The connected part of the Lorentz group SO(3,1) and the group of Möbius transformations, described by the group of 2x2 complex matrices S with unit determinant (SU(2)) with S and -S identified, are mathematically identical.

This means that the positions of stars (assumed infinitely-far-away) on the celestial sphere are transformed by a Möbius transformation when viewed from a Lorentz transformed coordinate system. The rotation part of these transformations are easily visible if you go out on a clear night, and spin around. The boost transformation part is not equally easy to detect, it is a consequence of stellar aberration.

So, the Riemann sphere and the celestial sphere are more-or-less the same thing. This was discovered by James Terrell and Roger Penrose in the late 1950's. It means that a spherical object on the sky do not look Lorentz contracted when observed from a very-fast-moving frame, although it can be described as such. The difference is that one for the former has to take into account the finite speed of light transmitting an image of the object being looked at. Hence, what is seen should look Terrell-rotated, not Lorentz contracted.

http://www.relativitybook.com/wiki/Terrell-Penrose_controversy

These issues are, now, homework problems: http://www.math.wustl.edu/~feres/Math496F15/Math496F15HW02Sol.pdf

But one should separate ambiguous language from unambiguous mathematical statements.

In particular, these issues don't have anything at all to do with the night sky-the term ``celestial sphere'', unfortunately, is used to denote a completely different object in the context of the Lorentz group, which doesn't have anything to do with celestial mechanics, that refers to space and not spacetime.

Let us   consider Minkowski space  E^{1,n} and a reduced  Lorenentz transformation  L of this space. Define PL  by    PL(t ,x_1,...,x_n)=(1 , x_1/t,...,x_n/t). It seems that 

(A)   PL is Mobius

transformation  of the   sphere ${(1 ,x_1,...,x_n) :  x_1^2 + ... +x_n^2=1} onto itself.

What   are  good references  related to (A).

It seems  that  the homework problems(by Renato Feres): http://www.math.wustl.edu/~feres/Math496F15/Math496F15HW02Sol.pdf

suggested   by  prof  Nikolis   are useful  for the introduction to the subject.

There are opinions that ''a re-definition of the original length and time units'', related to Lorentz transformation, in is not only mathematically but also physically inconsistent as it essentially means that there is no basis for any measurement anymore (see for example http://www.physicsmyths.org.uk/lightspeed.htm).

Here we try to give some reasons for ''re-definition of the original length''.

Consider 1-dim situation. Let $x_1$ and $x_2$ are coordinate of points $A$ and $B$ in $S$.

Obs2 in S' which is moving by velocity $v$ wrt to $S$ measures time $t_1$ the S-clock at $A$ and arrives at $B$ at time $t_2$ (according the S-clock at $B$).

By definition of velocity $v$ then $x_2-x_1= v(t_2-t_1)$. By the homogeneity of space and time, this def is independent from the starting time $t_1$ and $x_1$.

By symmetry, Obs1 in $S$ measures $x_2'-x_1'=- v(t_2'-t_1')$.

What is generally accepted opinion about this subject?

See also  Project Relativity and question:

What is new about derivation of Lorentz transformation?

Miodrag> There are opinions

Yes, the web is filled with opinions. The general opinion of Dr. Thomas Smid (behind physicsmyths) is that everything in science is wrong. What incredible luck that, nevertheless, science works!

Think of all the science on which the modern web is built; isn't it parodic that this rather well-functioning infrastructure is used to propagate opinions that modern science is fundamentally wrong?

If you want to understand special relativity, I recommend a two-step approach:

1) learn how to solve a system of 2 equations in 2 unknowns:

ax + by = A

cx + dy = B

Once you have mastered this skill (easy, but *do it first*) go to the second step:

2) Take a good modern textbook of relativity, and read it from the first page to the last. Do all calculations in detail. And if some statement appears contradictory, work on it under the assumption that most probably you are misunderstanding something.

Having done this, you can now attempt to see where the Smid's of this world go wrong. It is illuminating. In the case of the web page you give a link to, the problem is directly related to the author's lack of ability to solve two linear equations in two unknowns. With this technical baggage, it is easy to understand. Without it, you will easily get confused.

Timeline for such a project: first step should take one or two days, the second, depending on the time you choose to dedicate to it and on your previous exposure to physics, 2-10 days. Not much for understanding one of mankind's great intellectual triumphs.

Dear followers

Kåre Olaussen,Stam Nicolis and  F. Leyvraz,

Thanks for your answers.

I am a mathematician (and not a physicist), with background in Complex Analysis.

I understand that  physicists will rightfully have a different perspective.

We try to make a mathematical model which is motivated by STR.

Let us consider two inertial frames  $S$

and   $S'$, where the second one moves with

a speed   $v$, along the  x-axis, with respect to the first one. The co-ordinates and time

in the-frame will be denoted by  $x$  and  $t$, and in the frame

$S'$, they will be denoted

with $\xi $  and  $\tau$.

We consider an interesting  thought experiment.

Let us consider 

the propagation of a light signal emitted by a moving light source (system S')  and subsequently reflected back to the light source by a co-moving mirror), which would be in the rest frame connected to the corresponding times of the reflection at the mirror at time   $t_1$  (at point x_1)   in S  (at time  \tau_1 in S') and return to the light source at time $t_2$ ( at point x_2    in S;  at time \tau_2 in S')).  In the literature   we can find that   $t_1$ and $t_2$ are given by      by the equations

(1)       $t_1= x'/(c-v)$

(2)       $t_2= x'/(c-v) + x'/(c+v)$

where x' is the (fixed) distance between the light source and the reflecting mirror .(Einstein use the notation  $x' = x - vt$).

Also we have,

(3)       $ \tau(0,t_2)= 2  \tau(x',t_1)$

where the start values have been set to zero here for simplicity, and also the y and z components not been written.

Whether here   there is the inconsistency with  the Lorentz transformation?

Suppose that  $O=O'$  at $\tau=t=0$.

By Lorentz transformation    $\tau_1= \gamma(t_1- x_1 v/c^2)$. Since by Invariant principle 

(4)    $x'= c\tau_1$

and  $x_1=ct_1$,

we find $x'= c\tau_1= \gamma(t_1- x_1 v/c^2)=\gamma(t_1- c t_1 v/c^2) =\gamma t_1 (c-v)$.

(4)   should be corrected by an appropriate factor as   F. Leyvraz   has been  observed.  So there is no   the inconsistency with the Lorentz transformation.

You use the same letter for different concepts: x' as occurring after eqs. (1-2) means the (fixed) distance between the two mirrors *as measured in S*. On the other hand, when you invoke the ``invariance principle'' you say

x' = c tau_1

This is incorrect unless x' is now taken to mean the distance between the mirrors as measured in the reference frame S', which differs from the other by an appropriate factor.

However, if you wish to analyse a ``paradox'' properly, the correct way is *always* to try first to identify *events*. When you speak about objects, even mirrors and the like, you always run the risk of committing some kind of error involving simultaneity. So identify a finite set of events which articulate the story you are telling. Then find the coordinates if the events in one frame. You will find that they naturally and uniquely transform into another.

The crucial point guaranteeing consistency is tht the Lorentz transformation with velocity v has an inverse expressed by the Lorentz transform with velocity -v.

In the case of your story, the events are best identified as follows (the coordinates are given as (x, t) in the rest frame S):

E1: (0,0): the light beam starts at the position of the first mirror

E2: (cL/(c-v), L/(c-v)): the light beam hits the second mirror, which initially was at a distance L from the first in the S frame, and moving with velocity v away from the beam.

E3: The light beam hits the first mirror for the second time. The time is determined as you claim, and the position is the velocity v times that time, since that is how the first mirror is moving:

(v[L/(c-v)+L/(c+v)], L/(c-v)+L/(c+v)) = (2vcL/(c^2-v^2), 2cL/(c^2-v^2))

Now use the Lorentz transform to pass these events to the S' frame, which moves at velocity v, and you will get a completely coherent narrative, namely the beam goes between two mirrors at rest and take an equal time to go one way and back. Further, that time is related to the length between the mirrors in S', which I call L', by

tau = L'/c.

Thanks.

My  story is motivated by   Einstein's  famous 1905 paper On the Electrodynamics of Moving Bodies.   In §2 of it, He says ''If we place $x' = x - vt$, it is clear that a point at rest in the system k must

have a system of values x', y, z, independent of time. We first define $\tau$ as a

function of x', y, z, and t. To do this we have to express in equations that $\tau$ is

nothing else than the summary of the data of clocks at rest in system k, which

have been synchronized according to the rule given in § 1.''

It seems  to me  that we need here some explanations concerning notations.

Set  E2=(x_1,t_1). By   \xi  Einstein  denotes the space coordinate in ''moving'' frame.  Then by Lorentz transformation,

\xi= \gamma x'=\gamma (x_1-vt_1) and therefore t_1= \gamma x'/(c-v).

Perhaps I am missing here something. In F. Leyvraz  answer   notation x'=L.

No. I just answered the problem, without confusing historical references. You have two mirrors, moving at a common velocity v. They are separated in the rest frame by a distance L. Then the events have the coordinates I describe. If you believe otherwise, please propose a set of (x,t) coordinates for the three events, in the notation I suggest.

I refused to take your notation, because it turned out to be confusing. But nothing can depend entirely on notation.

Thus, to justify E2, we have the following: the light beam's motion is given by

xL(t) = ct

and the second mirror's is given by

xM(t) = L + vt

These two positions are equal when

t = L/(c-v)

and they are then both equal to

x(E2) = cL/(c-v)

So the coordinates of E2 are

(cL/(c-v), L/(c-v))

The rest of the computation proceeds accordingly.

A careful reading of Einstein's paper will also reveal no error. But what he is doing is quite a bit harder, since at that point in the paper he is *deriving* the LT. We already have it, so it is quite enough to compute event coordinates and then transform. What Einstein is doing in that part of the paper is to show that the LT is the *only* way of reaching coherence. It is correct, but quite a bit more challenging.

Further, this is very far from being the simplest derivation of the LT. There are plenty of others, which are much more straightforward.

@F. Leyvraz

I am agree  with your derivation. At  first glance, I did not follow carefully  Einstein's paper and try to adjust his notations. I understand that Einstein     is *deriving* the LT in the paper.

As far as remember He  did  not  justify E2  and E3 (but you did now).  I can follow your notation.  Thanks.

A quick derivation of the LT, just as a comment. The transformation must be linear (justification: if not, it maps inertial motion onto non-inertial, which is observable). By the second postulate, if x=ct, then x'=ct'. From this follows that

x' - ct' = lambda(v) (x - ct)

Similarly for x=-ct. This also implies x'=-ct', and hence

x' + ct' = mu(v) (x - ct)

The requirement that the inverse transformation should correspond to the same transformation, with -v, and of course, with the + and - signs interchanged, implies

lambda(-v) = mu(v)

mu(-v) = lambda (v)

But clearly, using pure algebra, the inverse transformation  is given by

lambda(-v) = 1/lambda(v)

from which follows

lambda(v) = 1/ mu(v)

And from this, together with the definition of velocity, everything follows.