text:
\section{Introduction}

Corrections to Special Relativity (SR) coming from quantum gravity effects have been predicted and theoretically explored by string theory and quantum gravity research since more than one decade~\cite{qugra}. Experimental tests have also been carried out in search of residual effects at low energies of these high-energy violations of SR~\cite{Mattingly:2005re,Coleman:1998ti}, putting strong constraints to Lorentz violating coefficients in the Standard Model Extension (SME)~\cite{Colladay:1998fq}, a generalization of the Standard Model that allows for violations of Lorentz and CPT symmetry in the framework of a local effective field theory (EFT). The SME, inspired by what happens in the context of string theory, provides a dynamical scenario which assumes that the Lorentz invariance violation (LIV) arises from spontaneous symmetry breaking in a more fundamental theory with Lorentz covariance. This implies that microcausality and the usual conservation laws of energy and momentum are expected to hold in the low energy effective theory. However, since the vacuum breaks Lorentz invariance, this EFT is formulated in a certain system of reference, and the relativistic principle (equality of the dynamical laws derived from this EFT for different inertial observers) does no longer hold.
There are many experiments and observations which are sensitive to the existence of a preferred reference frame and that imply very strong constraints on the LIV~\cite{Kostelecky:2008ts}.

Doubly Special Relativity (DSR) emerged~\cite{dsr} as an alternative way to consider violations of Lorentz invariance. Lorentz symmetry is \emph{deformed} instead of \emph{broken}, in a way that it is still possible to formulate observer-independent laws of physics.
In this case it is much more difficult to find observable effects of the deviations from SR.
The characteristic deformation of DSR is encoded in an energy-momentum dispersion relation which depends on a high-energy (or short-distance) scale which is invariant in the same sense that the speed of light is a relativistic invariant, together with new (deformed) Lorentz transformations which preserve the form of the dispersion relation. In particular, since the presence of a new energy scale in the modified dispersion relation does not necessarily require to deform the rotations, DSR usually considers a deformation only in the transformation laws under boosts, which, owing to their noncompact character, offer less stringent bounds than the usual rotational invariance.\footnote{This is an heuristic argument that is explicitly seen in the constraints one gets for rotational and nonrotational invariant parameters in the SME, see e.g.~\cite{Kostelecky:2008ts}. Up to our knowledge there are no such type of studies in the case of a deformed symmetry, but considering a rotational invariant deformation is a common practice in DSR theories and will prove to be an important algebraic simplification in the analysis presented here.}

Then, the generators of the deformed Lorentz transformations still satisfy the ordinary Lorentz algebra, but are represented non-linearly in momentum space~\cite{AmelinoCamelia:2002an}. This non-linearity is a source of complication in the multi-particle sectors of DSR theories, since one can no longer define the total momentum of two particles as the sum of the individual momenta if it has to transform under boosts as the momentum of a single particle. This makes necessary the introduction of a deformed composition law which will depend on the new scale of the theory, and evidences that the DSR framework goes beyond the EFT paradigm, where usual composition laws apply.

The fact that DSR cannot be embeded in the SME makes it difficult to provide it with a dynamical formulation. Initially, only kinematic considerations in momentum space guided the explorations of DSR effects in quantum gravity phenomenology. In fact the space-time structure underlying the theory was suspected from the start to be rather non-trivial, for example by containing some fundamental noncommutativity~\cite{KowalskiGlikman:2002jr}. More recently, it became clear that nonlocal issues were an important ingredient of DSR models~\cite{nonlocality}. This is another fact reflecting that the DSR proposal goes beyond the EFT framework. Then, in Ref.~\cite{relativelocality} it was introduced the concept of `relative locality' in terms of a non-trivial geometry of momentum space. A spacetime with relative locality seems a natural candidate for a DSR spacetime, but the proposal in Ref.~\cite{relativelocality} was set forth in a rather general way, without implementing a relativity principle.
In fact not every geometry of momentum space is compatible with the relativity principle. An attempt in this direction was presented in Ref.~\cite{locandrelpple}, where it was remarked that an additive conservation law describes an interaction which is local in spacetime, and non-linear corrections to this law cause the locality property to be lost for a general observer. This means that DSR cannot be local in the same sense as SR, and in fact the description of interactions that was given in Ref.~\cite{relativelocality} under the relative locality notion can be used for a DSR theory which is implemented in terms of the so-called auxiliary variables~\cite{Judes:2002bw}, which are non-linear mappings between a physical momentum (transforming according to deformed Lorentz boosts) and an auxiliary momentum which transforms linearly~\cite{locandrelpple}.

However, the use of auxiliary variables is not enough to describe a generic DSR theory, as we will later clarify in the present paper. This makes the connection between DSR and the possible geometry of relative-locality momentum spaces more involved. Since the latter primarily results in modifications of the on-shell relation and of the law of momentum conservation~\cite{relativelocality}, the problem can be more generally formulated as the connection between a theory which, while including a new high-energy scale, implements a relativistic principle (with deformed Lorentz transformations) and the possible modifications of both the dispersion relation and the composition law when the theory is such that it reduces to SR at energies much lower than the new ultraviolet scale. This was the main viewpoint adopted in Ref.~\cite{AmelinoCamelia:2011yi}, where a `golden-rule' for the `DSR-compatibility' of a given geometry of momentum space (in fact, for the compatibility between the implementation of a relativity principle and a given modified dispersion relation and composition law) was derived. This golden-rule was found by considering two physical processes that cannot occur in a theory beyond SR endowed with a relativity principle: photons cannot decay into electron-positron pairs (since this reaction is forbidden in SR, it would imply the existence of a threshold depending on the ultraviolet scale, which is self-contradictory because the energy of the photon can be tuned above or below this observer-invariant threshold with an appropriate boost) and it must always be possible for a photon of any energy to produce electron-positron pairs in interactions with some sufficiently high-energy photons (if it were not so, this would imply again an energy threshold for the switch-off of this reaction, and then the same argument as before applies). However, Ref.~\cite{AmelinoCamelia:2011yi} expressed doubts whether the obtained golden-rule to be satisfied by a modified dispersion relation and a composition law to be DSR-compatible was not only necessary, but also a sufficient condition.

The main result of the present manuscript is a derivation of the constraints that the principle of relativity imposes on the dispersion relation and the composition law in a theory beyond SR. As in Ref.~\cite{AmelinoCamelia:2011yi}, we will work in a scenario of rotational invariant (polynomial) modifications to SR to the leading order in $M$, the scale of new physics. We will consider an implementation of deformed boost transformations such that the ordinary Lorentz algebra is still satisfied. This will allow us to obtain simple relations between the coefficients which parameterize the leading deviations to SR, to which presumably quantum gravity phenomenology will be most sensitive~\cite{dsrfacts}. In doing so, we will re-derive the golden-rule of Ref.~\cite{AmelinoCamelia:2011yi} in a completely different way and will establish that it is not a sufficient condition for the DSR-compatibility of a modified dispersion relation and composition law. The relations we will provide allow one to identify the most generic set of deviations from SR that are still compatible with the relativity principle, at first order in $1/M$.

The outline of the paper is as follows. In Section II we will give the general forms of the modified dispersion relation and composition law for a two-particle system both in the case $1+1$ dimensions, which we will treat separately because of its simplicity, and in the more relevant $3+1$ case. Then in Section III we will examine the constraints that the relativity principle imposes on them, deriving the set of relations that must exist between the coefficients in the dispersion relation and in the composition law to be compatible with a relativity principle. We will also check these relations in specific examples.

There are some points in common between this work and Ref.~\cite{asr}, where a systematic approach to SR compatible with a relativistic principle was presented also at leading order of Planckian effects, referring to that framework as Asymptotic Special Relativity (ASR). That previous work tried to be more general in the sense of including arbitrary functions of energy or momentum instead of simple polynomials in the corrections to SR, but in fact was more restrictive than the present analysis because it only considered the definition of auxiliary (energy and momentum) variables as a way to go beyond SR. That this is not sufficiently general will be clear in Section IV, which therefore establishes that a criticism of DSR that is often heard (that it is just SR rewritten in another set of variables) is unfounded.

Finally, in Section V we will argument how the present work can be generalized beyond the two-particle case and beyond the $M^{-1}$ order and give some concluding remarks.

\section{Modified dispersion relation and composition law}

We will consider a departure from SR which can be expanded in powers of momenta and the inverse of an ultraviolet scale $M$. If one relates these departures with a quantum spacetime structure associated with quantum gravity fluctuations it is natural to identify the ultraviolet scale $M$ with the Planck mass but more general cases could be considered.
We will assume that all the energies are much smaller than the scale $M$ so that the dominant effect of the corrections to SR kinematics are on the first order terms in the $1/M$ expansion.  We will also restrict all the discussion to the case where there are no departures from rotational symmetry.

We will consider the kinematic analysis of reactions with no more than two particles in the initial or final state. As we will comment at the end of the work, there does not seem to be any obstruction to extend the present analysis to reactions involving more than two particles in the initial or final state and to higher orders terms in the $1/M$ expansion.

There are two ingredients in the generalized kinematics: a modification of the energy-momentum relation of a particle (modified dispersion relation) and a modification of the composition law of two momenta (and the associated modification of the energy-momentum conservation law).

\subsection{$1+1$ dimensional case}

Let us start by considering the simpler case of a generalized kinematics in $1+1$ dimensions.
In order to make easier a comparison with a rotational invariant $3+1$ dimensional generalized kinematics, we will impose in the present case invariance under the parity transformation $p_0 \to p_0$, $p_1 \to - p_1$.
With this restriction, the general form for the composition of two momenta $p$, $q$ is
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0 + \frac{\beta_1}{M} p_0 q_0 + \frac{\beta_2}{M} p_1 q_1 {\hskip 1cm}   \left[p \oplus q\right]_1 \,=\, p_1 + q_1 + \frac{\gamma_1}{M} p_0 q_1 + \frac{\gamma_2}{M} p_1 q_0
\label{cl1+1}
\end{equation}
where $\beta_1$, $\beta_2$, $\gamma_1$, $\gamma_2$ are dimensionless coefficients and we are implementing the condition
\begin{equation}
p \oplus q|_{q=0} \,=\, p {\hskip 1cm}  p \oplus q|_{p=0} \,=\, q
\label{cl0}
\end{equation}
on the composition law.

The general form for the dispersion relation of a particle is given by
\begin{equation}
C(p) \,=\, p_0^2 - p_1^2 + \frac{\alpha_1}{M} p_0^3 + \frac{\alpha_2}{M} p_0 p_1^2 \,=\, \mu^2\,.
\label{dr1+1}
\end{equation}
The generalization of the SR kinematics is parameterized by the six dimensionless coefficients $\beta_1$, $\beta_2$, $\gamma_1$, $\gamma_2$, $\alpha_1$, $\alpha_2$. We want to determine what are the conditions that these coefficients have to satisfy in order to have a generalized kinematics compatible with the relativity principle (absence of a preferred reference frame).

\subsection{$3+1$ dimensional case}

In $3+1$-dimensions the general form of the composition law of two momenta compatible with rotational invariance takes the form
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0 + \frac{\beta_1}{M} \, p_0 q_0 + \frac{\beta_2}{M} \, \vec{p}\cdot\vec{q} {\hskip 1cm}   \left[p \oplus q\right]_i \,=\, p_i + q_i + \frac{\gamma_1}{M} \, p_0 q_i + \frac{\gamma_2}{M} \, p_i q_0
+ \frac{\gamma_3}{M} \, \epsilon_{ijk} p_j q_k
\label{cl3+1}
\end{equation}
and the particle dispersion relation is given by
\begin{equation}
C(p) \,=\, p_0^2 - \vec{p}^{\,2} + \frac{\alpha_1}{M} \, p_0^3 + \frac{\alpha_2}{M} \, p_0 \vec{p}^{\,2} \,=\, \mu^2 \,.
\label{dr3+1}
\end{equation}
All the difference with the 1+1 dimensional case is that one has an additional (seventh) dimensionless coefficient ($\gamma_3$) in the composition law.
Note that this term would be absent in a parity invariant generalized kinematics in $3+1$ dimensions: without it, $[p \oplus q]_i$ would go to $-[p \oplus q]_i$ under the transformation $p_0 \to p_0$, $\vec p \to -\vec p$, $q_0 \to q_0$, $\vec q \to -\vec q$.
However, we will maintain this term in the $3+1$ dimensional case for a more general discussion.

\section{Relativity principle}

Let us see how a relativity principle can be consistently implemented when one has a generalized composition law of momenta and a generalized dispersion relation.

\subsection{$1+1$ dimensional case}

In the $1+1$ dimensional case all one needs is the transformation under a boost (parameterized by $\xi_1$) of a two particle system with momenta $p$, $q$. The main ingredient in the implementation of the relativity principle is that the transformation in general does not act separately on the momenta of each particle. After the transformation the momenta of the particles will be $T^L_q(p)$, $T^R_p(q)$ with subindex $q$ on $T^L$ indicating that the transformed momentum of the particle that had a momentum $p$ can depend on the momentum of the other particle, and similarly for the subindex $p$ in $T^R$. The upper indexes $L$, $R$ indicate the possibility to have a different transformation laws (noncommutativity) for the two momenta.

The condition (\ref{cl0}) on the composition law allows us to derive the transformation under a boost of a one particle system $p \to T(p)$ from the transformation law of a two particle system
\begin{equation}
T(p) \,=\, T^L_0(p) \,=\, T^R_0(p) \,,
\end{equation}
where the equality comes from the assumption that a one-particle system is equivalent to a two-particle system in which one of the particles has zero momentum.
The general form of the boost transformation of a one particle system at order $1/M$ can be expressed in terms of three dimensionless parameters $\lambda_1$, $\lambda_2$, $\lambda_3$ as
\begin{eqnarray}
\left[T(p)\right]_0 &=& p_0 + p_1 \xi_1 + \frac{\lambda_1}{M} \, p_0 p_1 \xi_1 \nonumber \\
\left[T(p)\right]_1 &=& p_1 + p_0 \xi_1 + \frac{\lambda_2}{M} \, p_0^2 \xi_1 + \frac{\lambda_1 + 2\lambda_2 + 3\lambda_3}{M} p_1^2 \xi_1
\label{Tlambda}
\end{eqnarray}
where a choice of coefficients has been made in order to make easier the comparison of the $1+1$ and $3+1$ dimensional cases.

The invariance of the dispersion relation (\ref{dr1+1}) under a boost transformation requires that
\begin{equation}
C(T(p)) \,=\, C(p) \,.
\label{drinv}
\end{equation}
This fixes the dimensionless coefficients $\alpha_1$, $\alpha_2$ in the modified dispersion relation in terms of the three parameters $\lambda_1$, $\lambda_2$, $\lambda_3$ of the boost transformation~(\ref{Tlambda}):
\begin{equation}
\alpha_1 \,=\, -2 \,(\lambda_1 + \lambda_2 + 2\lambda_3) {\hskip 1cm}
\alpha_2 \,=\, 2 \, (\lambda_1 + 2\lambda_2 + 3\lambda_3)\,.
\label{alpha(lambda)}
\end{equation}

The general form of the boost transformation of a two particle system at order $1/M$ is given by
\begin{equation}
T^L_q(p) \,=\, T(p) + {\bar T}^L_q(p) {\hskip 1cm} T^R_p(q) \,=\, T(q) + {\bar T}^R_p(q)
\end{equation}
with
\begin{equation}
\left[{\bar T}^L_q(p)\right]_0 \,=\, \frac{\eta_1^L}{M} \, q_0 p_1 \xi_1 + \frac{\sigma_1^L}{M} \, q_1 p_0 \xi_1 {\hskip 1cm} \left[{\bar T}^L_q(p)\right]_1 \,=\, \frac{\sigma_2^L}{M} \, q_0 p_0 \xi_1 + \frac{\sigma_3^L}{M} \, q_1 p_1 \xi_1 \,,
\label{TbarL1+1}
\end{equation}
\begin{equation}
\left[{\bar T}^R_p(q)\right]_0 \,=\, \frac{\eta_1^R}{M} \, p_0 q_1 \xi_1 + \frac{\sigma_1^R}{M} \, p_1 q_0 \xi_1 {\hskip 1cm} \left[{\bar T}^R_p(q)\right]_1 \,=\, \frac{\sigma_2^R}{M} \, p_0 q_0 \xi_1 + \frac{\sigma_3^R}{M} \, p_1 q_1 \xi_1 \,.
\label{TbarR1+1}
\end{equation}
But the invariance of the dispersion relation of each of the particles under a boost transformation requires that
\begin{equation}
C\left({\bar T}^L_q(p)\right) \,=\, C\left({\bar T}^R_p(q)\right) \,=\, 0 \,.
\label{drinv2}
\end{equation}
This implies that
\begin{equation}
\sigma_1^L \,=\, \sigma_3^L \,=\, \sigma_1^R \,=\, \sigma_3^R \,=\, 0 {\hskip 1cm}
\sigma_2^L \,=\, \eta_1^L {\hskip 1cm} \sigma_2^R \,=\, \eta_1^R
\end{equation}
and then one has
\begin{eqnarray}
\left[{\bar T}^L_q(p)\right]_0 &=& \frac{\eta_1^L}{M} \, q_0 p_1 \xi_1 {\hskip 1cm}
\left[{\bar T}^L_q(p)\right]_1 \,=\, \frac{\eta_1^L}{M} \, q_0 p_0 \xi_1 \nonumber \\
\left[{\bar T}^R_p(q)\right]_0 &=& \frac{\eta_1^R}{M} \, p_0 q_1 \xi_1 {\hskip 1cm}
\left[{\bar T}^R_p(q)\right]_1 \,=\, \frac{\eta_1^R}{M} \, p_0 q_0 \xi_1
\label{Tbar1+1}
\end{eqnarray}
so that finally one has only two new dimensionless parameters $\eta_1^L$, $\eta_1^R$ in the boost transformation of a two particle system.

The last step in the implementation of the relativity principle is the requirement that
\begin{equation}
T(p\oplus q) \,=\, T^L_q(p) \oplus T^R_p(q) \,.
\label{clinv}
\end{equation}
This guarantees the invariance under boosts of the energy-momentum conservation in the decay of one particle into two. It is a straightforward algebraic problem to determine the relations that Eq.~(\ref{clinv}) implies between the parameters of the boost transformations and the dimensionless coefficients in the composition law of momenta:
\begin{align}
\beta_1 &= 2 \,(\lambda_1 + \lambda_2 + 2\lambda_3) &
\beta_2 &= -2 \lambda_3 - \eta_1^L - \eta_1^R \nonumber \\
\gamma_1 &= \lambda_1 + 2\lambda_2 + 2\lambda_3 - \eta_1^L &
\gamma_2 &= \lambda_1 + 2\lambda_2 + 2\lambda_3 - \eta_1^R \,.
\label{clT1+1}
\end{align}

It is clear then that the boost transformation of the one- and two-particle system, determined
by the $\lambda$ and $\eta$ coefficients,  fix completely the momentum composition law ($\gamma$ and $\beta$ coefficients). DSR models, for which the deformed boost transformation of the one particle system is well defined, usually contain an ambiguity in the energy-momentum conservation law for particle processes. We see that, in the context of the $1/M$ expansion that we are dealing with, this ambiguity corresponds to the deformation in the boost of the two particle system ($\eta_1^L$ and $\eta_1^R$ coefficients) that one may consider. Once the transformation laws for the one particle and two particle systems are given, both the dispersion relation and the composition law are fixed.

On the other hand, one could consider the composition law, which gives the conservation laws in an interacting system, as more fundamental than the transformation laws from a physical point of view. Then we see that, for a given composition law ($\beta$ and $\gamma$ coefficients) there is a one-parameter family of deformed transformation laws that can implement a relativity principle. Defining $\lambda=(\lambda_1+\lambda_2)/2$, then the coefficients that define these transformation laws are
\begin{alignat}{2}
\lambda_1 &= \frac{3\beta_1}{4}+\frac{\beta_2}{2}-\frac{\gamma_1}{2}-\frac{\gamma_2}{2}+\lambda \quad \quad \quad \quad
\lambda_2 &= -\frac{3\beta_1}{4}-\frac{\beta_2}{2}+\frac{\gamma_1}{2}+\frac{\gamma_2}{2}+\lambda \quad \quad \quad \quad
\lambda_3 &= \frac{\beta_1}{4}-\lambda \nonumber \\
\eta_1^L &= -\frac{\beta_1}{4}-\frac{\beta_2}{2}-\frac{\gamma_1}{2}+\frac{\gamma_2}{2}+\lambda \quad \quad \quad \quad
\eta_1^R &= -\frac{\beta_1}{4}-\frac{\beta_2}{2}+\frac{\gamma_1}{2}-\frac{\gamma_2}{2}+\lambda \quad \quad \quad \quad &
\end{alignat}

When one combines these relations with the expressions (\ref{alpha(lambda)}) of the coefficients in the modified dispersion relation in terms of the parameters of the boost transformations one gets
\begin{equation}
\alpha_1 \,=\, - \beta_1 {\hskip 2cm} \alpha_2 \,=\, \gamma_1 + \gamma_2 - \beta_2 \,.
\label{RP}
\end{equation}
The relativity principle implemented through (\ref{drinv}), (\ref{drinv2}), (\ref{clinv}) fixes the modification in the dispersion relation in terms of the modification in the composition law. This remark has never been made in the DSR literature before and is one of the main results of our work.

A consequence of the choice of a modified dispersion relation consistent with a modified composition law is the relation
\begin{equation}
\alpha_1 + \alpha_2 + \beta_1 + \beta_2 - \gamma_1 - \gamma_2 \,=\, 0 \,.
\label{goldenrule}
\end{equation}
This is just the `golden-rule' derived in Ref.~\cite{AmelinoCamelia:2011yi} as a consequence of the incompatibility of the relativity principle with the existence of an energy threshold for the decay of one particle into two (``no-photon-decay-switch-on constraint'') and with the existence of an energy threshold for the energy of one particle in the production of two particles by two particles (``no-pair-production-switch-off constraint''). In fact in Ref.~\cite{AmelinoCamelia:2011yi}  it is conjectured, based on a few examples, that the golden rule~(\ref{goldenrule}) is not only a necessary but also a sufficient condition for the compatibility of a departure from SR with the relativity principle.

The systematic derivation of the modification in the dispersion relation induced by a modification in the composition law and the compatibility with the relativity principle gives an alternative derivation of the golden rule independent of considerations of thresholds in particular reactions. It also shows that the golden rule (\ref{goldenrule}) \underline{is not a sufficient condition} for a relativistic kinematics but just a combination of the two relations (\ref{RP}) that fix the modified dispersion relation compatible with a given composition law.

\subsection{Some examples}

We can check explicitly that the examples considered in Ref.~\cite{AmelinoCamelia:2011yi} to test the validity of the golden rule and its possible sufficiency are particular cases of the implementation of the relativity principle as discussed in this section. In the first example, know in DSR literature as `DSR1'~\cite{Bruno:2001mw}, one has a boost transformation\footnote{We are identifying $1/M$ with the invariant length $\ell$; this is just a convention since all the relations are invariant under a  simultaneous rescaling of all the dimensionless coefficients, together with a corresponding rescaling of $M$.}
\begin{equation}
\left[T(p)\right]_0 \,=\, p_0 + p_1 \xi_1 {\hskip 1cm} \left[T(p)\right]_1 \,=\, p_1 + p_0 \xi_1 +
\frac{1}{M} p_0^2 \xi_1 + \frac{1}{2 M} p_1^2 \xi_1 \,.
\end{equation}
This corresponds in our notation to
\begin{equation}
\lambda_1 \,=\, 0 {\hskip 1cm} \lambda_2 \,=\, 1 {\hskip 1cm} \lambda_3 \,=\, - \frac{1}{2} \,. \label{lambdaDSR1}
\end{equation}
The modified dispersion relation corresponds to
\begin{equation}
C(p) \,=\, p_0^2 - p_1^2 + \frac{1}{M} p_0 p_1^2\,,
\end{equation}
that is,
\begin{equation}
\alpha_1 \,=\, 0 {\hskip 1cm} \alpha_2 \,=\, 1 \,. \label{alphaDSR1}
\end{equation}
The choice (\ref{lambdaDSR1})  and (\ref{alphaDSR1})  is compatible with the relations (\ref{alpha(lambda)}). For the composition law one has
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0 + \frac{1}{M} p_1 q_1 {\hskip 1cm}
\left[p\oplus q\right]_1 \,=\, p_1 + q_1 + \frac{1}{M} p_0 q_1 + \frac{1}{M} p_1 q_0
\end{equation}
so that
\begin{equation}
\beta_1 \,=\, 0 {\hskip 1cm} \beta_2 \,=\, 1 {\hskip 1cm} \gamma_1 \,=\, 1 {\hskip 1cm} \gamma_2 \,=\, 1 \,.
\end{equation}
One can easily check that the relations (\ref{RP}) are satisfied and then one has a modified dispersion relation compatible with the modification in the composition law. One can also check that the relations (\ref{clT1+1}) between dimensionless coefficients in the composition law and parameters in the boost transformation are satisfied with
\begin{equation}
\eta_1^L \,=\, \eta_1^R \,=\,0 \,.
\end{equation}
Then one has in this case
\begin{equation}
{\bar T}^L_q(p) \,=\, {\bar T}^R_p(q) \,=\, 0 {\hskip 1cm} T(p\oplus q) \,=\, T(p)\oplus T(q)
\end{equation}
and the boost transformation of the two particle system reduces to an independent boost transformation on each of the particles.

The second example considered in Ref.~\cite{AmelinoCamelia:2011yi} corresponds to
\begin{equation}
\left[T(p)\right]_0 \,=\, p_0 + p_1 \xi_1 - \frac{1}{M} p_0 p_1 \xi_1 {\hskip 1cm} \left[T(p)\right]_1 \,=\, p_1 + p_0 \xi_1 +
\frac{1}{M} p_0^2 \xi_1 + \frac{1}{M} p_1^2 \xi_1 \,,
\end{equation}
and then
\begin{equation}
\lambda_1 \,=\, -1 {\hskip 1cm} \lambda_2 \,=\, 1 {\hskip 1cm} \lambda_3 \,=\, 0 \,.
\end{equation}
The modified dispersion relation is
\begin{equation}
C(p) \,=\, p_0^2 - p_1^2 + \frac{2}{M} p_0 p_1^2\,,{\hskip 1cm} \alpha_1 \,=\, 0 {\hskip 1cm} \alpha_2 \,=\, 2\,,
\end{equation}
and the modified composition composition law in this example is
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0  {\hskip 1cm}
\left[p\oplus q\right]_1 \,=\, p_1 + q_1 + \frac{1}{M} p_0 q_1 + \frac{1}{M} p_1 q_0\,,
\end{equation}
so that
\begin{equation}
\beta_1 \,=\, 0 {\hskip 1cm} \beta_2 \,=\, 0 {\hskip 1cm} \gamma_1 \,=\, 1 {\hskip 1cm} \gamma_2 \,=\, 1\,.
\end{equation}
Also in this example the relation (\ref{RP}) is satisfied. The relations (\ref{clT1+1}) lead also in this case to
\begin{equation}
\eta_1^L \,=\, \eta_1^R \,=\, 0 {\hskip 1cm} {\bar T}^L_q(p) \,=\, {\bar T}^R_p(q) \,=\, 0 {\hskip 1cm} T(p\oplus q) \,=\, T(p)\oplus T(q)
\end{equation}
and then to a trivial boost transformation for the two particle system as in the first example.

The other two examples considered in Ref.~\cite{AmelinoCamelia:2011yi} are inspired by studies of the $\kappa$-Poincar\'e Hopf algebra.
The third example has an additive composition law of energies as in the previous example but the composition of momentum is noncommutative:
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0  {\hskip 1cm}
\left[p\oplus q\right]_1 \,=\, p_1 + q_1 + \frac{1}{M} p_0 q_1\,,
\end{equation}
so that
\begin{equation}
\beta_1 \,=\, 0 {\hskip 1cm} \beta_2 \,=\, 0 {\hskip 1cm} \gamma_1 \,=\, 1 {\hskip 1cm} \gamma_2 \,=\, 0 \,.
\end{equation}
The boost transformation acting on a particle is given by
\begin{equation}
\left[T(p)\right]_0 \,=\, p_0 + p_1 \xi_1 {\hskip 1cm}
\left[T(p)\right]_1 \,=\, p_1 + p_0 \xi_1 + \frac{1}{M} p_0^2 \xi_1 + \frac{1}{2M} p_1^2 \xi_1\,,
\end{equation}
and then
\begin{equation}
\lambda_1 \,=\, 0 {\hskip 1cm} \lambda_2 \,=\, 1 {\hskip 1cm} \lambda_3 \,=\, -\frac{1}{2} \,.
\end{equation}
The modified dispersion relation compatible with this boost transformation is
\begin{equation}
C(p) \,=\, p_0^2 - p_1^2 + \frac{1}{M} p_0 p_1^2 \,, {\hskip 1cm} \alpha_1 \,=\, 0 {\hskip 1cm} \alpha_2 \,=\, 1
\end{equation}
and once more these coefficients of the modified dispersion relation are compatible with the relativity principle constraints (\ref{RP}). From the equations (\ref{clT1+1}) which give the modified composition law in terms of the boost transformations we can see that the transformation of a two particle system is given in this case by
\begin{equation}
\eta_1^L \,=\, 0 {\hskip 1cm} \eta_1^R \,=\, 1
\end{equation}
and then
\begin{equation}
T(p\oplus q) \,=\, T(p) \oplus T^R_p(q) {\hskip 1cm} \left[{\bar T}^R_p(q)\right]_0 \,=\, \frac{1}{M} p_0 q_1 \xi_1 {\hskip 1cm}  \left[{\bar T}^R_p(q)\right]_1 \,=\, \frac{1}{M} p_0 q_0 \xi_1\,,
\end{equation}
so that we have a nontrivial transformation of the two particle system. Note that in general one has from (\ref{clT1+1}),
\begin{equation}
\gamma_1 - \gamma_2 \,=\, \eta_1^L - \eta_1^R
\end{equation}
and then a noncommutativity ($\gamma_1\neq \gamma_2$) in the momentum composition law automatically implies a nontrivial boost transformation of the two particle system.

The last example in Ref.~\cite{AmelinoCamelia:2011yi} corresponds to an unmodified dispersion relation ($\alpha_1=\alpha_2=0$) and an additive composition law of energies. The constraints from the relativity principle (\ref{RP}) require $\gamma_1=-\gamma_2$ so that one has
\begin{equation}
\left[p\oplus q\right]_0 \,=\, p_0 + q_0 {\hskip 1cm}
\left[p\oplus q\right]_1 \,=\, p_1 + q_1 + \frac{1}{2M} p_0 q_1 - \frac{1}{2M} p_1 q_0 \,.
\end{equation}
The boost transformation of a particle state is given by
\begin{equation}
\left[T(p)\right]_0 \,=\, p_0 + p_1 \xi_1 + \frac{1}{2M} p_0 p_1 {\hskip 1cm}
\left[T(p)\right]_1 \,=\, p_1 + p_0 \xi_1 + \frac{1}{2M} p_0^2
\end{equation}
so that
\begin{equation}
\lambda_1 \,=\, \frac{1}{2} {\hskip 1cm} \lambda_2 \,=\, \frac{1}{2} {\hskip 1cm} \lambda_3 \,=\, - \frac{1}{2}
\end{equation}
and
\begin{equation}
\eta_1^L \,=\, 0 {\hskip 1cm} \eta_1^R \,=\, 1\,,
\end{equation}
which corresponds to the same boost transformation for the two particle system as in the previous example.

The fact that in all four examples the golden rule (\ref{goldenrule}) is satisfied is directly a consequence of the relativity principle present in all the examples. It is very easy to introduce a modification in the dispersion relation or the composition law in each of the four examples such that the golden rule (\ref{goldenrule}) is still satisfied but not the two relations (\ref{RP}) between the coefficients in the dispersion relation and the coefficients in the composition law: in all these cases one would have a modification of the SR kinematics consistent with the golden rule but not with the relativity principle, signaling the presence of a preferred reference frame.

\subsection{$3+1$ dimensional case}

The general form the transformation of one particle at order $1/M$, which now depends on three parameters $\vec{\xi}$,  is in this case
\begin{eqnarray}
\left[T(p)\right]_0 &=& p_0 + (\vec{p} \cdot \vec{\xi}) + \frac{\lambda_1}{M} \, p_0 (\vec{p} \cdot \vec{\xi}) \nonumber \\
\left[T(p)\right]_i &=& p_i + p_0 \xi_i + \frac{\lambda_2}{M} \, p_0^2 \xi_i + \frac{\lambda_3}{M} \, {\vec p}^{\,2} \xi_i + \frac{\lambda_4}{M} \, p_i ({\vec p} \cdot {\vec \xi}) + \frac{\lambda_5}{M} \, p_0 \epsilon_{ijk} p_j \xi_k \,.
\end{eqnarray}
In the case of $3+1$ dimensions one has an additional restriction for $T$ to correspond to a boost transformation. In order to reproduce the Lorentz algebra one has a condition on the composition of two transformations $T^{(1)}$, $T^{(2)}$ with parameters ${\vec \xi}^{\,(1)}$,  ${\vec \xi}^{\,(2)}$:
\begin{eqnarray}
&&\left[T^{(2)}\left(T^{(1)}(p)\right) - T^{(1)}\left(T^{(2)}(p)\right)\right]_0 \,=\, 0 \nonumber \\
&&\left[T^{(2)}\left(T^{(1)}(p)\right) - T^{(1)}\left(T^{(2)}(p)\right)\right]_i \,=\,  ({\vec p} \cdot {\vec \xi}^{\,(1)}) \, {\xi^{(2)}}_i  -   ({\vec p} \cdot {\vec \xi}^{\,(2)}) \, {\xi^{(1)}}_i \,.
\label{LaT}
\end{eqnarray}
This requires that
\begin{equation}
\lambda_5 \,=\, 0 {\hskip 1cm} \lambda_4 \,=\, \lambda_1 + 2\lambda_2 + 2\lambda_3
\end{equation}
and the boost transformation of a one particle system is then
\begin{eqnarray}
\left[T(p)\right]_0 &=& p_0 + (\vec{p} \cdot \vec{\xi}) + \frac{\lambda_1}{M} \, p_0 (\vec{p} \cdot \vec{\xi}) \nonumber \\
\left[T(p)\right]_i &=& p_i + p_0 \xi_i + \frac{\lambda_2}{M} \, p_0^2 \xi_i + \frac{\lambda_3}{M} \, {\vec p}^{\,2} \xi_i + \frac{(\lambda_1 + 2\lambda_2 + 2\lambda_3)}{M} \, p_i ({\vec p} \cdot {\vec \xi}) \,.
\label{T3+1}
\end{eqnarray}
The invariance of the modified dispersion relation (\ref{dr3+1}) under boosts fixes the coefficients $\alpha_1$, $\alpha_2$
\begin{equation}
\alpha_1 \,=\, -2 \,(\lambda_1 + \lambda_2 + 2\lambda_3) {\hskip 1cm}
\alpha_2 \,=\, 2 \, (\lambda_1 + 2\lambda_2 + 3\lambda_3)
\label{alpha(lambda)3+1}
\end{equation}
which are exactly the expressions found in the $1+1$ dimensional case.\footnote{This is the reason for the choice of coefficients in $1+1$ dimensions.}

For a transformation of a two particle system we have the same general structure of the $1+1$ dimensional case with
\begin{eqnarray}
\left[{\bar T}^L_q(p)\right]_0 &=& \frac{\eta_1^L}{M} \, q_0 ({\vec p} \cdot {\vec \xi}) + \frac{\sigma_1^L}{M} \, p_0 ({\vec q} \cdot {\vec \xi}) + \frac{\eta_2^L}{M} \, ({\vec p} \wedge {\vec q}) \cdot {\vec \xi} \nonumber \\
\left[{\bar T}^L_q(p)\right]_i &=&  \frac{\eta_3^L}{M} \, q_i ({\vec p} \cdot {\vec \xi}) + \frac{\sigma_2^L}{M} \, p_i ({\vec q} \cdot {\vec \xi}) + \frac{\eta_4^L}{M} \, q_0 \epsilon_{ijk} p_j \xi_k + \frac{\sigma_3^L}{M} \, ({\vec p} \cdot {\vec q}) \xi_i + \frac{\sigma_4^L}{M} \, p_0 \epsilon_{ijk} q_j \xi_k + \frac{\sigma_5^L}{M} \, p_0 q_0 \xi_i
\end{eqnarray}
and similar expresions for ${\bar T}^R_p(q)$. The invariance of the dispersion relation when we replace $p$ by $T^L_q(p)$ requires that $C({\bar T}^L_q(p))=0$. This implies that
\begin{equation}
\sigma_1^L \,=\, \sigma_2^L \,=\,  0 {\hskip 1cm} \sigma_3^L \,=\, - \eta_3^L {\hskip 1cm} \sigma_4^L \,=\,  \eta_2^L {\hskip 1cm} \sigma_5^L \,=\, \eta_1^L \,,
\end{equation}
so that we have
\begin{eqnarray}
\left[{\bar T}^L_q(p)\right]_0 &=& \frac{\eta_1^L}{M} \, q_0 ({\vec p} \cdot {\vec \xi}) + \frac{\eta_2^L}{M} \, ({\vec p} \wedge {\vec q}) \cdot{\vec \xi} \nonumber \\
\left[{\bar T}^L_q(p)\right]_i &=& \frac{\eta_1^L}{M} \, p_0 q_0 \xi_i + \frac{\eta_2^L}{M} \,  p_0 \epsilon_{ijk} q_j \xi_k + \frac{\eta_3^L}{M} \, \left[q_i ({\vec p} \cdot {\vec \xi}) - ({\vec p} \cdot {\vec q}) \xi_i\right] + \frac{\eta_4^L}{M} \, q_0 \epsilon_{ijk} p_j \xi_k
\end{eqnarray}
(and a similar expression for ${\bar T}^R_p(q)$) replacing (\ref{Tbar1+1}) in $1+1$ dimensions. We have three new coefficients $\eta^L_2$, $\eta^L_3$, $\eta^L_4$  that multiply terms vanishing identically in $1+1$ dimensions (the 1+1 dimensional limit can be taken by putting all the $2$ and $3$ components of $\vec p$, $\vec q$ and $\vec \xi$ to zero). But there is still another condition to have a relativistic kinematics. The transformation $p \to T^L_q(p)$ has to be consistent with the Lorentz algebra. This last condition implies that
\begin{equation}
\eta_3^L \,=\, \eta_1^L {\hskip 1cm} \eta_4^L \,=\, - \eta_2^L\,,
\end{equation}
so that finally we have just two  additional coefficients ($\eta_2^L$, $\eta_2^R$)  in the $3+1$  boost transformations of a two particle system
\begin{eqnarray}
\left[{\bar T}^L_q(p)\right]_0 &=& \frac{\eta_1^L}{M} \, q_0 ({\vec p} \cdot {\vec \xi}) + \frac{\eta_2^L}{M} \, ({\vec p} \wedge {\vec q}) \cdot{\vec \xi}
{\hskip 1cm} \left[{\bar T}^R_p(q)\right]_0 = \frac{\eta_1^R}{M} \, p_0 ({\vec q} \cdot {\vec \xi}) - \frac{\eta_2^R}{M} \, ({\vec p} \wedge {\vec q}) \cdot {\vec \xi}
\nonumber \\
\left[{\bar T}^L_q(p)\right]_i &=& \frac{\eta_1^L}{M} \, p_0 q_0 \xi_i + \frac{\eta_2^L}{M} \,  \left[p_0 \epsilon_{ijk} q_j \xi_k  - q_0 \epsilon_{ijk} p_j \xi_k\right] +
\frac{\eta_1^L}{M} \, \left[q_i ({\vec p} \cdot {\vec \xi}) - ({\vec p} \cdot {\vec q}) \xi_i\right] \nonumber \\
\left[{\bar T}^R_p(q)\right]_i &=& \frac{\eta_1^R}{M} \, p_0 q_0 \xi_i - \frac{\eta_2^R}{M} \,  \left[p_0 \epsilon_{ijk} q_j \xi_k  - q_0 \epsilon_{ijk} p_j \xi_k\right] +
\frac{\eta_1^R}{M} \, \left[p_i ({\vec q} \cdot {\vec \xi}) - ({\vec p} \cdot {\vec q}) \xi_i\right]\,.
\label{Tbar3+1}
\end{eqnarray}

The last step in the implementation of the relativity principle is to enforce the invariance under boosts of the energy-momentum conservation in the decay of one particle into two
\begin{equation}
T(p\oplus q) \,=\, T^L_q(p) \oplus T^R_p(q)\,.
\end{equation}
A straightforward algebra (just more tedious than in the $1+1$ dimensional case) allows to determine the five dimensionless coefficients $\beta_1$, $\beta_2$, $\gamma_1$, $\gamma_2$, $\gamma_3$ of the $3+1$ dimensional composition laws (\ref{cl3+1}) in terms of the parameters $\lambda_1$, $\lambda_2$, $\lambda_3$, $\eta_1^L$, $\eta_1^R$, $\eta_2^L$, $\eta_2^R$  of the boost transformations (\ref{T3+1}), (\ref{Tbar3+1}). The results are
\begin{alignat}{3}
\beta_1 &= 2 \,(\lambda_1 + \lambda_2 + 2\lambda_3) \quad\quad &
\beta_2 &= -2 \lambda_3 - \eta_1^L - \eta_1^R \quad\quad & \nonumber \\
\gamma_1 &= \lambda_1 + 2\lambda_2 + 2\lambda_3 - \eta_1^L \quad\quad &
\gamma_2 &= \lambda_1 + 2\lambda_2 + 2\lambda_3 - \eta_1^R \quad\quad & \gamma_3 = \eta_2^L - \eta_2^R
\label{clT3+1}
\end{alignat}
which are the same results obtained in the $1+1$ dimensional case for $\beta_1$, $\beta_2$, $\gamma_1$, $\gamma_2$ in terms of  $\lambda_1$, $\lambda_2$, $\lambda_3$, $\eta_1^L$, $\eta_1^R$ and an additional expression for the coefficient $\gamma_3$ (which does not appear in $1+1$ dimensions) as a function of the new ($3+1$ dimensional) parameters in the boost transformations $\eta_2^L$, $\eta_2^R$. Since the expressions for the coefficients $\alpha_1$, $\alpha_2$ in the dispersion relation as a function of the parameters $\lambda_1$, $\lambda_2$, $\lambda_3$ in the one particle boost transformations are also the same as in $1+1$ dimensions one concludes that the derivation of the modified dispersion relation from a modified conservation law in $1+1$ dimensions is also valid in $3+1$ dimensions without any change. The only additional ingredient when going to $3+1$ dimensions is that there is a new additional source of noncommutativity in the momentum composition law ($\gamma_3\neq 0$) together with the one already present in $1+1$ dimensions ($\gamma_1-\gamma_2\neq 0$). As in the case of $1+1$ dimensions one sees that a noncommutative momentum composition law requires a nontrivial boost transformation of the two particle system (${\bar T}^L_q(p) \neq 0$ or ${\bar T}^R_p(q) \neq 0$).
In this case, there is a two-parameter family of transformation laws for the one particle and two particle systems which are compatible with a given composition law in a relativistic theory.

\section{Generalized kinematics versus SR kinematics}

Lacking a complete consistent dynamical framework incorporating the previous discussion of the generalized relativistic kinematics, a question arises whether the generalized kinematics reduces to SR in a nontrivial (e.g. nonlinear) choice of energy-momentum variables. In fact it is not clear whether one has a freedom for such a choice of variables or if the variables used in the modified expression of the composition law have a physical dynamical content. Letting aside this crucial question one can formally analyze how the composition law of momenta varies when one makes a general change of energy-momentum variables at order $1/M$.

In $1+1$ dimensions one can introduce new variables $P_0$, $P_1$ through
\begin{equation}
p_0 \,=\, P_0 + \frac{\delta_1}{M} P_0^2 + \frac{\delta_2}{M} P_1^2 {\hskip 1cm} p_1 \,=\, P_1 + \frac{\delta_3}{M} P_0 P_1 \,.
\label{aux}
\end{equation}
At the same time, the previous change of variables allows us to give an interpretation to the symbols $\left[P\oplus Q\right]_0$ and $\left[P\oplus Q\right]_1$. Since $(p\oplus q)$ is also a momentum, applying Eq.~(\ref{aux}) to it gives
\begin{eqnarray}
\left[p\oplus q\right]_0 &=& \left[P\oplus Q\right]_0 + \frac{\delta_1}{M} \left[P\oplus Q\right]_0^2 + \frac{\delta_2}{M} \left[P\oplus Q\right]_1^2 \,, \nonumber \\
\left[p\oplus q\right]_1 &=& \left[P\oplus Q\right]_1 + \frac{\delta_3}{M} \left[P\oplus Q\right]_0 \left[P\oplus Q\right]_1 \,.
\end{eqnarray}
Then the composition law (\ref{cl1+1}), when rewritten in terms of the new variables, takes the form
\begin{eqnarray}
\left[P\oplus Q\right]_0 &=& P_0 + Q_0 + \frac{(\beta_1-2\delta_1)}{M} P_0Q_0 + \frac{(\beta_2-2\delta_2)}{M} P_1Q_1 \nonumber \\
\left[P \oplus Q\right]_1 &=& P_1 + Q_1 + \frac{(\gamma_1-\delta_3)}{M} P_0Q_1 + \frac{(\gamma_2-\delta_3)}{M} P_1Q_0 \,.
\end{eqnarray}

In the case of a commutative momentum composition law ($\gamma_1=\gamma_2$) it is possible to choose new variables by
\begin{equation}
\delta_1 \,=\, \beta_1/2 {\hskip 1cm} \delta_2 \,=\, \beta_2/2 {\hskip 1cm} \delta_3 \,=\, \gamma_1 \,=\, \gamma_2
\label{delta}
\end{equation}
so that the composition law of momenta is additive in the new variables.

For the dispersion relation~(\ref{dr1+1}), Eq.~(\ref{aux}) gives
\begin{equation}
\mu^2 =   P_0^2 - P_1^2
+ \frac{2\delta_1+\alpha_1}{M} P_0^3 + \frac{2(\delta_2-\delta_3)+\alpha_2}{M}  P_0 P_1^2\,.
\end{equation}
For the new variables with an additive composition law, relations~(\ref{delta}) allow us to write
\begin{equation}
\mu^2 =  P_0^2 - P_1^2
+ \frac{\beta_1+\alpha_1}{M} P_0^3 + \frac{\beta_2-\gamma_1-\gamma_2+\alpha_2}{M}  P_0 P_1^2\,,
\end{equation}
and then the dispersion relation takes the standard unmodified SR form as a consequence of the relativity principle Eq.~(\ref{RP}).

Therefore, for a commutative momentum composition law,
it is possible to change variables to an ``auxiliary energy-momentum'' which behaves just as the SR energy-momentum. In this case,
the possibility of a generalization of SR kinematics is based on the physical dynamical meaning of the choice of energy-momentum variables. However, if one has a noncommutative momentum composition ($\gamma_1\neq \gamma_2$), then the physical content of the generalized kinematics is manifest.

The previous results can be extended to the $3+1$ dimensional case. We have
 \begin{equation}
p_0 \,=\, P_0 + \frac{\delta_1}{M} P_0^2 + \frac{\delta_2}{M} {\vec P}^{\, 2} {\hskip 1cm} p_i \,=\, P_i + \frac{\delta_3}{M} P_0 P_i
\end{equation}
and
\begin{eqnarray}
\left[P\oplus Q\right]_0 &=& P_0 + Q_0 + \frac{(\beta_1-2\delta_1)}{M} P_0Q_0 + \frac{(\beta_2-2\delta_2)}{M} ({\vec P} \cdot {\vec Q}) \nonumber \\
\left[P \oplus Q\right]_i &=& P_i + Q_i + \frac{(\gamma_1-\delta_3)}{M} P_0Q_i + \frac{(\gamma_2-\delta_3)}{M} P_iQ_0 + \frac{\gamma_3}{M} \epsilon_{ijk} P_j Q_k\,, \nonumber \\
\mu^2 &=&   P_0^2 - |\vec P|^2
+ \frac{2\delta_1+\alpha_1}{M} P_0^3 + \frac{2(\delta_2-\delta_3)+\alpha_2}{M}  P_0 |\vec{P}|^2\,.
\label{aux3+1}
\end{eqnarray}

In the case of a commutative composition of momenta ($\gamma_1-\gamma_2=\gamma_3=0$) the same choice of new variables as in $1+1$ dimensions~(\ref{delta}),
together with the relativity principle~(\ref{RP}),
leads to unmodified composition laws and dispersion relations so that the physical content of the generalization of the kinematics rests on the noncommutativity of the momentum composition or on the dynamical content of the choice of energy-momentum variables.

The message of this section can be rephrased in the language of the geometry of momentum
space as introduced in Refs. \cite{relativelocality}: we have basically asked when a composition law can be described by a flat connection on a Minkowski momentum space, but in a generalized coordinate set. The generality of the answer is limited by the choice we have made throughout this
paper to work at first order in $M^{-1}$. At this order one is not sensitive
to the curvature of momentum space (which can enter only at order $M^{-2}$).
There is still space for a torsion and nonmetricity of momentum space, which are first-order effects. Our result was, unsurprisingly, that our momentum space is Minkowski if the torsion is zero (in fact the torsion measures the noncommutativity of the composition law and $\gamma_1-\gamma_2=\gamma_3=0$ imply that it is zero).
Since in this case the relativity principle~(\ref{RP}) sends the dispersion relation in Eq.~(\ref{aux3+1}) to
$\mu^2=P_0^2- |\vec{P}|^2$, we conclude that the case of a commutative composition law is compatible with a geometry of momentum space which is Minkowski with the metric connection.

\section{Concluding remarks}

In this work we have studied the constraints that the relativity principle imposes on the dispersion relation and the composition law in a
(rotational invariant) theory beyond SR. In particular, we have shown that the composition law fixes completely the dispersion relation but not the transformation laws, so that in $3+1$ dimensions there is a two-parameter family of transformation laws compatible with a given composition law. On the other hand, giving the transformation laws for the one particle and two particle systems, fixes everything.

It is easy to see that, at the order $1/M$ for which we have derived these results, there is no further freedom in the definition of the transformation laws and the composition laws for a system of more than two particles. The restrictions
\begin{equation}
p\oplus q \oplus r|_{r=0}=p\oplus q \quad \quad p\oplus q \oplus r|_{q=0}=p\oplus r \quad \quad p\oplus q \oplus r|_{p=0}=q\oplus r
\end{equation}
fix completely the composition law of three momenta at order $1/M$ in terms of the composition law of two momenta:
\begin{align}
[p\oplus q \oplus r]_0 & = p_0+q_0+r_0+\frac{\beta_1}{M}(p_0 q_0+p_0r_0 + q_0r_0)+\frac{\beta_2}{M}(p_1 q_1+p_1r_1 + q_1r_1) \nonumber\\
[p\oplus q \oplus r]_1 & = p_1+q_1+r_1+\frac{\gamma_1}{M}(p_0 q_1+p_0r_1 + q_0r_1)+\frac{\gamma_2}{M}(p_1 q_0+p_1r_0 + q_1r_0)
\end{align}
and also it fixes the implementation of the boost transformation in the three particle system in terms of the boost transformation of the two particle system
\begin{equation}
\{p,q,r\} \to \{T_{q,r}^{(1)}(p),T_{p,r}^{(2)}(q),T_{p,q}^{(3)}(r)\}
\end{equation}
with
\begin{equation}
T_{q,r}^{(1)}(p)=T(p)+\bar{T}_{q,r}^{(1)}(p) \quad \quad
T_{p,r}^{(2)}(q)=T(q)+\bar{T}_{p,r}^{(2)}(q) \quad \quad
T_{p,q}^{(3)}(r)=T(r)+\bar{T}_{p,q}^{(3)}(r)
\end{equation}
and
\begin{equation}
\bar{T}_{q,r}^{(1)}(p)=\bar{T}_q^L(p)+\bar{T}_r^L(p) \quad \quad
\bar{T}_{p,r}^{(2)}(q)=\bar{T}_p^R(q)+\bar{T}_r^L(q) \quad \quad
\bar{T}_{p,q}^{(3)}(r)=\bar{T}_p^R(r)+\bar{T}_q^R(r).
\end{equation}
This result obviously extends to the $3+1$ dimensional case.

In terms of the geometric interpretation of the ``relative locality'' proposal \cite{relativelocality}, the above results are a consequence of the
fact that all the novelty that can enter in three-particle vertices is contained in the possibility of a nonassociativity of the composition law, which encodes the curvature of the connection (its Riemann
tensor). But at first order in $1/M$ there is no distinction between
a flat and a curved connection, and therefore all the properties of three- (or several-) particle
systems are encoded by those of the two-particle system.

One can also try to generalize the discussion of the $1/M$ relativistic kinematics to the next order in the power expansion in $1/M$. The number of coefficients for the composition law, dispersion law and transformation laws gets much higher, complicating the algebra, but there is not any obstruction for the analysis. In this case one has a composition law for two particles and a new composition law for three particles which is no longer fixed by the two particle law. In the geometric picture one would say that the (unspecified) components of the curvature enter in the composition law.

It could be interesting to establish the correspondence between the present framework for relativistic kinematics based on a modification of the momentum composition law and a framework based on symmetry algebras generalizing the Poincar\'e symmetry algebra of SR. The derivation of boost transformations of multiparticle systems and modified energy-momentum conservation laws from a $\kappa$-Poincar\'e Hopf algebra presented in Ref.~\cite{Gubitosi:2011ej} should correspond to a particular case of the framework presented in this paper.

A modification of the momentum composition (and conservation) law, requires a new implementation of translational symmetry and a generalization of the SR notion of spacetime. It is an open question how a physically meaningful spacetime should be introduced in a generalized relativistic kinematics. There are even doubts~\cite{AmelinoCamelia:2012qw} on the possible redundancy of such a notion.

Finally, in the paper we gave some hints on the interpretation of some of our results in terms of the geometry of momentum space. It would be very interesting to translate all  the consistency conditions we found for a  dispersion relation and a composition law of momenta  to be compatible with the relativity principle into conditions on the geometry of momentum space. The first task would be to calculate the relationship that holds, at first order in $1/M$, between our coefficients $\alpha,\beta,\gamma$ and the components of all the geometric tensors that define momentum space: the metric,
the torsion and the nonmetricity (as we already remarked, the curvature is unspecified at that order).
Then one would need to ``mod out'' by diffeomorphisms, or general coordinate transformations.
Hopefully one should be able to end up with a simple characterization of the most generic geometry of momentum space that is compatible with the relativity principle.
This issue as well as the study of new examples of generalized kinematics constructed from the general framework presented in this work is presently under investigation.

\section*{Acknowledgments}
This work is supported by CICYT (grant FPA2009-09638) and DGIID-DGA (grant 2011-E24/2).

redpajama_set_name: RedPajamaArXiv


New Delhi: An angry mob on Saturday attacked a Jawed Habib salon in Uttar Pradesh's Unnao district and vandalised the shop over the use of a caricature of Hindu gods in one of his advertisements.
The activists of the Hindu outfit also threatened the owner to wind up his business or face consequences, following which a police complaint was filed, according to media reports.
A police team visited the salon and took some eyewitness account of the attack.egional Secretary of the Hindu Jagran Manch Vimal Dwivedi said sacrilege of Hindu gods would not be tolerated at any cost. "We will not allow the salon to operate," he said.
The controversial ad in question carried the caption 'Gods too visit JH Salon' that led to a case being registered against the hairstylist in Hyderabad.


Flora Duffy could not have asked for a better race scenario as a sheer dominating performance captured her the 2018 MS Amlin World Triathlon Bermuda crown in front of thousands of cheering fans.
The gold earned on home soil was her first World Triathlon Series win of the season and first for the inaugural event as she won with a marginal victory of 96 seconds in the first Olympic-Distance race of the season.
Second place then went to Great Britain's Vicky Holland after she won in a photo-finish sprint on the line with USA's Katie Zaferes who then took the bronze.
"I don't think this moment is going to sink in for a while, but what I will say when it does is that this will probably be one of the highlights of my career. I mean this is incredible, the streets were lined, and it was just a phenomenal atmosphere, it was incredible. It was hard for me to control my emotions throughout that whole 10km, but with like 9.5km to go I just thought I need to smile and celebrate and soak this in," Duffy said of her hometown-hero win.
Collecting the leading position in the swim, bike and run, Duffy's win also meant she became the first woman in WTS history to win a race after being the leader in all three disciplines, which included a 40-kilometre solo bike ride.
"That wasn't my plan, I wanted to get a small breakaway, that is why I pushed in the swim. Going up corkscrew I wasn't really sure where everybody was, I knew Kirsten (Kasper) was there, I attacked it because that was my plan and I thought she would come with me, but she didn't. So I thought, well I guess I am going 40km solo," she added.
Meanwhile, it was an overall historical day on the island as the debut 2018 MS Amlin World Triathlon Bermuda saw the nation of Norway sweep the men's podium for the first-time in WTS history.
In only his third World Triathlon Series elite race of his career, Casper Stornes dominated his way to claim Gold, while compatriots Kristian Blummenfelt and Gustav Iden finished off the Silver and Bronze to complete the sweep.


text:
package com.facebook.buck.shell;

import com.facebook.buck.util.immutables.BuckStyleTuple;
import com.google.common.collect.ImmutableList;

import org.immutables.value.Value;

@Value.Immutable
@BuckStyleTuple
interface AbstractWorkerJobParams {
  @Value.Parameter
  ImmutableList<String> getStartupCommand();
  @Value.Parameter
  String getStartupArgs();
  @Value.Parameter
  String getJobArgs();
}

redpajama_set_name: RedPajamaGithub


text:
National Leagues are a really good learning process – Steve Diamond
December 11, 2018/
Under : Interviews , National 1
Director of Rugby Steve Diamond insists the lower tiers are a key part of player development at Sale Sharks.
Things are on the up at the AJ Bell Stadium.
On Saturday night, Sale Sharks secured a thumping win in the European Challenge Cup and their qualification for the knockout stages is almost guaranteed.
A convincing 50-24 victory over Bordeaux Begles in France means the Sharks maintained their perfect record in Pool 3, but also their momentum is continuing to build.
It's starting to look a lot like Christmas ? as our Sale Sharks light up the pitch and deliver an early present https://t.co/4G2vtc1QTW
— #rugbysalesharks (@RugbySaleSharks) December 9, 2018
Since the first European window, Sale have lost on just two occasions and in that time, there have been some standout individual performances.
Chris Ashton, Denny Solomona and Byron McGuigan have grabbed the headlines in recent weeks, but Sale's young stars have also shone as Director of Rugby Steve Diamond continues to hand opportunities to some of the club's emerging talent.
The Curry twins [Tom and Ben] as well as the James brothers [Sam and Luke] are past examples of Sale willing to promote youth and this season, it has certainly been a plus point for the club as a whole.
"For anybody who knows what we do at Sale, we have a massive development pathway and I always give lads opportunities," Diamond said. "If they are good enough, they are old enough in my book. Arron Reed, Kieran Wilkinson, Cameron Redpath, to name just three, are outstanding players meaning they will always get a chance."
But, it is not just through the academy where players can gain opportunities to showcase their abilities for Sale.
The club's link with Sale FC has continued to grow and this season, the likes of Kieran Wilkinson, Arron Reed, Connor Doherty, Nic Dolly and Matt Sturgess have all appeared in National One.
Added to this, Gus Warr and Sam Dugdale have both gained game time at National Two North side Fylde during the the campaign.
Congratulations to our 8 England U20 Six Nations squad members! ?
That's 25% of the squad ?
Well done! ? @bevan_rodd @nicdolly1 @connor13doherty @lukeejamess12 @arronreed1 @cam_redpath999 @warr_gus @wilkyxwilkinson #SharksFamily #SharksByNature pic.twitter.com/kICC7dtCTX
— Sale Sharks? (@SaleSharksRugby) November 26, 2018
"One of the flaws in the academy system is kids are just playing against their own age groups all the time while Sale FC and the National Leagues offers them an opportunity to play against men," Diamond admitted. "They are mature players who have been playing for 10-15 years and it is a really good learning process for the younger players."
Diamond is not the first high profile name in the sport to praise the National Leagues with World Rugby Chairman Bill Beaumont highlighting how this level of rugby can benefit the game across the board.
In terms of results, Sale FC currently sit 11th in National One and will be aiming to end a three-game losing streak when they take on Ampthill and District this weekend while the Sharks also occupy the same position in the Gallagher Premiership.
Steve Diamond's men are embroiled in a remarkable fight in the bottom half of the table, but he feels if his side can continue to produce encouraging performances between now and the end of the year, Sale can kick on heading into 2019.
He added: "It is a good competition the Challenge Cup and we are top at the group at the minute. They [Bordeaux] are doing well in the French league put we just want good performances and it hopefully should leave us in a good position going into the last two pool stage matches.
"We have got something out of every game in the last few fixtures. We had a poor start to the season, but we seemed to have rectified that so hopefully it can be a big momentum builder for us."
Sale will be aiming to wrap up qualification for the next round of the Challenge Cup when they host Bordeaux in the reverse fixture on Saturday afternoon.
Steve Diamond was speaking at a 'Cash for Kids' Mission Christmas event in Sale as the club helped to support the drive to make sure that disadvantaged children in Manchester have presents to open at Christmas.
Anyone wishing to donate gifts to Cash for Kids' Mission Christmas appeal this year can do so through designated community drop off points, with more than 300 drop offs across the country.
For more information, visit: http://www.cashforkids.uk.com/mission/
National 1Sale FCSale SharksSteve Diamond

redpajama_set_name: RedPajamaCommonCrawl


text:
New Jersey's Cory Booker ends his presidential campaign
Booker said he didn't see a path to victory without more money
Posted Jan 13, 2020 11:08 AM
Simone Pathé
@sfpathe
Democrats try to expand House battlefield by targeting six more districts New Hampshire's Kuster backs Buttigieg Impeachment managers all represent safe Democratic seats
New Jersey Sen. Cory Booker is ending his presidential bid. (Caroline Brehman/CQ Roll Call file photo)
New Jersey Sen. Cory Booker ended his presidential campaign on Monday, saying he was out of money.
"Our campaign has reached the point where we need more money to scale up and continue building a campaign that can win — money we don't have, and money that is harder to raise because I won't be on the next debate stage and because the urgent business of impeachment will rightly be keeping me in Washington," Booker said in a statement.
The former Newark mayor, who made a name for himself over the years as a social-media savvy politician, campaigned on love and unity but failed to qualify for recent debates and languished at the bottom of polls.
Booker was one of two African-American senators who ran for the Democratic nomination for president in 2020. California Sen. Kamala Harris ended her bid in December. With Booker's departure, former Massachusetts Gov. Deval Patrick is the only black candidate left in the race.
Booker made his political experience in Newark, his adopted hometown, a central theme throughout his campaign.
He announced his candidacy in there in February 2019 and featured it in his paid advertising. Despite mixed reviews of his tenure as the city's mayor, Booker had quickly earned the backing of every Democrat from New Jersey in the House last year, including his previous opponents.
On the debate stage, Booker frequently talked about being the only candidate who chose to live in a low-income inner-city community. His policy proposals — criminal justice reform, as well as a national gun licensing and a "baby bonds" program — reflected his experience in that community.
"I showed that you can actually fight and win, but not stoop to the levels and tactics and dirty tricks of the opposition," Booker told CQ Roll Call in 2019 about his experience going up against the city's political machine and how that prepared him to take on President Donald Trump.
Elected to the Senate in a 2013 special election, Booker is running for a second full term this year. In 2018, Democratic Gov. Phil Murphy signed a law that allowed Booker to appear on the primary and general election ballots for president and lower office. If the eventual Democratic nominee were to chose Booker as a running mate, Booker could still appear on both ballots.
Newark civil rights advocate Lawrence Hamm has filed with the Federal Election Commission to challenge Booker in the Democratic Senate primary in June. The chair of the People's Organization for Progress, Hamm is also the chairman of Vermont Sen. Bernie Sanders' presidential campaign in New Jersey.
Doug Steinhardt, the state Republican chairman, issued a statement saying Booker's agenda of "socialized healthcare and out of control taxes" was fueling the state's affordability crisis.
"If he expects to waltz back to New jersey into open arms, he is absolutely wrong," Steinhardt said.
Inside Elections with Nathan L. Gonzales rates the Senate race Solid Democratic.
Topics: 2020 at-the-races democrats elections includephoto politics primaries Business california Campaign Finance Campaigns Cory Booker democrats Donald J. Trump Elections Executive Branch House Massachusetts media New Jersey polling Republicans Senate Special Election Vermont Washington POLI


text:
package com.microsoft.azure.management.logic;

import com.fasterxml.jackson.annotation.JsonProperty;

/**
 * Error reponse indicates Logic service is not able to process the incoming
 * request. The error property contains the error details.
 */
public class ErrorResponse {
    /**
     * The error properties.
     */
    @JsonProperty(value = "error")
    private ErrorProperties error;

    /**
     * Get the error value.
     *
     * @return the error value
     */
    public ErrorProperties error() {
        return this.error;
    }

    /**
     * Set the error value.
     *
     * @param error the error value to set
     * @return the ErrorResponse object itself.
     */
    public ErrorResponse withError(ErrorProperties error) {
        this.error = error;
        return this;
    }

}


text:
Higher raw material cost impacts Kansai Nerolac performance
Bavadharini K S | Updated on May 03, 2018 Published on May 03, 2018
Price increases have offered some respite
The stock of Kansai Nerolac is down marginally by about 1 per cent in trade today. Despite strong growth in segments such as decorative, industrial and powder coatings, the steady increase in raw material costs for the company has impacted its margins and bottom line.
The company's revenue growth of 17 per cent YoY (net of excise duty) can be attributed to the strong recovery in the decorative and industrial paint segment. The company has reported double digit growth in all its segments. The company is the largest player in the decorative segment, which contributes nearly 55 per cent of its revenue, while the industrial segment contributes 45 per cent (chunk from automotive coatings).
At the operating level, the company's margins were impacted due to the rise in crude oil prices. Titanium dioxide, a derivative of crude oil, for instance, witnessed a sharp price increase. The raw materials cost (as a percentage of sales) has increased by 6 percentage points YoY to 64 per cent in the March quarter (on revenues net of excise). Nonetheless, the company's operating margins were down by a lower 2 percentage points, thanks to the company's sustained efforts to affect a price increase, which to an extent, has aided margins.
Nonetheless, the dip in operating margins has impacted earnings -- profits declined by about 9 per cent YoY during the March quarter.
Going ahead, the volatility in crude prices is a cause for concern as a chunk of its raw material is in the form of crude derivatives. Though this increase in raw materials costs can hurt margins, price hikes in the decorative segment can cushion the fall to some extent.
The company recently entered into agreement to acquire Marpol, a pioneer in the powder coating business. Given Kansai Nerolac's strong market presence in this segment, the acquisition of Marpol will boost its market position and improve its customer base.
Kansai Nerolac Paints Ltd
Justice B L Bhat to continue as officiating NCLAT chief till March 31
Finmin looks at BIC model after RBI raises concern over zero coupon bonds for PSBs recap
CPI (M) to hold nationwide all-party meets to support protesting farmers
NCLAT
personal investing


A unique opportunity for students to create a COMET! Find out first-hand about these mysterious intergalactic bodies We turn your students into Astronomers who visit the Oort Cloud to better understand how a comet is created! What better way to understand comet behavior than a true hands-on experience!
Astronomers will be amazed as comets begin to appear right before their eyes! They will recognize how "cool" a comet really is and the important roll the sun plays in a comet's rotation.


text:
Bilkent Theses
Theses - Department of Chemistry
Dept. of Chemistry - Master's degree
Molten salt assisted self-assembly (MASA) : synthesis, characterization and solar cell performances of mesoporous silica-CdSe and titania-CdSe thin films
Embargo Lift Date: 2016-07-21
Han, Ahmet Selim
Dağ, Ömer
A series of solutions of a salt ([Cd(H2O)4](NO3)2), a polymerizing agent (Si(OCH3)4, TMOS or Ti(OC4H9), TTBO), two surfactants (cetyltrimethylammonium bromide, CTAB and 10-lauryl ether, C12H25(OCH2CH2)10OH, C12EO10) have been prepared and used as the molten salt assisted self-assembly precursors by only changing the Cd(II)/surfactant mole ratios for the preparation of mesoporous films. The thin films were prepared in two steps: in the first step, the titania particles (titania particles are typically 20-25 nm (P25)), dispersed in ethanol, have been spin coated over various substrates and annealed at 450oC and in the second step, the clear solutions given above (salt, CTAB, C12E10, silica or titania source and ethanol) were drop casted or spin coated over the titania (P25) films and calcined at 450oC. Slow calcination of the films (with an increment of 1 oC/min), starting from the melting point of the salt (around 65oC) to 450 oC has produced the mesoporous silica/titania-cadmium oxide (CdO) thin films denoted as meso-CdO-SiO2-P25 and meso-CdTiO3-P25. The clear solutions were also used to make the thin films without P25 and denoted as meso-CdTiO3. The films were then exposed to a H2Se atmosphere at 100oC for 30 min, and the samples were denoted as meso-CdSe-SiO2-P25, meso-CdSe-TiO2-P25 and meso-CdSe-TiO2. The silica sample was further treated with a dilute HF solution (etching process) that results silica free meso-CdSe-P25. The characterization of the materials produced in this thesis was made by using XRD, FT-IR spectroscopy, UV-VIS, Raman, EDX and Solar Measurement techniques. Silica samples have greater amount of CdSe than titania samples according to the Raman and EDX data which implies that silica samples are more reactive. Also in both silica and titania, the samples with a Cd/surfactant mole ratio of 6 have the greatest amount of CdSe. Fluorine doped SnO2 (FTO) has been used as a transparent conductive substrate for the preparation of the anode electrode for the solar measurements. According to the solar measurements, the silica and titania samples on P25 show greater efficiency than the titania samples without P25 and generally have similar efficiencies (the efficiency is the ratio of the electrical output of a solar cell to the incident energy in the form of sunlight). The most efficient samples are generally the samples prepared using Cd/surfactant mole ratios equal to 6 and 8. In order to increase the efficiency, 3rd row transition metal cations such as Mn(II), Fe(III), Cu(II) and Co(II) are doped into above samples. The efficiency has been increased by 10 % in the silica and 30 % in the titania samples upon doping with 15% Mn(II). The solar cell characteristics of the electrodes were tested using the following parameters; by changing the Cd(II)/surfactant mole ratio (many samples were prepared and used for this purpose), by changing the aging of the electrolyte (Na2S/S8), by using multiple coating the MASA solution in the preparation stage of the electrodes, by doping the samples with various transition metal ions, and finally by modifying the surface of the electrode by coating with ZnS or CTAB. Each of these parameters has an effect on the cell performance. The effect of each parameter is monitored by measuring the I-V characteristics of the cells and found out that the best results were obtained from the electrodes made using 6 or 8 Cd/surfactant mole ratio in MASA system, freshly prepared Na2S/S8 electrolyte, coating of the MASA solution twice on electrode, doping Mn(II) cations with 0.15 Mn(II)/Cd(II) ratio and using CTAB for coating anode electrode.
MASA process
mesoporous CdO-SiO2
mesoporous CdTiO3
mesoporous CdSe-TiO2
Quantum dot sensitized solar cells
Dept. of Chemistry - Master's degree 129


My friend Karen and I have a lot in common, and one of the things I enjoy about our friendship is we both love a bargain.
When you live in a city that costs an arm and both legs to live in, if you're smart, you're constantly looking for a deal. Unless you're rich. And if so, stop reading; this article isn't for you!
Last Friday, Karen and I had a spa day. Sounds fancy, right? Let me assure you, I am not fancy. I'm just high maintenance.
The bargain-hunters that we are, we found a great Groupon deal that provided us with a 60-minute massage AND a 45-minute facial for $55. Unbelievable, right?
So off to the spa we went. Our deal was at Mama Spa which happens to be in the Lower East Side. Usually I avoid the Lower East Side (LES to us New Yorkers) like the plague. The LES is supposed to be this uber-hip neighborhood, but whenever I'm there, I'm scared I'll catch herpes or accidentally get addicted to heroin. I admit it; I'm too much of a princess for the uber-hip neighborhood.
However, Mama Spa is a pearl among ashes. Located on Allen Street, you have to buzz to get in. And once you're in, you're transported to a beautiful, tranquil spa. Karen was waiting for me in an oversized leather chair, which hopefully made her forget I was a couple minutes late.
We started out with the massages. Wendy led me to a room where I got nice and comfy and pretended I wasn't about to be naked in front of a perfect stranger. This wasn't my first massage; I knew the drill. Wendy came back into the room, dimmed the lights, and began to work. I was ready for a good nap.
I wasn't able to turn my mind off (not Wendy's fault) and nap, but I was feeling tension leave my body. Why is it us women can't turn our minds off? Or is it just me? Wendy kept working her magic fingers. I hurt my back a few years ago in an impromptu furniture-rearranging night where I convinced myself I can do it all by myself. I did do it, but my back paid the price. Wendy relieved years of muscle tension. God bless her. That was probably the best massage of my life. At one point, I wondered if Wendy's fingers had multiplied a few times over. Her technique was flawless. Karen and I both walked out feeling fantastic.
Next, we had our facials. I have no idea what was placed on my face. All I know is I walked out looking ten years younger, maybe five. I had been feeling run down and haggard lately, and my facial totally revived me. I had a glow for a couple of days after. (Gentlemen, this is when you tell me I always look wonderful.) New York summers can do a number on your face, so it's always nice to wash away the nasty three months of humidity at the end of it all. Karen and I had perfect timing as Fall has started to nip in.
So why am I making you all jealous of my experience? Well, one because I wanted to relive it, but more importantly, I'm going to give you the chance to do the same.
That's right devoted New York Minute reader, you are getting a great deal for a 60-minute massage. I loved my massage so much that I asked Mama Spa if they would give my readers a deal, and they said YES! There's no voucher or code or fancy download. We're doing this the old-school way.
Ladies and Gents, to get a 60-minute massage for $39 (half the price!!!) at Mama Spa, call 212-780-1913 and utter this secret phrase: "Sarah from New York Minute sent me." You have from today (September 14th, 2012) to next Friday (September 21st, 2012) to schedule your massage. Don't be stingy; share this with your friends!
See, we're always looking out for you at New York Minute!
If you have a business you'd like New York Minute to check out, contact us here.
I am going to do this AGAIN…who's with me?
If I lived in New York, I would jump on this deal!


An armchair with a very special seat design for people who like to be unique. Its cantilever base gives it a great comfort. MySoul offers many options: single units with footstools, double units, and an amazing double chaise longue. The model is available with high wooden legs, Nordik style.


Maya ot Raya's fine lace & gem & beads jewelry is delicate and hand crafted with great care to last long time, however with normal wear proper care is required to keep it looking good.
A great fact about my lace jewelry is that the lace is made from special metallized thread so it is extraordinarily durable.
I hope that you'll find these guidelines helpful and if you have any further questions don't hesitate to contact me.
Here are some basic guidelines to consider when wearing my jewelry and any other jewels as well.
Remove Jewelry During Tasks – When performing manual tasks, remove your jewelry to prevent physical damage or exposure to chemicals or cleaning fluids.
Don't Wear Jewelry In Swimming Pools and Spas – Chlorinated water can react with the metals found in jewelry causing color changes and even structural damage.
Remove Your Jewelry Before Bathing – Remove all jewelry before showering or cleaning. Soap can cause a film to form, making it appear dull and dingy. By preventing the formation of this film you immediately reduce the occasions of servicing.
Contact Sports and Jewelry Don't Mix– Hard blows during sports can damage jewelry not to mention the people involved. All jewelry should be removed before play begins.
Preserve from long exposure on sun light or exessive heat – Just as the sun's harmful rays can damage our skin, over time and in excess, it can also fade and weaken some gemstones, such as amethyst, kunzite, topaz, and pink conch-shell cameos. Pearls and other delicate materials, like your lace jewels, will bleach under extreme exposure to light. Other gems, especially amber, can darken over time when exposed to too much light. Excessive heat and sudden temperature changes may also fracture the gem. Heat can easily remove the natural moisture some gems need to keep their beauty. Pearls, for instance, can dry out, crack and discolor. Opals will turn white or brown, develop tiny cracks, and might lose their play-of-color.
Use Jewelry Polishing Cloths for Best Results – Brush your lace jewels with a dry very soft make up brush to clean the dust and after this, brush them again but with wet make up brush to remove the rest of the dirt. Allow your lace jewels to dry before you store them.
Beaded jewelry can be brushed with dry and soft brush to remove the dust away, after this, brush the jewels again but this time with wet and soft brush to remove the rest of the dirt. Allow your beaded jewels to dry before you store them.
Soft gems, such as pearls, can easily be scratched. Use an unused makeup brush and warm, soapy water. Lay the pearls on a towel to dry. The wet string can stretch—and attract dirt—so don't touch a string of pearls until they are completely dry. Pearls worn every few days should be restrung once a year.
It is recommended to clean most colored gems with warm water, mild soap (no detergents), and a soft brush. A pulsed-water dental cleaning appliance and a soft, lint-free cloth can also be used. Be sure to stop the sink's drain or use a rubber mat in case the stone comes loose from its setting.
Polish the metal components of your jewels with polishing cloth for best results. You can use a svelte or an old diaper, but the professional cloths are the best. Use of tissue or paper towels can cause scratches because of fibers in these products.
Anyone that has a collection of jewelry can appreciate how difficult it can often be to keep it organized.
About Tarnish and Storage – I frequently use Sterling plated or Sterling jewelry components in my work. If you are wearing your jewelry very frequently, generally tarnish will not be a problem. Occasional polishing with a jewelry cloth will keep your jewelry shiny. But if you are not wearing your jewelry all the time it is easier to store it in a way that inhibits tarnish.
Clear, clean, plastic bags that have no holes and seal tightly are perfect to keep air and moisture off the Sterling. You will have to polish it much less, if at all. It is best to add to the sealed bag a piece of paper, a desiccant or silica gel pack (often found in new shoe boxes), or even a small amount of rice rolled into paper or fine netting. This will absorb any moisture that may lead to tarnishing. After you have placed the jewelry in the bag and made sure it is flat, gently press as much air out as possible.
I know it is not particularly glamorous but it totally works and is worth it when you like to wear your jewelry without worrying whether it has been polished.
Keep Your Jewelry Secure – Store your jewelry in a container and prevent pieces from moving around. While fabric-lined jewelry boxes are ideal, this can be as simple as using a shoebox and pieces of fabric. To store your jewelry, wrap your items, clean + dry, in a soft cloth or bag. Keep your items separate while storing to avoid tangling and scratching.
When Travelling with Jewelry, Use a Case – Travelling can be tough on jewelry, so it's wise to use a travel case to protect your favorite pieces. Traditionally made of fabric or leather, a jewelry travel case can carry jewelry of all types and easily rolls up to a size that's easily stored in most suitcases, makeup cases or business attaches. Clearly, a case like this is a good investment to protect your jewelry on the road.


text:
Al "The Bronx Beauty" Singer (September 6, 1909 – April 20, 1961) was an American boxer who won the world lightweight championship in 1930.

Early life and career
Singer was born in a tenement on Broome Street, part of the Jewish section in New York's Lower East Side on September 6, 1909.  He was one of four sons and a daughter born to  an ambitious ladies' garment entrepreneur who would keep his large family in America's middle class.  Singer, an all round athlete who loved basketball, was discovered by boxing trainers Harry Drucker and Hymie Caplan after his tenth amateur bout which included a great showing in his first outing against the New York State Bantamweight champion, Jimmy Cruze. As an amateur boxer, he won the Metropolitan AAU featherweight title.   Singer had unsurpassed grace and a studied style in the ring, and he could punch with authority, but unlike the great lightweight Benny Leonard, to whom he was often compared, he could not take a strong punch and was considered to have a "glass jaw".  His quick rise to fame, and equally quick demise could be explained by these competing characteristics.

Quick rise to boxing prominence
Debuting as a professional in Brooklyn with a knockout of Jim Reilly on July 2, 1927, Singer went undefeated in his first two years as a professional.  His affectionate New York fans dubbed him "The Bronx Beauty" for both his looks and exceptional grace in the ring.  Though Singer was young and had stayed far distant from crime, his exceptional winning record caught the eye of the New York mob, who hoped to bring him to a championship as quickly as possible. Most reference books on Singer noted that in 1928, two men came to his training camp and asked to meet with his manager Harry Drucker.  Accepting a ride with the men, Drucker was never heard from again, and Singer came under the influence of the mob, who many believed fixed a few of his early fights. He performed exceptionally well against Lou Moscowitz at Madison Square Garden and Pete Zivic at St. Nicholas in late 1928, knocking them out in five and six rounds respectively. In his first real feature match on September 14, 1928, Singer and the exceptionally skilled Tony Canzoneri, former holder of the featherweight world title, battled to a 10-round draw, satisfying an impressed audience and spotlighting Singer as a potential feather or lightweight contender. In his career, Singer won 61 of 72 pro fights (25 by KO), drawing twice, and losing nine.

In February and March 1929, Singer faced former 1927 world bantamweight champion Bud Taylor at Madison Square Garden, winning in a fourth round disqualification and a ten-round points decision.  In Singer's win by disqualification, Taylor landed a low left hook to the body, 52 seconds into the fourth, that caused Singer to drop to his knees in pain. The bout was fast, furious, and close in the opening rounds. In their ten-round bout on March 15, before 20,000 at the Garden, the fighting was close, but Taylor's attempts to end the bout with hooks to the body were not adequate to slow down Singer, who was permitted to land a few rights to the jaw of Taylor, gaining essential points in the closely scored bout.

In a non-title bout on July 24, 1929, before a crowd of around 29,000, Singer defeated Frenchman Andre Routis, reigning NBA world featherweight champion, in an impressive second-round technical knockout at Ebbetts Field in Brooklyn.  Singer first floored the Frenchman with a count of five from a right to the jaw preceded by a right left combination, that was followed by a count of three with another right after Routis resumed boxing.  On his third trip to the canvas, the referee called a technical knockout.

Loss to Kid Chocolate, August, 1929
On August 29, 1929, Singer lost an important twelve round bout with Black Cuban boxer Kid Chocolate, at the Polo Grounds in New York with 50,000 in attendance. It was a swift, clean fight that featured no knockdowns, but studied technique and style.  In the fifth and the eighth, Chocolate landed two powerful rights to the head of Singer that nearly staggered him, but he stayed on his feet.  In the second and eleventh, Singer boxed Chocolate around the ring, but at the end of each round Chocolate answered with a flurry of his own. Singer appeared better in the early rounds, and was the aggressor particularly in the second, fourth and fifth.  In the seventh, eighth and ninth, Chocolate rallied, using his right with enough effect to gain a points advantage from which Singer could not recover.  Chocolate clearly took the twelfth, conceding the tenth by a hair to Singer.  The decision was unpopular with the New York crowd, who may have been aware that Chocolate effectively blocked a large number of the blows that Singer appeared to land.  Chocolate let Singer take the lead, slowly wearing him down, and using his best defense against him.

Capturing the world lightweight championship against Sammy Mandell, July 1930

Singer captured the World Lightweight Championship on July 17, 1930, before a crowd of 35,000 at Yankee Stadium, with a first-round knockout (1:46) of champion Sammy Mandell. Though both boxers started the match boxing cautiously, Singer dropped Mandell with a left hook to the jaw in less than a minute of fighting.  Three more times Mandell went down to the mat for short counts, scarcely able to raise his hands in defense.  The final knockdown came with a crushing one two punch to the jaw.  Mandell had defended his four-year claim to the lightweight crown only a few times, and claimed that the weight loss required to make weight had sapped his strength.

Devastating loss to Jimmy McLarnin, September 1930

Against the better judgement of many, Singer accepted a non-title bout with boxing sensation Jimmy McLarnin before a crowd of around 25,000 on September 2, 1930.  McLarnin flattened Singer in the third round with a right to the jaw. Although Singer fought well in the opening round, dropping his opponent for an instant from a right, he foolishly decided to fight toe to toe with the stronger McLarnin in the third round. McLarnin had at least a five-pound advantage, and a small but significant advantage in reach.   A lightning right to the jaw dropped Singer for a count of nine.  When he arose and went into a brief clinch, McLarnin threw a left hook to the jaw to counter a right from Singer that missed, and Singer went down for the count.  A Doctor at the scene, who inspected Singer, said the blow may have affected his upper spine and caused a momentary partial paralysis, not uncharacteristic of a knockout.

Upset loss of the world lightweight championship to Tony Canzoneri, November 1930

He surrendered the title on November 14, 1930 to Tony Canzoneri in a knockout only 1:06 into the first round.  Many believe Singer should have taken more than two months to recover from the bruising loss he took from McLarnin the previous September.  The upset came as Singer was designated the early betting favorite at 12-5.  Coming out of a clinch, Canzoneri threw a left hook to Singer's mouth, and then one to his chin which snapped back his head, and put him down for the count.  Singer was unable to come to his feet without collapsing.

Recovering from his loss of the title on June 18, 1931, Singer defeated Lew Massey before a crowd of 6,395, at Madison Square Garden in a ten-round points decision.  Observing how he threw jabs and punches nearly at will, the Associated Press gave Singer eight of ten rounds, allowing Massey only the seventh when he switched to infighting.  Singer scored with hard rights in the tenth. Other reporters considered the boxing fairly close, noting that with his reach, height, and weight advantage, Singer  should have won more decisively.

Loss to world featherweight champion Battling Battalino, December 1931

Singer lost decisively to Battling Battalino, reigning New York State Athletic Commission (NYSAC) world featherweight champion, before a crowd of 17,000 at New York's Madison Square Garden on December 11, 1931.  Battalino may have bobbed and weaved away from Singer's early volleys in the first, and was likely shaken by a blow or two, but his counterattack was devastating to Singer.  Though Singer started strong very early in the opening round, he was soon put down three times by Battalino, and went down in the second from a series of rights to the chin for a count of seven.  After he rose, he was chased around the ring by Battalino who put him down again with a right to the chin for a count of four, just before the referee justifiably called the bout a technical knockout and helped move Singer to his corner.   Singer's retreat from Battalino after he was first put down, his inability to defend himself, and his struggle to mount an attack, indicated a boxer who had, at least for the moment, lost most of his physical faculties.  Both boxers were above the featherweight limit preventing any chance of the match being a featherweight title match.  The match ended any of Singer's championship aspirations.

Attempted comeback, October–November 1935
Four years later, Singer tried a comeback from October–November 1935 under a new manager, Leon See, winning four straight matches, but against strictly second rate opponents.  He achieved a fourth-round technical knockout against Buckey Keyes on October 14, flooring his opponent in the second and third rounds.  In a final victory, Singer retired after defeating Johnny Toomy on November 11 in a fifth-round technical knockout.

Life after boxing

Singer served in WWII, fighting a number of exhibitions for the soldiers.  He was stationed at Camp Upton in New York but received a medical discharge due to eye trouble.  He married the former show girl, Billy Boze in 1947.  Having earned around a quarter million from boxing in his lifetime, Singer tried various businesses in his thirties and forties including sales, restaurants, real estate, and theater, owning several Cabarets.  He was appointed to a judgeship by the New York State Boxing Commission in 1955.

On April 20, 1961, he died of a heart attack in his New York City apartment at age 51. Singer, who was Jewish, was inducted into the International Jewish Sports Hall of Fame in 2006.

Professional boxing record

See also
Lineal championship
List of lightweight boxing champions
List of select Jewish boxers

References

External links

Cyber Boxing Zone bio and record
Jewish Sports bio

1909 births
1961 deaths
Boxers from New York City
Lightweight boxers
World lightweight boxing champions
World boxing champions
Jewish American boxers
Jewish boxers
American male boxers
20th-century American Jews

redpajama_set_name: RedPajamaWikipedia


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