You may look at http://arxiv.org/pdf/1309.0660.pdf

I acknowledge to everybody for helpful discussion. Especially, I thanks Ondrey for recommending the paper that seems to be related to my question.

Robert Low: I agree to your opinion. I am intersted in the deformed scalr multiplication such as v(+)v(+)v ....(+)v = n(*)v, where (+) is Einstein addition, (*) is a deformed scalar multiplication. I wonder whether infinit sum of infinitesimal boost gives a finite boost under the Einstein addition. If it is true, Is it possible to replace the Lorenz tr with

the x (-) v (*) t in terms of new scalar multiplication and Einstein addition without the square root? How do you think about this?

It's useful to keep in mind what we want to do. When we add (or subtract) two velocities, we obtain a quantity-and it *is*, indeed, a velocity. When we try to multiply two velocities, we should ask what we want to do with the result. One way of multiplying two velocities to obtain something useful is the dot product, which gives a number, (the cosine of) the angle between the two velocity 3-vectors. This is invariant under rotations-not, however, under boosts. We can, of course, compute a relatvistic invariant, by adding to this the product of the two time-like components.

The ``cross product'' can, also, be defined for velocity 3-vectors, but this vector isn't a particularly useful quantity-first of all, it isn't a velocity, of course, and it doesn't tell us something useful, whether about physics or mathematics. The ``cross product'' of two 4-vectors isn't a 4-vector, for instance.

It's actually impossible. The Lorentz group has two Casimirs: one is the mass, the other is the spin of the particle. So you do need to know what its spin is, to describe its interactions in a relativistically invariant way.

Of course it is: you need the Dirac algebra-this is what allows you to define spinors, if you want to describe electrons, that have spin.

I think that too much belief for hero such as Newton, Planck , Einstein is frequently an obstacle to renew physics. We have reponsibility for renewining physics. In my opinion, Nature is not bewutiful. But Physicsts dream that Nature is beatiful , so they believe that the real physical model can be decribes in terms of the beutiful linear theory such as Huygence principle , Lie algebra, planck formula, Schroedinger eq etc. But, In my opinion , Nature does not seem to be beautiful. I think that we have much courage to deny long lasting belief such as QM , spin, special relativty, etc for succsse in future physics .

What happens if we change the Einstein addition into

v_1 (+) v_2 = ( v_1 + v_2 )/ ( 1 + q v_1 v_2 ) , c=1 ?

Here q >0 is a deformation parameter. This addition is associative.

When q =0 , Newtonian

When q =1 , Einsteinian

When 0

Dear Professors!

Let me be simple. I have a problem with the initial question. Summing velocity with velosity you obtain velocity, -- the same physical units. Myltiplying velocity by velocity, you do not obtain velocity units. This is not quite as in group theory where 'ab" belongs to the same set (of physically WHAT?) as 'a' and 'b'.

By the way, what is your criticism re my article considering the equality of the average electrical and magneic energies in EM wave? You can find it in the Research Gate. (I am fraid that I have two stations there; at least at one of them it is surely posted).

Yours, Emanuel.

Do you mean multiplication by a scalar or what?

I mean scalar product giving kinetic energy

I think that two multiplications deserved to be discussed: One is scalar times vector and another is scalar product between two vectors.

Despite the name is "addition of velocity" really addition? Velocity is relative, e.g. the velocity of object B relative to object A, let's write: A->B. If I have velocity A->B and velocity C->D for different obects A, B, C, and D, does it make sense to "add" A->B and C->D? It seems to me that really what we do is "compose" velocities, we do not add them. We may compose velocites if we have A->B and B->C to obtain A->C.

Therefore it seems to me that the question about "multiplication" of velocities is actually without meaning.

Other comments have addressed the representation of velocity as a vector but it seems to me that discussing addition and multiplication of vectors is to confuse the concept of velocity with just one of it's possible representations.

True- these are just words, until we give them a meaning. The issue is that, starting from two velocities, how can we obtain another velocity in a way that satisfies certain properties. What we call this operation then matters only insofar as it can be identified with another operation (e.g. addition) that we're familiar with and which, also, is useful for manipulating two velocities to obtain a velocity.

Not just words. "velocity is relative" has a meaning (in physics). "addition" has a meaning (in mathematics). I think the issue is representation. Is addition (of vectors) a good representation for composition of velocities?

Yes, because velocities do behave as vectors-not, necessarily, as 3-vectors, but as 4-vectors.

Representing velocity A->B as a vector v does not faithly represent the concept of relative motion, i.e. we can add vectors when we can not meaningfully compose velocities. This is quite separate from the issue of exactly how we "add" vectors.

The statement that velocity is a polar 4-vector completely specifies its properties. That historically concepts were learned in a certain way doesn't mean that, now that we do know better, we should keep on talking about them that way.

Suppose we have three relative velocities, A->B, B->C, and C->D for different objects A, B, C, and D. We may compose these velocities is two different ways to give a relative velocity A->D

A->C with C->D

or

A->B with B->D

I propose that the following is a required property of velocity: that these two ways of composing three relative velocities gives the same result. If this were not the case then we would have to admit that the relative velocity A->D is not well defined.

@Stam Nicolis can you show that the properties of what you call a "polar 4-vector" satisfies this property? How do you represent composition of velocities?

Of course, that's why we say that the Lorentz transformations form a group. This is textbook stuff. You have to pay attention to some technical issues, but this fact is a starting point.

I am not sure exactly what this has to do with your original claim concerning "polar 4-vector" quantities but I will try to be more technical.

Given objects A and B we might wish to associate the relative velocity A->B with the Lorentz transformation L_AB that maps velocities relative to A into velocities relative to B. But I claim that in general

L_CD L_AC ≠ L_BD L_AB ≠ L_AD

So this does not satisfy the required property.

The point William Page makes is a good one. It was made a long time ago in Silberstein: Relativity 1914 (available on Wikipedia)

This has, precisely to do with the fact that, if you try to perform Lorentz transformations with relative velocities that aren't parallel, you have to be a bit careful. But, once more, this is textbook stuff, a typical homework problem.

Of course-but this is hardly front page news, but standard textbook stuff, that's, hopefully, taught at introductory physics courses.

It is certainly not textbook stuff because you won't find the answer in any book, The proper definition of relative velocity in more than one dimension was never treated in the standard theory of relativity. Strange, the theory of relativity never properly defined relative motion.

Of course it is-the Lorentz group is reviewed in many textbooks on quantum field theory and the subject is assigned in homework assignments, cf.

http://insti.physics.sunysb.edu/~siegel/620.html

http://epx.phys.tohoku.ac.jp/~yhitoshi/particleweb/particle.html

You are not understanding. What I am saying is that if two particles A and B move with general velocities in 3 dimensions, there is no formula in standard special relativity theory to find the relative velocity of A and B (or B and A)

(The discussion has changed subject) It's a standard homework exercise in Lorentz transformations, which can be solved with the tools in the links I gave (for instance). What is NOT well-defined (i.e. cannot be given an invariant meaning) is the relative velocity of two systems in *curved* space-time, where Lorentz invariance is a *local* symmetry.

The matrix method does not determine a velocity because of matrix multiplication. Also relative velocity A to B is not the negative of the relative velocity B to A. It is similar to the Mocanu paradox. The matrix methods you reference look impressive but lead to paradoxes. That is the point I am making.

No, there aren't any paradoxes and the phrase ``the matrix method does not determine a velocity because of matrix multiplication'' does not make sense. The composition rule for the Lorentz group is well known and in all textbooks that deal with this subject. The only ``paradoxes'' ensue when trying to define relative velocity, or, more generally, comparing 4-vectors, in *curved* spacetime. There, relative velocity cannot be consistently defined, in general. In flat spacetlme it always can. The reason is that Lorentz invariance is *global* in the latter case and *local* in the former. This implies, in particular, that parallel transport of a vector depends on the path in the former case and leads to the aforealluded ``paradoxes''. Flatness of spacetime implies that parallel transport of a vector cannot depend on the path and thus cannot lead to any paradox.

Are you familiar with what is called the Mocano paradox and if you are, are you prepared to accept it as a paradox? It arises from repeated Lorentz transformation and so involves matrix multiplication which brings with it a rotational effect. That also happens with the problem I discuss. The resulting transformation is the product of a pure Lorentz transformation (boost which defines a velocity) and a rotation. Isn't it rather inappropriate for a relative velocity to have a rotation? Exactly contrary to what you say I think it needs a curved space to rescue the situation by parallel transport. I hope I have made myself clear. You say that relative velocity can always be defined in flat space-time. Exactly how? No generalities please!

Actually I'd have permuted the phrases: ``it falls out of the mathematics naturally and it explains (thus) Thomas precession (among other issues).''

Stam Nicolis wrote: "The issue is that, starting from two velocities, how can we obtain another velocity in a way that satisfies certain properties. " I pointed out that one of the required properties of this operation is that it be associative. Stam says this is solved "as a standard homework exercise" by the use of Lorentz transformations. I claim that in spite of nearly 100 years of such "homework exercises" this dogma is wrong. In the general case of non-colinear velocities the "Einstein addition", for example as derived by V. Fock and presented in numerous text books is not associative.

I know of only two authors who have considered this problem in detail, unfortunately they are not well known and only one of them has published a textbook on the subject. Tomas Matolcsi in "Spacetime Without Reference Frames", provides a complete and consistent presentation of both Newtonian space time and relativistic space time. He defines relative velocity and the rules for composition of velocities and shows how it is related to (but not solved by) Lorentz Transformations. Zbigniew Ozewicz has published numerous articles on this subject (I am listed as a minor co-author on one of these). He has independently derived that same composition of velocities and shown that it is associative.

Also Abraham Ungar considers the non-associativity of Einstein addition of velocities (as give by Fock) as motivation for studying non-associative not commutative group-like structures from mathematical perspective.

V. Fock, The theory of space, time and gravitation, The Macmillan Co., New York, 1964.

Tamás Matolcsi, Spacetime without reference frames, Akadémiai Kiadó, 1993

http://www.cs.elte.hu/~matolcsi/

https://www.researchgate.net/profile/Zbigniew_Oziewicz/

The Lorentz transformations define a group, that's associative and, in 3+1 dimensions, in general, non-commutative. This is mathematically consistent and leads to predictions confirmed by experiment. From the group structure we can deduce how the *parameters* transform in turn (and vice versa). Nothing else is required.

Of course the Lorentz transformations form a group. The mistake you are making is to take if for granted that this is sufficient to ensure that composition of relative velocities is associative.

But the subjhect of this question refers to "Einstein's addtion of velocities". If this is not well defined then isn't that a problem? In fact that was the reason why I replied to this question in the first place.

Stam Nicolis also wrote: "... in *curved* spacetime. There, relative velocity cannot be consistently defined, in general. In flat spacetlme it always can." However the relative velocity defined by Matolcsi and Oziewicz depends only on the metric (dot product) and is well defined even in the general case.

A velocity is a 4-vector, therefore its parallel transport depends on the path-that's how curvature can be defined. It doesn't matter on what it can depend on, it matters that it's a *vector* and not a *scalar*. *Any* 4-vector has this property.

I agree that vectors have this property. How is that relevant to the definition of relative velocity?

A relative velocity is a 4-*vector*.

I agree that relative velocity can be represented as a 4-vector, not withstanding comments I made earlier in this thread about composition. How is parallel transport related to the consistent definition of relative velocity as a 4-vector?

Once more: If a 4-vector is defined at point P1 and parallel transported to point P2, the 4-vector at point P2 depends on the path used to parallel transport the vector from P1 and P2, not only on the points P1 and P2. Therefore, taking different paths, different vectors can be obtained. This is the hallmark of curved space-time. Therefore the relative velocity of two frames at points P1 and P2 is not uniquely defined, since the velocity of P2, viewed from P1 depends on the path used to parallel transport the velocity in question to P1. This isn't the case for flat space-time, where parallel transport does not depend on the path, but only on the points P1 and P2.

These statements hold for *any* 4-vector, not just velocity. A nice discussion is in

http://arxiv.org/pdf/gr-qc/9712019.pdf p.63-65.

Relative velocity is not an intrinsic property of an object (or frame), it is a relationship between one object and another. I do not know what you mean by "parallel transport the velocity in question".

Hmm... Carroll really did write: "But two particles at different points on a curved manifold do not have any well-defined notion of relative velocity — the concept simply makes no sense" (page 64). So Carroll would claim that it is not possible to define the relative velocity of a distant object confined to the surface of the Earth because the Earth is curved?

@Charles Francis it is not clear ot me what problem you are trying to solve. Yes Lorentz transforms do have useful applications. Yes the combination of Lorentz boosts is associative but not closed. My claim is just that they are not good for the definition of relative velocity.

``is not possible to define the relative velocity of a distant object confined to the surface of the Earth because the Earth is curved?'' It isn't the curvature of the Earth that's relevant, it's the curvature of *spacetime*. To a very good approximation, the spacetime aroud the Earth *is* flat. (For certain applications, like GPS, however, you *do* need general relativistic corrections.)

Nobody's claiming that Lorentz transformations are good at defining relative velocity-for curved spacetime manifolds-since it can't be defined at all, in that case, in general. They're perfectly good, on the other hand, for *flat* spacetime manifolds, from the group structure of Lorentz transformations.

@Stam Nicolis I disagree with you on both points.

Let me be specific (paraphasing Matolcsi): Associate with every object p, q, etc. a time-like vector field denoted by the same letter, g(p,p) = g(q,q) = -1, etc. where g is the metric tensor. The world line/path of an object is an integral curve of this vector field where tangents to the curve are time-like vectors of the field. Define the velocity of q relative to p as the space-like vector field

v_qp = q/-g(p,q) - p

The composition of relative velocities

v_rq v_qp = v_rp

is well defined and associative. No where do we need Lorentz transformations or parallel transport to define relative velocity.

@Robert Low Thank you. Yes v_qp is in the three dimensional space-like subspace of vectors orthogonal to p

E_p = { x | p.x=0 }

i.e. the space of all velocities relative to p.

This still makes since in curved space-time. Adapting your notation where p is a particle and P_s is the time-like tangent vector at some point s on the worldline of p, consider the congruence of the worldlines of all such possible particles at rest (relative velocity = 0) relative to p. This is the vector field associated with p. All particles at rest relative to p have the same associated field. There is no need to compare four-vectors at different points of space-time.

II have a way to define frame independent 4 velocities which I gave at a conference in 2004. I meant to work up to talking about this before but started off on the wrong foot and got a bit muddled. If you are interested it is online ar www..space-lab.ru/

PIRT_VII-XII.php?lang=eng Select PIRT IX It is called "Relative Motion and Hyperbolic Geometry in Special Relativity" It is easy to describe the basic idea.

@Robert Low Associated with each object is a time-like vector field P called the observer. The integral curves of this vector field, i.e. solutions of the differential equation

r'_p = P(r)

are worldlines. Yes there are some technical details which are addressed at length by Matolcsi.

One does not need to compare points on different worldlines. Relative velocity is define in terms of observers (vector fields).

@Charles Francis There is no frame dependence. That is one of the main points of Matolcsi's book. I think that you should not expect to be able to understand something based only on a few paragraphs written here by a third party (me). I highly recommend consulting the references that I gave earlier. If you don't have access to Matolcsi's book then a web search will at least show a few articles that can be downloaded including at least one in arXiv. Oziewicz has articles available for download here on ResearchGate.

Sorry to butt in again into your conversation but relative velocity is not the right quantity to be looking for in a space-time with a general metric as in general relativity. You cant define velocity itself. But if you work with rapidity the answer comes out fairly easily. You can get a triangle of rapidities and hence define a relative rapidity. But I suppose you will ignore this remark and continue with your conversation. Good luck! PS Minkowski space, having a pseudo metric is not flat (Euclidean). It is like a cylinder, flat in one direction and curved in others.

@Charles Francis thankyou very much for taking a quick look through our little article. The abstract actually says that velocity is "reference-dependent". The word frame (which might be considered a synonym of coordinate system) is carefully avoided. The meaning of reference-dependent is elaborated on page 6. I would welcome further critical comments but since this is off topic in the context of the current question I think it would be more appropriate to continue only under the Discussion tab of that paper.

Robert Low: Thank you for your interest. You raise two points (1) The usual argument about the Riemann-christoffel tensor vanishing is only partly true. For a space with positive definite metric .its vanishing implies space locally Euclidean. For a pseudo Euclidean space vanishing implies the space is again locally pseudo Euclidean which takes a hyperbolic metric. This is the case with relativitity. Proper time supplies an extra direction (2) The basic result is that rapidities have a 'triangle of velocities' in hyperbolic space. This dates back onver a hundred years to Varicak, Robb and Borel

More recently Ungar and myself have written about it all my approach is different to that of Ungar with whom I disagree. I do prove the result in the paper quoted above and also in the paper at the PIRT 2002 conference which you can also access online there.