In other threads it was suggested that kinetic energy and potential energy compete for curvature. Kinetic energy was said to bend space backward compared to gravity. LaGrange function is well accepted as a factor in determining how celestial bodies move. Action S is calcu;ated for all the possible movements and the smallest S identifies the motion. In GR the calculations predict gravity opposing light and slowing light when approaching.
Questions of gravitational red shift arise which seems to say gravity attracts light.
Dicke supported a competing theory which predicts light losing energy as it departs from a star.
If Dicke is correct then GR is in doubt.
In LaGrange Function Why Is Potential Energy Subtracted From Kinetic Energy?
``Why" questions depend on the assumptions, in order for answers to be meaningful. The relative sign between the kinetic and the potential energy, in the expression of the Lagrangian in the non-relativistic approximation, is fixed by the assumption that the potential energy is bounded from below.
The statements about curvature are wrong and the whole issue doesn't have anything to with GR, where the distinction between kinetic and potential energy isn't well-defined, anyway.
Stam Nicolis
Thank you for answer. I notice that LaGrange Function is used to predict non relativistic movement of planets and moons, when conservation of energy allows more than one set of movements.
Even in non relativistic systems, curvature of gravity is still acting on the several bodies. Einstein wrote that when gravity must be considered GR applies and several rules are changed. High speed relativity is not required for GR to apply.
I agree with you that high speed relativity does not uniquely define energy in coordinate transformations, since sequence of events is not the same for all observers.
Although other explanation may explain why. The minus sign implies that kinetic energy has reverse curvature and decreases the affect of gravity in correctly predicting motion of celestial bodies in complicated systems.
I believe GR applies even to low speed movements in the LaGrangian because SR and Newtonian Mechanics do not resolve the complicated system.
As always your comments are welcome and helpful.
Jerry Decker ,
Kinetic energy is operatively defined as the work of a force to raise the speed of a body from 0 to v, provided that it is measured in an inertial frame.
The Lagrangian is L=(T-V) where V is an energy which does not involve the macro-kinetics. L is the classic Lagrangian where T is the classic approximation of the relativistic kinetic energy.
Lagrange did not use a variational principle (ACTION) to arrive at the equations of analytical mechanics: Euler-Lagrange equations. Hamilton did it some decades later.
In the relativisitc case after 1905 it was made a choice.
The derivative of L with the speed corresponds to the momentum in classic lagrangian T=1/2mv2.
For the relativisic kinetic energy T =(gamma-1)E0 it emerged that its derivative in speed was not anymore the relativistic momentum.
A choice was made in order to make the derivative of L in the speed to be equal to the relativistic momentum: L = - E0 gamma-1 . Such quantity is Lorentz invariant but negative definite although bounded from below since its minimum value is -E0 .
A serious discontinuity between L classic an L relativistic occurs, to my opinion which is a source of several issues....
Jerry Decker
Not sure how I got here, anyway I'm quite interested in a broader interpretation of the action principle, e.g. such as in the line of De Broglie (1924): “Nevertheless, action is a very abstract notion, and as a consequence of much reflection on light quanta and the photoelectric effect, we have returned to statements on energy as fundamental, and ceased to question why action plays a large role in so many issues”
Strictly, V is not always bounded from below (Stam Nicolis), e.g. in a wave, so that is not automatically explanatory. At the same time, the Hamiltonian is biased toward macro kinetics - ref. Stefano Quattrini, but that is not always a problem. As I see it the Lagrangian is the approach of choice in a - potentially - geometrized spacetime scenario. But I'm not sure whether this idea is presently fully exploited.
If the potential energy isn’t bounded from below, the space of states isn’t well defined. For mechanical waves the potential energy is bounded from below and for electromagnetic waves, also, in fact. What’s forgotten are the boundary conditions at infinity.
In any event, what the Lagrangian provides is a variational principle for obtaining the equations of motion, on the one hand, in a way that’s ``more convenient" for ``complicated" systems and for obtaining the conserved quantities in a way that shows their relations to symmetries more clearly. In the process, what are the ``sensible" signs comes out, by itself.
If you take the potential energy in the Lagrangian with the ``plus" sign, it will appear in the energy with the ``minus" sign.
As I understood it, boundedness is more or less an ad hoc requirement in speculative theory, such as the fabricated quartic term in a scalar field self-interaction with btw. a negative mass term. If the theory is sound there are no +/- infinities. I believe proper geometrization avoids ill behavior caused by defining field modes only; internal and external symmetries are then related.
Dear Frank van den Bovenkamp ,
thanks for the citation of DeBroglie, it is amazing that such scientists did not understand the profound meaning and implications of such a principle.
going back to the Lagrangian proposed in SR, L = - E0γ-1
the following LR= γ-1 E0(γ-1) is equivalent with E0 = mc2 and E0(γ-1) is the relativistic kinetic energy emerging always from the Kinetic energy theorem but derived from the work of F=dP/dt with p=mvγ.
LR is much better
a) LR is positive definite
b) LR is still related to Kinetic energy and reduces to 1/2mv2 exactly, at low speeds
c) LR follows still a Least action principle not a action principle.
Stam Nicolis
I believe you are correct about sign of kinetic energy in Lagrange. Some researchers now use Lagrange function or density in stress energy tensor, making curvature smaller but not out of observed limits. In this case the tensor gets a minus sign to off set the minus in Lagrange.
Observation cover cases of gravity potential being dominate. Max Born allowed kinetic energy to dominate in other cases that need to be tested.
With much new science proposed in related threads, it could all change after new tests are developed.
Of much concern is local variations in light speed. Einstein Born, and Tolman wrote light speed slows dowm in gravity fields.
Braun and Dicke wrote LS speeds up when approaching gravity field considering mass equivalent of energy,
Bergmann said LS doesn't change, only the perception of some observers. This view is widely taught.
The answer is important for high speed in deep space. If Born is correct then star travel may become possible, but not if Dicke is correct. Black holes event horizons are said to have light speed of zero by Einstein, or 2c by Dicke. Zero seems to agree that light cannot escape.
More recent publications have favored the Einstein view because of black hole calculations. Classical mechanics favors Dicke which must agree in a limit. For mechanics to agree with Einstein a sign must be changed in the force equation. Then classical mechanics can give two exact answers, one exactly the same as Dicke, and one exactly the same as Einstein and Tolman.
Max Born is especially interesting as the only one who gave results to light speed in terms of gUV curvatures, Einstein notation, but alloying speed to increase locally.
Bergmann is also compelling until it is realized his argument was not supported by any thing other than his opinions.
Thank you for reply.
Once more: The ``correct'' signs are those that lead to a phase space of finite volume per degree of freedom, as deduced from the Hamiltonian constraint.
In flat spacetime, where time translations can be globally defined, there's no problem. In curved spacetime, things are less obvious.
Stefano Quattrini
It seems that De Broglie indeed foresaw the use to this day, of the action principle as a mathematical convenience, while, as hinted by Stam Nicolis, more congenial with Noether's Theorem. As you can read on wiki, De Broglie pioneered the idea of hidden thermodynamics, and "Up to his final work, he appeared to be the physicist who most sought that dimension of action which Max Planck, at the beginning of the 20th century, had shown to be the only universal unity (with his dimension of entropy).".
As for your LR= γ-1 E0(γ-1), it seems to imply more or less the opposite of what γ is supposed to imply. And generally new physics is never just a matter of reshuffling a few parameters. Where and why exactly do you find a problematic discontinuity between Lclassic and Lrelativistic ?
Frank van den Bovenkamp
infact the Lagrangian I proposed LR= γ-1 E0(γ-1)
is not the relativistic Lagrangian used in SR, which is LSR =γ-1 E0 .
LR is just the high energy version of LC=1/2mv2 , it does not suffer with the discontinuity as LSR does.
There’s no discontinuity in the Lagrangian for the free relativistic particle, when taking the non-relativistic limit.
Mass should accumulate dark matter in excess of the local gravity field unless the excess is decomposed by quantum exchange of the field with the mass. Decomposition seems likely, but decray products are not obvious other than possibility of microwaves.
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