I think it is possible, but the main fact is role of the "angles", which quiet seems hard to articulate.  

What do you mean by a lightlike plane, that is different from a plane spanned by a time(like) axis and a space(like) axis?

So, your generators has the form

L_a = K_a + i J_a, and N_a = K_a - i J_a  

for a=1,2,3. Here K are boost generators, J rotation generators. The i may or may not be there, depending on conventions.

The Lie algebra of the L's and N's closes separately to two commuting SU(2) algebras. Hence, using transformations by two different L's or two different N's cannot give you all Lorentz transformations (because they can be rewritten as the transformation by a single L or N, neither of which gives you everything).

However, I feel pretty confident that a transformation by one properly chosen L (exponentiated) and one properly chosen N (exponentiated) is sufficient.  In fact, the finite dimensional irreducible representations of the Lorentz group can be labelled by their values of L-spin and N-spin.

Dear Professor Olaussen, 

Thank you very much on your kind reply. I will try to choose properly  L and N.

Akira> split physics community into two mutually hostile groups

Your post is off-topic relative to the question, but deserves a couple of comments anyway:

1. There is absolutely no split in the physics community regarding "modern" (post-1900) physics. Just check the textbooks used, and physics courses taught at (say) the 1000 best ranked universities in the world, or physics articles published in scientific journals with any recognizable merit. The relevant discussions are about extensions of this established physics.

But it is true that a development like the internet, made possible by "modern" physics, has also made it possible for many people to demonstrate the Dunning-Kruger effect to a global audience. To the uninformed, this may skew the perception of the real situation.

2. Regarding the MM-experiment, the situation you describe does not correspond to the experiment, where the mirrors are at rest in an inertial frame moving with velocity v relative to the (hypothetical) ether. Hence the light round-trip time, calculated according to non-relativistic physics, becomes

T_tot = L/(c+v) + L/(c-v) = (2L/c)/[1-(v/c)**2]

The situation you analyze corresponds to the case where the mirrors are moving together with the ether, as observed from the frame above.

Akira ~

It is not good for your health to stay up this late.

Akira> You are the Physicists.

I have a PhD in mathematics from MIT.

Akira ~

It is not good for your health!

Akira ~ Where did your v come from?

I meant the v in your analysis of the MM experiment.

(Most people I meet in science have an interesting ethnic background; this tells me that talent is uniformly distributed)

I think the question is not well posed. What is a rotation in a lighttlike plane? Does this mean that it is the identity in some complementary plane? This cannot exist unless it is the identity itself.

If you mean that a lightlike plane is just invariant, then any boost in the e_1e_4 plane is itself such a lightlike rotation: It has n = e_1+e_4 as eigenvector and keeps e_2 fixed.

Best regards

Jost Eschenburg

Dear Professor Kanda, 

Thank you very much on your comment. I am not expert in physics, but I am interested in your explanation. Can you please explain the nature of speed v,  since you said that the apparatus of length d move with speed v. I do not understand why you can sum the speed c and the speed v. 

Best regards, 

Emilija Nesovic

Dear Emilija;

Many thanks four explaining your question. I think that the answer is yes at least for the boosts: They can be obtained as a composition of two rotations about lightlike planes.

I guess that these rotations are precisely those elements of the Lorentzian group which fix precisely one lightlike direction R.n (the direction of the lightlike fixed vector n). In other words, for such Lorentzian transformation A, there is no second likelike vector n' (which is not a multiple of n) such that  An' = tn'  for some real number t.

(Proof: Otherwise, the timelike plane (n,n') would be preserved, hence fixed (since n is fixed), thus A is a rotation in the spacelike plane perpendicular to (n,n'), but this must be trivial since A has also a spacelike fixed vector.)

Now we pass to another model of the Lorentzian group: It acts on the 2-dimensional sphere S^2 as the group of conformal (Moebius) transformation; the sphere is the boundary of the light cone modulo scalar multiples. Conformal transformations with just one fixed point are precisely planar translations when S°2 is identified with the plane R^2 via stereographic projection whose center (north pole) is the fixed point of the given conformal map. I would rather view this conformal map as a vector field on S^2 by transplanting the translation vector to S^2 by the stereographic projection; the given conformal map belongs to the flow of this vector field.

Now consider two such vector fields v,w, where v has its zero in the south pole and w in the north pole. It is enough to consider the great circle S^1 \subset S^2 through the north pole and south poles in the direction of the translation vector in the plane. Then we see that the conformal vector field v+w has two zeros (in the "east" and "west" poles on S^1, so to speak). Hence v+w correspond to a conformal map (the flow of v+w) with two fixed points, and this corresponds to a boost in Minkowski space. Vice versa, any boost has two fixed points which can be (after conformal transformation) considered as north and south pole of the 2-sphere and the above construction applies.

Clear enough?

Best regards

Jost

Dear Professor Jost, 

Thank you very much on your explanation  and a very nice example. It is very good  to know that the boosts can be obtain as a composition of two rotations about   two lightlike planes.  This is a new view of the boosts as rotations in mathematical sense.  But in physics, we have to be careful where we can use them.  

I agree with prof. Akira that the constancy of the speed of light is  the only physical assumption which match well with many theories, without   any serious and clear  proof.   

Best regards, 

Emilija

Dear Emilija,

I think that the answer is yes again. We can show this for the Lorentz group  G in 2+1 dimensions already. We use the model of G as the isometry group of the hyperbolic plane (the Poincare disk). The rotations about light-like planes are those hyperbolic isometries with just one fixed point at the infinite boundary. They "rotate" the boundary but one point (the fixed point). Take two of these "rotations" with same orientations but with opposite fixed points. The composition of these two hyperbolic isometries has no fixed point at the boundary, hence (by Brouwer) there is a fixed point on the open disk which means that we obtained a hyperbolic rotation about this point.

Understandable? I can add a figure if needed.

Best regards

Jost

I think postings should be time-stamped with their local time of entry. To alert readers when thoughts were thought at 2AM or 03:35AM.

Dear Professor Akira, 

In attachment please find one interesting paper about one  existing dilemma in physics. 

Best regards, 

Emilija

Dear professor Akira, 

In the above paper, the basic steps are done by means of  the mathematical logic. So, too many  mathematical logic and its appropriate use (and not only logic),  are really necessary for physicists  in order to clarify and predict their science. 

Best regards, 

Emilija

Dear Emilija,

I am glad if my comments were helpful. It is interesting to compare the two constructions, for boosts and for rotations; they are almost identical, but in the second case, the two "rotations" at infinity (with one fixed point at infinity) add up and produce an isometry without fixed points at infinity, while in the first case they are "subtracted" (opposite rotation sense) and thus their composition has two fixed points at infinity.

Many thanks for your interesting question and best regards

Jost

Dear professor Akira, 

Below  is an answer about the nature of gravity in physics, posted in Researchgate. I suppose it is a usual situation in physics, which we mathematicians can not adopt  easily. 

Best regards, 

Emilija

"Both Newton's and Einstein's Theory are not "true," they are simply models used to describe and predict phenomena. Any model is neither true nor false, but only more or less useful for the purpose of scientific systematization, explanation and prediction. Newton's theory of universal gravitation is useful for everyday gravitational phenomena; Einstein's theory of space-time curvature was used to predict the apparant shift of a star close to the solar disk, and to explain the optical effects called gravitational lensing. No model can account for the "nature" of something, this is the nature of science. No theory can be proven, but only disproven".

An old question that is still fresh: Is gravity a Newtonian force or Einstein space-time curvature?. Available from: https://www.researchgate.net/post/An_old_question_that_is_still_fresh_Is_gravity_a_Newtonian_force_or_Einstein_space-time_curvature#view=557a98d25e9d972c038b45bb [accessed Jun 16, 2015].