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@@ -0,0 +1,674 @@
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+version or of any later version published by the Free Software
+Foundation.  If the Program does not specify a version number of the
+GNU General Public License, you may choose any version ever published
+by the Free Software Foundation.
+
+  If the Program specifies that a proxy can decide which future
+versions of the GNU General Public License can be used, that proxy's
+public statement of acceptance of a version permanently authorizes you
+to choose that version for the Program.
+
+  Later license versions may give you additional or different
+permissions.  However, no additional obligations are imposed on any
+author or copyright holder as a result of your choosing to follow a
+later version.
+
+  15. Disclaimer of Warranty.
+
+  THERE IS NO WARRANTY FOR THE PROGRAM, TO THE EXTENT PERMITTED BY
+APPLICABLE LAW.  EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT
+HOLDERS AND/OR OTHER PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY
+OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO,
+THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
+PURPOSE.  THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM
+IS WITH YOU.  SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF
+ALL NECESSARY SERVICING, REPAIR OR CORRECTION.
+
+  16. Limitation of Liability.
+
+  IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING
+WILL ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MODIFIES AND/OR CONVEYS
+THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING ANY
+GENERAL, SPECIAL, INCIDENTAL OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE
+USE OR INABILITY TO USE THE PROGRAM (INCLUDING BUT NOT LIMITED TO LOSS OF
+DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
+PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
+SUCH DAMAGES.
+
+  17. Interpretation of Sections 15 and 16.
+
+  If the disclaimer of warranty and limitation of liability provided
+above cannot be given local legal effect according to their terms,
+reviewing courts shall apply local law that most closely approximates
+an absolute waiver of all civil liability in connection with the
+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+                     END OF TERMS AND CONDITIONS
+
+            How to Apply These Terms to Your New Programs
+
+  If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+  To do so, attach the following notices to the program.  It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+    TP_latex
+    Copyright (C) 2019  Guyot
+
+    This program is free software: you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation, either version 3 of the License, or
+    (at your option) any later version.
+
+    This program is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with this program.  If not, see <http://www.gnu.org/licenses/>.
+
+Also add information on how to contact you by electronic and paper mail.
+
+  If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+    TP_latex  Copyright (C) 2019  Guyot
+    This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+    This is free software, and you are welcome to redistribute it
+    under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License.  Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+  You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+<http://www.gnu.org/licenses/>.
+
+  The GNU General Public License does not permit incorporating your program
+into proprietary programs.  If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library.  If this is what you want to do, use the GNU Lesser General
+Public License instead of this License.  But first, please read
+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
diff --git a/ModeleLaTeX_TP.gnuploterrors b/ModeleLaTeX_TP.gnuploterrors
new file mode 100644
index 0000000..a00b52d
--- /dev/null
+++ b/ModeleLaTeX_TP.gnuploterrors
@@ -0,0 +1,25 @@
+iter      chisq       delta/lim  lambda   a             b            
+   0 1.4700000000e+02   0.00e+00  1.40e+01    1.000000e+00   1.000000e+00
+   1 1.6935544886e+01  -7.68e+05  1.40e+00    7.830590e-01   9.780369e-01
+   2 1.3496165132e+01  -2.55e+04  1.40e-01    6.629727e-01   1.501074e+00
+   3 1.3285717162e+01  -1.58e+03  1.40e-02    6.251403e-01   1.695706e+00
+   4 1.3285714286e+01  -2.16e-02  1.40e-03    6.250000e-01   1.696429e+00
+iter      chisq       delta/lim  lambda   a             b            
+
+After 4 iterations the fit converged.
+final sum of squares of residuals : 13.2857
+rel. change during last iteration : -2.16481e-07
+
+degrees of freedom    (FIT_NDF)                        : 4
+rms of residuals      (FIT_STDFIT) = sqrt(WSSR/ndf)    : 1.82248
+variance of residuals (reduced chisquare) = WSSR/ndf   : 3.32143
+
+Final set of parameters            Asymptotic Standard Error
+=======================            ==========================
+a               = 0.625            +/- 0.1552       (24.84%)
+b               = 1.69643          +/- 0.7761       (45.75%)
+
+correlation matrix of the fit parameters:
+                a      b      
+a               1.000 
+b              -0.969  1.000 
diff --git a/ModeleLaTeX_TP.pdf b/ModeleLaTeX_TP.pdf
new file mode 100644
index 0000000..e120738
Binary files /dev/null and b/ModeleLaTeX_TP.pdf differ
diff --git a/ModeleLaTeX_TP.ps b/ModeleLaTeX_TP.ps
new file mode 100644
index 0000000..6615037
--- /dev/null
+++ b/ModeleLaTeX_TP.ps
@@ -0,0 +1,13388 @@
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+cleartomark
+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman7-Regular
+%!PS-AdobeFont-1.0: LMRoman7-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman7-Regular known{/LMRoman7-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman7-Regular)readonly def
+/FamilyName(LMRoman7)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -102 def
+/UnderlineThickness 34 def
+end readonly def
+/FontName /LMRoman7-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-483 -292 1562 1124}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman6-Regular
+%!PS-AdobeFont-1.0: LMRoman6-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman6-Regular known{/LMRoman6-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman6-Regular)readonly def
+/FamilyName(LMRoman6)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -88 def
+/UnderlineThickness 31 def
+end readonly def
+/FontName /LMRoman6-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-515 -298 1647 1125}readonly def
+currentdict end
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+cleartomark
+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman12-Bold
+%!PS-AdobeFont-1.0: LMRoman12-Bold 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman12-Bold known{/LMRoman12-Bold findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman12-Bold)readonly def
+/FamilyName(LMRoman12)readonly def
+/Weight(Bold)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -175 def
+/UnderlineThickness 66 def
+end readonly def
+/FontName /LMRoman12-Bold def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-476 -289 1577 1137}readonly def
+currentdict end
+currentfile eexec
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+%%CreationDate: 16th September 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 694 194 0
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+%%BeginFont: LMSans10-Bold
+%!PS-AdobeFont-1.0: LMSans10-Bold 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
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+17 dict begin
+/FontInfo 9 dict dup begin
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+{restore}if
+%%EndFont 
+%%BeginFont: LMRomanCaps10-Regular
+%!PS-AdobeFont-1.0: LMRomanCaps10-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRomanCaps10-Regular known{/LMRomanCaps10-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRomanCaps10-Regular)readonly def
+/FamilyName(LMRomanCaps10)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -146 def
+/UnderlineThickness 40 def
+end readonly def
+/FontName /LMRomanCaps10-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-496 -290 1501 1100}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMMathItalic8-Regular
+%!PS-AdobeFont-1.0: LMMathItalic8-Regular 1.200
+%%CreationDate: 16th September 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 556 156 0
+%%EndComments
+FontDirectory/LMMathItalic8-Regular known{/LMMathItalic8-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.003)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMMathItalic8-Regular)readonly def
+/FamilyName(LMMathItalic8)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle -14.0362 def
+/UnderlinePosition -117 def
+/UnderlineThickness 36 def
+end readonly def
+/FontName /LMMathItalic8-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-24 -250 1110 750}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMMathSymbols8-Regular
+%!PS-AdobeFont-1.0: LMMathSymbols8-Regular 1.200
+%%CreationDate: 16th September 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 556 156 0
+%%EndComments
+FontDirectory/LMMathSymbols8-Regular known{/LMMathSymbols8-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.003)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMMathSymbols8-Regular)readonly def
+/FamilyName(LMMathSymbols8)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle -14.0362 def
+/UnderlinePosition -117 def
+/UnderlineThickness 36 def
+end readonly def
+/FontName /LMMathSymbols8-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-30 -955 1185 779}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman8-Regular
+%!PS-AdobeFont-1.0: LMRoman8-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman8-Regular known{/LMRoman8-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman8-Regular)readonly def
+/FamilyName(LMRoman8)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -117 def
+/UnderlineThickness 36 def
+end readonly def
+/FontName /LMRoman8-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-456 -292 1497 1125}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMMathItalic12-Regular
+%!PS-AdobeFont-1.0: LMMathItalic12-Regular 1.200
+%%CreationDate: 16th September 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 833 233 0
+%%EndComments
+FontDirectory/LMMathItalic12-Regular known{/LMMathItalic12-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.003)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMMathItalic12-Regular)readonly def
+/FamilyName(LMMathItalic12)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle -14.0362 def
+/UnderlinePosition -175 def
+/UnderlineThickness 44 def
+end readonly def
+/FontName /LMMathItalic12-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-31 -250 1026 750}readonly def
+currentdict end
+currentfile eexec
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+%%EndFont 
+%%BeginFont: LMMathSymbols10-Regular
+%!PS-AdobeFont-1.0: LMMathSymbols10-Regular 1.200
+%%CreationDate: 16th September 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 694 194 0
+%%EndComments
+FontDirectory/LMMathSymbols10-Regular known{/LMMathSymbols10-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.003)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMMathSymbols10-Regular)readonly def
+/FamilyName(LMMathSymbols10)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
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+/UnderlinePosition -146 def
+/UnderlineThickness 40 def
+end readonly def
+/FontName /LMMathSymbols10-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
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+readonly def
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+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-29 -960 1116 775}readonly def
+currentdict end
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+cleartomark
+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman12-Regular
+%!PS-AdobeFont-1.0: LMRoman12-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman12-Regular known{/LMRoman12-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman12-Regular)readonly def
+/FamilyName(LMRoman12)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -175 def
+/UnderlineThickness 44 def
+end readonly def
+/FontName /LMRoman12-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-422 -280 1394 1127}readonly def
+currentdict end
+currentfile eexec
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+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman12-Italic
+%!PS-AdobeFont-1.0: LMRoman12-Italic 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 806 194 0
+%%EndComments
+FontDirectory/LMRoman12-Italic known{/LMRoman12-Italic findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman12-Italic)readonly def
+/FamilyName(LMRoman12)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle -14.0362 def
+/UnderlinePosition -175 def
+/UnderlineThickness 44 def
+end readonly def
+/FontName /LMRoman12-Italic def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-449 -289 1358 1125}readonly def
+currentdict end
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+cleartomark
+{restore}if
+%%EndFont 
+%%BeginFont: LMRoman17-Regular
+%!PS-AdobeFont-1.0: LMRoman17-Regular 2.004
+%%CreationDate: 7th October 2009
+% Generated by MetaType1 (a MetaPost-based engine)
+% Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).
+% Supported by CSTUG, DANTE eV, GUST, GUTenberg, NTG, and TUG.
+% METATYPE1/Type 1 version by B. Jackowski & J. M. Nowacki
+% from GUST (http://www.gust.org.pl).
+% This work is released under the GUST Font License.
+% For the most recent version of this license see
+% This work has the LPPL maintenance status `maintained'.
+% The Current Maintainer of this work is Bogus\l{}aw Jackowski and Janusz M. Nowacki.
+% This work consists of the files listed in the MANIFEST-Latin-Modern.txt file.
+% ADL: 805 195 0
+%%EndComments
+FontDirectory/LMRoman17-Regular known{/LMRoman17-Regular findfont dup/UniqueID known{dup
+/UniqueID get 0 eq exch/FontType get 1 eq and}{pop false}ifelse
+{save true}{false}ifelse}{false}ifelse
+17 dict begin
+/FontInfo 9 dict dup begin
+/version(2.004)readonly def
+/Notice(Copyright 2003--2009 by B. Jackowski and J.M. Nowacki (on behalf of TeX USERS GROUPS).)readonly def
+/FullName(LMRoman17-Regular)readonly def
+/FamilyName(LMRoman17)readonly def
+/Weight(Normal)readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition -252 def
+/UnderlineThickness 60 def
+end readonly def
+/FontName /LMRoman17-Regular def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 0 /.notdef put
+readonly def
+/PaintType 0 def
+/FontType 1 def
+/StrokeWidth 0 def
+/FontMatrix[0.001 0 0 0.001 0 0]readonly def
+%/UniqueID 0 def
+/FontBBox{-400 -286 1338 1125}readonly def
+currentdict end
+currentfile eexec
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+%\documentclass[11pt,a4paper]{article}
+\documentclass[12pt,a4paper,twoside]{scrartcl} % scrartcl est la classe d'articles de Koma-script
+% qui gère nativement le format A4 ; utilisez article pour un format de papier américain
+\usepackage[utf8]{inputenc} % gestion d'utf8
+\usepackage[francais]{babel} % gestion de la langue française
+\usepackage[T1]{fontenc} % 
+\usepackage{amsmath}	% symboles mathématiques
+\usepackage{amsfonts} % fontes pour les symboles mathématiques
+\usepackage{amssymb} % jeu de symboles suppélémentaires
+\usepackage{graphicx} % pour mettre des images
+\usepackage{caption} % pour des légendes
+\usepackage{lmodern}	% fonte spéciale
+\usepackage{subcaption} % pour les subfigures et les subtables
+\usepackage{float} % pour placer exactement les figure (utilisation de H)
+\usepackage{url} % pour une bonne gestion des urls
+\usepackage{siunitx} % pour la gestion des unités
+\usepackage{latexsym}
+\usepackage{keyval}
+\usepackage{ifthen} % pour permettre des tests (avec gnuplot)
+\usepackage{moreverb}
+\usepackage[siunitx]{gnuplottex} % pour permettre l'utilisation de gnuplot avec siunitx
+\graphicspath{{./images/}} % nécessaire pour l'import des fichiers eps_tex dans un sous répertoire
+\usepackage{endnotes}
+\usepackage{multirow}
+\usepackage{tabularx}
+
+\author{Vincent Guyot}
+\title{Relativités}
+
+\begin{document}
+\renewcommand{\tablename}{Tableau}
+\maketitle
+\tableofcontents
+\newpage
+
+\begin{quotation}
+\og C’est le développement de l'aptitude générale à penser, juger et travailler de façon autonome qui doit toujours rester au premier plan des préoccupations, et non l'acquisition de connaissances spécialisées \fg{} A. Einstein\endnote{Einstein A. dans \cite[p. 209]{T}.}
+\end{quotation}
+
+\section{Introduction}
+Le but de cet exposé est de présenter l'évolution de deux idées fondamentales de l'histoire de la physique~: celle de relativité et celle d'expace.
+
+Pour cela, nous reviendrons à Galilée pour la relativité et à Newton pour l’espace.
+Puis, nous verrons comment, et à partir de quoi, la relativité restreinte d'Einstein s'est développée. Alors, nous pourrons comprendre la relativité générale et le changement radical qu'elle a provoqué dans notre conception de l’espace.
+
+Cela nous permettra d'aborder enfin, en guise de conclusion, l’une des conséquences
+les plus spectaculaires de cette dernière : le modèle standard de l'Univers très primitif, plus communément appelé “modèle du Big Bang”.
+
+\section{Préliminaires}
+Comme on le sait, pour Galilée, le mouvement n’a de sens que par rapport à un autre
+corps qui en est privé~:
+\begin{quotation}
+\og Quand il [Aristote] écrit que tout ce qui se meut, se meut sur quelque chose d’immobile, je [Salviati] me demande s’il n’a pas voulu dire que tout ce qui se meut se meut respectivement à quelque chose d'immobile, cette dernière proposition ne soulevant aucune difficulté, alors que la première en soulève beaucoup \dots \fg\endnote{Passage de \cite{A}, reproduit dans \cite[pp. 11 et 18]{B} (le mot \og respectivement \fg{} est mis en évidence par moi).}
+\end{quotation}
+
+Pour Einstein, cela remonte aux Grecs :
+\begin{quotation}
+\og Depuis le temps des Grecs on saït bien que pour décrire le mouvement d’un corps on doit le rapporter à un autre corps. Le mouvement d’un véhicule est décrit par rapport au sol, celui d’une planète par rapport à l'ensemble des étoiles fixes visibles. En physique, les corps auxquels les mouvements sont rapportés dans l'espace sont appelés systèmes de coordonnées. Les lois de la mécanique de Galilée et de Newton ne
+peuvent être formulées qu'en employant un système de coordonnées. \fg\endnote{Passage de \cite[p. 13]{C} (lui-même de \og Qu'est-ce que la théorie de la relativité ?\fg, publié dans le London Time le 29 nov. 1919).}
+\end{quotation}
+
+De nos jours, on appelle le “corps” auquel on rapporte le mouvement~: “référentiel”, et les axes gradués qui nous permettent de repérer les positions : “système de coordonnées”.
+
+Pour bien comprendre ce que signifie l’idée de relativité, nous devons d’abord préciser ce que l’on entend par homogénéité, isotropie et “invariance des lois” en tout point de l'espace.
+
+Il existe plusieurs types d’invariance des lois~:
+\begin{description}
+\item [L'invariance par translation dans le temps] est chose connue. Il s’agit du fait que les lois de la physique persistent dans le temps, passé ou futur, que la marche des phénomènes physiques, à des instants différents de leur observation (dans les
+mêmes conditions), est la même. On parle de \og l’homogénéité du temps\fg.
+\item [L'invariance par translation dans l’espace] est la plus notoire. Il s’agit du fait que les lois de la physique sont les mêmes, pour deux référentiels issus l’un de l’autre par une translation (à Paris et à Marseille, par exemple). On parle alors de
+\og homogénéité de l'espace\fg.
+\item [L'isotropie de l'espace] est \og le dernier exemple simple, et qui ne concerne que l’espace, parce qu'il a plusieurs dimensions, c’est l'invariance par changement d'orientation ; autrement dit : il n'y a pas, dans l’espace, de direction absolue. \fg\endnote{\cite[p. 59]{D}.}
+\end{description}
+
+Homogénéité et isotropie sont ce qu'on appelle des \emph{symétrie} de l’espace (Relevons qu'il en existe d’autres (permutation de particules identiques, miroir, \dots) et qu’à chacune d'entre elles correspond une loi de conservation. Par exemple, à l’homogénéité du tomps correspond la loi de conservation de l'énergie, à celle de l’espace, la loi de conservation de l'impulsion et à l’isotropie, la loi de conservation du moment cinétique\endnote{Voir \cite[pp. 107 et 222]{E}. Pour les personnes fortement intéressées par la physique, la physique quantique et ses propriétés de symétries, je conseille l'ouvrage cité ci-dessus.}).
+
+Mais remarquons bien qu'il s’agit de symétries par rapport aux lois de la physique.
+Qu'est-ce que cela veut dire ? Prenons l'exemple du choc élastique entre deux objets. Ce processus obéit à la loi de conservation de l'énergie qui s'exprime dans un référentiel R par~:
+
+\[\frac{1}{2}\cdot M_1\cdot v_1^2 + \frac{1}{2}\cdot M_2\cdot v_2^2 = \frac{1}{2}\cdot M_1\cdot w_1^2 + \frac{1}{2}\cdot M_2\cdot w_2^2\]
+où v désigne la vitesse avant le choc et w celle après, et où les indices représentent les particules.
+
+Dans un référentiel R’, en translation uniforme V par rapport à R, l'expression numérique des vitesses est différente de celle dans R puisqu'on l’exprime par :
+
+\[{v'}_{1,2}=v_{1,2}-V\;\;\text{et}\;\;{w'}_{1,2}=w_{1,2}-V\]
+
+mais, le principe de conservation conserve sa forme particulière :
+
+\[\frac{1}{2}\cdot M_1\cdot {v'}_1^2 + \frac{1}{2}\cdot M_2\cdot {v'}_2^2 = \frac{1}{2}\cdot M_1\cdot {w'}_1^2 + \frac{1}{2}\cdot M_2\cdot {w'}_2^2\]
+L'invariance n’est donc pas numérique, mais formelle (on parle aussi de covariance). C'est la forme de l'équation de conservation de l'énergie qui est préservée par les symétries (en tant que transformations) dont nous venons de parler (voir annexe II).
+
+Mais les lois de la physique ne sont pas invariante sous toute transformation. Un
+exemple cher à Galilée\endnote{\cite[pp. 113-114]{J}.}, puisque c'est lui qui l’a découvert, est celui de la transformation d'échelle. Les lois qui régissent le vol dans l’air d’une maquette d’Airbus, par exemple, sont différente de celles qui régissent le vol (dans l’air) d’un Aïrbus grandeur nature. Pour étudier le vol d'un avion, les ingénieur le savent bien, il faut avoir recours à des grandeurs sans dimension qui permettent “d'immuniser” l'analyse du fait de la non-invariance des lois par changement d'échelle.
+
+Ainsi donc, l’invariance des lois en tout point de l’espace doit être comprise comme
+une invariance dans la forme des lois sous l’action de transformations qui opèrent un changement de référentiel\endnote{\cite[p. 111]{J} : \og C'est Poincaré qui eut l'idée d'analyser ce qu'on peut faire dans une équation sans la modifier, c'est lui qui le premier attira l'attention sur les symétries des lois de la physique\fg.}.
+
+\section{Le principe de relativité de Galilée}
+Le texte suivant, très connu et déjà maintes fois cité~:
+\begin{quotation}
+\og Enfermez-vous avec un ami dans la cabine principale à intérieur d'un grand bateau et prenez avec vous des mouches, des papillons et d’autres petits animaux volants. Prenez une grande cuve d’eau avec un poisson dedans, suspendez une bouteille qui se vide goutte à goutte dans un grand récipient en dessous d'elle. Avec le bateau à l'arrêt, observez soigneusement comment les petits animaux volent à des vitesses égales vers tous les côtés de la cabine. Le poisson nage indifféremment dans toutes les directions, les gouttes tombent dans le récipient en dessous, et si vous lancez quelque chose à votre ami, vous n'avez pas besoin de le lancer plus fort dans une direction que dans une autre, les distances étant égales, et si vous sautez à pieds joints, vous franchissez des distances égales dans toutes les directions. Lorsque
+vous aurez observé toutes ces choses soigneusement (bien qu'il n'y ait aucun doute que lorsque le bateau est à l'arrêt, les choses doivent se passer ainsi ), faites avancer le bateau à l'allure qui vous plaira, pour autant que Ja vitesse soit uniforme [c'est-à-dire constante] et ne fluctue pas de part et d'autre. Vous ne verrez pas le moindre changement dans aucun des effets mentionnés et même aucun d'eux ne vous permettra de dire si le bateau est en mouvement ou à l'arrêt \dots\fg\endnote{\cite{A}, extrait de la 2ème journée, traduit par M. Gruber, professeur à l'Ecole polytechnique de Lausanne dans \cite[p. 10.5]{I}.}
+\end{quotation}
+
+nous mène maintenant à considérer le \emph{principe de relativité de Galilée}.
+
+Pour ce faire, suivons concrètement l'exemple ci-dessus. Il s'agit d'un bateau en
+translation à vitesse constante par rapport à la terre. Galilée nous dit que le mouvement de ce bateau est \og comme nul \fg{}\endnote{\cite[p. 10]{T}.}? pour les mouvements des divers corps qu'il envisage, c'est-à-dire qu'il n'y a pas de différence entre la description de ces mouvements faite sur la terre et celle faite sur le navire.
+
+Pour Galilée, il existe donc des référentiels dont le mouvement est « comme nul »,
+c'est-à-dire que les lois du mouvement sont dans ceux-ci inaffectées par leur mouvement propre (le mouvement du référentiel lui-même). Ces référentiels sont les référentiels se déplaçant l’un par rapport à l'autre avec une vitesse de translation constante.
+
+Ainsi Newton ct Einstein énonceront successivement le principe de relativité de Galilée de la façon suivante :
+\begin{quotation}
+\og Les mouvements relatifs des corps enfermés dans un espace quelconque sont les mêmes que cet espace soit immobile, ou qu'il se mouve le long d'une ligne droite, sans rotation. \fg\endnote{Cité sans référence par F. Balibar, dans \cite[p. 26]{B}. La traduction donnée par Balibar doit être celle de Mme du Chastellet. Elle est imprécise car l'uniformité du mouvement n'y est pas explicitement formulée. Pour comparaison, voici la traduction de Mme M.-F. Biarnais dans \cite[p.50]{U}} pour Newton et,
+
+\og Etant donné deux référontiels en translation uniforme l’un par rapport à l’autre, les lois auxquelles sont soumis les changements d'états des systèmes physiques restent les mêmes, quel que soit le référentiel auquel ces changements sont rapportés.\fg\endnote{Ibid, p.27, du célèbre article d'Einstein \cite{F} de 1905.} ou
+
+\og Si K' est relntivoment à K un système de coordonnée qui effectue un mouvement uniforme sans rotation, les phénomènes de la nature se déroulent, relativement à K', conformément aux mêmes lois générales que relativement à K\fg\endnote{\cite[p. 21]{G}.} pour Einstein.
+\end{quotation}
+
+Pour bien comprendre toute l’extension de ce principe et notamment son appellation
+actuelle de \og principe de relativité restreinte\fg, il faut en lier l’expression donnée par Galilée à une formulation mathématique précise~: la transformation de Galilée\endnote{Elle n'est pas due à Galilée lui-même. Son appellation vient du fait qu'elle est la formulation mathématique de l'idée de relativité de Galilée.}.
+
+\subsection{Transformation de Galilée}
+Soient les deux référentiels S et S’, en translation uniforme l’un par rapport à l’autre, suivants :
+\begin{figure}
+\centering
+\def\svgwidth{10cm}
+\input{images/eps/referentiels.eps_tex}
+\caption{Référentiels}
+\end{figure}
+
+On a l’équation suivante~:
+\[\overrightarrow{r}(x',y',z',t')=\overrightarrow{r}(x-vt,y,z,t)\]
+
+On peut ainsi exprimer la transformation de Galilée pour passer de S à S' par~:
+\begin{align}\label{eq:transgalilee}
+x'&=x-vt\nonumber\\
+y'&=y\nonumber\\
+z'&=z\nonumber\\
+t'&=t
+\end{align}
+
+ou sous forme matricielle~:
+\begin{equation*}
+\begin{pmatrix}
+x'\\
+y'\\
+z'\\
+t'
+\end{pmatrix}=
+\begin{pmatrix}
+1&0&0&-v\\
+0&1&0&0\\
+0&0&1&0\\
+0&0&0&1
+\end{pmatrix}\cdot
+\begin{pmatrix}
+x\\
+y\\
+z\\
+t
+\end{pmatrix}
+\end{equation*}
+
+Le contenu de la matrice permettant de passer des anciennes coordonnées (non primées) aux nouvelles (primées) est caractéristique d'une transformation de Galilée en ce que, si le temps intervient dans la transformation des coordonnées d'espace (\(x' = x - v\cdot t\)), ces dernières n'interviennent pas dans la transformation du temps.
+\begin{itemize}
+\item Remarquons que la transformation présentée ci-dessus est une transformation de
+Galilée particulière. Celle-ci se fait en effet le long de l'axe x du référentiel S. En fait, on devrait parler des transformations de Galilée ou donner l'expression la plus générale les représentant.
+\item Remarquons aussi (voir annexe II) que l'équation \(x'=x - v_{ref}\cdot{t}\) implique ce que l’on appelle le fhéorème d’addition des vitesses~:
+\begin{equation}\label{eq:transvitgalilee}
+v =v'+ v_{ref}
+\end{equation}
+\item Contrairement à ce que dit son nom, qui est un hommage à Galilée, on doit cette transformation à Euler.
+\end{itemize}
+
+Toute la signification du principe de relativité de Galilée tient dans l’invariance des lois du mouvement par une transformation de Galilée.
+
+Il s’agit de relativité, puisqu'il existe une infinité de référentiels équivalents (en translation uniforme les uns par rapport aux autres) pour la description des lois du mouvement, c’est-à-dire par rapport auxquels les lois du mouvement sont formellement identiques.
+
+Mais il faut bien comprendre à quel poïint cette idée de relativité est générale. En effet, soit un référentiel R à partir duquel on détermine la classe A des référentiels en translation uniforme par rapport à lui. Prenons alors un référentiel R’ qui ne soit pas en translation uniforme par rapport à R et appelons B la classe des référentiels qui sont par rapport à lui en translation uniforme. Le principe de relativité affirme alors~:
+\begin{itemize}
+\item que les lois sont (formellement) identiques dans tous les référentiels de la classe A et
+\item que les lois sont (formellement) identiques dans tous les référentiels de la classe B.
+\end{itemize}
+
+Il ne dit rien d’une quelconque identité des lois entre les référentiels des deux classes. Nous savons que les lois dans À sont différentes de celles dans B (voir annexe O). Ainsi donc, les lois ne sont pas univoquement déterminées et certaines d’entre elles sont même sans doute plus simples que d’autres.
+
+C’est en ce sens que la relativité de Galilée est restreinte. En effet, les référentiels envisagés ne sont pas quelconques. Ce sont les référentiels dans lesquels les lois de la nature sont les plus simples. Il faut entendre par là ce que Galilée voulait dire par \og mouvements « comme nuls »\fg. On peut comprendre ces mots de deux façons différentes~:
+\begin{itemize}
+\item le mouvement de translation uniforme d’un référentiel par rapport à un autre est
+comme nul pour la description, sous forme de lois, du mouvement de l’objet \og test \fg{} considéré~: les lois sont formellement identiques quand elles sont exprimées dans deux référentiels en translation uniforme l’un par rapport à l’autre.
+\item le mouvement du référentiel lui-même n’a pas d'influence sur celui de l'objet
+\og test\fg, en ce sens qu’il n’exerce aucune action sur lui.
+\end{itemize}
+
+Comme nous l’avons vu, le premier sens définit l’idée de relativité. Le second, quant à lui, restreint cette idée à une classe particulière de référentiels : les référentiels inertiels. Un référentiel sera dit inertiel si, dans celui-ci, la loi d'inertie (voir ci-dessous la première loi de Newton) est valable.
+
+Le mouvement d’un objet n’est en effet indépendant du référentiel dans lequel il est décrit, que si ce référentiel est inertiel, car, dans ce cas, un objet isolé (de tous les autres objets qui l’environnent) ne subit par définition absolument aucune action (aucune force), et en particulier de la part du référentiel lui-même.
+
+Cette relativité restreinte est donc celle du sens commun. En effet, nous ne nous imaginons habituellement l'identité des lois qu'entre référentiels inertiels. Les lois du “ping pong” sont les mêmes à terre ou dans un train se déplaçant en ligne droite sans accélération. Qu'elles soient les mêmes entre un train accéléré et un référentiel en translation uniforme par rapport à celui-ci, nous ne le concevons même pas.
+
+Remarquons enfin une propriété de la transformation de Galilée nécessitée par la relativité. Supposons donné un référentiel inertiel. La seconde loi de Newton (voir ci-dessous) et l'invariance des lois pour les référentiels inertiels impliquent le fait que s'il n'existe aucune force dans ce référentiel, il en sera de même dans tous les autres (voir annexe III). Et par conséquent, tous les autres référentiel, issus par transformation de Galilée de ce référentiel inertiel, seront inertiel : le principe d'inertie est préservé par toute transformation de Galilée.
+
+La relativité restreinte de Galilée suppose donc l’équivalence formelle des lois du mouvement entre référentiels inertiels. Encore faut-il être très précis. On peut concevoir, comme on l’a vu, des référentiels inertiels ou non. La relativité de Galilée les concevra inertiels, mais pas la relativité générale d'Einstein. Mais, comme on le verra par la suite, on peut aussi concevoir des référentiels obéissant ou pas à une transformation de Galilée. La relativité restreinte de Galilée obéira aux transformations de Galilée, mais pas la relativité restreinte d’Einstein. On peut résumer cela comme suit :
+
+\begin{table}
+\centering
+\begin{tabular}{cc|c|c|c}
+  & \multicolumn{3}{c}{Référentiels} & \\
+  & \multirow{5}{*}{\rotatebox[origin=c]{90}{de Galilée}} & Inertiels & \multicolumn{2}{c}{Non-inertiels} \\ \cline{3-4}
+  \multirow{8}{*}{\rotatebox[origin=c]{90}{Transformation}} 	& & & & \\
+    & & Mécanique classique & Mécanique des systèmes accélérés & \\
+    & & Relativité restreinte de Galilée & & \\
+    & & & & \\ \cline{2-5}
+    & \multirow{5}{*}{\rotatebox[origin=c]{90}{de Lorentz}} & & & \\
+    & & Relativité restreinte d'Einstein & Relativité générale d'Einstein \\
+    & & Électromagnétisme & & \\
+    & & & & \\ \cline{3-4}
+    & & & \multicolumn{2}{c}{}
+\end{tabular}
+\caption{Tableau synoptique}
+\end{table}
+
+
+où la transformation de Lorentz est une transformation que nous retrouverons par la suite.
+
+Remarquons en substance que :
+\begin{itemize}
+\item le problème de savoir si Galilée a oui ou non pensé le principe d'inertie, s’il est intéressant du point de vue de l'émergence des idées\endnote{Voir \cite{H}.}, n’est en fait pas essentiel ici, où c’est la connaissance du principe de relativité de Galilée tel qu’il est formulé actuellement, qui va nous permettre de bien saïsir l’extension qu’en fera Einstein dans la relativité générale (voir annexe I).
+\item l'énoncé d’Einstein fait référence aux lois de la physique en général. En fait, à l'époque de Galilée, c'est des lois de la mécanique qu'il s'agissait. Nous verrons à quel point l’extension faite par Einstein du principe de relativité aux lois de la physique en général, a été cruciale pour la relativité restreinte.
+\item Enfin, qu’en raison de la présence du principe d'inertie dans le principe de relativité de Galilée, celui-ci est parfois dit \og de Galilée - Newton\fg.
+\end{itemize}
+
+On peut maintenant bien comprendre la définition actuelle du principe de relativité de
+Galilée qui postule l'existence de référentiels d'inertie et établit la relativité :
+
+\og
+\begin{itemize}
+\item Il existe des référentiels particuliers, appelés “référentiels d'inertie”, par rapport auxquels l'espace est homogène et isotrope et le temps est homogène ; en particulier tout corps \og isolé\fg a un mouvement rectiligne uniforme par rapport à un référentiel d'inertie.
+\item Les forces et les lois fondamentales de la mécanique sont les mêmes pour l’ensemble des observateurs en translation uniforme les uns par rapport aux autres (insistons sur le fait qu’en conséquence de la définition des transformations de Galilée, pour de tels observateurs, le principe d'inertie est valable) : les
+transformations de Galilée sont des symétries de la mécanique.
+\end{itemize}
+\fg\endnote{Voir \cite{I}.}
+
+Résumons nous pour conclure. Nous avons vu que la notion même de relativité implique la notion de transformations opérant des changements de référentiel. Le principe de
+relativité de Galilée nous dit qu'il existe une certaine classe de transformations entre référentiels se déplaçant en translation uniforme les uns par rapport aux autres (les transformations de Galilée), opérant des changements de référentiels d'inertie, par rapport auxquelles les lois de la mécanique sont formellement invariantes (voir annexe II). Il s'agit d'une autre symétrie que cclles déjà présentées (symétrie par rotation et par translation).
+Mais cela ne présuppose pas que celles de la physique en général, celles de l'électrodynamique (inconnue de Galilée) par exemple, le soient aussi. En fait, nous verrons que c'est Einstein qui va amener une telle généralisation.
+
+\section{L'espace absolu de Newton}
+Avant de passer à l'analyse proprement dite du concept d'espace chez Newton, il faut faire quelques rappels.
+
+\subsubsection{Rappels historiques}
+Newton (1642-1727) élabore un \og calcul des fluxions \fg, fondement du calcul différentiel et intégral (les “fluxions” de Newton ne sont rien d'autre que les \og dérivées \fg{} de Leïbnitz, élaborées à la même époque).
+
+Il écrit son ouvrage fondement de la mécanique classique : “Principes mathématiques de la philosophie naturelle” en 1687 où il énonce ses fameuses trois lois :
+\begin{enumerate}
+\item \textbf{Lex Prima (énoncé de Newton) ou \og loi d'inertie \fg}~:
+\og Tout corps persévère dans l'état de repos ou de mouvement uniforme en ligne droïte à moins que
+quelque force n’agisse sur lui et ne le contraigne à changer d'état. \fg
+
+\textbf{Première loi (énoncé actuel)}~:
+\og La quantité de mouvement d’un point matériel reste constante au cours de l’évolution si et seulement si la résultante des forces qui agissent sur lui est égale à zéro :
+\begin{align*}
+&\overrightarrow{p}(t)=\overrightarrow{p}(t_o)\;\Leftrightarrow\;\sum_\alpha \overrightarrow{F}_\alpha(t)=0\;\;\forall\;t\\
+\text{avec}&\overrightarrow{p}(t)=m\cdot \overrightarrow{v}(t)\;\;\text{ ; }\overrightarrow{p}\text{ : quantité de mouvement}\;\overrightarrow{v}\text{ : vitesse}
+\end{align*}\fg
+
+\item \textbf{Lex Secunda (énoncé de Newton)}~:
+\og Les changements qui arrivent dans le mouvement sont proportionnels à la force motrice (FAt) et se
+font dans la ligne droite dans laquelle cette force a été imprimée. \fg
+
+\textbf{Seconde loi (énoncé actuel)}~:
+\og A chaque instant, la variation par unité de temps de la quantité de mouvement d’un
+point matériel est égale à la résultante des forces qui agissent sur lui :
+\[\frac{d}{dt}\overrightarrow{p}(t)=\overrightarrow{F}(t)\;\;\text{où}\;\;\overrightarrow{F}(t)=\sum_\alpha \overrightarrow{F}_\alpha (t)\]\fg
+
+\item \textbf{Lex Tertia (énoncé de Newton)}~:
+\og L'action est toujours égale et opposée à la réaction, c’est-à-dire que les actions de deux corps l’un sur
+l’autre sont toujours égales et de direction opposées. \fg
+
+\textbf{Troisième loi (énoncé actuel)}~:
+\og A tout instant et quelque soit le mouvement du système, le torseur (on appelle torseur
+un ensemble fini ou infini de vecteurs liés ou glissant, de même dimension, noté
+\[\tau=\{(P_\alpha,\overrightarrow{\xi}_\alpha)\}\]
+des forces intérieures est équivalent à zéro :
+\[\sum_\alpha \overrightarrow{F}_\alpha^{int} = 0\;\;\text{et}\;\;\sum_\alpha \overrightarrow{OP}_\alpha\wedge\overrightarrow{F}_\alpha^{int.} = 0\]\fg
+\end{enumerate}
+
+Remarque~: Dans le cas particulier où le système est formé de deux points matériels, nous retrouvons l'énoncé de Newton qui est plus habituellement nommé \og principe de l’action et de la réaction \fg.
+
+De plus il établit la loi de la gravitation universelle~:
+\[\overrightarrow{F}_{A\rightarrow B}=-G\cdot\frac{M_A\cdot M_B}{|\overrightarrow{AB}|^3}\cdot \overrightarrow{AB}\]
+où \(G\) est la constante de la gravitation, \(M_A\) et\(M_B\) sont les masses gravifiques des corps A ct B et \(\overrightarrow{AB}\) est le vecteur liant A à B.
+
+Remarquons que cette loi n’est pas déduite des trois première lois et qu’elle figure ainsi dans la théorie de Newton au rang d’axiome.
+
+\bigskip
+Avant de commencer, répétons-nous encore une fois, tant cela est important.
+
+\medskip
+Nous avons précédemment dit que que le principe de relativité de Galilée discriminait,
+en quelque sorte, les référentiels pour lesquels les lois du mouvement sont identiques. Il nous disait aussi que ce n’est que par rapport à des référentiel d’inertie, que nous devons considérer les lois du mouvement. La loi d'inertie divise donc l’ensemble des référentiels en deux sous-ensembles : les référentiels inertiels \og autorisés\fg{} et ceux qui ne le sont pas. L'exemple suivant met bien en évidence ce problème de l’exigence de la validité de la loi d'inertie~:
+\begin{quotation}
+Comme Newton, supposons qu’un corps loin de toute matière soit un système isolé. II n’y a alors aucune action extérieure possible sur ce corps. En particulier, aucune force ne s'exerce sur lui. Si maintenant, par rapport à un référentiel R, son mouvement est rectiligne, il ne peut l'être aussi par rapport à un référentiel R’, en rotation par rapport à R. Et ce, malgré le fait qu'aucune force ne s'exerce sur lui.
+\end{quotation}
+
+Pour Newton, le mouvement d’un corps doit être rectiligne s’il ne subit aucune force,
+car, pour lui, la loi d’inertie est une des lois générales de la nature. Elle doit donc être toujours valable. Le problème de fond est ainsi celui du statut de la première loi. Il existe des référentiels dans lesquels le mouvement d’un corps, sur lequel aucune force n'est exercée, n’est pas une ligne droite. Ces référentiels ne doivent donc pas être utilisés pour décrire correctement le mouvement de ce corps. Aïnsi, pour faire de la première loi une loi universelle, il faut postuler l’existence d’un espace absolu dans lequel elle est valable et ne considérer les lois du mouvement que relativement à un référentiel immobile par rapport à celui-ci. Ainsi Newton écrira~:
+\begin{quotation}
+\og Quant à ceux [les termes] de temps, d'espace, de lieu et de mouvement, ils sont connus de tout lo monde ; mais il faut remarquer que pour n'avoir considéré ces quantités que par leurs relations à des choses sensibles, on est tombé dans plusieurs erreurs. Pour les éviter, il faut distinguer le temps, l'espace, le lieu et le mouvement en absolus et relatifs, vrais et apparents, mathématiques et vulgaires \dots
+
+L'espace absolu, sans relation aux choses externes, demeure toujours similaire et immobile \dots
+
+L'espace relatif est cette partie ou dimension mobile de l'espace, laquelle tombe sous noa éens par la relation aux corps, et que le vulgaire confond avec l’espace immobile. \fg\endnote{\cite[p. 40]{U}.}
+\end{quotation}
+
+Ce n'est que dans l’espace absolu qu'il y a relation entre force et accélération (validité de la seconde loi), ce n'est que par rapport à celui-ci que la force (celle qui est due à l'interaction avec d'autres objets matériels et que Newton nomme force imprimée) est cause de l'accélération~:
+\begin{quotation}
+\og Les causes par lesquelles on peut distinguer le mouvement vrai du mouvement relatif sont Isa forces imprimées dans les corps pour leur donner le mouvement : car le mouvement vrai d'un corps re peut être produit ni changé que par les forces imprimées à ce corps même, au liéu que son mouvement relatif peut être produit et changé, sans qu'il éprouve l'action d'aucune force ; il suffit qu'il y ait des forces qui agissent sur les corps par rapport auxquels on le considère, puisque ces corps étant mus, la relation dans laquelle consiste le repos ou le mouvement relatif change.\fg\endnote{Ibid, p. 41.}
+\end{quotation}
+
+Ainsi se résout le problème mentionné ci-dessus du système isolé. Ainsi s'explique
+aussi le terme de \og force d'inertie \fg. En effet, la cause du mouvement inertiel (mouvement de translation uniforme d'un objet sur lequel aucune force imprimée n’agit) ne peut être que l’espace absolu, puisque c’est le seul \og objet\fg{} en présence duquel se trouve le corps matériel. Newton nomme alors cette action de l’espace absolu \og force inertielle (insita) \fg, bien que cette action ne produise pas d’accélération et ne soit donc pas une force. Newton la définit comme \og la force qui réside dans la matière \fg{} et \og par laquelle tout corps persévère de lui-même dans son état actuel de repos ou de mouvement uniforme en ligne droite \fg\endnote{Ibid, p. 41.}.
+
+\section{Avant la relativité, un débat : l'éther}
+
+Toute la physique prérelativiste est donc basée sur la notion de référentiel d'inertie et sur le fait que les lois du mouvement, les lois de la mécanique, sont identique dans deux référentiel en translation uniforme l’un par rapport à l’autre.
+
+À la base des interrogations qui ont amené la création de la relativité restreinte, se
+trouve le problème de la vitesse de la lumière.
+
+Rappelons que la lumière est une onde et qu'avant l'avènement de la relativité res-
+treinte, on pensait que toute onde devait avoir un support matériel. Ainsi, de la même manière que les vagues (des ondes aquatiques) ont pour support les molécules d’eau, ou que les sons (des ondes de pression) ont pour support les molécules d’air, la lumière (des ondes électromagnétiques) devait avoir pour support quelque chose que l’on appelait \og l’éther \fg.
+
+D'autre part, on savait depuis 1676 avec Olaus Roemer (1644-1710) que la vitesse de la
+lumière dans le vide est une constante c (\SI{327000}{\kilo\metre\per\second} alors, au lieu de \SI{299792}{\kilo\metre\per\second} aujourd’hui). Roemer compare le mouvement des satellites de Jupiter prévu par la théorie de Newton avec l'observation et constate qu'ils semblent \og avoir tantôt huit minutes d'avance, tantôt huit minutes de retard \dots \fg\endnote{Cité sans référence par F. Balibar dans \cite[p. 96]{B}.}, \og Roemer, faisant confiance à la loi de la gravitation, en arriv[e] à l'intéressante
+conclusion que la lumière met un certain temps à voyager des lunes de Jupiter à la terre, et que lorsque nous regardons les lunes, nous ne les voyons pas là où elles sont maintenant, mais là où elles étaient il y a un certain temps, celui que met la lumière pour arriver ici \fg\endnote{Ibid, p. 98.} (voir Annexe IV).
+
+Rapidement des questions relatives à l'éther se sont posées. Il était intéressant
+notamment de savoir si l’éther était affecté par le mouvement des corps. Trois possibilités se présentaient :
+\begin{enumerate}
+\item L’éther est totalement entraîné dans le mouvement des corps.
+\item Une partie seulement de l’éther est entraîné dans leur mouvement.
+\item L’éther pénètre les objets, mais n’est pas entraîné par eux.
+\end{enumerate}
+
+Dans le premier cas, on ne peut alors détecter le mouvement d’un objet par rapport à
+l'éther, ce qui en fait une notion plutôt vague. En effet, la vitesse de la lumière étant constante par rapport à l'éther (la vitesse d’une onde est toujours définie par rapport à son support), elle est donc la même, dans l'éther mobile entraîné par la terre, que dans l’éther immobile. Or, l'éther ne se manifeste que dans les propriétés de propagation de la lumière pour lesquelles il a été imaginé. On ne peut done, dans ce cas détecter, le mouvement d’un corps par rapport à l'éther.
+
+Dans les second et troisième cas, grâce au théorème d'addition des vitesses (voir ci-
+dessus la transformation de Galilée) on le peut. En effet, supposons que la vitesse de la lumière par rapport à un référentiel K soit c. Selon un référentiel K’ en translation par rapport à K à la vitesse v, la vitesse de la lumière doit avoir, d'après ce théorème, une valeur \(c'\) telle que: \(c'=c\pm v\) différente de c (plus ou moins selon la direction de propagation).
+
+Plusieurs expériences (\og aberration de la lumière \fg, de Fizeau, de Hæk,
+\og électrodynamique des corps en mouvement\fg) montrerons qu’en tous les cas l’éther ne doit être que partiellement entraîné. Dans l’expérience de Fizeau, par exemple, il s'agissait de la mesure de la vitesse de la lumière dans un courant d’eau. Selon le principe d’additivité des vitesses ci-dessus, la vitesse de la lumière c' dans un courant d’eau de vitesse v devait être :
+\[c=c/n\pm v\]
+où n est l’indice de réfraction de l’eau et donc c/n la vitesse de la lumière dans celle-ci,
+
+Or, Fizeau n'obtint pas cette formule de composition des vitesses, mais la suivante~:
+\[c' = c/n \pm (1 - 1/n2)v\]
+qui s’expliquait par l'entraînement partiel de l’éther (\(0 < 1 - 1/n^2 < 1\)).
+
+Mais cette expérience (comme celle de l’aberration de la lumière) ne décrivait pas un
+mouvement absolu par rapport à l’éther, mais celui relatif de l’eau par rapport au référentiel du laboratoire.
+
+Pour mettre en évidence le mouvement absolu d’un corps par rapport à l’éther, il fallait donc une \og expérience interne\fg\endnote{Ibid, p. 100.}, c’est-à-dire \og que la source et l’observateur participent au mouvement étudié.\fg\endnote{\cite[p. 26]{J}.}. Que l’éther ne soit pas ou ne soit que partiellement entraîné, il devait néanmoins être possible de constater une différence de vitesse de la lumière selon que l’on se
+propage ou non dans sa direction. Pour ce faire, on utilisa les grandes variations de la vitesse de la terre autour du soleil dans l’expérience décrite ci-dessous~:
+
+\begin{figure}
+\centering
+\input{images/eps/MichelsonMorley.eps_tex}
+\caption{Expérience de Michelson et Morley}\label{fig:michelsonmorley}
+\end{figure}
+
+il s'agit d’une source envoyant un rayon de lumière sur deux miroirs placés à angle droit l’un par rapport à l’autre, comme le montre la figure \ref{fig:michelsonmorley}.
+
+Si le dispositif est au repos par rapport à l’éther, le temps mis par la lumière pour revenir à la source doit être le même dans les deux bras.
+
+Si par contre l’ensemble du dispositif est en translation à vitesse constante selon l’axe source-miroir B, on peut montrer\endnote{Ibid.} que les temps mis par la lumière (qui se propage dans l’éther immobile à la vitesse c) pour aller au
+miroirs A et B doivent être différents.
+
+L'ensemble de l'expérience doit pouvoir tourner sur elle-même de façon à repérer la direction du mouvement par rapport à l’éther.
+
+Cette expérience a été réalisée par Michelson et Morley en 1887 (leur dispositif étant
+cependant quelque peu plus complexe pour des raisons techniques), puis de très nombreuses fois par la suite.
+
+Or, pour quelque vitesse de la terre sur son orbite autour du soleil que ce fut, en
+quelque endroit sur terre que ce fut (en particulier, si l'éther n'avait du être que partiellement entraîné, la vitesse de la lumière auraît du être fonction de l'altitude), jamaïs on n’observa une différence dans les temps \og de vol\fg des rayons lumineux.
+
+Pour interpréter ce fait et conserver la notion d’éther, Fitzgerald et Lorentz ont, indépendamment, émis l'hypothèse en 1893 qu'il se produisait une contraction du bras parallèle au déplacement. Lorentz alla jusqu'à faire intervenir une dilatation du temps, à établir une localité des coordonnées d'espace et de temps des référentiel galiléens (un système de coordonnées spécifique attaché à chacun de ceux-ci) et à proposer une transformation de référentiel d'inertie qui tienne compte de ces faits. Ce fut la transformation de Lorentz sur laquelle, on va le voir, Einstein construisit sa relativité restreinte.
+
+\section{L'espace relatif d'Einstein, la théorie de la relativité restreintes}
+Les considérations précédentes, antérieures à la relativité, montrent que l’idée était
+\og dans l'air\fg, les bonnes questions étant posées. Quelle fut la clé qui permit d'expliquer les faits observés avec simplicité et sans hypothèses ad hoc et de résoudre adéquatement les contradictions ? C’est ce que nous allons voir maintenant en entrant de plein pied dans les fondements conceptuels de la théorie de la relativité restreinte d'Einstein.
+
+Rappelons encore une fois le contenu essentiel du principe de relativité restreinte de
+Galilée. Il tient en deux points~:
+\begin{itemize}
+\item \og Il existe une équivalence de tous les systèmes d'inertie (en mouvement rectiligne et uniforme à partir de l'un d’entre eux), pour la description des lois du mouvement. \fg\endnote{\cite[p. 23]{K}.}
+\item La relation entre ces systèmes d'inertie est la transformation de Galilée. Il en
+résulte la règle classique de composition des vitesses qui permet d'établir que la vitesse de la lumière est différente dans deux systèmes galiléens distincts.
+\end{itemize}
+
+Or, ainsi que nous l’avons vu, il n’est pas possible de mettre en évidence expérimentalement cette différence. Le second point du principe de Galilée doit donc être modifié. Einstein le remplace par :
+\begin{itemize}
+\item \og Dans le vide, la lumière se propage de façon isotrope. Sa vitesse est une constante universelle c.\fg\endnote{Ibid.}
+\end{itemize}
+
+Il faut bien comprendre que cette substitution implique l’abandon de la transformation
+de Galilée et qu’il est nécessaire alors d’en trouver une autre qui soit en relation réciproque avec la nouvelle proposition. D'autre part, cette substitution est fondamentale, car elle va avoir deux effets essentiels~:
+\begin{enumerate}
+\item Elle va permettre de rendre les équations de l’électrodynamique (théorie de
+la dynamique des phénomènes électromagnétiques) invariantes (on dit aussi covariantes) par la nouvelle transformation. Ce qu’elles n'étaient pas par transformation de Galilée.
+\item Elle va entraîner une relativisation de certains notions fermement établies
+de la mécanique newtonienne et une modification de ses axiomes et théorèmes.
+\end{enumerate}
+
+La nouvelle transformation est alors le résultat direct des deux postulats. On la déduit de l'expression mathématique de la constance de la vitesse de la lumière posée comme identique dans deux référentiels d'inertie.
+
+Mais, Einstein va plus loin. On va voir qu’il ramène celle-ci à la reconnaissance du
+caractère relatif de l’espace et du temps, tirée d’une analyse très claire et précise de la notion de simultanéité. C'est pourquoi, il renverse la déduction mathématique de la transformation à partir des deux principes de la relativité, pour considérer la simultanéité comme première et en tirer l’invariance de la vitesse de la lumière~:
+\begin{quotation}
+\og Ne pouvons nous pas supposer des changemonts tels dans le rythme do l'horloge en mouvement et dans la longueur de la barre en mouvement que la constance de la vitesse de la lumière suivra directement ces suppositions ? En effet, nous le pouvons.\fg\endnote{\cite[pp. 33-35]{L}.}
+\end{quotation}
+
+\subsection{La notion de simultanéité}
+
+À l'origine de la relativité se trouve donc la simultanéité. La simultanéité de deux événements voisins ne pose pas de problèmes. C'est donc celle de deux événements éloignés qu'il faut examiner. Pour comprendre ses propriétés, Einstein commence par en donner une définition opératoire~:
+\begin{quotation}
+\og On mesure la droite AB [entre les deux événements] \dots et l’on place au milieu de cette droite M un observateur muni d'un appareil (par exemple deux miroirs inclinés à \SI{90}{\degree}) qui lui permet d'observer simultanément les deux points A et B. S'il aperçoit les [événements] en même temps, ils sont simultanés.\fg\endnote{\cite[p. 155]{M}.}
+\end{quotation}
+
+Cette définition est conventionnelle. Elle permet de donner un sens exact à la simultanéité de deux événements sans impliquer le nouveau postulat : l'hypothèse de la constance de la lumière. En effet, elle définit une simultanéité où il n’est pas exigé que la lumière se propage avec la même vitesse de A à M que de B à M. Exiger cela nécessiteraït de pouvoir le vérifier et donc de disposer d’un moyen de déterminer un temps absolu\endnote{Ibid.} (je passe rapidement sur ce point qui n’est pas fondamental. Einstein lui-même n’a pas toujours défini la simultanéité indépendamment du postulat de la constance de la vitesse de la lumière\endnote{\cite[p. 175]{N}.}). À partir
+de cette définition de la simultanéité, Einstein montre que celle-ci est relative au référentiel utilisé. La démonstration d'Einstein est si simple et lumineuse qu’il serait malvenu ici de ne pas la reproduire tel quel~:
+\begin{quotation}
+\og Supposons un train très long se déplaçant sur [une voie ferrée] avec une vitesse constante v dans la direction indiquée sur la figure \ref{fig:simultaneitetrain}.
+
+\begin{figure}[h]
+\centering
+\input{images/eps/simultaneitetrain.eps_tex}
+\caption{La notion de simultanéité}\label{fig:simultaneitetrain}
+\end{figure}
+
+Les voyageurs de ce train auront avantage [à] se servir du train comme corps de référence rigide (système de coordonnées), auquel ils rapporterons tous les événements.Tout événement qui a lieu le long de la voie ferrée a aussi lieu en un point déterminé du train. La définition de la simultansité peut aussi être formulée exactement de la même façon par rapport au train que par rapport à la voie. La question suivante se pose ainsi tout naturellement~:
+
+Deux événements (par exemple les deux éclairs A et B), qui sont simultanés \emph{par rapport à le voie}, sont-ils aussi simultanés \emph{par rapport au train} ? Nous montrerons tout à l'heure que la réponse doit être négative.
+
+Quand nous disons que les éclairs A et B sont simultanés par rapport à la voie ferrée nous entendons par là que les rayons issus des points A et B se rencontrent au milieu M de la distance A-B située sur la voie. Mais aux événements A et B correspondent des endroits A[‘] et B['] dans le train. Soit M le milieu de la droite A[']-B['] du train en marche. Ce point M' coïncide bien avec le point M à l'instant où se produisent les éclairs (vus du talus), mais il se déplace sur le dessin vers la droite avec la vitesse v. Si un observateur dans le train assis en M n'était pas entrainé avec cette vitesse, il resterait d'une façon permanente en M et les rayons lumineux issus de A et de B l’atteindraient simultanément, c'est-à-dire que ces deux rayons se rencontreraient au point où il se trouve. Mais en réalité il court (vu du talus) vers
+le rayons de lumière venant de B, tandis qu'il fuit devant celui qui vient de A. Il verra, par conséquent, le rayon de lumière qui vient de B plus tôt que celui qui vient de A. Les observateurs qui se servent du train comme corps de référence doivent donc arriver à la conclusion que l'éclair B s'est produit antérieurement à l'éclair A. Nous aboutissons ainsi au résultat important suivant~:
+
+Des événements qui sont simultanés par rapport à la voie ferrée ne sont pas simultanés par rapport au train et inversément (relatitité de la simultanéité). Chaque corps de référence (système de coordonnées) a son temps propre : une indication de temps n'a de sens que si l'on indique le corps de référence auquel elle se rapporte\fg\endnote{\cite[p. 31]{G}.}
+\end{quotation}
+
+Relativité de ln simultanéité donc, impliquant la relativité du temps. Mais aussi, ja relativité des distances. En effet, la vitesse de la lumière est indépendante du référentiel. Le temps, lui, ne l'est pas. Ainsi, on peut dire en substance que la distance, comme produit de la vitesse de la lumière par le temps, ne peut être que relative (être précis ici serait trop long sans avoir recours à la transformation de Lorentz ci dessous).
+
+On pourrait à ce stade, comme le dit Einstein, postuler \og des changements tels dans le rythme de l'horloge en mouvement et dans la longueur de la barre en mouvement que la constance de la vitesse de la lumière suivra directement ces suppositions\fg. Ces changements et la constance de la vitesse de la lumière étant en relation réciproque, c'est l'inverse que l'on fait en pratique : on postule l'invariance de la vitesce de la lumière et on en tire une transformation de référentiel qui implique la relativité du temps et des longueurs. Einstein dans \og La relativité\fg\endnote{\cite[p. 156]{M}.} donne une dérivation simple de la transformation de Lorentz. Nous nous contenterons ici du résultat.
+
+\subsubsection{Transformation de Lorentz}
+
+Les hypothèses de départ sont les mêmes que celles de la transformation de Galilée donnée par l'éqation \ref{eq:transgalilee}, page \pageref{eq:transgalilee}. La
+transformation explicite est la suivante~:
+
+\begin{align}\label{eq:translorentz}
+x'&=\frac{x-v\cdot t}{\sqrt{1-v^2/c^2}}\nonumber\\
+y'&=y\nonumber\\
+z'&=z\nonumber\\
+t'&=\frac{t-v\cdot x/c^2}{\sqrt{1-v^2/c^2}}
+\end{align}
+
+ou, sous forme matricielle~:
+
+\begin{equation*}
+\begin{pmatrix}
+x'\\
+y'\\
+z'\\
+t'
+\end{pmatrix}=
+\begin{pmatrix}
+\frac{1}{\sqrt{1-v^2/c^2}}&0&0&\frac{-v}{\sqrt{1-v^2/c^2}}\\
+0&1&0&0\\
+0&0&1&0\\
+\frac{-v/c^2}{\sqrt{1-v^2/c^2}}&0&0&\frac{1}{\sqrt{1-v^2/c^2}}
+\end{pmatrix}\cdot
+\begin{pmatrix}
+x\\
+y\\
+z\\
+t
+\end{pmatrix}
+\end{equation*}
+
+Remarquons trois choses :
+\begin{itemize}
+\item si vest petit par rapport à la vitesse de la lumière c, les termes \(v^2/c^2\) sont proches de zéro et on retrouve la transformation de Galilée. La transformation de Lorentz est donc une généralisation de la transformation de Galilée à des vitesses proches de celle de la lumière,
+\item en appliquant cette transformation, on peut voir facilement que l'équation de propagation de la lumière selon l’axe x : \(x=c\cdot t\) est formellement invariante,
+c'est-à-dire s'écrit comme : \(x'= c\cdot t\)~;
+\item il suit de la transformation de Lorentz une nouvelle loi d’addition des vitesses~:
+\begin{equation}\label{eq:transvitlorentz}
+v'=\frac{v+v_{ref.}}{1+v\cdot v_{ref.}/c^2}
+\end{equation}
+avec les mêmes notations que pour la transformation de Galilée. Cette équation est à comparer avec l'équation \ref{eq:transvitgalilee}, page \pageref{eq:transvitgalilee}.
+\end{itemize}
+
+\medskip
+À partir de la transformation de Lorentz, on peut facilement montrer la contraction
+des longueurs et la dilatation du temps dans des référentiels galiléens (la transformation de Lorentz est une transformation qui relie des référentiels galiléens, c’est-à-dire des référentiels en translation uniforme les uns par rapport aux autres) inertiels.
+
+\subsection{La contraction des longueurs}
+Une règle d’une longueur L de un mètre est une règle dont l’origine correspond à la position \(x_{origine}=0\) et dont l’extrémité correspond à \(x_{extrémite}=1\) dans le référentiel, mettons R, dans lequel on la mesure (la règle est au repos dans celui-ci). Sa longueur dans un référentiel R’ en translation uniforme v (selon l’axe x) par rapport à R est calculée, au temps \(t_1'\), comme suit~:
+
+on a~:
+\begin{align*}
+x_{o,e}&=\frac{x_{o,e}-v\cdot t_1}{\sqrt{1-v^2/c^2}}\\
+t_1'=\frac{t_1-v\cdot x_{o,e}/c^2}{\sqrt{1-v^2/c^2}}\;&\Rightarrow\;t_1=t_1'\cdot \sqrt{1-v^2/c^2}+\frac{v\cdot x_{o,e}}{c^2}
+\end{align*}
+
+et donc~:
+\begin{align*}
+L'&= x'_{extrémité}-x'_{origine}\\
+&=\frac{x_e-v\cdot t_1}{1-v^2/c^2}-\frac{x_o-v\cdot t_1}{1-v^2/c^2}\\
+&=\frac{x_e-v\cdot [t_1'\cdot \sqrt{1-v^2/c^2}+v\cdot x_e/c^2]}{1-v^2/c^2}-\frac{x_o-v\cdot [t_1'\cdot \sqrt{1-v^2/c^2}+v\cdot x_o/c^2]}{1-v^2/c^2}\\
+&=\frac{x_e\cdot (1-v^2/c^2)-v\cdot t_1'\cdot \sqrt{1-v^2/c^2}}{\sqrt{1-v^2/c^2}}-\frac{x_o\cdot (1-v^2/c^2)-v\cdot t_1'\cdot \sqrt{1-v^2/c^2}}{\sqrt{1-v^2/c^2}}\\
+&=(x_e-x_o)\cdot \frac{1-v^2/c^2}{\sqrt{1-v^2/c^2}}=L\cdot \frac{1-v^2/c^2}{\sqrt{1-v^2/c^2}}=1\cdot \frac{1-v^2/c^2}{\sqrt{1-v^2/c^2}}\\
+&=\sqrt{1-v^2/c^2}
+\end{align*}
+
+Donc \(L'=\sqrt{1-v^2/c^2}\) ou, plus généralement :
+\begin{equation}\label{eq:contrlong}
+L’=L\cdot \sqrt{1-v^2/c^2}
+\end{equation}
+
+Remarquons que~:
+\begin{itemize}
+\item Le terme \(\sqrt{1-v^2/c^2}\) est inférieur à un. Il y a donc contraction des longueurs.
+\item Il vaut 1 pour \(v=0\). Les longueurs donc sont les mêmes si les référentiels sont au repos l'un par rapport à l’autre.
+\item Il vaut 0 pour \(v=c\). À la vitesse de la lumière, la longueur L' de la règle (dans le référentiel se déplaçant à la vitesse de la lumière) est donc nulle. La vitesse de la lumière est une vitesse limite.
+\item On parle quelquefois de longueur propre pour la longueur L de la règle au repos dans le référentiel R.
+\end{itemize}
+
+Einstein effectue, dans \og La relativité\fg\endnote{\cite[pp. 24-25]{O}.}, la transformation inverse en cherchant la longueur dans le référentiel R d’une règle au repos dans le référentiel R'. Il obtient :
+\[L=\sqrt{1-v^2/c^2}\]
+
+ce qui signifie qu'il y a aussi contraction. En fait, il y a contraction pour tout objet en mouvement par rapport au référentiel à partir duquel on considère ce mouvement. Cette contraction est donc \og réciproque\fg\endnote{Ibid, pp. 34-36.}, c’est-à-dire, à lieu tant pour un observateur dans R que dans R’. Cette contraction est donc différente de celle de Fitzgerald-Lorentz qui avait lieu pour tout objet en mouvement par rapport à l’éther. En effet, plaçcons nous dans un référentiel (appelons-le R') auquel est attaché une règle. Quand celui-ci est au repos par rapport à l'éther, la règle a une certaine longueur, mettons L. Avec Fitzgerald et Lorentz, quand R’ est en mouvement (toujours par rapport à l’éther) la règle se contracte et sa longueur devient inférieure à L. Avec Einstein, ce n’est que vu depuis un autre référentiel (on peut choisir ici l’éther, par exemple) que la règle est plus courte. \og Il s’agit d'un effet apparent (mais non illusoire) purement observationnel\fg\endnote{\cite[p. 135, appendice 1]{G}.}.
+
+\subsection{La dilatation du temps}
+Reprenons l’exemple d'Einstein dans \og La relativité\fg\endnote{\cite[p. 45]{G}.}. Soit une horloge à l’origine \(x=0\) de R’ (en translation uniforme v par rapport à R) et soit \(t’=0\) et \(t'=1\) deux battements successifs de celle-ci. On a~:
+\[x'=0=\frac{x-v\cdot t}{\sqrt{1-v^2/c^2}}\;\Rightarrow\;x=v\cdot t\]
+
+D'où, pour chaque battement~:
+\begin{align*}
+t'=0&=\frac{t-v\cdot x/c^2}{\sqrt{1-v^2/c^2}}=\frac{t-v^2\cdot t/c^2}{\sqrt{1-v^2/c^2}}\\
+&=t\cdot \frac{1-v^2/c^2}{\sqrt{1-v^2/c^2}}\;\Rightarrow\;t=0\\
+t’=1&=t\cdot \frac{1-v^2/c^2}{\sqrt{1-v^2/c^2}}=t\cdot \sqrt{1-v^2/c^2}\;\Rightarrow\\
+t&=\frac{1}{\sqrt{1-v^2/c^2}}
+\end{align*}
+
+L'intervalle de temps dans le référentiel R est donc de \(1/\sqrt{1-v^2/c^2}\) qui est légèrement plus long que dans R' (où il vaut 1). On a donc une dilatation du temps.
+
+Comme pour les distances, la dilatation se produit aussi dans l'autre sens.
+
+On parle de temps propre \(\Delta\tau\) pour désigner le temps de l'horloge au repos dans le référentiel R'. La relation liant l'intervalle de temps \(\Delta r\), mesuré dans R, de l'horloge au repos dans R', à l'intervalle de temps propre directement mesuré dans R' est alors~:
+\begin{equation}
+\Delta t=\frac{\Delta \tau}{1-v^2/c^2}
+\end{equation}
+
+Il faut insister sur le fait que le retard des horloges n'est pas dû à une quelconque influence \og mécanique \fg{} du mouvement. Le principe de relativité nous garantissant l'identité des lois dans R et R', les horloges ont strictement le même fonctionnement. La différence de temps vient de la définition même de l'intervalle de temps, et non de la mécanique des horloges.
+
+\subsection{L'espace-temps}
+En mécanique classique, l'intervulle de distance \(dx^2\) (un \(\Delta x\) rendu infiniment petit et construit à partir du théorème de Pythagore en géométrie euclidienne par : \(\Delta x^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2\)) et, indépendamment, celui du temps \(dt^2\) (\(\Delta t^2=(t_2-t_1)^2\)), sont préservés par une transformation de Galliée. On dit que \(dx^2\) et \(dt^2\) sont deux invariants de la transformation de Galilée.
+
+Dans la transformation de Lorentz, la coordonnée de temps est liée aux coordonnées d'espace. Il s'en suit que ni l'intervalle de temps, ni celui de distance n’est préservé par une transformation de Lorentz. Les éléments infinitésimaux \(dx^2\) et \(dt^2\) ne sont donc pas des invariants de celle-ci.
+
+On peut montrer\endnote{\cite[pp. 44-45]{K} ou \cite[p. 157]{M}.} que la quantité~:
+\begin{equation}\label{eq:ds2}
+ds^2 = -c^2\cdot dt^2 + (dx^2 + dy^2 + dz^2)
+\end{equation}
+
+est bien, elle, un invariant de Lorentz. On l'appelle intervalle d’espace-temps, puisque s’y trouvent réunies les coordonnées d'espace et de temps. Son invariance fonde la nécessité de ne plus considérer le temps indépendamment de l'espace puisque \og l'espace est différent pour tous les observateurs [:] le tomps également [:] mais l'espace-temps est le même pour tous \fg\endnote{\cite[p. 44]{K}.}
+
+L'intervalle d’espace-temps peut être positif, nul ou négatif. À chaqu'un de ces cas correspond une partie bien distincte de l'espace-temps. Pour la représenter, considérons une onde lumineuse dans un plan bidimengionnel issue d’un point P. Celle-ci va se propager dans toutes les directions du plan de façon circulaire. Si on reporte en troisième coordonnées le temps, on obtiendra un cône. C’est la représentation dite \og diagramme d'espace-temps\fg{} (voir la figure \ref{fig:conedelumiere}).
+\begin{itemize}
+\item L'intérieur du cône (passé ou futur) est le domaine de représentation des trajectoires des corps massiques, ne pouvant atteindre la vitesse de la lumière, leur tracé ne peut être sur la nappe du cône.
+
+L'intérieur du cône est aussi un domaine où existe la possibilité de relalions causales. En effet, il est en principe possible d'atteindre un événernent dans le futur, puisque il faut pour cela une vitesse inférieure (à la limite égale) à celle de la lumière.
+\item L'extérieur du cône ne peut être, lui, causalement atteint puisqu'il faudrait pour cela une vitesse supérieure à celle de la lumière. Contrairement aux événements situés à l'intérieur du cône, deux événements de l’ailleurs sont antérieurs, simultanés ou postérieurs l’un de l’autre en fonction du référentiel dans lequel on les considère.
+
+Par exemple, la Terre le 22 février 1993 à 15 heure et le Soleil à la même date
+et à la même heure seront dans l’ailleurs l’un de l’autre. L'explosion du Soleil à cette heure précise n'aura, à 15 heure, aucun effet sur la Terre. Ce n'est que quelques minutes plus tard, le temps que sa lumière ne nous parvienne plus, que nous devrons allumer nos radiateurs.
+
+\begin{figure}[h]
+\centering
+\input{images/eps/conedelumiere.eps_tex}
+\caption{Cône de lumière}\label{fig:conedelumiere}
+\end{figure}
+
+\end{itemize}
+
+\subsection{Géométries et espaces}
+La modification des relations entre le temps et l’espace, introduite par la transformation de Lorentz, implique donc une vision étendue de l’espace tridimensionnel de la physique newtonienne. Des positions de ce dernier, on passe à des événements (un événement est non seulement délimité dans l’espace mais aussi dans le temps) localisés dans un espace quadridimensionnel.
+
+Pour bien comprendre ce que la relativité générale va apporter de nouveau, il faut décrire rapidement les différentes structures d'espace que les mathématiques mettent à disposition de la physique. Le tableau \ref{tab:geoesp} les présente sous une forme résumée.
+
+\begin{table}
+\caption{Géométries et espaces}\label{tab:geoesp}
+\medskip
+\begin{tabularx}{\textwidth}{@{\extracolsep{\fill}}p{0.3\textwidth}Xp{0.3\textwidth}}
+Les événements physiques localisés dans l'espace et le temps forment un ensemble. ce sont les points d'un & \centering\(\longmapsto\) & Espace-temps\\
+&\centering \(\downarrow\)&\\
+Chaque point a un voisinage d'événements possibles & \centering\(\longmapsto\)\newline structure topologique & Espace topologique\\
+&\centering \(\downarrow\)&\\
+On peut référer les événements à l'aide de 4-coordonnées sur des cartes & \centering\(\longmapsto\)\newline structure de variété & Variété de dim. 4\\
+&\centering \(\downarrow\)&\\
+Il existe des champs de vecteurs, satisfaisant des équations différentielles & \centering\(\longmapsto\)\newline structure différentiable & Variété continue r fois différentiable, \(r\geq 3\)\\
+&\centering \(\downarrow\)&\\
+Il existe une notion de parallèlisme de deux vecteurs & \centering\(\longmapsto\)\newline structure affine (connection) & Variété affine\\
+&\centering \(\downarrow\)&\\
+On peut mesurer l'éloignement de deux événements. Il existe une flèche du temps et la vitesse de la lumière est finie & \centering\(\longmapsto\)\newline structure métrique & Espace de Riemann (métrique quelconque) Espace de Minkowski (métrique pseudo-rimannienne)
+\end{tabularx}
+\end{table}
+
+\bigskip
+L'espace de la relativité restreinte est un espace de Minkowsky. En particulier, on peut donc mesurer l'éloignement entre deux événements. Comment ? Nous l'avons vu, grâce à l'intervalle d’espace-temps :
+
+\[ds^2=-c^2\cdot dt^2+(dx^2+dy^2+dz^2)\]
+
+De quoi s'agit-il exactement ?
+
+Pour mesurer une distance, du point de vue mathématique, on utilise la notion de
+norme. On sait depuis le lycée (du moins peut-on se le rappeler) que la norme d’un vecteur est donnée par :
+
+\[\Vert\overrightarrow{a}\Vert^2 =\overrightarrow{a}\cdot \overrightarrow{a}\]
+
+Si on remplace le vecteur ci-dessus par le quadrivecteur \[\overrightarrow{ds} =(i\cdot c\cdot dt, dx, dy, dz)\] où \(i = \sqrt{-1}\), on peut effectuer le produit scalaire comme on en a l'habitude, pour obtenir précisément l'intervalle d'espace temps. On reconnait donc maintenant dans la seconde partie du
+\(ds^2\) le produit scalaire d’un petit vecteur de composantes \((dx, dy, dz)\).
+
+Or, la manière d’effectuer le produit scalaire (c'est-à-dire \(dx^2 + dy^2 + dz^2\) et non \(dx^2 + 8.dy^2 + dx.dz\), par exemple) est caractéristique de la géométrie euclidienne. En posant \(dw = ic\cdot dt\) on obtient un élément de longueur \(ds^2\) formellement identique à celui qui vaut en
+géométrie euclidienne tridimensionnelle : \(ds^2 = dw^2 + dx^2 + dy^2 + dz^2\). C’est pourquoi l’espace de la relativité restreinte est euclidien (on dit parfois pseudoeuclidien car la norme, le \(ds^2\), peut devenir négatif) tout en étant quadridimensionnel.
+
+Pour comparaison, l'expression de l'élément de longueur d’un espace bidimensionnel sphérique est:  \(ds^2=R^2\cdot (\sin^2(a)\cdot d\varphi^2 +d\alpha^2)\) où \(\alpha\) et \(\varphi\) sont les paramètres permettant de
+repérer les points sur une sphère de rayon R. Il s’agit de la norme du vecteur \(\overrightarrow{ds} = (dp,da)\) effectuée à l’aide d’une \og multiplication\fg{} (le · ci-dessus) propre à un espace sphérique~:
+
+\[\Vert\overrightarrow{ds}\Vert^2=\overrightarrow{ds}\cdot
+\begin{pmatrix}
+\sin^2(\alpha)&0\\
+0&1
+\end{pmatrix}\cdot\overrightarrow{ds}\;\;\text{ou}\;\;ds^2=\sum_{\mu,\nu}g_{\mu\nu}\cdot dx^\mu\cdot dx^\nu\]
+
+Le point trouve ici une expression plus générale sous la forme d’une matrice caractéristique que l'on représente par le symbole \(g_{uy}\).
+
+\section{La théorie de la relativité générale}
+Comme nous l'avons donc vu précédemment, tant la relativité de Galilée que celle
+d'Einstein est restreinte à la considération des lois de la physique à partir de référentiels privilégiés qui sont inertiels (ou, du moins, décrétés comme tels).
+
+D'autre part, les réflexions d'Einstein sur la simultanéité ont permis d'expliquer beaucoup de résultats que le recours à l’éther n'avaient pu rendre compréhensibles. Ce dernier devint alors caduc, en même temps que la notion d'espace absolu.
+
+\smallskip
+Or, un certain nombre de phénomènes relatifs à des systèmes accélérés, et donc non inertiels, menaçaient de faire revenir l'absolu écarté par la relativité restreinte. Ils sont caractéristiques des propriétés particulières des référentiels non inertiels, propriétés qui sont à l'origine de la non invariance formelle des lois entre les référentiels inertiels et ceux qui ne le
+sont pas. Voyons les deux plus célèbres des expériences qui révèlent ces phénomènes.
+
+\subsection{Le pendule de Foucault}
+
+\newpage
+\begin{thebibliography}{99}
+	\bibitem{A}
+	  Galilée,
+	  \emph{Dialogue concernant les deux plus grands systèmes du monde}.
+	  1632.
+	\bibitem{B}
+	  Balibar F.,
+	  \emph{Galilée, Newton lus par Einstein}.
+	  PUF 3ème éd. 1990.
+	\bibitem{C}
+	  Einstein A.,
+	  \emph{Einstein, Conceptions scientifiques}.
+	  Champ Flammarion 1920.
+\bibitem{D}
+	  Collectif,
+	  \emph{La symétrie aujourd’hui}.
+	  coll. Point, éd. Seuil 1989.
+\bibitem{E}
+	  Cohen-Tannoudji G. et Spiro M.,
+	  \emph{La matière-espace-temps}.
+	  folio essais, Fayard 1986.
+\bibitem{F}
+	  Einstein A.,
+	  \emph{Sur l’électrodynamique de corps en mouvement}.
+	  Analen der Physik.. 1905.
+\bibitem{G}
+	  Einstein A.,
+	  \emph{La relativité}.
+	  Petite bibliothèque Payot 1956.
+\bibitem{H}
+	  Koyré A.,
+	  \emph{Galilée et la loi d'inertie, Etudes galiléennes II}.
+	  Histoire de la pensée No 854, Hermann 1939.
+\bibitem{I}
+	  Gruber C.,
+	  \emph{Cours de mécanique}.
+	  Ecole Polytechnique Fédérale de Lausanne.
+\bibitem{J}
+	  Feynman R.,
+	  \emph{La nature de la physique}.
+	  R. Feynman, Point Sciences, éd. du Seuil 1980.
+\bibitem{K}
+	  Mavridès S.,
+	  \emph{La relativité}.
+	  Que sais-je, PUF 1988.
+\bibitem{L}
+	  SmithJ. H.,
+	  \emph{Introduction à la relativité}.
+	  éd. dirigée par J. M. Levy-Leblond, \oe InterEditions, Paris 1979.
+\bibitem{M}
+	  Tonnelat M.-A.,
+	  \emph{La relativité}.
+	  M.-A. Tonnelat, dans \og La science contemporaine, II, Le XXe siècle\fg Histoire générale des sciences, PUF 1964.
+\bibitem{N}
+	  Einstein A. et Infeld L.,
+	  \emph{L'évolution des idées en physique}.
+	  Champs Flammarion 1933.
+\bibitem{O}
+	  Einstein A.,
+	  \emph{Quatre conférence sur la théorie de Ia relativité}.
+	  Gauthier-Villars 1971.
+\bibitem{P}
+	  Hawking S.,
+	  \emph{Une brève histoire du temps}.
+	  Flammarion 1989.
+\bibitem{Q}
+	  Tonnelat M.A.,
+	  \emph{Histoire du principe de relativité}.
+	   Flammarion 1971.
+\bibitem{R}
+	  Misner C. W., Thorne K. S. et Wheeler J. A.,
+	  \emph{Gravitation}.
+	  W. H. Freeman and Company, San Francisco 1973.
+\bibitem{S}
+	  Brisson L., Meyerstein F. W.,
+	  \emph{Inventer l'Univers}.
+	  Les Belles Lettres, Paris 1991.
+\bibitem{T}
+	  Einstein A.,
+	  \emph{Albert Einstein, Science, Ethique, Philosophie}.
+	  Seuil, 1991.
+\bibitem{U}
+	  M.-F. Biarnais,
+	  \emph{?},
+	  ?.
+\end{thebibliography}
+
+\newpage
+\theendnotes
+\newpage
+
+\section{Annexes}
+\appendix
+
+
+\section{Annexe 1}
+
+
+\section{Annexe 2}\label{annexe2}
+
+
+\end{document}
\ No newline at end of file
diff --git a/README.md_ b/README.md_
new file mode 100644
index 0000000..91dd540
--- /dev/null
+++ b/README.md_
@@ -0,0 +1,11 @@
+# TP_latex
+
+Un modèle en latex pour faire des rapports de physique. Ce modèle est auto-descriptif : en le lisant, vous apprendrez à le faire
+fonctionner. Il présente l'utilisation de tout ce qui est nécessaire pour faire un rapport de physique. En particulier :
+- une structure de rapport de physique
+- des équations
+- des tableaux côte à côte
+- des schémas avec du texte LaTeX
+- des graphes avec GnuPlot
+
+Il est utilisé au lycée comme introduction à LaTeX.
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+           style="stroke-width:0.26458332;font-size:3.88055556px">B'</tspan></text>
+      <text
+         id="text5492"
+         y="24.186211"
+         x="58.529369"
+         style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+         xml:space="preserve"><tspan
+           style="stroke-width:0.26458332;font-size:3.88055556px"
+           y="24.186211"
+           x="58.529369"
+           id="tspan5490"
+           sodipodi:role="line">M'</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+         x="48.152168"
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+         id="text5496"><tspan
+           sodipodi:role="line"
+           id="tspan5494"
+           x="48.152168"
+           y="24.186211"
+           style="stroke-width:0.26458332;font-size:3.88055556px">A'</tspan></text>
+    </g>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="94.710625"
+       y="68.888092"
+       id="text5712"><tspan
+         sodipodi:role="line"
+         id="tspan5710"
+         x="94.710625"
+         y="68.888092"
+         style="stroke-width:0.26458332">v</tspan></text>
+    <g
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+       transform="translate(11.722625,62.029561)">
+      <text
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+         style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+         xml:space="preserve"><tspan
+           style="stroke-width:0.26458332;font-size:3.88055556px"
+           y="24.186211"
+           x="71.791824"
+           id="tspan5616"
+           sodipodi:role="line">B'</tspan></text>
+      <text
+         xml:space="preserve"
+         style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+         x="58.529369"
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+         id="text5622"><tspan
+           sodipodi:role="line"
+           id="tspan5620"
+           x="58.529369"
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+           style="stroke-width:0.26458332;font-size:3.88055556px">M'</tspan></text>
+      <text
+         id="text5626"
+         y="24.186211"
+         x="48.152168"
+         style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+         xml:space="preserve"><tspan
+           style="stroke-width:0.26458332;font-size:3.88055556px"
+           y="24.186211"
+           x="48.152168"
+           id="tspan5624"
+           sodipodi:role="line">A'</tspan></text>
+    </g>
+    <text
+       id="text5716"
+       y="90.870964"
+       x="98.716568"
+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       xml:space="preserve"><tspan
+         style="stroke-width:0.26458332"
+         y="90.870964"
+         x="98.716568"
+         id="tspan5714"
+         sodipodi:role="line">v</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="103.36018"
+       y="19.778755"
+       id="text8342"><tspan
+         sodipodi:role="line"
+         id="tspan8340"
+         x="103.36018"
+         y="19.778755"
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+         sodipodi:role="line"
+         x="103.36018"
+         y="25.070421"
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         id="tspan8344">A, B, A' et B'</tspan></text>
+    <text
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+       y="41.49371"
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+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       xml:space="preserve"><tspan
+         id="tspan8348"
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         y="41.49371"
+         x="103.139"
+         sodipodi:role="line">Le signal lumineux issu de B</tspan><tspan
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         y="46.785378"
+         x="103.139"
+         sodipodi:role="line"
+         id="tspan8354">atteint M' avant M</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="103.139"
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+       id="text8360"><tspan
+         id="tspan8358"
+         sodipodi:role="line"
+         x="103.139"
+         y="62.776665"
+         style="stroke-width:0.26458332;font-size:3.88055556px">Les deux signaux lumineux</tspan><tspan
+         sodipodi:role="line"
+         x="103.139"
+         y="68.068329"
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         id="tspan8364">atteignent M simultanément</tspan></text>
+    <text
+       id="text8370"
+       y="84.880241"
+       x="103.139"
+       style="font-style:normal;font-weight:normal;font-size:4.23333311px;line-height:1.25;font-family:sans-serif;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       xml:space="preserve"><tspan
+         id="tspan8368"
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         y="84.880241"
+         x="103.139"
+         sodipodi:role="line">Le signal lumineux issu de A</tspan><tspan
+         style="stroke-width:0.26458332;font-size:3.88055556px"
+         y="90.171906"
+         x="103.139"
+         sodipodi:role="line"
+         id="tspan8374">atteint M' après celui issu de B</tspan></text>
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+       xml:space="preserve"
+       id="flowRoot9267"
+       style="fill:black;fill-opacity:1;stroke:none;font-family:sans-serif;font-style:normal;font-weight:normal;font-size:16px;line-height:1.25;letter-spacing:0px;word-spacing:0px"><flowRegion
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+         id="flowPara9273"></flowPara></flowRoot>  </g>
+</svg>
diff --git a/test.txt b/test.txt
new file mode 100644
index 0000000..7b1c996
--- /dev/null
+++ b/test.txt
@@ -0,0 +1,30 @@
+1	1	0.1	1	2
+2	4	0.4	2	4
+3	11	0.2	3	6
+4	18	0.5	4	8
+5	26	0.3	5	10
+6	31	0.1	6	12
+
+
+10	16	5
+20	45	10
+30	53	15
+40	89	20
+50	110	25
+60	135	30
+70	140	35
+80	155	40
+90	170	45
+100	200	50
+
+
+10	15	1
+20	14	1
+30	15	1
+40	13	1
+50	15	1
+60	15	1
+70	14	1
+80	15	1
+90	14	1
+100	14	1
\ No newline at end of file