Can we discuss this?

[37818]

Nor does any mathematician on the planet.

I understand that. It is not the standard.

The fact that an equation can be written in a form in which there is a discreet answer which in equation has what, if taken by itself is indeterminate. Now in the equation, it functions as that value. It does. Saying it does not. Does not make it not so.

Bottom line: From what is regarded as a correct understanding. I am wrong. Fine. Then I'm wrong on it. I just honestly do not think so.

I'm open to further discussion. What is true and what is not. It is true my thinking on this is considered wrong.

Math is not a matter of opinion. Yet where a disagreement exists, there is opinion.

[shunyadragon]

I understand that. It is not the standard.

The fact that an equation can be written in a form in which there is a discreet answer which in equation has what, if taken by itself is indeterminate. Now in the equation, it functions as that value. It does. Saying it does not. Does not make it not so.

Bottom line: From what is regarded as a correct understanding. I am wrong. Fine. Then I'm wrong on it. I just honestly do not think so.

I'm open to further discussion. What is true and what is not. It is true my thinking on this is considered wrong.

Math is not a matter of opinion. Yet where a disagreement exists, there is opinion.

Follow the lead of sylas.

[37818]

Follow the lead of sylas.

And what do you mean by this?

[pancreasman]

And what do you mean by this?

I think he means, as I do, that Sylas' analysis is right and yours is not.

[37818]

I would like to thank you Leonhard for giving this solution:

Now what I would like to know. Not just from you, but from anyone who thinks they can explain.

What I would like to know is why that is wrong and sqrt(1 - v^2/c^2) = sqrt(1 - 2GM/(rc^2)) is really correct. Yielding v = sqrt(2GM/r). Based on the Schwarzschild radius. R=2GM/c^2

[37818]

I think he means, as I do, that Sylas' analysis is right and yours is not.

OK. So because I'm wrong, I'm not to be allowed to explain why I do not think I am incorrect in my thinking? That I should not discuss this any further because I am wrong. I'm not to be allowed to resolve this error? I'm just to go on convinced that I'm right, when I am not? And not really seeing where I am wrong. I would like to separate what is true and what is not. Again, I understand, my view is not the accepted view. But that does not cut it. The issue is not about opinions.

[sylas]

Added in edit. I stuffed up this post because I didn't read carefully. Sorry! I'll leave it here for reference. Basically, Leonhard solved a different equation you had given, equating a Newtonian potential energy and a relativistic kinetic energy, which is not likely to mean much. Relativity doesn't use potential energy in the same way as Newtonian physics. I think you are asking about escape velocities. I'll put together a better reply soonish.

I would like to thank you Leonhard for giving this solution:

The essential point is... solution to WHAT? Leonhard solved an equation you had given in which you equated (for some reason that was never explained) a gravitational dilation and a velocity dilation. It's not a useful equation as far as I can see, and the velocity calculated from it isn't representing anything useful either that I can see. You can solve the equation, but it was an equation that never had any defined problem or context. Leonhard asked the critical question -- a question that was never answered.

The problem is 37818, what on earth is this velocity?

Added in edit. Oops. I goofed in the above description; the equation that equated time dilation and gravitational dilation was easy to solve, and has as its solution the escape velocity of a particle: v = sqrt(2GM/r). This equation is valid for escape velocity in both relativity and Newtonian physics. The equation which Leonhard solved for you is

mv^2/2sqrt(1-v^2/c^2) = GMm/r

It looks to be an attempt to equation a kind of relativistic kinetic energy with a Newtonian potential energy. Relativity doesn't use the same potential energy concepts as Newtonian physics, so this equation was pretty meaningless from the start.

Looking at this now I think I can guess better at what you are thinking of; but I still think it would help to give a plain statement of the problem you are trying to solve. Are you trying to derive escape velocity?

This whole discussion went off the rails in the first page because terms and equations were introduced from nothing, without any indication of what problem was being discussed. The original point was concerning photons, with v as the photon frequency; and asked about frequency shifts as photons come out of a gravitational well. It was a well defined problem, and all the variables represented aspects of that problem, and the correct solution was given on the first page. Since then, there's been no defined context for the additional velocity variable that suddenly showed up from left field.

Then, in post #4, we suddenly had v being used to represent a velocity; but without ever indicating what particular velocity was being discussed. It appeared from nowhere as this:

So if we were to write: 1/sqrt(1 - v^2/c^2) = 1/sqrt(1-2GM/rc^2)

You need to step back and state clearly the question you are asking, and what the variables represent. I presume v is the velocity of a particle moving in the gravitation field of a mass M, at a distance r. But what velocity? Circular orbit velocity? Escape velocity? The equation doesn't apply for either of those cases.

If you can define a problem and relate your variables to the problem, then we can sensibly look at solutions.

Cheers -- sylas

[pancreasman]

OK. So because I'm wrong, I'm not to be allowed to explain why I do not think I am incorrect in my thinking? That I should not discuss this any further because I am wrong. I'm not to be allowed to resolve this error? I'm just to go on convinced that I'm right, when I am not? And not really seeing where I am wrong. I would like to separate what is true and what is not. Again, I understand, my view is not the accepted view. But that does not cut it. The issue is not about opinions.

No, you are totally allowed to ask why you are wrong and have it explained to you. That's perfectly alright.

I regret to say my suspicions get aroused in this kind of thread that no amount of explanation will satisfy you. Of course I may be wrong.

[JonathanL]

Well said, sir. I only have a bachelor's degree in mathematics myself. I think I tend to respond a little tersely to a certain kind of commentary on a subject matter that I know a little something about from someone who expresses an 'opinion' that is out of left field. I once used to be patient and even tempered but several long exchanges with Magellan on electromagnets and dynamics have corrupted me. I plead the fifth, failing that, I'll drink a fifth.

I have no idea what you just said, but I liked the way you worded it, so you get an amen.

[firstfloor]

[Leonhard]

I would like to thank you Leonhard for giving this solution:



Now what I would like to know. Not just from you, but from anyone who thinks they can explain.

What I would like to know is why that is wrong and sqrt(1 - v^2/c^2) = sqrt(1 - 2GM/(rc^2)) is really correct. Yielding v = sqrt(2GM/r). Based on the Schwarzschild radius. R=2GM/c^2

Okay I think I've figured out a proper way to interpret this equation, and I think its actually kinda interesting, however I want to point out that its based on using the newtonian equation for potential energy in a gravitational well, and using the relativistic result for kinetic energy, and applying them at the same time. The way potential energy is treated is going to break down for black holes due to the extreme mass and tiny radius.

However, it can straightforwardly be interpreted at a correction to the Newtonian escape velocity owing to the relativistic increase of inertia (physicists call this 'stiffness'), showing that the escape velocity is actually slightly smaller than what's predicted by Newtonian mechanics.

Which is kinda cool.

So as long as those two assumptions apply we can go ahead straightforwardly. In order to better understand what's going on, I'm going to recast the equation into an approximation:



The first square roots contains the classical escape velocity, and the other contains the correction coming from the term for the special relativistic kinetic energy. This has the dimension of unity, but for the Earth its only different from 1.0 by (5.22*10^-10), and (2.00*10^6) for the densest dwarf sun I was able to find on a quick search.

Where there's a real difference, is when you try to calculate the difference for the escape from a neutron star.

Using the classical Newtonian formula you get 0.85c, but with this correction you get 0.5c, which is probably closer to the true value.

I'd love to see it compared to the true calculation using the full General Theory of Relativity, as I think there's other gravitational effects, but its hard to guess how large they would be.

But viewing the equation as a relativistic corrected escape velocity, its actually kinda interesting.

I want to point out again 37818, that this result can't be used as an argument that light can always escape from black holes. The assumptions underpinning it will break down at those conditions.

[Teallaura]

This is a bit of a non sequitur. The triviality of your (incorrect) belief isn't actually linked to the supposed triviality of the actual difference. You could believe any number of things that are grossly inaccurate, but those inaccurate beliefs may never make a difference at any point in your life.

Not really - they are the same in that both are meaningless distinctions - if the one matters then the other should as well.

It's not really winning if the other person is still convinced they're right but can't be bothered to get into it.

Except that isn't what happened. The thing bothers me and there's something about it that strikes me as different. I am not arguing that it actually IS different - in fact I conceded that I'd treat 0.999... as one in an equation so I don't see a valid objection here. I'll go further - in my opinion a good mathematician should keep things as simple as possible (that's what they taught me in school - and college!) so if I were to encounter 0.999... in an equation, I'd simply write it as 1 thereafter. How much more of a concession could you want?

I didn't argue that the thing was truly different - I simply said I wouldn't concede true equality because it bothers me. For all other purposes - and certainly in actual use - I would treat them as the same.

No, they won - they just wouldn't admit it.

[Teallaura]

I'm not upset at all, Teal. Honest. I am a mathematician and all, however; and I concede that if you are not, then it makes no difference at all.

I don't really get what you saying on the subject matter; but this particular thread is an interesting case of how one might deal with technical subjects (maths, physics) in a case where people, and the thread originator in particular, seem to think it is just a matter of opinion, personal preference, philosophy, feeling, utility, or whatever.

I don't mean to cause offense; but in my experience in these discussions I think it is best to be quite definite on the subject details while avoiding personal insult. My major comment is that the debate over whether 1 and 0.999... are different or not all invariably because novices with the technical side don't make a distinction between the number and the writing that denotes a number.

If people don't want to discuss the subject I don't mind in the least. I don't demand that everyone understand the point. But for people who DO want to discuss the subject, then I'll contribute -- and with a definite claim to real expertise and understanding of the specific topic at hand.

Cheers -- sylas

All I said was that it bothers me. I conceded that the darn thing has no real difference - heck, even if a miracle happened and God dumped a proof in my head that there actually IS a difference some googleplex or so digits out I don't see that it would matter - and I'd still treat the darn thing as 1.

I doubt I have the patience to read through a proof that they are truly equal - and I wasn't disputing it as fact. It just bugs me. For my own purposes I'll think about it. I won't debate it - it's a stupid thing to debate since I concede the functional equality at the very least and I'm not even saying I'm sure there is a difference.

It only came up because I mistook it for 0.999 which is also treated frequently as 1 - but isn't. That much I did get right.

Then everyone has to tell me that I just have to accept the thing - which I wasn't even disputing. Fine. Post a link to the danged proof and I'll at least try to read it.

But it still bothers me.


Edit: I wasn't saying anything about the subject matter. By that point it was a total derail. My only legit point in this thread was that it's stupid to post that you're not gonna argue with the OP - either engage it, as you did, or ignore it, but don't dogpile just because you (general) can.

[Teallaura]

FYI: My apologies to Bible - didn't mean to derail your thread on a meaningless side note.

[Zymologist]

All I said was that it bothers me. I conceded that the darn thing has no real difference - heck, even if a miracle happened and God dumped a proof in my head that there actually IS a difference some googleplex or so digits out I don't see that it would matter - and I'd still treat the darn thing as 1.

I doubt I have the patience to read through a proof that they are truly equal - and I wasn't disputing it as fact. It just bugs me. For my own purposes I'll think about it. I won't debate it - it's a stupid thing to debate since I concede the functional equality at the very least and I'm not even saying I'm sure there is a difference.

It only came up because I mistook it for 0.999 which is also treated frequently as 1 - but isn't. That much I did get right.

Then everyone has to tell me that I just have to accept the thing - which I wasn't even disputing. Fine. Post a link to the danged proof and I'll at least try to read it.

But it still bothers me.


Edit: I wasn't saying anything about the subject matter. By that point it was a total derail. My only legit point in this thread was that it's stupid to post that you're not gonna argue with the OP - either engage it, as you did, or ignore it, but don't dogpile just because you (general) can.

FWIW, it annoyed me too and so I googled a proof of it. This one seemed to be the simplest:

1/9 = 0.111...
9 * 1/9 = 9 * 0.111...
Therefore, 1 = 0.999...