The link above doesn't work, apparently. This is what Grok calls a share link:
I believe Osiak's alternative relativistic energy formula is the correct one
- Thread starter Dave Lush
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Grok (at the link above) says that Osiak relativity can't be right because it violates Noether's theorem:
"Standard E follows from Noether's theorem (spacetime translations → P^μ conservation, with E = p⁰ c)."
But before it says that, it also says this:
"The four-momentum remains standard (P^μ = m γ (c, v)), so p⁰ = γ m c (temporal momentum) is conserved, but the "energy" (now decoupled from p⁰ c) is E = (p⁰ c)² / (2 m c²)."
I have a lot of textbooks, including the Rindler 2006 one it mentions (although I cited the older Intro to Special Relativity Rindler book in my paper) and various of them (Rindler and Jackson in particular) show 4-momentum conservation without linking the temporal momentum to the energy.
A point I try to make is that breaking energy conversation is not the disaster people seem to think it is, because the same thing that is conserved in Osiak relativity, i.e., the temporal component of (4-)momentum, is also conserved in Einstein relativity and so can set pair-creation thresholds. Osiak relativity says it needs to be halved, but there is already the Thomas factor of a half in the spin-orbit coupling doing the same thing as the Osiak factor of half would do. I have been arguing since before Osiak wrote his paper (see here: https://arxiv.org/abs/1108.4343 ) that the Thomas precession cannot account for the spin-orbit coupling anomaly because it's an observer-dependent relativistic-kinematic effect. As I note, Bergstrom identified it as the cause of the magnetic force, which notably doesn't do any work (it acts perpendicular to velocity) and so can't have an energy effect. It's actually much more important than currently appreciated because just as the magnetic force can be understood as an anti-coriolis force, a force similar to the strong force, that is already present in Maxwell-Lorentz electrodynamics, can be derived as an anti-centrifugal force due to Thomas precession.
Grok claims that Thomas's analysis that the rate of precession of the electron spin is halved compared to if it wasn't orbiting a proton or nucleus somehow proves beyond doubt that the interaction energy must be similarly halved. In reality many human beings have doubted that's what it implies. Muller wrote a paper claiming to show that there's a torque associated with it. That paper has to be true for the Thomas factor to apply to energy. My point is that it's not so firmly established that it's not worth considering doing a simple test that can determine which is the correct explanation of the factor of a half that everybody agrees has to be applied. Also, Thomas precession never appears explicitly in the Dirac theory, which nonetheless obtains the correct coupling. This is to be expected if Osiak relativity is true, as I explain here: https://www.academia.edu/142910127/Dirac_Quantum_Theory_Using_Alternative_Relativity
"Standard E follows from Noether's theorem (spacetime translations → P^μ conservation, with E = p⁰ c)."
But before it says that, it also says this:
"The four-momentum remains standard (P^μ = m γ (c, v)), so p⁰ = γ m c (temporal momentum) is conserved, but the "energy" (now decoupled from p⁰ c) is E = (p⁰ c)² / (2 m c²)."
I have a lot of textbooks, including the Rindler 2006 one it mentions (although I cited the older Intro to Special Relativity Rindler book in my paper) and various of them (Rindler and Jackson in particular) show 4-momentum conservation without linking the temporal momentum to the energy.
A point I try to make is that breaking energy conversation is not the disaster people seem to think it is, because the same thing that is conserved in Osiak relativity, i.e., the temporal component of (4-)momentum, is also conserved in Einstein relativity and so can set pair-creation thresholds. Osiak relativity says it needs to be halved, but there is already the Thomas factor of a half in the spin-orbit coupling doing the same thing as the Osiak factor of half would do. I have been arguing since before Osiak wrote his paper (see here: https://arxiv.org/abs/1108.4343 ) that the Thomas precession cannot account for the spin-orbit coupling anomaly because it's an observer-dependent relativistic-kinematic effect. As I note, Bergstrom identified it as the cause of the magnetic force, which notably doesn't do any work (it acts perpendicular to velocity) and so can't have an energy effect. It's actually much more important than currently appreciated because just as the magnetic force can be understood as an anti-coriolis force, a force similar to the strong force, that is already present in Maxwell-Lorentz electrodynamics, can be derived as an anti-centrifugal force due to Thomas precession.
Grok claims that Thomas's analysis that the rate of precession of the electron spin is halved compared to if it wasn't orbiting a proton or nucleus somehow proves beyond doubt that the interaction energy must be similarly halved. In reality many human beings have doubted that's what it implies. Muller wrote a paper claiming to show that there's a torque associated with it. That paper has to be true for the Thomas factor to apply to energy. My point is that it's not so firmly established that it's not worth considering doing a simple test that can determine which is the correct explanation of the factor of a half that everybody agrees has to be applied. Also, Thomas precession never appears explicitly in the Dirac theory, which nonetheless obtains the correct coupling. This is to be expected if Osiak relativity is true, as I explain here: https://www.academia.edu/142910127/Dirac_Quantum_Theory_Using_Alternative_Relativity
Another thing Grok says, is that the idea that the time-reversed electromagnetic field is imaginary is ad hoc, i.e. just made up to support the hypothesis of antimatter as time-reversed matter. It's actually necessary if the electromagnetic field temporal momentum density is allowed to be negative, because the temporal component (interpreted as the energy density in Einstein relativity) is the sum of the squares of the electric and magnetic field density. (This is standard electromagnetic theory.) In Einstein relativity temporal momentum as energy is not allowed to be negative, but in Osiak relativity there's no justification for saying it's positive definite. Grok also scoffs at this idea, asking how can this cause real (i.e., not imaginary) motion of a charge. I was puzzled about this too for a while (a year or so ago) until I realized that Gauss's law implies that imaginary fields require existence of imaginary charge. (The Gauss law is the Maxwell equation that the integrated divergence of the electric field is equal the charge, so if the field is imaginary, so is the charge.) Also, it's required (as explained in the paper) that whether a charge appears imaginary is determined by whether it's in the future or past of the observer. So the real or imaginary character of a charge is observer-dependent. It all follows from, and so is fully consistent with, the well-known invariance of electrodynamics under time inversion. The positron charge being of the same sign as the electron's but imaginary also follows. Hence, if we have an imaginary charge moving in an imaginary field, the net effect is a real force that's opposite to the force on a charge of the same sign in a real field. That is, it's an apparent reversal of the charge sign. But, turns out (as I show), the helicity of cyclotron motion in a magnetic field is opposite to what occurs if the charge sign is simply inverted. That's why Feynman didn't figure this out 80 years ago. It's readily detectable, once you know about it, but not on a cloud-chamber photograph, which is effectively a time-exposure.
I decided that Grok had a point, that I didn't do a good job of justifying why Osiak relativity is correct and Einstein relativity incorrect, at least with respect to how to calculate particle energy, so I decided I would step back and try to understand the issue better. Personally I find Osiak's argument that Einstein relativity violates its own principle of Lorentz invariance of physical law pretty convincing that it would be worth experimentally testing, if the experiment were as cheap and easy as it is, but that doesn't mean there isn't more to say about it.
I decided to consider a test case from standard relativity problems, of the 1-g accelerating rocketship and its acceleration profile as observed from Earth. Usually this problem is about how the 1 g rocket can go to the center of the Milky Way and back in 40 ship years, and about what its mass-to-payload ratio has to be, but I was interested in its acceleration to thrust profile looked like on earth. It's already available in the problem solution so it's really almost trivial if you can do the problem to get that the acceleration, just like in the Osiak integral is the Lorentz factor gamma to the fourth power times the ship-frame acceleration. So after a few years of ship time, the acceleration seen on earth is vastly larger than 1 g. That's standard relativity, and follows from the acceleration being a second time derivative of position, so is affected by time dilation and Lorentz-Fitzgerald contraction, each twice, to obtain the fourth power of gamma acceleration as seen on earth.
It's also standard relativity that the space part of the four-force, the Minkowski force, scales exactly the same way. In four-vector notation, the Minkowski EOM is just F = m A, and so according to it, the force and acceleration remain balanced according to earth-based observers. However, Einstein had to redefine the force to be f = dp/dt (three vectors) which reduces to f = ma in the non-relativistic limit, in order to preserve energy conservation. What this amounts to is that earth-based observers see the force become much less than m a as the rocket becomes relativistic. So what causes the extra acceleration? I challenge anyone to explain it.
I think it might be plausible to find an explanation if the acceleration was less than the force, what with all the time dilation and space contraction going on, but explaining the force being less than the acceleration is a different matter.
Essentially this is already implicitly in the integrals I posted above, but I can post the math for this when I finish writing it up in latex. I'm doing an even better version of it, an electron being accelerated by a uniform static electric field, for my paper. This case is better because there is no question of whether it's physically realizable and of how do you maintain the thrust and what about the changing rocketship mass etc. Also there is an explicit covariant statement of the electromagnetic force, that incorporates an additional gamma factor compared to the pre-relatavistic Lorentz force. The gamma factor is what allows the force to grow wth acceleration, according to the stationary observer. Also it's interesting and simplifying that the electric field is invariant to Lorentz boosts in the field direction, so the electron always sees the same field as the lab people see. But, the gamma factor on the Lorentz force can be canceled to get f = dp/dt, thus breaking f = m a.
So, does anybody want to try to explain how the force can be wayyy less than the acceleration?
The statement that force needs to be defined as f = dp/dt is in Goldstein's classical mechanics textbook, as well as Rindler Intro to SR, and I think his other book too.
I decided to consider a test case from standard relativity problems, of the 1-g accelerating rocketship and its acceleration profile as observed from Earth. Usually this problem is about how the 1 g rocket can go to the center of the Milky Way and back in 40 ship years, and about what its mass-to-payload ratio has to be, but I was interested in its acceleration to thrust profile looked like on earth. It's already available in the problem solution so it's really almost trivial if you can do the problem to get that the acceleration, just like in the Osiak integral is the Lorentz factor gamma to the fourth power times the ship-frame acceleration. So after a few years of ship time, the acceleration seen on earth is vastly larger than 1 g. That's standard relativity, and follows from the acceleration being a second time derivative of position, so is affected by time dilation and Lorentz-Fitzgerald contraction, each twice, to obtain the fourth power of gamma acceleration as seen on earth.
It's also standard relativity that the space part of the four-force, the Minkowski force, scales exactly the same way. In four-vector notation, the Minkowski EOM is just F = m A, and so according to it, the force and acceleration remain balanced according to earth-based observers. However, Einstein had to redefine the force to be f = dp/dt (three vectors) which reduces to f = ma in the non-relativistic limit, in order to preserve energy conservation. What this amounts to is that earth-based observers see the force become much less than m a as the rocket becomes relativistic. So what causes the extra acceleration? I challenge anyone to explain it.
I think it might be plausible to find an explanation if the acceleration was less than the force, what with all the time dilation and space contraction going on, but explaining the force being less than the acceleration is a different matter.
Essentially this is already implicitly in the integrals I posted above, but I can post the math for this when I finish writing it up in latex. I'm doing an even better version of it, an electron being accelerated by a uniform static electric field, for my paper. This case is better because there is no question of whether it's physically realizable and of how do you maintain the thrust and what about the changing rocketship mass etc. Also there is an explicit covariant statement of the electromagnetic force, that incorporates an additional gamma factor compared to the pre-relatavistic Lorentz force. The gamma factor is what allows the force to grow wth acceleration, according to the stationary observer. Also it's interesting and simplifying that the electric field is invariant to Lorentz boosts in the field direction, so the electron always sees the same field as the lab people see. But, the gamma factor on the Lorentz force can be canceled to get f = dp/dt, thus breaking f = m a.
So, does anybody want to try to explain how the force can be wayyy less than the acceleration?
The statement that force needs to be defined as f = dp/dt is in Goldstein's classical mechanics textbook, as well as Rindler Intro to SR, and I think his other book too.
Are you sure? As the ship nears the speed of light in the earth frame, surely its acceleration as measured on Earth must decrease.
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No Ghost In This Shell
Registered Member
Haven’t read the thread in detail, but I’d like to comment on this particular point.
In my mind this isn’t so much a matter of conservation, but rather of definition - the very definition of what the term “energy” actually physically means comes from Noether’s theorem. And the answer here is clear - a particular continuous symmetry, being spacetime translation symmetry - leads to a particular and unique conserved quantity, the energy-momentum tensor (NB. not just energy!). And that’s it. In cases where there is no such symmetry, this quantity is not meaningfully defined. Everything else - the momentum 4-vector in SR, the above quoted energy-momentum relations etc - ultimately come from this tensor. But this tensor isn’t an arbitrary choice, it arises uniquely from Noether, so I don’t see how this alternative Osiak version can possibly be compatible with Noether, as it would need to arise from an energy-momentum tensor that has a different form. It’s essentially a quantity that has a different physical meaning.
I know, right! That's what I was expecting.
I have a book of solved relativity problems by Lightman, et al., (Princeton University Press 1975,) and I coded the solution up in matlab, and it reproduces the trajectory velocity versus tau and t, and the ship and earth times. I didn't code the mass budget one; I don't think I need to. Then I decided to look at the acceleration, which is not part of the stated problem, but is pretty much a required step in solving it. My inuition was telling me the same as yours but actually it blows up as (gamma)^4. When I saw it blows up I looked again at the four -acceleration, and sure enough it says the same thing. My intuition was just blocking me from seeing it.
I guess if I post this on Physics Forums they will say that's when you do get to use the Minkowski equation, and the explanation will make no sense to me. Or else they will take down the post and possibly ban me.
Temporal component of four-momentum is proved to be a conserved quantity by the same argument (in old textbooks) that proves conservation of "relativistic" mass. So Noether's theorem can be applied to momentum and temporal momentum, with or without writing in terms of Einstein energy. Osiak relativity says it's incorrect to interpret the four-momentum vector as the energy-momentum four-vector. This allows energy to be not conserved but still have particle creation thresholds as observed, except you have to scale things by a half. That just happens to resolve the spin-orbit coupling anomaly without invoking Thomas precession, which frees the Thomas precession to truly important in other contexts.
No Ghost In This Shell
Registered Member
The momentum 4-vector has a very precise definition - it is the total flux integral of momentum through a space-like hypersurface, and as such comes directly from the energy-momentum tensor, which in turn comes directly from Noether:
$$P^{\mu}=\int_{\Sigma} \Sigma_{\nu} T^{\nu \mu}$$
This isn’t a matter of interpretation, it’s the unique quantity associated with spatial translation invariance. You can choose to work with a different quantity, but then that quantity necessarily is not going to have the same physical meaning. That was my point. So it’s fine to work with Osiaks expression, but one needs to remember that difference in physical meaning.
The momentum four-vector including its definition are identical in Einstein and Osiak relativity. The components are identical as
$$ (\gamma m c, \gamma m v_x, \gamma m v_y, \gamma m v_z) \equiv (p_0, p_x, p_y, p_z) $$
So, everything that is true about it in Einstein relativity is still true in Osiak relativity. This includes that the temporal component is conserved. However, only in Einstein relativity can it be identified as an energy-momentum four vector, because only in Einstein relativity is the energy defined as
$$ E \equiv \gamma m c^2 = c p_0 $$
In Osiak relativity the energy definition is derived from the covariant Minkowski equation of motion F = m A, while in Einstein relativity it's required to define force as change in relativistic three-momentum. Using the Minkowski EOM obtains that
$$ E = \frac{\gamma^2 m c^2}{2} \neq c p_0 $$
In summary, Einstein and Osiak relativity agree that the spatial and temporal components of the four-momentum are separately conserved.
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Certainly there is vast evidence that the temporal component of four-momemtum is conserved over time. There is no need to counter that because both Osiak and Einstein relativity versions are in agreement about this.
There is a separate test proposed to determine which relativity version is correct, based on detecting the existence of positrons prior to their creation event. It was discussed in more detail in this thread: https://www.sciforums.com/threads/t...r-casually-and-regularly.166855/#post-3771702
Academia.edu has created some discussion about it. I merged some of it to make a video here:
They also created some videos, but they are not as long as the podcasts or "fireside chats". They are also posted on my YouTube channel.
All of the Academia In Depth podcasts are now on spotify. This one is about the proposed experiment specifically:
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No Ghost In This Shell
Registered Member
But that was precisely my point - only in Einstein relativity is the definition actually compatible with Noether’s Theorem, because it emerges directly from the energy-momentum tensor. Osiak essentially defines a different quantity with a different physical meaning - it does not and cannot come from Noether.
Sergej Materov
Registered Member
As far as I understand, Osiak rejects Planck's equations of motion, the equivalence principle and ignores the theorem Noether and that relativistic energy is not conserved in particle collisions and $$E_0=\frac{m_0c^2}{2}$$But then in reality nuclear reactors would produce half as much energy and the annihilation energy of a positron and an electron will not correspond to the rest mass energy of the electron for each gamma quantum.
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No Ghost In This Shell
Registered Member
Just to add to what I posted above - in Einstein relativity, the definition of “energy” is independent of, and more fundamental then, forces and kinematics. Even before relativity, we already knew that the conserved charge for time-translation invariance is
$$Q=\int d^{3}xT^{00}$$
And independently, we know from Hamiltonian mechanics that
$$H=\int d^{3}x\mathcal{H}$$
Therefore, the 00-component of T necessarily has the physical meaning of energy, and thus the same must be true for the 0-component of P. So this isn’t just an arbitrary interpretation, it’s pretty much non-negotiable. Thus is why Osiak’s version cannot have the same physical meaning of “energy”. This is also more fundamental than any kinematics.
I think Osiak relativity is consistent with the equivalence principle. It does however allow gravitational mass to be different from inertial mass, if ordinary matter is a composite of matter and antimatter, as in the Harari and the Shupe preon models.
As I have already said several times I think, Osiak relativity is consistent with Noether's theorem because it agrees with Einstein relativity on the definition of four-momentum where the temporal component is p_0 = (gamma) m c. Since in Einstein relativity E = p_0 c, and c is unity in appropriate units, one can replace the word "energy" with the words "temporal momentum" in any correct claim about energy, and still be correct according to Einstein relativity. Osiak relativity then agrees with these claims, as does Noether's theorem. Osiak relativity has a different energy expression though that doesn't doesn't equate with temporal momentum and so can't be part of the four-momentum, and isn't conserved.
I use the term quasi-energy or Einstein energy to refer to the temporal component of four momentum because it's still useful in Osiak relativity as it is the quantity that sets thresholds for pair creation, for example. However there is the factor of a half. I view it as simply resolving the spin-orbit coupling anomaly without the Thomas precession, which is a weak explanation for that factor of a half. (That frees Thomas precession to become hugely important in another way.) Also I think Einstein jumped to a false conclusion when he equated the photon energy with the Planck spectrum energy quantum. Planck's energy quanta are associated with standing wave modes of a cavity oscillator, while photons carry momentum so it takes two photons to make the Planck quantum. So for a photon, E = h nu / 2.
Dark Matter as the Difference of Inertial and Gravitational Mass
It is shown how two tentative theories, one of Osiak proposing necessary corrections to Einstein's relativistic energy expressions, and another of Harari and Shupe supposing leptons and quarks have internal structure, when combined might explain
www.academia.edu
No Ghost In This Shell
Registered Member
Yes.
The standard formalism of GR uses a (1,1) representation of the Lorentz group, being symmetric tensors - so of course you are going to be off by a factor of 2 when trying to couple the equations of motion to a particle that has intrinsic half-integer spin. This problem arises not from a failure of the theory, but from an inappropriate choice of formalism. To avoid this issue, all you need to do is pick a different representation of the Lorentz group, such as (½,½), being covariant bispinors, and formulate the same theory of GR in terms of those objects (bispinor tetrad formalism). When you do this, the issue is simply not there anymore.
If you want to retain tensors (greater mathematical simplicity), then another way to do it is to choose a connection other than Levi-Civita on your spacetime manifold. This introduces non-vanishing torsion, which couples naturally to half-integer spin.
Either way, re-defining the physical meaning of an existing concept is not a good way to do this, in my opinion.
No Ghost In This Shell
Registered Member
Correction to previous post: the correct representation for covariant bispinors should have been (½,0)+(0,½). (½,½) are in fact Lorentz 4-vectors.
Apologies for the mistake.
Apologies for the mistake.
I have been arguing since at least 2024 (see https://arxiv.org/abs/1609.04446) that the 00 component of the EM stress tensor represents the temporal momentum rather than energy. Letting it be temporal momentum makes the EM field complex, with imaginary fields propagating backwards in time. This can explain the complex character of the Schroedinger wavefunction, as shown at the link. Then also, a positron as a time-reversed electron has an imaginary negative charge and is influenced only by imaginary EM fields. This reproduces observations on cloud-chamber photographs but differs from conventional electrodynamics in straightforwardly observable ways. Hence my proposed experimental test.
Seems to me energy is also properly temporal momentum in Hamiltonian mechanics. Re-interpreting the Hamiltonian this way in quantum theory is consistent with the Dirac equation, and provides new insights as I explain, for example, here: https://ijqf.org/archives/7386
Hamiltonian mechanics predates relativity by a lot, and prior to the existence of relativity, no such distinction could be made. Energy and the temporal component of four momentum reduce to the same quantity in the non-relativistic limit. Einstein relativity assumes temporal momentum and energy are the same thing, without justification, and that forces a redefinition of energy to be rate of change of momentum, from the classical definition of energy as work done. That leads to a paradox that force must be less than acceleration in highly-relativistic situations.
I would rather do the experimental test that can settle the issue, than to argue the minutia ad infinitum.