and avoid the problem alltogether?

Because \phi and \theta are independent coordinates. So your hack is illegal.

Kev's post #8 is in agreement with all the references I can find on the Schwartzschild metric, and the algebra checks out. What, in your opinion, is the right metric and definitions?

Nope, it doesn't. Look it up.

 

[starthaus]

espen180, can you settle this? When you originally wrote the OP, were you starting a new thread at the time or were you just responding to a prior thread that had been started by Dmitry67?

Not relevant. What is relevant is that post 8 by kev is wrong. For a list of errors see here.

 

[espen180]

Because \theta and \phi are independent coordinates.

But you have spherical symmetry, and since the choice of the \theta axis is arbitrary, you can always define a new single coordinate which represents the total angular distance traversed, right?

Nope, it doesn't. Look it up.

I don't have a book handy to look it up in. I can only observe that other PF members like JesseM seem to have given him their support.

 

[starthaus]

But you have spherical symmetry, and since the choice of the \theta axis is arbitrary, you can always define a new single coordinate which represents the total angular distance traversed, right?

Nope. Like I said, you need to read about Schwarzschild metric and Schwarzschild coordinates.
Contrary to your beliefs, \thetaand \phi are not interchangeable.

I don't have a book handy to look it up in.

Google is your friend. Try "Schwarzschild metric", "Schwarzschild coordinates"

I can only observe that other PF members like JesseM seem to have given him their support.

This is not a scientific criterion.I know that you are a big fan of kev's from other encounters but this is not a scientific criterion either. I posted kev's errors in the thread where he did his derivation.

 

[espen180]

Nope. Like I said, you need to read about Schwarzschild metric and Schwarzschild coordinates.
Contrary to your beliefs, thetaand \phi are not interchangeable.

Google is your friend. Try "Schwarzschild metric"
This is not a scientific criterion. I posted kev's errors in the thread where he did his derivation.

I realize that the two independent angle coordinates have different definitions, but you must also realize that there is no preferred coordinate systems.

Let me elaborate on the contraction. Define new angle coordinates \text{d}\theta^2 + \sin^2\theta\text{d}\phi^2=\text{d}\Sigma^2 + \sin^2\Sigma\text{d}\Omega^2 and define the orientation of this new coordinate system such that \Sigma = \frac{\pi}{2}. Since keeping the same angular orientation of the coordinates is not an issue due to spherical symmetry, there is no problem defining \text{d}\theta^2 + \sin^2\theta\text{d}\phi^2=\text{d}\Omega^2

As for the Schartzschild metric, I know it by heart. Kev is using the correct metric

c^2\text{d}\tau^2=c^2\left(1-\frac{r_s}{r}\right)\text{d}t^2-\frac{1}{1-\frac{r_s}{r}}\text{d}r^2-r^2\text{d}\theta^2-r^2\sin^2\theta\text{d}\phi^2

If you read pervects original derivation you would know that kev was working with a particle in orbit around the equator, where \theta=\frac{\pi}{2}. But he would not need to. He could just have used the nagle contraction explained above to shift the coordinates such that motion around the equator was realized.

 

[starthaus]

As for the Schartzschild metric, I know it by heart. Kev is using the correct metric

c^2\text{d}\tau^2=c^2\left(1-\frac{r_s}{r}\right)\text{d}t^2-\frac{1}{1-\frac{r_s}{r}}\text{d}r^2-r^2\text{d}\theta^2-r^2\sin^2\theta\text{d}\phi^2

Good for you.

If you read pervects original derivation you would know that kev was working with a particle in orbit around the equator, where \theta=\frac{\pi}{2}. But he would not need to. He could just have used the nagle contraction explained above to shift the coordinates such that motion around the equator was realized.

So what? his derivation is wrong just the same.

 

[espen180]

So what? his derivation is wrong just the same.

You are missing the point. Your original claim that kev is using the wrong metric is false. Now that we have established that there is nothing wrong with the definitions, please point to the spesific place the error occurs, and preferably propose the correct result is its place.

 

[JesseM]

Not relevant.

You seemed to think it was relevant before when you said "Do you now understand what my objection is to your citing the inappropriate material for answering Dmitry7's OP?" The problem is that rather than sticking to a single criticism, you keep changing your line of attack, never really admitting that you made any mistakes in your previous attacks, as if you somehow believe that as long as you can show kev was wrong in some way, you have "won", even if the way you finally decide he is wrong had not even occurred to you at the moment you started attacking his post. Your latest criticism in post #41 about angles isn't any better than your previous attacks. The Schwarzschild metric is spherically symmetric, so although the fact that \theta only ranges from 0 to \pi means you can't have a full orbit with constant r and \phi, the time dilation equation is only talking about the instantaneous rate a clock is ticking relative to a clock at infinity over an infinitesimally short section of its orbit. It is certainly possible to have a circular orbit which for one half of the orbit has \theta varying from 0 to \pi while r has a constant value of R and \phi has a constant value of \pi/2 (so for any infinitesimal section of an orbiting object's worldline whose endpoints lie on this half of the orbit, dr and d\phi would be 0), while the other half of the orbit also has \theta varying from 0 to \pi and r having a constant value of R, but now with \phi having a constant value of -\pi/2 (so for any infinitesimal section of an orbiting object's worldline whose endpoints lie on this half of the orbit, dr and d\phi would still be 0). kev's derivation would work just fine in this case.

 

[starthaus]

You seemed to think it was relevant before when you said "Do you now understand what my objection is to your citing the inappropriate material for answering Dmitry7's OP?" The problem is that rather than sticking to a single criticism, you keep changing your line of attack, never really admitting that you made any mistakes in your previous attacks, as if you somehow believe that as long as you can show kev was wrong in some way, you have "won", even if the way you finally decide he is wrong had not even occurred to you at the moment you started attacking his post. Your latest criticism in post #41 about angles isn't any better than your previous attacks.

His derivation is a hack and you've been doing your darnest to defend it. Why is it so difficult for you to admit that it is wrong?

The Schwarzschild metric is spherically symmetric, so although the fact that \theta only ranges from 0 to \pi means you can't have a full orbit with constant r and \phi, the time dilation equation is only talking about the instantaneous rate a clock is ticking relative to a clock at infinity over an infinitesimally short section of its orbit. It is certainly possible to have a circular orbit which for one half of the orbit has \theta varying from 0 to \pi while r has a constant value of R and \phi has a constant value of \pi/2 (so for any infinitesimal section of an orbiting object's worldline whose endpoints lie on this half of the orbit, dr and d\phi would be 0), while the other half of the orbit also has \theta varying from 0 to \pi and r having a constant value of R, but now with \phi having a constant value of -\pi/2 (so for any infinitesimal section of an orbiting object's worldline whose endpoints lie on this half of the orbit, dr and d\phi would still be 0). kev's derivation would work just fine in this case.

Please read here. Sorry, but no matter how hard you may try, \omega is not \frac{d\theta}{dt}
We are talking about rigotous derivations, not about hacks, right?

 

[espen180]

Please read here.
We are talking about rigotous derivations, not about hacks, right?

Everything in that post has been adressed above.

Regarding "hacks", I would like to hear your definition of one, and why you think using algebra is "against the rules" if they don't conform to your rules (also state those rules, please).

 

[starthaus]

Everything in that post has been adressed above.

Regarding "hacks", I would like to hear your definition of one, and why you think using algebra is "against the rules" if they don't conform to your rules (also state those rules, please).

You mean using algebra badly? Like in truncating the metric by missing non-null terms? You claimed that you knew the metric by heart.
Like in using the wrong definition of angular speed?

 

[JesseM]

His derivation is a hack and you've been doing your darnest to defend it. Why is it so difficult for you to admit that it is wrong?

But currently your only basis for saying it's wrong is the argument in post #41. Regardless of whether that argument is valid, can you not admit that all your previous unrelated arguments which had nothing to do with \phi vs. \theta were on the wrong track?

Please read here.
We are talking about rigotous derivations, not about hacks, right?

Yes, I already did read that post, it's the same argument as the one I was responding to when I referred to "Your latest criticism in post #41 about angles". It is perfectly "rigorous" to consider a circular orbit which has constant r=R and constant \phi=\pi/2 for one half, and constant r=R and constant \phi=-\pi/2 for the other half, so that for any infinitesimal section of the object's worldline on either half, dr = d\phi = 0; do you deny that such an orbit should be physically possible in the Schwarzschild spacetime? It may be true that for the purposes of a derivation, it might be a bit more "elegant" to consider a different orbit where r and \theta remain constant for the whole orbit, but there's nothing physically wrong or non-rigorous about the way kev did it.

 

[starthaus]

Yes, I already did read that post, that's exactly what I was responding to above. It is perfectly "rigorous" to consider a circular orbit which has constant r=R and constant \phi=\pi/2 for one half, and constant r=R and constant \phi=-\pi/2 for the other half, so that for any infinitesimal section of the object's worldline on either half, dr = d\phi = 0; do you deny that such an orbit should be physically possible in the Schwarzschild spacetime? It may be true that for the purposes of a derivation,

What about the missing terms in \phi? What about the v=r\frac{d\theta}{dt}. Wouldn't it be easier for you to admit that you are backing the wrong formulas rather than patching in all kinds of special pleads? I gave you the correct general formula, it does not agree with kev's formula. I gave you the general derivation, it does not agree with the pervect/kev derivation. Can you at least decide which is right and which is wrong?

it might be a bit more "elegant" to consider a different orbit where r and \theta remain constant for the whole orbit, but there's nothing physically wrong or non-rigorous about the way kev did it.

Isn't this the problem that needs to be solved? Isn't this the problem I solved at post 2?

 

[espen180]

In #2 you gave the metric, which of course is at the heart of all the results suggested in the thread. Personally I am unsure where the problem is. For my part, I choose dr/dt=0 and theta=pi/2 at the beginning of the derivation, but the argument here was that keeping these zero is not neccesary. The calculation is just as valid, for example at the apogee of the particle's trajectory, is what I think JesseM meant.

The other disputes have been focused on individual pieces of the derivation, like how to treat the metric or how to define certain variables.

 

[JesseM]

What about the missing terms in \phi?

It's true that kev did not write out the full metric, but given that he was assuming an orbit where for any infinitesimal segment you'd have d\phi = 0, those extra terms would disappear anyway so this wouldn't affect his final results. And kev never claimed he was starting from the full metric, he said in post #8 that he was "Starting with this equation given by pervect", and pervect had already eliminated terms that went to zero.

What about the v=r\frac{d\theta}{dt}.

What about it? That would appear to be an equation for Schwarzschild coordinate velocity (as opposed to kev's 'local velocity') for an object in circular orbit with varying \theta coordinate, as with the type of orbit I described--again, do you agree that the type of orbit I described is a physically valid one? If you agree there would be a valid physical orbit with that type of coordinate description (with \phi having one constant value for half the orbit and a different constant value for the other half), do you disagree that the above equation would be the correct coordinate velocity for an object in this orbit?

Wouldn't it be easier for you to admit that you are backing the wrong formulas rather than patching in all kinds of special pleads?

I think you don't understand what http://www.nizkor.org/features/fallacies/special-pleading.html is, the fact that I and others respond to each of your various arguments with counterarguments, resulting in you continually abandoning your previous arguments in favor of new arguments you have invented on the spot, does not qualify as "special pleading". Yes or no, do you acknowledge that the arguments you made against kev's derivation prior to the new argument you've made in the posts here and here were flawed?

I gave you the correct general formula, it does not agree with kev's formula.

kev's formula is not intended to be a "general" one for arbitrary motion, it deals specifically with the case of an object in circular orbit. And since the OP was asking about whether total time dilation was a sum of gravitational and velocity-based time dilation, I thought it would be interesting to point out that for this specific case, total time dilation was actually a product of the two (whereas your more general formula does not relate in any obvious way to the formulas for gravitational and velocity-based time dilation)

I gave you the general derivation, it does not agree with the pervect/kev derivation.

Do you deny that the general formula would reduce to the specific formulas found by pervect/kev in the specific case they were considering, namely an infinitesimal section of a circular orbit where the radial coordinate and one of the two angular coordinates are constant?

Can you at least decide which is right and which is wrong?

If a general formula reduces to a more specific formula under the specific conditions assumed in the derivation of the specific formula, I'd say that both are right.

 

[starthaus]

whereas your more general formula does not relate in any obvious way to the formulas for gravitational and velocity-based time dilation

Are you even reading what I am writing? Can you re-read posts 2,6,39,47?