Is E=MC² the optimal description of nature?

[swansont]

2 hours ago, Frogton said:

“Dimensionally inconsistent” was not a good choice of words, as by definition it implies in-correctness. I should have confined myself to giving you your due for spotting the issue with so little to go on.

No, it was a good choice.

 

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There may be some dark corner of your mind panicking about the catastrophic possibility that the Frogton Universal Force Law might be correct, but the title of the thread is “Is E=MC² the optimal description of nature?”

No, since the units don’t work, I have no worry whatsoever.

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Thank you for rushing in where Swansont feared to tread, with that monstrous equation. But you are doing what I object to, by trying to impress with jargon.

To demonstrate that you understand nature, and are not just repeating what you have read, derive that equation, explain how it relates to E=MC², and reveal the true meaning of 'petitio principii'.

 

Why should Markus have to derive in equation? It’s not his thread. It’s yours, and you refuse to provide the derivation of your equation.
 

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You need to consider just the first 3 paragraphs of my post together, and say what you disagree with. I do not have unlimited time to reply to everything.

You shouldn’t waste it posturing, then.

[Phi for All]

3 hours ago, Frogton said:

You need to consider just the first 3 paragraphs of my post together, and say what you disagree with. I do not have unlimited time to reply to everything.

!

Moderator Note

You need to provide the derivation of your equation in your next post, or the thread will be closed. The membership deserve to know if you aren't arguing in good faith.

 

[Markus Hanke]

14 hours ago, Frogton said:

To demonstrate that you understand nature, and are not just repeating what you have read, derive that equation, explain how it relates to E=MC²

Sure, there are many ways to do this. Momentum is defined to be the derivative of the Lagrangian with respect to velocity, i.e. it describes how kinetic energy relates to relative velocity, as a function of rest mass - this is simply a generalisation of the good old p=mv, as we all know it from our school days:

\[p^{\mu } =\frac{\partial L}{\partial v^{\mu }} =mu^{\mu }\]

with the Lagrangian for relativistic motion being

\[L=-mc\int ds=-mc\int \sqrt{\eta _{\mu \nu } dx^{\mu } dx^{\nu }}\]

Energy, on the other hand, is defined as

\[E=p^{\mu } v_{\mu } -L\]

which is in essence a restatement of the fact that energy is the conserved quantity that arises from time-translation invariance in spacetime (see Noether‘s theorem for formal proof). Put these together and rearrange to get

\[E=\sqrt{m^{2} c^{4} +p^{2} c^{2}}\]

as stated above. You could also simply look at the general form of a 4-momentum vector, and see immediately that its temporal and spatial parts are related as above. This would be a standard way to derive this, but there are many, many other ways to do this, both informally and in very formal ways.

For massive particles at rest you have m<>0 and p=0 - insert into the above to get

\[E=mc^2\]

as requested. You can also derive all of this from first principles - the symmetries of Minkowski spacetime - in a very formal way by using Noether‘s theorem; this gives you the energy-momentum tensor as a conserved quantity, and from its vanishing divergence you can derive the above expression as well.

The problem here isn‘t any of the above, because you can find all of this in pretty much any standard undergrad text on Special Relativity. This is basic stuff, and it is not in contention. The problem though is that your response to this will be along the lines of: “You don’t understand anything, you are just parroting what you have read!”.

Am I right?

Edited December 31, 2020 by Markus Hanke

[studiot]

13 hours ago, Frogton said:

Thanks. It is fantastic to have finally found somebody who appreciates my sense of humour.

What a pity you spoilt your post by reverting to type at the end. Also a shame that you are not an etymologist, because then you could have ascertained the origin of the name Isaac Frogton, and reached the opposite conclusion.

Are you ?  But +1 anyway.

I repeat my observation that this is just a wind up.

isaac, I've rumbled your game.

So [game] - on

The game being

[Insert your favourite reptile] - on

My favourite reptile is

[Komodo drag] - on

or perhaps

[Pyth] -on

Your move I believe.

 

Edited December 31, 2020 by studiot

[joigus]

On 12/30/2020 at 1:23 PM, Frogton said:

If you could share your solution that would be very interesting.

I won't, that's for my notebook for the time being. The equation doesn't make sense anyway. I'm having fun with the mathematical problem. I've re-derived the eqs. for hyperbolic motion, which I had forgotten, in case I need to explain them later.

Your mathematical problem can be solved under assumptions of which you have provided no information.

You haven't clarified whether \( T \) is proper time or coordinate time. Your equation makes dimensional sense if \( c=1 \) and all velocities are dimensionless. But you haven't said that, creating a lot of confusion. It's implied on your blog, though --see below--, when you say,

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[...] and distance is measured in light-seconds [...]

If you choose velocities as dimensionless, you're OK, because a t-dependent parameter that appears in solving the eq. is,

\[ \left[\frac{FT}{M}\right]=MLT^{-2}\times T\times M^{-1}=LT^{-1} \]

Another thing you haven't clarified is whether force, velocity and acceleration are collinear. I have assumed that, because you should first tackle that one before getting into other cases.

I've had to go through different hypotheses, like constant 4-force (derivative of 4-momentum with respect to proper time), constant rate of energy supply; \( T \) being proper time or \( T \) being coordinate inertial time. I've tried to study all the cases you could fork into.

Something you should understand is that there is no universal character to how a particle moves under given circumstances. The law of motion is not a universal law under different assumptions for \( F \); it's just how that particle is moving under given circumstances. The key assumption seems to be \( F = \) constant. That is not universal. That just represents how you decide to push the particle.

On your blog,

advertising link removed by moderator

where I've finally been able to take a look at your "derivation", I've been able to spot a couple more mistakes. There's no such thing as Lorentz mass dilation formula. And saying that energy is always force times distance is grossly mistaken. Also mistaken is assuming that momentum rate of change can be taken to be \( \frac{MV}{T} \). Neither can anything depend on \( \frac{V-V_0}{T-T_0} \) --finite differences-- in the equation of motion, because of well-known symmetry properties. This question of symmetry properties I leave for Markus, because he's the one dealing with that aspect on this thread.

[Frogton]

On 12/30/2020 at 2:39 PM, joigus said:

No. But people have. I remember a conversation with J.M.R. Parrondo years ago. I loved Feynman's work, but he didn't seem to find it so fascinating, for some reason. He was busy finding flaws in it. Apparently he found one in the chapter on thermodynamics and the ratchet, and that's what led him to Parrondo's paradox. My take on it is that even in error, Feynman was incredibly inspiring.

I like Feynman because he tries to make nature understandable, but that inevitably leads to errors which books that merely quote equations avoid.

 

On 12/30/2020 at 3:36 PM, studiot said:

my valuable time.

If your time was valuable you would not be on here sneering.

On 12/31/2020 at 4:01 AM, joigus said:

advertising link removed by moderator

where I've finally been able to take a look at your "derivation", I've been able to spot a couple more mistakes. There's no such thing as Lorentz mass dilation formula. And saying that energy is always force times distance is grossly mistaken. Also mistaken is assuming that momentum rate of change can be taken to be MVT . Neither can anything depend on V−V0T−T0 --finite differences-- in the equation of motion, because of well-known symmetry properties. This question of symmetry properties I leave for Markus, because he's the one dealing with that aspect on this thread.

I only read the thread today since my last post, because I had other things to do with my valuable time.

Actually people posting on here should enjoy my book, because just about everything they have said, is said by a character in my book.

Interesting that you started as my harshest critic, but are apparently now the only person interested. You posted my blog several days ago, but according to Wordpress I have had only 1 visitor.

The point is that special relativity is based on the 3 Lorentz transformations, that is all that is needed for the principle of relativity to hold. You can easily derive the E² equation from the Einstein??? mass dilation formula, but only if you start off by assuming E=MC².  However why go out of your way to make things more complicated?

The thread is about to be closed, but it is boring anyway because I pretty much know what people are going to say before they say it. But anybody can always contact me via my blog.

[Phi for All]

31 minutes ago, Frogton said:

The thread is about to be closed, but it is boring anyway because I pretty much know what people are going to say before they say it. But anybody can always contact me via my blog.

!

Moderator Note

What blog?