series \vspace*{-1.678902345in} {Deviation equations and\\[1ex] weak equivalence principle\\[1ex] in spaces with affine connection }% main title % \\ \vspace{0.22ex} \\ \vspace{1.1ex} {\LARGE }% subtitle
Bozhidar Z.\ Iliev \thanks{Laboratory of Mathematical Modeling in Physics
% Some connections between the deviation equations and weak equivalence % principle are investigated. %
gr-qc0605008.tex
%**	This is a pure LaTeX 2e file [article class]
%**	Filename:	de-wep.tex [Deviation Equations and Weak equivalence
%**				    Principle ...]
%**	Last update: May 1, 2006
%**     Final version: 1984
%**	Title:	Deviation equations and weak equivalence principle
%**		in spaces with affine connection
%**	Purpose: Rigorous approach to the weak equivalence principle

%**	Author (and responsible for all errors):

%***************************************************************************%
%**                                                                       **%
%**	 Article style of BOZHIDAR ZAKHARIEV ILIEV: 		 	  **%
%**                                                                       **%
%**	 Laboratory of Mathematical Modeling in Physics			  **%
%**	 Institute for Nuclear Research and Nuclear Energy                **%
%**	 Bulgarian Academy of Sciences                                    **%
%**	 Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria           **%
%**	 E-mail address: bozho@inrne.bas.bg				  **%
%**      URL: http://theo.inrne.bas.bg/~bozho/	        		  **%
%**                                                                       **%
%**                                                                       **%
%**			(May be used freely by everybody)                 **%
%**                                                                       **%
%***************************************************************************%

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\typeout{"Deviation equations and weak equivalence principle}
\typeout{in spaces with affine connection"}
\typeout{by Bozhidar Zakhariev Iliev}
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%**     BEGINNING of LaTeX2e style package of BOZHIDAR ZAKHARIEV ILIEV    **%
%**	Filename: bozhomac.sty                                            **%
%**	Author (and responsible for all errors):                          **%
%***************************************************************************%
%**                                                                       **%
%**	  BOZHIDAR ZAKHARIEV ILIEV:					  **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
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%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
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%**                                                                       **%
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%***************************************************************************%
%**     BEGINNING of LaTeX2e logo package of BOZHIDAR ZAKHARIEV ILIEV     **%
%**	Filename: bozhlogo.sty                                            **%
%**	Author (and responsible for all errors): Bozho                    **%
%***************************************************************************%
%**                                                                       **%
%**       This file defines the commands \bozho and \BOZHO                **%
%**       which are the logos of BOZHIDAR ZAKHARIEV ILIEV:                **%
%**                                                                       **%
%**	  Laboratory of Mathematical Modeling in Physics		  **%
%**	  Institute for Nuclear Research and Nuclear Energy               **%
%**	  Bulgarian Academy of Sciences                                   **%
%**	  Boul. Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria          **%
%**	  E-mail address: bozho@inrne.bas.bg                              **%
%**       URL: http://theo.inrne.bas.bg/~bozho/		       		  **%
%**                                                                       **%
%**                                                                       **%
%**		    (DON'T USE WITHOUT PERMISSION)			  **%
%**                                                                       **%
%***************************************************************************%

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\bibitem{bp-DE+Sava}
Bozhidar~Z. Iliev and S.~Manov.
\newblock Deviation equations in spaces with affine connection.
\newblock JINR Communication P2-83-897, Dubna, 1983.
\newblock In Russian, 14~pp.\\ http://arXiv.org e-Print archive, E-print No.\
  math.DG/0512008, December 1, 2005. In English.

\bibitem{Heyde}
P.~von~der Heyde.
\newblock The equivalence principle in the ${U}_4$ theory of gravitation.
\newblock {\em Lettare al {N}uovo {C}imento}, 14(7):250--252, 1975.

\bibitem{MTW}
C.~W. Misner, K.~S. Thorne, and J.~A. Wheeler.
\newblock {\em Gravitation}.
\newblock W.~H. Freeman and Company, San Francisco, 1973.
\newblock Russian translation in 3 volumes: Mir, Moscow, 1977.

\bibitem{Mitskevich}
N.~V. Mitskevich.
\newblock {\em Physical fields in general theory of relativity}.
\newblock Nauka, Moscow, 1969.
\newblock In Russian.

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{\huge   Deviation equations and\\[1ex] weak equivalence principle\\[1ex]
	 in spaces with affine connection        }% main title
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\author{
Bozhidar Z.\ Iliev
\thanks{Laboratory of Mathematical Modeling in Physics,
Institute for Nuclear Research and \mbox{Nuclear} Energy,
Bulgarian Academy of Sciences,
Boul.\ Tzarigradsko chauss\'ee~72, 1784 Sofia, Bulgaria}
\thanks{E-mail address: bozho@inrne.bas.bg}
\thanks{URL: http://theo.inrne.bas.bg/$\sim$bozho/}
}


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\date{
 \vspace{2.27ex}\ShortTitle{Deviation equations and equivalence principle}	\\[0.27ex]
 \vspace{3.27ex}
\small
%
	\begin{tabular}{r@{$\colon\to~$}l}
%  \vspace{0.09ex} Basic ideas	& 1983	\\[0.09ex]
% \vspace{0.09ex} Began		&	\\[0.09ex]
% \vspace{0.09ex} Ended		&	\\[0.09ex]
%
% \vspace{0.09ex} Initial typeset& 1983 	\\[0.09ex]
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 \vspace{0.27ex} http://arXiv.org e-Print archive No. & gr-qc/0605008
								\\[0.27ex]
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% \vspace{0.27ex} Submitted to		&	\\[0.27ex]
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\vspace{0.27ex} Published in	       %	&
Proceedings  of the 6-th Soviet (USSR) Gra\-vity Conference\\
``Modern  theoretical and experimental problems of relativity theory\\
and gravitation'', Univ. "Friendship between nations"\\
Moscow, 3-5 July 1984,  Moscow, 1984, p.~227 (In Russian).	\\[0.27ex]
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 \vspace{0.27ex}\Subject{General relativity}	\\[2.27ex]
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 \vspace{0.27ex}\MSC[2001]{53B05, 83C99, 53B50}	%	\\[0.27ex]
&
 \vspace{0.27ex}\PACS[2003]{02.40.Sf, 04.90.+e} %	\\[0.27ex]
	\end{tabular} \\[1.27ex]
\vspace{0.27ex}\KeyWords{Deviation equations, Weak equivalence
			principle}	\\[0.27ex]
}


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	 Deviation equations and weak equivalence principle}}

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% Some connections between the deviation equations and weak equivalence
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\vspace*{3ex}
	\begin{center}
	\begin{minipage}{0.6\textwidth}
\textbf{Abstract.}
Some connections between the deviation equations and weak equivalence
principle are investigated.\\[5ex]
	\end{minipage}
	\end{center}

	For any point $x_0$ in a space with affine connection $L_n$, there
exists generally anholonomic frame $(i_0)$ such that the connection
coefficients $\Gamma_{i_0j_0}^{k_0}$ vanish at $x_0$ in it,
$\Gamma_{i_0j_0}^{k_0}(x_0)=0$, but, in the general case, their derivatives
do not vanish, $\Gamma_{i_0j_0,l_0}^{k_0}(x_0)\not=0$
(see~\cite{bp-DE+Sava,Heyde}).  This system/frame is  called locally inertial
or locally Lorentzian.

	The weak equivalence principle states
that~\cite{MTW,Mitskevich,Heyde} in the above pointed frame in
``sufficiently small'' neighborhood of the point $x_0$ the laws/equations of
motion take a form identical (up to infinitesimal quantities) with the one in
(locally-)flat spacetime. Note, this can be exactly valid at the point $x_0$
and possibly along some path through it.

	It can be asserted that the (generalized) deviation
equation~\cite{bp-DE+Sava} is the rigorous mathematical equivalent to the
weak equivalence principle. Moreover, that assertion is valid on arbitrary
spaces $L_n$ with affine connection.

	To illustrate the above statement, consider the deviation equation of
two free, ``infinitesimally close'', and independent
particles~\cite{bp-DE+Sava}:
	\begin{equation}	\label{1}
\frac{\bar{D}V^k}{\od\tau}
=
R_{ijl}^{k} u^iu^j\xi^l + u^j \frac{\bar{D} (T_{jl}^{k}\xi^l)}{\od\tau} ,
	\end{equation}
where $\tau$ is the parameter of the trajectory (worldline) of one of the
particles (basic particle or observer) with tangent vector $u^i$,
$\frac{\bar{D}}{\od \tau}$ is the covariant derivative with respect to
$\tau$, $R_{ijl}^{k}$ is the curvature tensor,
\(
T_{jl}^{k}
:=  - 2 \Gamma_{[il]}^{k} - C_{il}^{k}
\)
with $C_{il}^{k}$ defining the commutators of the basic vectors of the
frames and $[\ldots]$ denoting antisymmetrization with coefficient
$\frac{1}{2}$, $\xi^i$ is the (infinitesimal) displacement vector of the
particles, and $V^k:=\frac{\bar{D}\xi^k}{\od \tau}$ is their relative
velocity.

	In (locally) flat  $L_n$ space, the equation~\eref{1} takes the form
	\begin{equation}	\label{2}
\frac{\bar{D}V^k}{\od\tau}
= u^j \frac{\bar{D}(T_{jl}^{k}\xi^l)}{\od\tau}
= T_{jl}^{k}  u^j  V^l
+ u^j  \xi^l\frac{\bar{D} T_{jl}^{k}}{\od\tau}
	\end{equation}
at every spacetime point and in any frame $(i)$.
The equations~\eref{1} and~\eref{2} in a locally Lorentzian frame $(i_0)$ at
$x_0$ read respectively
	\begin{gather}	\label{3}
\frac{ \bar{D}V^{k_0} }{\od\tau} \Big|_{x=x_0}^{(1)}
=
\Bigl[ \Bigl(
- 2 \Gamma_{i_0 (j_0,l_0)}^{k_0}  u^{i_0}  \xi^{l_0}
- C_{j_0l_0}^{k_0} V^{l_0}
+ \xi^{l_0} \frac{\bar{D} T_{j_0l_0}^{k_0}}{\od\tau}
\Bigr) u^{j_0} \Bigr] \Big|_{x=x_0}
\\			\label{4}
\frac{ \bar{D}V^{k_0} }{\od\tau} \Big|_{x=x_0}^{(2)}
=
\Bigl[ \Bigl(
- C_{j_0l_0}^{k_0} V^{l_0}
+ \xi^{l_0} \frac{\bar{D} T_{j_0l_0}^{k_0}}{\od\tau}
\Bigr) u^{j_0} \Bigr] \Big|_{x=x_0}
	\end{gather}

	Let us introduce the quantities
	\begin{equation}
A_{k_0}(x_0)
:=
\frac{ \bar{D}V^{k_0} }{\od\tau} \Big|_{x=x_0}^{(1)}
-
\frac{ \bar{D}V^{k_0} }{\od\tau} \Big|_{x=x_0}^{(2)}
=
[(\cdots)u^{j_0} \xi^{l_0}]|_{x=x_0} ,
	\end{equation}
which are generally not components of a vector.

	The weak equivalence principle states in the particular case that in
``sufficiently small'' neighborhood of $x_0$ (which is assumed to be on the
basic path) the r.h.s.\ of~\eref{3} and~\eref{4} must coincide, \ie
$A^{k_0}(x_0)=0$. Excluding some particular cases (like $\xi^i=\const u^i$),
this means that the particles coincide at the point $x_0$, \ie
$\xi^l_0(x_0)=0$. This result agrees with the fact that the weak equivalence
principle is a local statement. Therefore the quantities $A^{k_0}(x_0)$ are a
measure for the validity of the weak equivalence principle in a neighborhood
of any spacetime point.

	It is clear in the particular case, the weak equivalence principle
is completely contained in the deviation equation. Similar consideration
reveal that the weak equivalence principle is a consequence of the
generalized deviation equation (cf.~\cite{bp-DE+Sava}).





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>=>=>=>=>=>
 	   PUT HERE ANY ADDITIONAL APPLICABLE INFORMATION

							 <=<=<=<=<=<=<=<=<=<

=============================================================================
=============================================================================
>>>		I do not want to profit from the Science,                 <<<
>>		but the Science may profit from me!                        <<
>						    Bozho                   <
=============================================================================
=============================================================================

\\
Title: Deviation equations and weak equivalence principle
	 in spaces with affine connection
Authors: Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear
	Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria)
Comments: 3 LaTeX pages.
	The packages AMS-LaTeX and amsfonts are required.
	This is an English translation from the Russian original.
	For related papers, visit the "publication" pages at
	http://theo.inrne.bas.bg/~bozho/
Report-No:
Journal-ref: Proceedings  of the 6-th Soviet (USSR) Gravity Conference
	"Modern theoretical and experimental problems of relativity theory
	and gravitation", University "Friendship between nations",\\
	Moscow 3-5 July 1984, Moscow, 1984, p.~227 (In Russian).
Subj-class:
MSC-class: MSC numbers:  53B05 (Primary) 83C99, 53B50 (Secondary)
PACS numbers: 02.40.Sf, 04.90.+e
\\
Some connections between the deviation equations and weak equivalence principle
are investigated.
\\


gr-qc0605008.extracted_bib
\begin{thebibliography}{1}

\bibitem{bp-DE+Sava}
Bozhidar~Z. Iliev and S.~Manov.
\newblock Deviation equations in spaces with affine connection.
\newblock JINR Communication P2-83-897, Dubna, 1983.
\newblock In Russian, 14~pp.\\ http://arXiv.org e-Print archive, E-print No.\
  math.DG/0512008, December 1, 2005. In English.

\bibitem{Heyde}
P.~von~der Heyde.
\newblock The equivalence principle in the ${U}_4$ theory of gravitation.
\newblock {\em Lettare al {N}uovo {C}imento}, 14(7):250--252, 1975.

\bibitem{MTW}
C.~W. Misner, K.~S. Thorne, and J.~A. Wheeler.
\newblock {\em Gravitation}.
\newblock W.~H. Freeman and Company, San Francisco, 1973.
\newblock Russian translation in 3 volumes: Mir, Moscow, 1977.

\bibitem{Mitskevich}
N.~V. Mitskevich.
\newblock {\em Physical fields in general theory of relativity}.
\newblock Nauka, Moscow, 1969.
\newblock In Russian.

\end{thebibliography}