When analyzing a moving body with respect to a moving observer we can use the Galilean or Lorentz transformations. As we approach the speed of light we know the Lorentz transformation gives the correct results. At low speeds they are "supposed" to be the same, but they do not converge to exactly the same result. There is a slight discrepancy between the two. The question is: what is the physical intuition behind this difference and what is it telling us?
They are equivalent. And it's known how to show this, cf.
http://www.pnas.org/content/39/6/510.full.pdf
The question doesn't have anything to do with physics, in fact, it's a purely mathematical statement, about how groups are related in a consistent way.
They are 'exactly equal' only when the speed is 'exactly zero'. Otherwise, there will be a small correction term in Galilean transformation. Mathematically, Lorentz transformation approaches to Galilean transformation as the speed between the observers approaches to zero.
True, when the speed approaches to zero, but we deal wit finite speeds in physics.
The orbiting speed of the Earth is cca 30000m/s, therefore, for clocks distanced cca 10^13 m (the Solar System diameter) we'll have a measurable synchronization error (vx/c^2), even if we put gamma = 1.
Galilean and Lorentz transformations are equivalent if we consider the limit of low speed and small space region.
Derek,
please explain in which situations something related to your question 'does not converge to exactly the same result'. I don't think that there is something of that kind when we consider limits v --> 0. Of course there are differences in the structural implications of Poincare-invariance and Galilei-invariance, n-body mechanics being a striking example.
In Galilean transformation the time is univesral constant. Time is the same in all inertial coordinates systems. Inertial coordinate systems are systems which are moving with a constant velocity.
Velocity v= v1 in the train and velocity on the ground v2 are related by
v2= v1 in the moving frame+velocity of moving frame. Therefore
x= x'+ v t
This is not true in the Lorentz transformation. When the velocities are comparable to the velocity of light then the coordiante transformations are related by when x is as seen on the groung and x' as seen on the moving frame are related by
x= (x '+ v t')/ sqrt(1- v²/c²)
time are related by
t =(t' + x'v/c²)/sqrt(1-v²/c²)
You can say the Lorentz transformations are universal and under approximation they are equivalent to Galiliean transformations. Please check the Lorentz transformation equations.
@Vaman Kulkarni,
they are not, since in the term x'v/c² (the synchronization error) we can have finite contributions if x' is big and v is small, but finite, as I commented before. Therefore, as Ivanov said, to have equivalence under approximations (not the limit) we have to consider a small space regions. This is not a trivial fact. Consider objects in the Milky Way galaxy, they are rather slow relative to the Earth, however, due to large distances, approximated LT will give results different than GT. The Lorentz transformations imply the time non-simultaneity, which cannot be approximated to zero at large regions of space. In fact, this becomes evident even at dimensions of the Solar system.
Dear Miroslav Josipovic ,
it would be a big issue if this happened for v
The paper by Stefano Quattrini contains a severe error:
for γ → 1, vt’= [vt- x +x] = vt, (should be added: where v=0)
hence t’=t (wrong conclusion since v=0 in the limit)
and x ‘= (x – vt) (should be added: = x since v=0 in the limit)
Comment: One cannot pass with quantity gamma to 1 forgetting to pass simultaneously with v to zero, unless c -> oo.
Domsta,
>
what are you inventing to support your totally wrong claim???
v has to be small but not zero it is what happens in experiments with classical approximation... also the walls know that for v
Stefano Quattrini
What does it mean "time and simultaneity in LT are relative" in this context? Just write down the Lorentz transformations for differences of coordinates and it immediately follows that you cannot ignore the synchronization error term if the space difference is big enough.
You cannot prove that transformations are equivalent using only speed, you have a counterexample. You must also assume a small region of space, this is a simple mathematical fact. Interesting, this question is well treated in Russian literature (I speak Russian).
And there is nothing strange in such an equivalence, because both transformations follow from the same three symmetries: relativity principle, isotropy, and homogeneity.
Miroslav Josipovic
in such case they would be unusable, since you cannot control the approximation.
by expressing the LT as
t'= γ-1t - vx'/c2,
x'=γ(x - vt)
for γ-->1 you can find
t'= t - vx'/c2,
x'=(x - vt)
which thing would say that you are right!!
but this is not the way to make the equations converge since it is a system of equations linked, you cannot calculate the limit one by one.
Mixing them up you will find
Preprint THE LORENTZ TRANSFORMATIONS AT LOW SPEEDS DO NOT REDUCE TO T...
In the case you are right, it would be the case to immediately resort to Tangherlini Trasnformations in order to avoid such inconvenience, but it is not the case yet..
Stefano Quattrini
Why "unusable"? Is there a limit on the size of coordinate systems out there? Based on what? And what do you mean by "control the approximation "? This is a simple high-school mathematics.
Miroslav Josipovic
t'= t - vx'/c2,
x'=(x - vt)
can you say in your problems previously when x' will be big or small to make
vx'/c2 negligible?
Like that it would be not usable in Physics...
maybe yes, but what about Physical application???
Dear Miroslav Josipovic
I think you will be interested in the answer I have given in another thread to Stefano Quattrini:
SQ, you are denying correctness of the limit procedure stated by me as follows,
JoaD:>> For c -> oo we have:
1. equation t'= gamma(t - vx/c2) approaches t'=t,
and independently
2. equation x'=gamma(x-vt) approaches x'=x-vt,
which form the GT.
Domsta, It Is confusing! It is difficult to understand what you said and what I said!!!
Special for Stefano Quattrini:
The following answer by Biswajoy Brahmachari
" For |v/c|
this is your total invention, never said that!!!
Where did I say that vx/c2 should be preserved??? YOU MUST BE KIDDING!!
Contrarily, I am objecting that SQ is neglecting vx/c^2.
This is the abbreviated form of my objection:
On one side SQ is accepting beta c for the coeficient at t in [ x' = x- beta c t ], on the other side he is neglecting beta/c for the coefficient at x in [ t' = t - beta x/c ].
Obviously this is inconsistency when keeping c=const > 0 and considering only first order tems wrt beta.
SPECIAL FOR JOAD,
For c -> oo we have:
1. equation t'= gamma(t - vx/c2) approaches t'=t,
and independently
2. equation x'=gamma(x-vt) approaches x'=x-vt,
which form the GT.
Domsta,
while t' = t - beta x/c is what can come out if gamma-->1, but x/c is large
from t'=gamma(t-xv/c2)
but what is this?? x' = x- beta c t
how can you write such nonsense x'=x-vt /c2 which does not even match with the dimension of [L] space????
the equation is x'=gamma(x-vt) where did you possibly get that NONSENSE??
THIS IS RIDICULOUS NOW YOU UNDERSTAND HOW RIDICULOUS it is???
SQ,
1. I was and still am stating that you were and are objecting the way when c->oo, without precisely stating that your objection was mathematical.
2. I was and am stating that you are objecting mathematical correctness of the result that for v/c -> 0 while c kept positive and constant, GT is not a limit of LT. Stop mixing things, OK?
Domsta,
>
PLEASE READ CAREFULLY THE POSTS WHICH ARE ADRESSED TO YOU, AND DO NOT PUT MAKE STRAWMAN GAMES to JUSTIFY YOUR TRICKS!!!!
How can you even think that I did not agree that for c-->oo the form of LT does not reduce to the form of GT, that is elementary math... it is even insulting..
do you understand what is obvious or you have the difficulty to understand it...
but who do you think you are dealing with?? I could solve that limit when I was 17 in highschool.. YOU MUST BE KIDDING???
I ALWAYS said that such operation was not possible PHYSICALLY!!!!!!
And the solution of that limit could also be found with gamma-->1 not necessarily with c-->oo.
Domsta,
>
?????????
v/c goes slower to 0 than v2/c2 which is the first term of the gamma, so the approximation v/c ->0 with v small at will but not identically 0 and c constant holds...
Dear all and Joachim Domsta
let's make a recap of the quarrel..at least we get something out of the confusion.
1) SQ: "In Physics the classical approximations are obtained for low speeds voo"
3) SQ:"since c is a constant and not a parameter, the only thing to do is to make v sufficiently small to allow voo LT reduce to GT but it is a problem if it is the only way"
5) Josipovic: given the LT in the alternative form t'= γ-1t - vx'/c2, x'=γ(x - vt)
"since in the term x'v/c² (the synchronization error) we can have finite contributions if x' is big and v is small, but finite, as I commented before. Therefore, as Ivanov said, to have equivalence under approximations (not the limit) we have to consider a small space regions. This is not a trivial fact. "
6) Josipovic: " One must also assume a small region of space, this is a simple mathematical fact. Interesting, this question is well treated in Russian literature (I speak Russian). "
Miroslav Josipovic can you provide the reference to the paper?
FROM the HP:
a) if Domsta is right in supporting the fact that LT do not reduce to GT for v small in general, but only for c->oo.
b) if SQ is right in supporting the fact that in Physics the only way to make the two get closer is to make v small, since we cannot act on the speed of light
then Josipovic is right as well as per 5) and 6)
from the equivalent form of LT
t'= γ-1t - vx'/c2, x'=γ(x - vt) v/c
“…Why are Galilean and Lorentz transformations not equivalent at low speeds?……”
- that is a little strange question. The transformatons indeed never are equivalent – and what?
As well as so the answer is quite evident – because of the Lorentz transformations are completely adequate to the reality in most cases, i.e. if corresponding systems of bodies constitute rigid systems,
- when the Galilean transformations never are completely adequate to the reality, including when the Lorentz transformations are valid.
When there is no any rational physical reason from which should follow something that would force the transformations to be equivalent.
That is another thing, that the Lorentz transformations – and so the SR - aren’t completely adequate to the reality if a system is the system of free bodies, as that, for example, J. S. Bell showed in the “Bell paradox”.
When, why, and to what extent, the transformations are adequate completely and not completely is shown in the Shevchenko-Tokarevsky’s informational physical model https://www.researchgate.net/publication/273777630_The_Informational_Conception_and_Basic_Physics DOI 10.5281/zenodo.16494, the paper https://www.researchgate.net/publication/317620440_About_some_conventions_in_mechanics DOI 10.5281/zenodo.1142628 is useful also.
Including, because of from, say, the rigorous Dongle objection to the SR [see, e.g. the SS post 3 days ago now in https://www.researchgate.net/post/Is_there_a_solid_counter-argument_against_Dingles_old_objection_to_Relativity_Theory#view=5e11ca96d7141b9b784daa82 ] ,
by the rigorous proof by contradiction rigorously follows that Matter’s spacetime is absolute, and so observation of an absolute motion and measurement of an absolute velocity is only a technical problem, which can be solved just when the Lorentz transformations aren’t completely valid, how that can be made – see https://www.researchgate.net/publication/259463954_Measurement_of_the_absolute_speed_is_possible DOI 10.5281/zenodo.48709
Happy New Year and Merry true X-mas coming!
Cheers
Stam Nicolis
for you it is sufficient to state that for c-->oo LT and GT , are equivalent or in other words LT reduce to GT, which is straightforward to demonstrate (TRIVIAL) and to a certain extent intuitive.
Unfortunately LT and GT continue not to be equivalent at all, at low speeds, as stated by Derek Abbott since low speeds not necessarily comes with c-->oo . And as also stated by Miroslav Josipovic the only solution is
t'= t - vx'/c2,
infact we cannot play with c, which is a constant, but the only thing we can do is to keep the speed sufficiently low to avoid the relativistic effects.
So v
Well, this is interesting.
1. If we start from LT with low speed limit, we cannot get GT, since c is finite and we can use the coordinate x large enough (this means that we can get a measurable synchronization error even with small speeds).
2. The only way to get GT from LT is to take the limit c -> Infinity (note that synchronization error disappears this way).
3. Both GT and LT (as well as pure rotations in 4D linear space) follow from three symmetries: relativity principle, isotropy, and homogeneity. In fact, we get general transformations with three characteristic numbers: -1, 0, and 1 (for pure rotations, GT, and LT). We choose LT because of experiments.
4. In geometric algebra of 3D Euclidean vector space (Cl3), it appears that we have three important types of multivectors that square to: -1 (complex numbers, rotations, spinors), 0 (dual numbers, GT), and 1 (hypercomplex numbers, LT). Yes, with geometric algebra, we get both GT and LT in 3D. In addition, it appears that the numbers that square to -1, 0, and 1 are important in quantum mechanics, as well (see Baez). Of course, quantum mechanics is naturally formulated in geometric algebra (Cl3), over real numbers and without the imaginary unit. So, if we correct problems with the cross product of vectors and introduce geometric product (due to Grassmann and Clifford), we get the formulas of the special relativity as a part of geometry of 3D Euclidean vector space (in fact, twice). This brings a new light on the special theory of relativity and quantum mechanics, with clear geometric interpretation. More information on this subject can be found in my new book (you get also a Mathematica implementation of Cl3) at
https://www.springer.com/gp/book/9783030017552
Well,
it is also interesting to know that Tangherlini Transformations do not suffer of such inconvenience:
t'= γ-1 t
x'=γ(x-vt),
for v
Stefano Quattrini
LT follow from three symmetries: relativity principle, isotropy, and homogeneity. This is a simple mathematical fact (the proof is in my book). The Tangherlini Transformations do not follow from these symmetries. This means that we have to have a strong physical reason to abandon the listed symmetries (which one). Personally, I do not believe in unisotropic universe, especially after my insights into the geometry of 3D Euclidean vector space in the light of geometric algebra. Without the symmetries, life of physicists will become extremely hard. But, this is just my intuition, we need experiments (I am not impressed with the stories about CMB).
The main concern in Physics are conservation laws. Symmetries have been artificially provided by the Lorentz group by forcing the speed of light to remain constant everywhere. They are as comfortable as detrimental since they hide the intrinsic asymmetric ity of evolution of processes and is at variance with thermodynamics second law.
“…The main concern in Physics are conservation laws…”
- that is indeed so; however that
“…Symmetries have been artificially provided by the Lorentz group by forcing the speed of light to remain constant everywhere…”
- - isn’t. Symmetries have not been artificially provided by the Lorentz group.
When the conservations laws, which quite really act in Matter, indeed,
since Matter is the simple logical informational system that exists and always constantly changes basing on simplest binary reversible logics [so the energy conservation law really follows from that interactions in Matter are reversible in time] in accordance with a rather small set of utmost universal laws/links/constants, and practically for sure basing on “hardware” of 4D fundamental logical elements (FLE), which have equal “sizes” in all dimensions of Matter’s absolute [5]4D Euclidian spacetime with metrics (cτ,X,Y,Z,ct), i.e. all what happens in Matter happens with “equal footing” in all quite “symmetrical” dimensions,
- exist and act be actualized/ in accordance with because of these symmetries, and the Lorentz group is adequate to the reality by the same reason, and because of every particle, and so every body, etc., is a 4D close-loop algorithm and so is some 4D gyroscope, what to the above adds the rotational symmetry.
Again, that really happens in Matter, and so the Lorentz transformations and the group mostly quite really adequate to the reality and so that
“….They are as comfortable as detrimental since they hide the intrinsic asymmetric ity of evolution of processes and is at variance with thermodynamics second law..…..”
- isn’t correct also. The conservation laws, and the symmetries are valid in any thermodynamic process on micro levels. That is another thing, that real processes in Matter proceed only in one direction in the ct-dimension [“true time” dimension], and in only both opposite directions in the cτ-dimension [“coordinate time” dimension, i.e. the “time what clocks measure”], that is fundamental property of the absolutely fundamental phenomenon/notion “Time”.
At that indeed, all/every processes in Matter are in depth absolutely fundamentally random because of the fundamental self-inconsistence of the phenomenon/notion “Change”, just therefore the QM exists; and if there is a system of essentially independent material items, the system’s configuration as a rule is random also; and so the system evolves in the direction of utmost probable state.
More see, for example, a few SS posts in the thread https://www.researchgate.net/post/Number_of_universe_dimensions#view=5e0f6d3636d2358d6078c837
More see the SS posts above and links in the posts.
Happy New Year and Merry true X-mas!
Cheers
Everything depends on the definition of equivalence. A problem at the start is that every LT depends on 2 parameters c and v while GT only on one of themm v. Next, some researchers prefere to calculate some distance (a value of a metric betweeen the linear mappings) or the distance between the images under the two mappings (then usually with the same c).
Whas is clear after the above discussion,
-- the Taylor expansion of LT(c,v) wrt the beta:=v/c (suggested by Miroslav Josipovic up to terms of the first order is not GT(v), unless v=0.
-- the limit of LT(c,v) as c -> oo equals GT(v)
-- the matrix norm of composition LT(c,v)\circ GT(-v) differs from 1 by a quantity of first order magnitude wrt beta:= |beta| only (if c is kept fixed), etc.
The most spectacular difference between LT an GT is that LT possesses diagonal algebraic equivalent (in other words - complete set of eigenvectors), but GT does not possess it (unless v=0.). Therefore, in particualr, you cannot obtain GT from LT by linear change of coordinates.
Usually equivalence between expression under certain conditions means that they give same results.
- Only the limit of LT(c,v) as c -> oo equals GT(v)
but this option is impossible to implement in experiments
- The Taylor expansion of LT(c,v) wrt the beta:=v/c (suggested by Miroslav Josipovic up to terms of the first order is not GT(v), unless v=0.
The first order Taylor expansion of LT is not GT unless v=0, but by applying it on x'=(x-vt) this has the consequence of x=x' which is not even a GT, so it is not viable the option of v->0.
- The condition of v
Statement.
Some parts of the above comment by Stefano Quattrini on my "previous post" do not present my scientific opinion. In particular I never stated that
- The condition of v
Quattrini,
1. In your last but one post you did NOT refer to other authors, but me. So you are simply laying when stating in the last post:
SQ: it was based also on what Josipovic said.
If you have made this by mistake, you would have already changed the last but one post.
2.Aren't you able to catch, that you are presenting false statement as if I was claiming them? This is a robbery and deserves application of severe warning: if you don't stop this ugly practice I would have to call you a criminal.
3. I hope this was for the last time that you refer to my achievements/ comments/ statements neglecting strict reference by link and/or by faithful quotation.
4. I have to frankly confese, that you are spoiling my opinion since
- you do not understand my claims which makes you feeling competent to change according to your WRONGLY seemed meaning
- moreover, you don't know that you don't understant my claims.
- and since you know a little anyway, my formulations after changes freely made by you are for an unexperienced people looking like true facts, and this is the greatest danger.
Domsta
A lot of answers here and i don't have time to read them all, so someone may have already offered what i do here. If you are just looking at motion, then the two transformations are virtually identical at slow motion. This is not true when transforming, for example, Maxwell's equations. A charge moving through a magnetic field does not have be going very fast to experience the force due to an electric field found by a Lorentz transformation of the magnetic field to the rest frame of the charge. The Galilean transformation fails here.
I think you better read them, since your post does not really have an added value now.
The Lorentz force here is due to the Lorentz transformation. A charge stationary in a magnetic field does not experience the Lorentz force. When it moves there is an electric field present in its rest frame due to the Lorentz transformation.
>
E=qv x B according to your opinion depends on the LT?
"Using Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics , Lorentz arrived at the correct and complete form of the force law that now bears his name"
I remind you that the LT were not derived by Lorentz, they were derived by Poincare' in 1905, as a modification of some relations found by Lorentz for EM fields.
it is due to the nature of electromagnetism not the LT.
I suppose you meant F = qv x B. The v x B is the electric field found from the LT. And with that, i'm not following this question any more.
Yes F= q vxB that is the Lorentz force.
It has nothing to do with LT. You must be kidding!!!!
As a matter of fact the most upvoted answer to this thread are both unsatisfactory:
a) STAM NICOLIS does not answer to the question but says they are equivalent. Since for c which goes to infinite LT reduce to GT, which means that they are not necessarily the same at low speeds, hence they are not equivalent.
b) Biswajoy Brahmachari does answer to the question saying that the first order expansion of LT is GT. ..it is enough to check it out by taking the first order expansion, the term vx/c2 stays, so he is wrong as well.
Congratulations
Dear Derek Abbott
the Physical difference between GT and LT at low speeds is represented by vx/c2 or better vx'/c2 as pointed out by Josipovic.
That term remains at low speeds and it represents the reduction/increment of the time (at the first order) taken by light to cross the distance x/c (or x'/c) which separates the source and the absorbed due to the fact that the body-absorber is in motion at speed v and not stationary (otherwise it would be x/c). It is strongly dependent on the Einstein Sync procedure at the base of the LT.
The vx'/c2 is the desync (at the first order) experienced by a clock in motion in regards to the clock which remains at rest, if they use the speed of light to communicate their time-stamps.
It is understandable that GT and LT can never be the same, even at low speeds, since GT do not use signals at the speed of light to communicated events between clocks.
Dear Stefano ~
The “Lorentz” force law was first derived not by Lorentz but by Maxwell in 1861. It is equation (77) in https://en.wikisource.org/wiki/On_Physical_Lines_of_Force.
Maxwell, of course, knew nothing about Lorentz transformations!
On the other hand, the force law F = q(E + v × B) can be derived from the electrostatic force law F = qE by performing a Lorentz transformation once we know the transformation laws of E and B.
Dear Eric,
the Lorentz Force is Lorentz Invariant, very different from affirming that q(vxB) is the force derivable from LT.
The problem with the Lorentz force when it is derived from a Lorentz transformation is the fact that it has an added Lorentz factor (or gamma factor). Ignoring the gamma factor, we can probably account for the appearance of vxB based on the work of Weber. The Lorentz transformation simply incorporates the ratio of electrostatic units of charge to electromagnetic units of charge. See Section III in this article,
Article The 1855 Weber-Kohlrausch Experiment (The Speed of Light)
But neither Weber nor Lorentz had the correct physical context within which to accurately connect the Coulomb Force to the vxB force.
This theory here by Purcell contradicts the Lorentz transformation because it converts the Lorentz force into the electrostatic force across two reference frames using the Lorentz contraction, and unlike with the Lorentz transformation, there is no Lorentz factor in the final result. The conversion is direct.
http://galileo.phys.virginia.edu/classes/252/rel_el_mag.pdf
So either Purcell is wrong, or Lorentz is wrong, or they are both wrong. They can't both be right.
Seems as is rather strange discussion in last series of posts about some textbook point, see, e.g., §38 in
https://detritus.fundacioace.com/pub/books/Landau%20L.D.%20%26%20Lifschitz%20E.M.-%20Vol.%202%20-%20The%20Classical%20Theory%20of%20Fields.pdf
-?
Cheers
In order to support comments by Francis Redfern and Eric Lord on the relation between E and B under LT within SRT, I have passed the considerations of the Einstein paper of 1905 and the Planck paper of 1906 which present the formal way required for getting the influence of changing the IRF. The result is not often presented in text books, and therefore I have put it into a preprint of my project SRT+gravity
Preprint SRTplusGRAVITY05rob - Interpretation of claim 2 of Einstein'...
Here I confine myself to state that the relativistic Biot-Savart Law (with Lorentz factor) can be derived by application of LT to electrostatic field of a point-charge at rest at the origin of the original IRF and the result is
B = q v \times E / gamma, . . . *)
where v is the speed of charge with respect to the moving IRF.
- - - - - - - - - - - - - - - - - -
*) there is a correction added 10 hours later, sorry, JoaD
Dr. Abbott, The two transformations only converge when the motion stops completely. Even at very slow speeds, there will still be a slight difference between them.
Dear Frederick ~
The gamma factor that enters when the Lorentz force is obtained by applying a Lorentz transformation to the electrostatic F = qE simply reflects the difference in the concept of "force" in Newtonian dynamics and in SR. That in turn comes from the difference in the definition of momentum: p = mv in the Newtonian limit but p = γmv in SR.
My last comment contains an excidentally created false statement, which correctly should sound as follows:
The relativistic Biot-Savart Law (i.e. with Lorentz factor) can be derived by application of LT to electrostatic field of a point-charge at rest at the origin (t,x)=(0,0)of the original IRF and the result is
B*(0,x*) = \gamma q v \times E(X(0,x*)) = \gamma-1 q v \times x* / |x*|3
where v is the speed of charge with respect to the new IRF* at the event with coordinates t*=0 and space position x*.
Eric, The gamma factor which arises during a Lorentz transformation would appear to represent a Doppler shift in the field patterns. It doesn't relate to the aspect of the transformation that makes −∇φ and −∂A/∂t in one frame appear as v×B in the other frame. That conversion seems to be due to another part of the maths that shares a strong element of similarity to that which we see in a rotational transformation, something like what produces the Coriolis force when a radial position vector is differentiated in polar coordinates. It’s this latter aspect of the Lorentz transformation that introduces the curl factor into A. Lorentz was trying to make Maxwell’s equations Lorentz invariant and he used an aether as his rest frame, but without specifying the physical details or how it operated. If only he had used Maxwell’s aether in particular, then he’d have been well on the way to reconciling his theory with how Maxwell derived equation (77) in his 1861 paper. The two theories could have been merged to some degree. This "rotational" part of the Lorentz transformation also imports the Weber constant from the 1856 experiment, as can be seen where it derives the equation B = 1/c2v×E.
As it stands though, the Lorentz transfomations appear to be fictitious, as like in the case of a rotating frame of reference. They don't contain the substance that is associated with Maxwell's version of F = qvxB where a charged particle moving in a magnetic field experiences an actual deflection. The Lorentz interpretation implies merely that vxB is just another way of looking at something from another frame of reference. Lorentz seemed to be using a hydrodynamical aether approach whereby A is current/momentum, while φ is potential energy/pressure, hence φv is also a current momentum. The Lorenz gauge (as in Ludvig Lorenz the Dane, as opposed to Lorentz the Dutchman himself) seems to be simply the equation of continuity of free electric fluid (aether) density, with Bernoulli's Principle being central to the conservation aspects. But instead of having a sea of tiny aether vortices like Maxwell where the elasticity and density introduce the speed of light, Lorentz has simply imported the speed of light from Maxwell's equations. And rather than considering the rational of Maxwell's equations within the context that Maxwell derived them, it seems that Lorentz was looking purely at ad hoc maths that would make Maxwell's equations Lorentz invariant. In other words, the Lorentz transformations are a mixture between fact and fallacy. They are also a mixture of 3D concepts such as curl, all packaged together to look like a 4D concept. But we all know that there is no curl operator in 4D.
I somehow agree with Fredrick approach, although it is clear that the aether as a "substance" conceived by Maxwell is wiped away by the behavior of the Longitudinal Doppler effect (LDE).
At the same time though the LDE is worth some clarifications.
The LDE is the frequency shift of the absorbed radiation at 0 angle between an emitter body and observer one.
Such frequency shift can be easily derived, as a first approximation, from the momentum of light. It is from a dynamic point of view a transmission of linear momentum (n*hv/c) from the emitter to the absorber having their own proper frequencies of absorption.
In accelerated motion between two equally accelerated bodies what occurs is not different from a LDE between two objects in relative motion, since light does not really care if a body changes its linear speed, due to the fact that the events of emission and absorption occur at constant speed in any case.
The idea according to which the rear body sees the radiation blueshifted has two not complementary interpretations, but mutually exclusive.
a) the the rear body has more momentum than the forward one, when the light is absorbed in the IRF where it was emitted:
light is provided an additional momentum which increases its frequency/energy in the reference frame of emission, hence the blue-shift.
b) the clock at the rear end ticks slower so it sees the radiation of the head blue-shifted
(a) nothing is changed in the emitter/absorber oscillators, but it is simply the dynamics which makes the photon have an higher freq.
(b) In the second case something "changed in the oscillators", so that the same radiation is "seen" different. These are just effects in v/c. They have nothing to do with v2/c2 which actually involve periods of clocks.
(b) Is the first detrimental consequence of the term vx'/c2 in LT which prevent them to reduce to GT at low speeds.
Stefano, I don't doubt what you are saying about longitudinal Doppler effect, but I don't see how its existence in anyway undermines Maxwell's sea of molecular vortices. I would fully expect there to be longitudinal Doppler shift. And it would be too much of a coincidence if Maxwell were able to obtain vxB at equation (77) in his 1861 paper from vortex hydrodynamics, and for Lorentz to also obtain this exact same term from his transformation equations, unless there was a strong physical linkage between the two theories.
And it's actually the vx'/c2 term in the Lorentz tranformations that I would want to keep, because that's the part that links it to Maxwell by converting −∂A/∂t and −∇φ into v×B. Also, I don't get your point about this being the term that prevents it from reducing to the Galilean transformations at low speeds. They never do converge until the speed is zero.
My guess is that it's the gamma factor that needs to be adjusted, possibly reduced to the binomial first order approximation so as to remove the absurd asymptotic implications. Lorentz seemed to have been drawn into the unnecessary quest to establish a symmetry that does not exist.
As for time dilation, it was a very bold prediction on Lorentz's part, but so long as we operate from a base aether rest frame, hence removing the clock paradox, the time dilation can then be understood in terms of how physical motion through the aether interacts with matter and retards all the internal atomic and molecular processes. In other words, time dilation is the microscopic extension of Bernoulli's Principle. Space and time are conserved in the same way that potential energy/pressure and kinetic energy/momentum are conserved. As a clock becomes more pressurized due to motion, everything slows down inside it.
Dr. Domsta, Yes, The Biot-Savart law can indeed be derived from the Lorentz transformations, but it can also be understood in terms of fine-grained angular momentum within Maxwell's molecular vortices. Therefore rather that ignoring Maxwell's work as many do, one should be seeking the commonality between Maxwell and Lorentz. That quest of course ended in 1905 when Einstein cast out all the ingredients. He did nevertheless restore it again in 1921 in a half-baked state, not fit for purpose. It was only with Dirac that it started to emerge again, but by then, it's linkage to electromagnetism had been lost and forgotten. See Section VII here on the Biot-Savart Law,
Article Ampère's Circuital Law and Displacement Current
"And it's actually the vx'/c2 term in the Lorentz tranformations that I would want to keep, because that's the part that links it to Maxwell by converting −∂A/∂t and −∇φ into v×B. "
That term allows the communication of momentum of light in fact, and is ok with the Longitudinal Doppler. It has nothing to do with periods of clocks though as it is wanted to mean. I do not mean at all that LT do not have any use, it is how they have been used in some case which is not appropriate.
Stefano, Correct. It is the gamma factor that is to do with time dilation, and not the vx'/c2 term. The vx'/c2 is good. In fact you can actually get to the Maxwell bits with this term without even involving the gamma factor at all. I'm not saying though that the gamma factor should be abolished altogether. More likely it should be adjusted to its binomial first order approximation to get rid of the absurd asymptotic implications. There is a lot of experimental evidence claimed in favour of the gamma factor, although that certainly doesn't mean that it has to be interpreted as per Einstein's vision. I attempted to address the slowing of clocks here in terms of conservation of energy and aether shear interaction, as well as using the binomial first order approximation,
Article Atomic Clocks and Gravitational Field Strength
The gamma factor is compulsory.
You can get the Doppler effect of the first order just from Galilean Transformations, same as it is found for acoustics.
The point is that the symmetry is compulsory to find the higher order longitudinal doppler.
Stefano, The gamma factor is not compulsory for the purposes of obtaining v×B from −∇φ and −∂A/∂t. Cut the gamma factor out of the transformation and you will still be able to do those same manoeuvres. Those are hydrodynamical manoeuvres acting on pressure, φ, and momentum/current A. The gamma factor only compresses the fields along their direction of motion.
Fredric,
the gamma factor is essential to explain many phenomena including description of experiments at high energy, the accelerators, p=mv*gamma hence E2=E02+(pc)2 , inelastic scattering etc.
That factor is unavoidable!!!!
Stefano, Yes, but it's not unavoidable as regards converting between the different electric force terms such as −∇φ , −∂A/∂t, and v×B, which is the bit that links Lorentz to Maxwell's aether. And with this aspect, the speed is only relevant as regards the magnitide of v×B. But with the aspect that relates to the gamma factor, speeds need to be in the order of the speed of light before they become significant.
Fredric,
what counts is the result of experiments (KAUFMANN to begin with),
this has some similarities with the discontinuity issue raised several times in the other thread which you ignored...
That factor is always present, that it is negligible in most cases depends only on the accuracy of measurement performed.
Stefano,
Yes there are some similarities, but those similarities need to be clarified. In the other thread you were talking about a continuity between the electrostatic force field, −∇φ, and the time-varying electromagnetic force field, −∂A/∂t. I was saying that only the latter is involved in the Poynting vector. Any continuity which exists between the two is related to Bernoulli’s Principle. The potential energy term, φ, is a pressure, while A is a momentum, or a current. And for that reason alone, you can see why only A would be involved in energy flow.
As regards the Lorentz transformation, it has its origins in a principle which looks like Bernoulli’s Principle, but this time as between space and time, as opposed to as between pressure and velocity. But when the Lorentz transformation is applied to electric and magnetic fields, it becomes even more like Bernoulli’s Principle and that’s where the gamma factor becomes important.
But I wasn’t talking about the gamma factor. I was talking about how the beta factor curls the A field and induces v×B. In other words, the link to Maxwell’s aether is not related to the gamma factor, but it does appear to be related to the beta factor.
Dear Stefano Quattrini ,
"The gamma factor is compulsory.
You can get the Doppler effect of the first order just from Galilean Transformations, same as it is found for acoustics."
And we can get the Doppler effect of " the second order" just from general Galilean transformation (Galilean transformation of "the second order"). Because the Doppler effect and relativity are the single phenomenon. And, "gamma factor" or "Lorentz factor" is nothing more than a term which we find to the law of cosines. To the longitudinal relativity (Doppler effect) "gamma factor" is vanished because the angle is zero and 180 degree, and in other case we have "gamma factor".
To @Stam Nicolis and to all, just a little summary.
For STAM it is sufficient to state that c-->oo to make LT reduce to GT, which is straightforward (TRIVIAL) and intuitive.
Unfortunately LT and GT are not equivalent at all, at low speeds, as stated by Derek Abbott since low speeds is not the same as setting c-->oo .
By writing the LT in the equivalent form
t'= γ-1t - vx'/c2, x'= γ(x-vt)
As also stated by Miroslav Josipovic the time transform for v
The difference between LT(v,c) and GT(v) [with |v| < c [and with c>0 the fixed constant speed of light], at event (x,t) ∈ R2 is given by the equations:
(1). . . . x'L - x'G = ( γ - 1) ( x - v ⋅ t) , . . . . . . (2). . . . t'L - t'G = ( γ - 1) ⋅ t - γ ⋅ v ⋅ x / c2 .
It seem to be reasonable to accept the following convention: For any T> 0 the spatial difference between the galilean GT(v) and the lorentzian LT(v,c) is qualitatively negligible in a region of events contained in the layer between instants -T and T whenever the second term of RHS(2) is not greater than the first term of RHS(2) as follows
(3) . . . . |γ ⋅ v ⋅ x / c2 | ≤ (γ - 1) ⋅ T
Then the implied necessary restriction for the space coordinates says the following
(4) . . . . |x| ≤ |v| ⋅ (γ -1) ⋅ T / [γ ⋅ β2 ] ≤ |v| ⋅ T . . . [independently of c (!)],
which confirms in a rigorous form that the assumed condition (3) defining negligibility cannot be obtained within the whole range of space coordinates.
Intuitively, position of the ends of a bar depend on the direction of the communicating signal (light, sound etc.) from the ends relative to the observer. If the bar is moving, the signal from the end farthest from the observer will take longer to arrive to the observer. So, when the signal from the near end arrives at the observer, the signal that is simultaneously arriving from the far end is when the bar far end was at a previous position. That is, the bar appears shorter if moving. The Galilean transform assume the signal has an infinite speed. The aerodynamics of bodies in air (sound waves) has the same phenomena.
Dear Joachim Domsta
I guess in this expression or yours: |x| ≤ |v| ⋅ (γ -1) ⋅ T / [γ ⋅ β2 ]
under the condition v/c 0
|x| ≤ |v| ⋅ 0 ⋅ T / [1 ⋅ β2 ] = 0
it holds only for |x| -> 0
Dear Stefano Quattrini ,
If v=0 then (3) is fulfilled for all x.
If 0 < |v| ≪ c then the length of range of x-s fulfilling condition (3), satisfies 2⋅|v|⋅T ≪ 2⋅c⋅T.
Consideration of v→ 0 concerns infinitely many cases, each with OWN set of x-s (!) , thus deriving as the consequence that x goes to zero is illegal.
If 0
Dear all,
from the following equivalent form of the LT
t'= γ-1t - vx'/c2, x'= γ(x-vt)
since in accelerators the distances in the reference frame of the collision which is always chosen as a RF, the term x'/c corresponds to a very small elapsed time since x' is a very short length, then vx'/c2 gets negligible.
In such case the transformations used become
t'= γ-1t
x'= γ(x-vt)
which are the Tangherlini Transformations
Dear Stefano Quattrini
>> x' is a very short length, then vx'/c2 gets negligible. In such case the transformations used become . . . .
Dear Joachim Domsta
yes, the neglibility is within the "economy of experiments in accelerators" where the scattering is concentrated in nanometers around the center of mass of the scattering masses, taken as the primed reference frame.
Nanometers/c means times of 10-9m/(108m/s) = 10-17 s, which is a very small time in the lab frame in comparison with the elapsed times in the "moving frame". The most precise atomic clock so far built is barely capable of accounting for 10-17 s, in addition the scattered particles are strongly time-delayed due to their intrinsic speed.
The time transform of TT and LT accounts for a difference of less than 10^-17 s, hence the transformation t'= γ-1t is more than enough to give account to what is happening.
One of the biggest difference between TT and LT, is that |v|/c
Dear Stefano Quattrini ,
sorry but I do not see the way you infer from
>>The time transform of TT and LT accounts for a difference of less than 10^-17 s, . . . . . hence the transformation t'= γ-1t is more than enough to give account to what is happening.
Dear Joachim Domsta ,
let's see the points of agreement!!!!!
one dimensional case
LT : t'= γ-1t - vx'/c2, x'= γ(x-vt)
TT: t'= γ-1t , x'= γ(x-vt)
a) The Difference: TT - LT
in the time transform is Diff= vx'/c2
no difference in the space transform
b) vx'/c2 it is an elapsed time
c) Diff = vx'/c2 = x/c * beta , IN ACCELERATORS since the primed frame is the center of mass of the colliding particles |x'| is of the order of nanometers, hence x/c is of order of 10-17 s. Diff is a fraction of 10-17 s, since beta is lower than unity.
d) hence Diff is non detectable in experiments in accelerators, is virtually absent.
e) As a matter of fact since Diff is negligible t'= γ-1t is the relation which gives a non negligible result to t' .
Dear Stefano Quattrini
>> t'= γ-1t is the relation which gives a non negligible result to t'.
Dear Joachim Domsta
please explain better this sentence..
The negligibility depends on the accuracy of experimental setup used, hence on the technology used. Technologically clocks cannot have an accuracy under 10^-17 s.
Since vx'/c2 is lower than that order of magnitude in accelerators, any experiment will fail to detect it.
in other words: no experiment is able detect a difference between the LT and TT in accelerators since since such difference is always lower than the capabilities of the instruments at disposal nowadays.
This is the information needed to state what is negligible or not.
So the accelerators, so far, tested the validity of the Tangherlini Transformations since the difference between TT and LT in such conditions is so small that no instrument can detect it.
Dear Stefano Quattrini
>> So the accelerators, so far, tested the validity of the Tangherlini Transformations since the difference between TT and LT in such conditions is so small that no instrument can detect it. xL comparable to t c
Joachim Domsta
As usual, what you just said reveals that your level of understanding is quite limited.
I am not available to go through explaining simple things to whom does not have the will or aptitude of understanding them.
Dear Stefano,
the order of comments was as follows:
PART I.
SQ: >> So the accelerators, so far, tested the validity of the Tangherlini Transformations since the difference between TT and LT in such conditions is so small that no instrument can detect it. Great! Therefore LT is aproximated by GT in the same way as TT is approximated by GT and conversly, the GT approximates with the same accuracy both TT and LT. As usual, what you just said reveals that your level of understanding is quite limited. I am still expecting an explanation what does have to do
[ SQ: ] The time transform of TT and LT accounts for a difference of less than 10^-17 s, ...
with the preference of TT expressed as follows:
[ SQ: ]... hence the transformation t'= γ-1t is more than enough to give account to what is happening.
Treat this question as a serious problem for me, please. I am not available to go through explaining simple things to whom does not have the will or aptitude of understanding them.
A pairwise comparison of the Lorentz (LT), Galilean (GT) and Tangherlini (TT) transformations of he R1+1 space-time
They are defined by the following formulas for the new (primed) coordinates by the original (unprimed) coordinates as follows"
LT(v,c) . . . . . .x'L = γ ⋅ ( x - v⋅ t) . . . . t'L = γ ⋅ ( t - v ⋅ x /c2)
GT(v) . . . . . . x'G = x - v⋅ t . . . . . . . . . t'G = t
TT(v,c) . . . . . x'T = γ ⋅ ( x - v⋅ t) . . . . t'T= t / γ
where γ = 1 / √ (1 - β2), β = v/c, -c < v < c, c >0.
For every possible pair {A,B} ⊂ {L,G,T} the best uniform estimates of the errors of the approximation of transformation AT by BT is expressed by the differences
(*) . . . . . . δxAB ≔ x'B - x'A . . . . . . δtAB ≔ t'B - t'A
and β (up to order 3 for small β), for the transformed region of the space-time with the original coordinates restricted as follows
(**) . . . . . . |x| ≤ D . . . . . . |t| ≤ D/c, for any D >0.
(LG appr) . . . .|δxLG|/D ≤ ½ \beta2 + ½ |β|3 . . . . . c⋅|δtLG|/D ≤ |β| + ½ β2 + ½ |β|3
(TG appr) . . . . .|δxTG|/D ≤ ½ \beta2 + ½ |β|3 . . . . . c⋅|δtTG|/D ≤ ½ β2
(LT appr) . . . . . . . . . . . .|δxLT|= 0 . . . . . . . . . . . . c⋅|δtLT|/D ≤ |β| + β2 + ½ |β|3
The critical term |β| appears for the LT apprroximation by both GT and TT which proves that neither GT nor TT approximate as well as the TT and GT approximate each other.
Moreover, since these are the best orders , for any velocity, any of the above approximations in an unbouded region (with D= ∞) is impossible.
By using the equivalent form of LT and by considering γ = 1/√(1 - β2), β = v/c
LT: t'= γ-1t - vx'/c2;______x'= γ(x-vt).
TT: t'= γ-1t;_____________ x'= γ(x-vt).
GT: t'= t;________________x'= x-vt.
1) classical Physics is found by |v|/c t'= t ; x'= x-vt : GT
Tangherlini at low speeds reduce to GT which is the correct classical Physics
------------
from LT: t'= γ-1t - vx'/c2; x'= γ(x-vt). --> t'= t - vx'/c2; x'= (x-vt). ???
Lorentz transformation at low speeds reduce to transformations which are not the ones used in classical mechanics, they still contain the speed of light.
------------------------------------------------------------------------------------------------------
2) When γ >> 1 due to high speeds and x'/c t'= γ-1t ; x'= γ(x-vt) :TT
Lorentz transformations and TT at high speeds and very short distances in the primed frame are the same.
____________________________________________________________________
3) for γ->1 and t >> vx'/c2 (very short distances in the primed frame such that that x'/c is not detectable by laboratory instruments)
LT: t'= γ-1t - vx'/c2; x'= γ(x-vt). --> t'= t ; x'= x-vt : GT
Lorentz transformations reduce to Galilean only for low speeds and very short distances in the primed frame which is not at all general.
CONCLUSION
------------------------
Tangherlini T. have the non trivial advantage over LT that,
a) whenever the speeds are sufficiently low such that γ->1 , they reduce to the transformations of classical mechanics.
b) In accelerators they can be used instead of LT.
Joachim Domsta
t'= γ-1t ; x'= γ(x-vt) what is the result of the two equations for γ->1 then?
I have said:
I.
JoaD: >> The critical term |β| appears for the LT apprroximation by both GT and TT which proves that neither GT nor TT approximate as well as the TT and GT approximate each other. (up to order 3 for small β) > Diff is a fraction of 10-17 s, since beta is lower than unity. > since Diff is negligible t'= γ-1t is the relation which gives a non negligible result to t'
SQ: >> t'= γ-1t ; x'= γ(x-vt) what is the result of the two equations for γ->1 then? 0 and γ v -> 0 and γ-1 -> 1
(**) t' = γ-1t ----> t' = t
(***) x'= γ(x-vt) ----> x'=x.
Domsta,
v2 goes to 0 faster than v, γ goes to 1 much faster than v goes to 0.
hence for sufficiently small v, γ is very close to 1
so for v/c which gets very small
t'= γ-1t ; x'= γ(x-vt) approaches very fast to t'= t ; x'= (x-vt) .
this is what counts,
while vx'/c2 does not get sufficiently small if x/c gets large, hence
we get stuck here t'= t - vx'/c2; x'= x-vt
FULL STOP
P.S. Do you want to see it numerically??
Dear Stefano Quattrini,
please check the theorem on limits of sequences to obtain that, independently of how fast gamma goes to 1 and v goes to 0, the product gamma \cdot v goes to zero.
Therefore, your inference
SQ: >> t'= γ-1t ; x'= γ(x-vt) approaches very fast to t'= t ; x'= (x-vt) .
I will do with sum of converging sequences which is simpler and closer to a numerical demonstration!!!!!
the power series of t'= γ-1t developed in the proximity of v/c is just
t'=t (1 - β 2/2 - β 4/8 - β 6/16 - 5 β 8/128 + O(x9)) (Taylor series)
for v small then β small then β 2/2 and the other terms go to zero faster and the espression approaches to
t'=t
while
x'= γ(x-vt)
x' = (1 + β 2/2 + β 4/8 + β 6/16 + 35 β 8/128 + O(x9)) x - vt (1 + β 2/2 + β 4/8 + β 6/16 + 35 β 8/128 + O(x9))
v gets small β gets small
1 + β 2/2 + β 4/8 + β 6/16 + 35 β 8/128 + O(x9) approaches to 1
although v gets smaller linearly , t gets larger (which is always the case, time advances constantly)
x' = x - vt
only for t constant and v getting smaller, the final result would be x' = x
but this is never the case.
SO
TT ---> GT for v/c getting smaller is simply right!!!
This inconsistent end of your way rigorously should use the differences deltatTG and delta x TG as they are given by my answers
It is certainly possible and rigorous to make a power series in the way I proposed.
again!!!!
x'= γ(x-vt) = γx- γvt then
x' = (1 + β 2/2 + β 4/8 + β 6/16 + ...) x - c β t (1 + β 2/2 + β 4/8 + β 6/16 + ...) ;
x' = (1 + β 2/2 + β 4/8 + β 6/16 + ...) x - ct (β + β 3/2 + β 5/8 + β 7/16 + ...) ;
x' /c = (1 + β 2/2 + β 4/8 + β 6/16 + ...) x/c - t (β + β 3/2 + β 5/8 + β 7/16 + ...)
for β getting smaller that expression approaches to its first approximation which is
x' /c = x/c - β*t
which does not degenerate into
x' /c = x/c
for β getting smaller,
since for t getting sufficiently large it goes into x'/c = x/c - T , where T= β*t is a finite number.
it is clear that this implication
> for v getting smaller, is false in general, unless t remains constant, the mistake is to assume a constant t which can never be.
t is a variable which gets larger and larger.
For how slow a body proceeds at a speed vslow , there will be always a sufficiently long time tlong that will allow it to go across a finite path length L such that
L = vslow tlong
hence x'=x+L never x'=x, (unless v is identically null so nothing can never move, which is not the case), v 0. v never 0, just its limit is.
The Galilean transformations are the first order approximation of the Tangherlini transformations!!!!!!!!!!!!
On the other hand it is true that the result of LT for v/c getting smaller is
t'= t - vx'/c2; x'= x-vt
since x'/c can be taken sufficiently large to keep vx'/c2 = β* x'/c from getting small.
(it is not certainly obtained from LT for c->oo which would give just t'= t ; x'= x-vt about which all the world agrees!!! ).
Dear Stefano Quattrini,
You are writing suggesting that I am opposing your claims,which is evidently not true. Please read my claims carefully once more. The problem probably lies in omitting by you the definition of deltas.
Thus,the lately formulated ( recalled by me) estimate means exactly the same as your claim that GT is the first order approximation of TT wrt small beta, since the difference is of order beta^2. Period.
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