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Corso di Relatività Generale
I Parte
Fondament= i di Geometria Differenziale e Relatività
Generale
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Summary of  Riemanian Ge= ometry and Vielbein formulat= ion
Manifolds
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Privileged observers and affine manifolds
Both Newtonian
Physics and
Special Relativity
have privileged
observers
=
Affine Manifold
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Affine Manifolds
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Curved Manifolds and Atlases
The intuitive idea = of an atlas of open= charts, suitably reformulated in mathematical terms,provides the = very definition of = a differentiable = manifold=
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Homeomorphisms
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file:///C:/1E4AD011/Relat1_1_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
Topology invariant under
homeomorphisms
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Homeomorphism
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Differentiable struc= ture
M1
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The axiom M2
Transition
functions
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The axiom M3
Differentiable Manifolds<= /span>
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This is a constructive definition
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Smooth manifolds
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Complex Manifolds
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Example the S= N sphere
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The stereographic projection
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file:///C:/1E4AD011/Relat1_1_files/slide0032.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
The transition funct= ion
There are just two open charts and the transition function is the following one
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charset="us-ascii" Corso di Relatività Generale II Parte
Calculus on Manifold= s
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Gluing Rules<= /div>
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Global Functions
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Towards ta= ngent spaces: Curves on a Manifold
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Tangent ve= ctors at a point
p = 2 M
Intuitively the  tangent in p at a curve that starts <= /span>from p is the curve’s initial direction
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quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
Example: t= angent space at
p = 2 S2
Let us make this intuitive notion mathematically precise
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Composed function
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Derivative =3D tange= nt vector
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Tangent vectors and derivations of algebras
Algebra of germs
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Derivations of algeb= ras
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Vectors  as differential operators
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Controvari= ant versus
covariant vectors=
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Covariance
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Parallel Transport
A vector field is = para= llel transported along a curve, when = it m= antains a constant angle with = the = tangent vector to the curve
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The difference between flat and curved manifolds
In a flat manifold, = = while transported, the vector is not rotated.
In a curved manifold= = it is  rotated:
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To see the= real effect of curvature we must consider.....= =
Parallel transport along LOOPS=
After transport = <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:133%;color:#333= 300'>along a loop, the vector does not = <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:133%;color:#333= 300'>come back to the original position but <= /span>= it is rotated of some angle. <= /div>
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La 1-forma di connessione
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Corso di Relatività Generale II Parte
Riferimenti e potenz= iali vettori
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Le funzioni di trans= izione
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Trasformazioni di gauge =3D cambio di trivializzazione locale
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a
b
g
On a sphere
The sum = of the internal angles of a triangle is = larger t= han 1800
This means that t= he curvature is posi= tive <= /b>
How are the sides= of the this triangle drawn?
They are arcs of maximal circles, namely geodesics =
for this manifold
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The hyperboloid: a space with <= span lang=3DIT style=3D'font-size:90%'>negative curvature and lorentzian <= span lang=3DIT style=3D'font-size:90%'>signature
<= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:117%;color:red'= >This surface is the locus of points satisfying  the equation <= /div>
Then we obtain the induc= ed metric= 3;
We can solve the equation parametrically by setting:
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file:///C:/1E4AD011/Relat1_1_files/slide0066.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
The met= ric: a rule to calculate the lenght of curves!!
A
B
A curve on the surface is described by giving t= he <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%;color:#0033= 00'>coordinates as functions of a single parameter<= /b> t<= /i>
This integral is a rule ! Any such rule is a  Gravitational Field!!!!
How long is this curve?
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Underlying our rule for lengths is <= span lang=3DIT>the induced metric:
Where a and q = are the coordinates of our space. This is a Lorentzian metric = and it is just induced by the flat Lorentzian metri= c in three dimensions:
using the parametric solution for= X0   ,  = X1   ,  X2
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What do particles do in a = = gravitational field?
Answer: They just go straight a= s in empty space!!!!
It is the concept of straight line that is modified by the <= span style=3D'position:absolute;top:68.25%;left:0%;width:84.78%;height:33.33%'>= presence of gravity!!!!
The metaphor of  Edd= ington’s sheet
summarizes  General Relativity.
In curved space straight lines are =
different  from straight lines in flat space!! &#= 13;
The red line followed by the ball falling in the throat is a straight line (geodesics). On the other hand space-tim= e is bendedweight of matter moving in= side it!
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The Methaphor as a M= ovie
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Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
What are the straight lines
They are the geodesics, curves that do not change length under small = deformations. These are the curves along which we = have parallel transported our vector= s
On a sphere geodesics are= = maximal circles
In the parallel transport the angle with = the tangent vector remains fixed. On <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%;color:blue'= >geodesics the tangent vector is transported parallel to its= elf.
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quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
Let us see what are the straight lines  = (=3Dgeodesics) on the Hyperboloid
Three diff= erent = types of <= /span>= geodesics<= /span>
Relativity
=3D Lorentz signature   - , +
time
space
•<= /span> ds2  < 0 space-like geodesics: cannot be <= /span><= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>followe= d by any particle (it would travel faster than light)
•<= /span> ds2  > 0 time-like geodesics. It is a possible worldline for a massive particle!
•<= /span> ds2  =3D 0 light-like geodesics.  It is a = <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>possible world-line for a massless particle like a photon=
Is the rule to calculate lengths
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Content-Type: image/gif R0lGODlhTwAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwBL AAYAgAAAAAAA/wIhhI+py+1/QoC02hvlBLr7D4biSJbmiaYpxraX5sZysx0FADs= ------=_NextPart_01C7992A.2795BFB0 Content-Location: file:///C:/1E4AD011/Relat1_1_files/slide0072.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
Deriving the geodesics from a va= riational principle
=
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The Euler Lagrange equations are
The cons= erved quantity p is, in = the time-like or null-like cases, the energy of the particle travelling on the geodesic
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Continuing...= =
= This procedure to obtain the differential equation of orbits extends from our &= #13;
= toy model in two dimensions to more realistic cases in four dimensions: <= /b>
= it is quite general
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Still continuing
Let us now study the shapes and properties of these curves
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Space-like
These c= urves lie on the hyperboloid and are space-l= ike. They stretch from megative to positive infinity= . They turn a little bit around <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>the thr= oat but they never make a complete <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>loop ar= ound it . They are characterized by <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>their inclination p.
This la= tter is a constant of motion, a first integral
&#= 13;
The sh= ape of geodesics is a consequence of our rule to calculate <= /span>the length of curves, namely of the metric=
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Time-like
Thes= e curves lie on the hyperboloid and <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>they= can wind around the throat. <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>They= never extend up to infinity. They <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>are = also labeld by a first integral of the <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>moti= on, = E, <= /span>that= we <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:83%'>can = identify with the energy
Here we see a possible danger for causality:
Closed time-like curves!
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file:///C:/1E4AD011/Relat1_1_files/slide0079.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
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R0lGODlhHQBNAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAc AEsAgQAAAAAAAKSkAMzMAALqhI+pGbHPmhQOrimyrvY0DW6d8YUmBwXmSnUqe6YvHKLITNdXztok v/LhgKIbMWg8xjzKJWDYbDCPEkmSZ01hfTsat9v7gp2jH7J83Ygtw7VLh2bA42m362W/Z/JsFb/f QnchJTj413eYMpD4ELBYeDHwCHkjyajgaEnpITm5man52RlKCeoJaXqKyTZKiimW6noD29rJ5fgV a2uD21jbipKZ+ztKKOxLXNxgi5xsyxzpTHwrPd1cXXyNLUuyDay9PezNzTmuWjm+pmvtEl627HwI X3s5D42aLUpOB3qJ7l8JYMBNswjegFAAADs= ------=_NextPart_01C7992A.2795BFB0 Content-Location: file:///C:/1E4AD011/Relat1_1_files/slide0080.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Corso di Relatività Generale II Parte
the Christoffel symb= ols are:
Where fr= om do they emerge and what is their meaning?
ANSWER:
They are the coefficients of an af= fine <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:117%;color:#663= 300'>connection, namely the proper mathematical concept underlying the <= /span><= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:117%;color:#663= 300'>concept of parallel transport.
Let us review the <= span lang=3DIT style=3D'font-family:"Times New Roman";font-size:117%;color:#009= 900'>concept of connection
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Connection= and covariant derivative
A connection is a map
From the= product of the
tangent = bundle with itself to
the tang= ent bundle
1
2
3
4
with defining properties:
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In a basis...
This defines the covariant derivative of a (controvariant) vector field
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Torsion and Curvature
Torsion Tensor
Curvature = Tensor <= /div>
The Riem= ann curvature tensor
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Corso di Relatività Generale II Parte
If we have a metric........
= = An affine connection, namely a rule for the parallel transport can be
= arbitrarily given, but if we have a metric, then this induces a canonical
= special connection:
THE LEVI CIVITA CONNECTION <= /b>
This connection is the one which emerg= es from the variational principle of geodesics!!!!!=
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(sld.mAdvDelay>0) && !parent.IsFramesMode() ) setTimeout("parent.GoToNextSld()",sld.mAdvDelay) } function GetObj(id) { if(g_supportsPPTHTML) return document.all(id); else return document.getElementById(id); } function SldHasTrans() { return SlideObj.style.getAttribute("filter")!=3D""= } function GetSldId() { return sId=3Dlocation.href.substring(location.href.la= stIndexOf('/')+1) } function HideMenu() { if( frames["PPTSld"] && PPTSld.document.all.item("ctx= tmenu") && PPTSld.ctxtmenu.style.display!=3D"none" ) { PPTSld.ctxtmenu.styl= e.display=3D'none'; return true } return false } function IsWin( name ) { return window.name =3D=3D name } function IsNts() { return IsWin("PPTNts") } function IsSldOrNts() { return( IsWin("PPTSld")||IsWin("PPTNts") ) } function SupportsPPTAnimation() { return( navigator.platform =3D=3D "Win32"= && navigator.appVersion.indexOf("Windows")>0 ) } function SupportsPPTHTML() { var appVer=3Dnavigator.appVersion, msie=3DappVer.indexOf("MSIE "), ver=3D0 if( msie >=3D 0 ) ver=3DparseFloat( appVer.substring( msie+5, appVer.indexOf(";",msie) ) ) else ver=3DparseInt(appVer) return( ver >=3D 4 && msie >=3D 0 ) } function _RSW() { if( !g_supportsPPTHTML || IsNts() || ( !g_scaleInFrame && (!IsWin("PPTSld") || !parent.IsFullScrMode()) ) ) return var padding=3D0; if( IsWin("PPTSld") && parent.IsFramesMode() ) padding=3D6 cltWidth=3Ddocument.body.clientWidth-padding cltHeight=3Ddocument.body.clientHeight-padding factor=3D(1.0*cltWidth)/g_origW if( cltHeight < g_origH*factor ) factor=3D(1.0*cltHeight)/g_origH newSize =3D g_origSz * factor if( newSize < 1 ) newSize=3D1 s=3DSlideObj.style s.fontSize=3DnewSize+"px" s.posWidth=3Dg_origW*factor s.posHeight=3Dg_origH*factor s.posLeft=3D(cltWidth-s.posWidth+padding)/2 s.posTop=3D(cltHeight-s.posHeight+padding)/2 if( g_scaleHyperlinks ) ScaleHyperlinks( factor ) } function _InitAnimations() { animRuntimeInstalled =3D ''+document.body.localTime !=3D 'undefined'; isFullScreen =3D (window.name =3D=3D "PPTSld") && !parent.IsFramesMode(); g_animUseRuntime =3D g_showAnimation && animRuntimeInstalled && !(isFullSc= reen && parent.IsSldVisited()); if( g_animUseRuntime ) { collSeq =3D document.all.tags("seq"); if( collSeq !=3D null ) { for(ii=3D0;ii numSlds ) gSldJumpIdx =3D numSlds; if ( gSldJumpIdx >=3D 0 ) { if ( gSldJumpIdx =3D=3D 0 ) gSldJumpIdx =3D 1; var jumpTo =3D parseInt(gSldJumpIdx); gSldJump =3D 0; gSldJumpIdx =3D ""; win.GoToSld( parent.GetSldList().mList[jumpTo-1].mSldHref ) } } } function _KDH() { if( event.keyCode =3D=3D 8 ) { event.returnValue =3D 0; parent.GoToPrevSld(); } } var g_OnMouseOver =3D 0, g_HideWhenNotPlaying =3D 0; function StopMedia( szName ) { if ( !g_animUseRuntime ) { rMovie =3D document.all.item( szName ); rDiv =3D rMovie.parentElement; rDiv.style.visibility =3D g_HideWhenNotPlaying ? "hidden" : "visible"; rMovie.style.visibility =3D "hidden"; if( g_animManager !=3D null ) { g_animManager.m_isPlaying =3D 0; g_animManager.Peek(); g_curAnim =3D null; } } } function PlayMedia( szName ) { if ( !g_animUseRuntime ) { rMovie =3D document.all.item( szName ); rMovie.style.visibility =3D "visible"; rMovie.Run(); if( g_animManager !=3D null ) g_animManager.m_isPlaying =3D 1; } } function PlaySound( href, loop ){ href=3D unescape(href); if( window.event ) window.event.cancelBubble =3D true; pptSound.src =3D href; pptSound.loop =3D loop ? -1 : 0; } function DocumentOnClick() { if( IsNts() || parent.HideMenu() ) return; if( ( g_allowAdvOnClick && (window.name=3D=3D"PPTSld") && !parent.IsFrames= Mode() ) || (event && event.keyCode=3D=3D32) ) { if( g_animUseRuntime && g_animMainSequence && g_animMainSequence.cangonex= t ) return; parent.GoToNextSld(); } } var g_supportsPPTHTML =3D SupportsPPTHTML(), g_scaleInFrame =3D 1, gId=3D""= , g_bgSound=3D"", g_scaleHyperlinks =3D false, g_allowAdvOnClick =3D 1, g_showInBrowser = =3D 0, gLoopCont =3D 0, gUseSldTimings =3D 1; var g_showAnimation =3D g_supportsPPTHTML && SupportsPPTAnimation() && ( (w= indow.name=3D=3D"PPTSld" && !parent.IsFramesMode()) || g_showInBrowser );va= r g_animManager =3D null; var g_animUseRuntime =3D false; var g_animItemsToHide, g_animInteractiveItems, g_animSlideTime; var g_animMainSequence =3D null; var ENDSHOW_MESG=3D"End of slide show, click to exit.", SCREEN_MODE=3D"Fram= es", gIsEndShow=3D0, NUM_VIS_SLDS=3D82, SCRIPT_HREF=3D"script.js", FULLSCR_= HREF=3D"fullscreen.htm"; var gCurSld =3D gPrevSld =3D 1, g_offset =3D 0, gNtsOpen =3D gHasNts =3D gO= tlTxtExp =3D 0, gHasNarration =3D 0, gOtlOpen =3D true window.gPPTHTML=3DSupportsPPTHTML() var gMainDoc=3Dnew Array(new hrefList("slide0001.htm",1,-1,1),new hrefList(= "slide0003.htm",1,-1,1),new hrefList("slide0014.htm",1,-1,1),new hrefList("= slide0015.htm",1,-1,1),new hrefList("slide0017.htm",1,-1,1),new hrefList("s= lide0016.htm",1,-1,1),new hrefList("slide0018.htm",1,-1,1),new hrefList("sl= ide0019.htm",1,-1,1),new hrefList("slide0020.htm",1,-1,1),new hrefList("sli= de0021.htm",1,-1,1),new hrefList("slide0022.htm",1,-1,1),new hrefList("slid= e0023.htm",1,-1,1),new hrefList("slide0024.htm",1,-1,1),new hrefList("slide= 0025.htm",1,-1,1),new hrefList("slide0026.htm",1,-1,1),new hrefList("slide0= 027.htm",1,-1,1),new hrefList("slide0028.htm",1,-1,1),new hrefList("slide00= 29.htm",1,-1,1),new hrefList("slide0030.htm",1,-1,1),new hrefList("slide003= 1.htm",1,-1,1),new hrefList("slide0032.htm",1,-1,1),new hrefList("slide0033= .htm",1,-1,1),new hrefList("slide0034.htm",1,-1,1),new hrefList("slide0035.= htm",1,-1,1),new hrefList("slide0036.htm",1,-1,1),new hrefList("slide0037.h= tm",1,-1,1),new hrefList("slide0038.htm",1,-1,1),new hrefList("slide0039.ht= m",1,-1,1),new hrefList("slide0040.htm",1,-1,1),new hrefList("slide0041.htm= ",1,-1,1),new hrefList("slide0042.htm",1,-1,1),new hrefList("slide0043.htm"= ,1,-1,1),new hrefList("slide0044.htm",1,-1,1),new hrefList("slide0045.htm",= 1,-1,1),new hrefList("slide0046.htm",1,-1,1),new hrefList("slide0047.htm",1= ,-1,1),new hrefList("slide0048.htm",1,-1,1),new hrefList("slide0049.htm",1,= -1,1),new hrefList("slide0050.htm",1,-1,1),new hrefList("slide0051.htm",1,-= 1,1),new hrefList("slide0052.htm",1,-1,1),new hrefList("slide0053.htm",1,-1= ,1),new hrefList("slide0054.htm",1,-1,1),new hrefList("slide0055.htm",1,-1,= 1),new hrefList("slide0056.htm",1,-1,1),new hrefList("slide0057.htm",1,-1,1= ),new hrefList("slide0058.htm",1,-1,1),new hrefList("slide0059.htm",1,-1,1)= ,new hrefList("slide0060.htm",1,-1,1),new hrefList("slide0012.htm",1,-1,1),= new hrefList("slide0008.htm",1,-1,1),new hrefList("slide0010.htm",1,-1,1),n= ew hrefList("slide0061.htm",1,-1,1),new hrefList("slide0062.htm",1,-1,1),ne= w hrefList("slide0063.htm",1,-1,1),new hrefList("slide0005.htm",1,-1,1),new= hrefList("slide0006.htm",1,-1,1),new hrefList("slide0009.htm",1,-1,1),new = hrefList("slide0007.htm",1,-1,1),new hrefList("slide0013.htm",1,-1,1),new h= refList("slide0064.htm",1,-1,1),new hrefList("slide0065.htm",1,-1,1),new hr= efList("slide0066.htm",1,-1,1),new hrefList("slide0067.htm",1,-1,1),new hre= fList("slide0068.htm",1,-1,1),new hrefList("slide0069.htm",1,-1,1),new href= List("slide0070.htm",1,-1,1),new hrefList("slide0071.htm",1,-1,1),new hrefL= ist("slide0072.htm",1,-1,1),new hrefList("slide0073.htm",1,-1,1),new hrefLi= st("slide0074.htm",1,-1,1),new hrefList("slide0075.htm",1,-1,1),new hrefLis= t("slide0076.htm",1,-1,1),new hrefList("slide0077.htm",1,-1,1),new hrefList= ("slide0078.htm",1,-1,1),new hrefList("slide0079.htm",1,-1,1),new hrefList(= "slide0080.htm",1,-1,1),new hrefList("slide0081.htm",1,-1,1),new hrefList("= slide0082.htm",1,-1,1),new hrefList("slide0083.htm",1,-1,1),new hrefList("s= lide0084.htm",1,-1,1),new hrefList("slide0011.htm",1,-1,1)); function FullScrInit() { g_allowAdvOnClick =3D GetCurSld().mAdvOnClk document.body.style.backgroundColor=3D"black" document.oncontextmenu=3Dparent._CM; document.onkeydown =3D _KDH; document.ondragstart=3DCancel document.onselectstart=3DCancel self.focus() } function Redirect( frmId ) { var str=3Ddocument.location.hash,idx=3Dstr.indexOf('#'), sId=3DGetSldId() if(idx>=3D0) str=3Dstr.substr(1); if( window.name !=3D frmId && ( sId !=3D str) ) { obj =3D GetObj("Main-File") window.location.href=3Dobj.href+"#"+sId return 1 } return 0 } var MHTMLPrefix =3D CalculateMHTMLPrefix(); function CalculateMHTMLPrefix() { if ( document.location.protocol =3D=3D 'mhtml:') { href=3Dnew String(document.location.href) Start=3Dhref.indexOf('!')+1 End=3Dhref.lastIndexOf('/')+1 if (End < Start) return href.substring(0, Start) else return href.substring(0, End) } return ''; } function GetTags(base,tag) { if(g_supportsPPTHTML) return base.all.tags(tag); else return base.getElementsByTagName(tag); } function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP= refix+GetHrefObj( gCurSld ).mNtsHref ) } function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() } function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() } function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() } function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ToggleOtlPane() { frmset=3Ddocument.all("PPTHorizAdjust") frm=3Ddocument.all("PPTOtl") if( gOtlOpen ) frmset.cols=3D"*,100%" else frmset.cols=3D"25%,*" gOtlOpen=3D!gOtlOpen frm.noResize=3D!frm.noResize UpdOtNavPane() } function ToggleNtsPane() { frmset=3Ddocument.all("PPTVertAdjust") frm=3Ddocument.all("PPTNts") if( gNtsOpen ) frmset.rows=3D"100%,*" else frmset.rows=3D"*,20%" gNtsOpen=3D!gNtsOpen UpdNtsPane() } function ClearMedia() { if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() { if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else return gCurSld } function GetNumSlds() { if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; PPTSld.location.replace(MHTMLPrefix+obj.mSldHref) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 PPTSld.location.replace(MHTMLPrefix+href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document doc.open() doc.writeln('


' + ENDSHOW_MESG + '

') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); function ImgBtn( oId,bId,w,action ) { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId =3D oId t.mBorderId=3D bId t.mWidth =3D w t.mIsOn =3D t.mCurState =3D 0 } function _IBSetA() { if( this.mIsOn ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _IBSetI() { if( this.mIsOn ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0 } } function _IBSetP() { if( this.mIsOn ) { obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 ) obj.style.posLeft+=3D1; obj.style.posTop+=3D1 } } function _IBSetD() { obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 ) obj.style.posTop=3D0 } function _IBSetE( state ) { var t=3Dthis GetObj( t.mBorderId ).style.visibility=3D"visible" if( state !=3D t.mIsOn ) { t.mIsOn=3Dstate if( state ) t.SetInactive() else t.SetDisabled() } } function _IBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( t.ChangeIcon ) { obj=3DGetObj(t.mObjId) if( t.ChangeIcon() ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth else obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-0)*t.mWidth } t.SetActive() } } function _IBUI( clr1,clr2,nextState ) { var t=3Dthis SetBorder( GetObj( t.mBorderId ),clr1,clr2 ) obj=3DGetObj( t.mObjId ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-obj= .style.posTop t.mCurState=3DnextState return obj } function TxtBtn( oId,oeId,action,chkState ) { var t=3Dthis t.Perform =3D _TBP t.SetActive =3D _TBSetA t.SetInactive=3D _TBSetI t.SetPressed =3D _TBSetP t.SetDisabled=3D _TBSetD t.SetEnabled =3D _TBSetE t.GetState =3D chkState t.UserAction =3D action t.ChgState =3D _TBUI t.mObjId =3D oId t.m_elementsId=3D oeId t.mIsOn =3D 1 } function _TBSetA() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gHiliteClr,gShadowClr,0,0 ) } function _TBSetI() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) } function _TBSetP() { if( this.mIsOn ) this.ChgState( gShadowClr,gHiliteClr,1,1 ) } function _TBSetD() { this.ChgState( gFaceClr,gFaceClr,0,0 ) this.mIsOn =3D 0 } function _TBSetE() { var t=3Dthis if( !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) else t.ChgState( gShadowClr,gHiliteClr,1,1 ) t.mIsOn =3D 1 } function _TBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( !t.GetState ) return if( t.GetState() ) t.SetPressed() else t.SetActive() } } function _TBUI( clr1,clr2,lOffset,tOffset ) { SetBorder( GetObj( this.mObjId ),clr1,clr2 ) Offset( GetObj( this.m_elementsId ),lOffset,tOffset ) } function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle= ft } function SetBorder( obj, upperLeft, lowerRight ) { s=3Dobj.style; s.borderStyle =3D "solid" s.borderWidth =3D 1 s.borderLeftColor =3D s.borderTopColor =3D upperLeft s.borderBottomColor=3D s.borderRightColor =3D lowerRight } function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] } function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() } function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() } function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() } function BtnOnUp() { b=3DGetBtnObj() if( b !=3D null ) b.Perform() else Upd() } function GetNtsState(){ return parent.gNtsOpen } function GetOtlState(){ return parent.gOtlOpen } function GetOtlTxtState(){ return parent.gOtlTxtExp } function NtsBtnSetFlag( fVal ) { s=3Ddocument.all.item( this.m_flagId ).style s.display=3D"none" if( fVal ) s.display=3D"" else s.display=3D"none" } function _BSetA_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etActive() } function _BSetI_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etInactive() } var gHiliteClr=3D"THREEDHIGHLIGHT",gShadowClr=3D"THREEDSHADOW",gFaceClr=3D"= THREEDFACE" var gBtnArr =3D new Array() gBtnArr["nb_otl"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOtlPan= e,GetOtlState ) gBtnArr["nb_otlElem"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOt= lPane,GetOtlState ) gBtnArr["nb_nts"] =3D new TxtBtn( "nb_nts","nb_ntsElem",parent.ToggleNtsPan= e,GetNtsState ) gBtnArr["nb_prev"]=3D new ImgBtn( "nb_prev","nb_prevBorder",30,parent.GoToP= revSld ) gBtnArr["nb_next"]=3D new ImgBtn( "nb_next","nb_nextBorder",30,parent.GoToN= extSld ) gBtnArr["nb_sldshw"]=3D new ImgBtn( "nb_sldshw","nb_sldshwBorder",18,parent= .FullScreen ) gBtnArr["nb_sldshwBorder"] =3D new TxtBtn( "nb_sldshw","nb_sldshwBorder",pa= rent.FullScreen,null ) gBtnArr["nb_sldshwBorder"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwBorder"].SetInactive =3D _BSetI_Border; gBtnArr["nb_sldshwText"] =3D new TxtBtn( "nb_sldshw","nb_sldshwText",parent= .FullScreen,null ) gBtnArr["nb_sldshwText"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwText"].SetInactive =3D _BSetI_Border; gBtnArr["nb_voice"] =3D new ImgBtn( "nb_voice","nb_voiceBorder",18,parent.= ToggleVNarration ) gBtnArr["nb_otlTxt"]=3D new ImgBtn( "nb_otlTxt","nb_otlTxtBorder",23,parent= .ToggleOtlText ) gBtnArr["nb_nts"].m_flagId=3D "notes_flag" gBtnArr["nb_nts"].SetFlag =3D NtsBtnSetFlag gBtnArr["nb_otlTxt"].ChangeIcon=3D GetOtlTxtState var sNext=3D"Next",sPrev=3D"Previous",sEnd=3D"End Show",sFont=3D"Arial",sAr= row=3D"Arrow",sFreeform=3D"Freeform",sRect=3D"Rectangle",sOval=3D"Oval" function ShowMenu() { BuildMenu(); var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y= =3DPPTSld.event.clientY+doc.scrollTop m =3D PPTSld.document.all.item("ctxtmenu") m.style.pixelLeft=3Dx if( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) ) m.style.pixelLeft=3Dx-m.scrollWidth m.style.pixelTop=3Dy if( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) ) m.style.pixelTop=3Dy-m.scrollHeight m.style.display=3D"" } function _CM() { if( !parent.IsFullScrMode() ) return; if(!PPTSld.event.ctrlKey) { ShowMenu() return false } else HideMenu() } function BuildMenu() { if( PPTSld.document.all.item("ctxtmenu") ) return var mObj=3DCreateItem( PPTSld.document.body ) mObj.id=3D"ctxtmenu" mObj.style.visibility=3D"hidden" var s=3DmObj.style s.position=3D"absolute" s.cursor=3D"default" s.width=3D"120px" SetCMBorder(mObj,"menu","black") var iObj=3DCreateItem( mObj ) SetCMBorder( iObj, "threedhighlight","threedshadow" ) iObj.style.padding=3D2 CreateMenuItem( iObj,sNext,M_GoNextSld,M_True ) CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld ) CreateSeparator( iObj ) CreateMenuItem( iObj,sEnd,M_End,M_True ) mObj.style.visibility=3D"visible" } function Cancel() { window.event.cancelBubble=3Dtrue; window.event.returnVa= lue=3Dfalse } function Highlight() { ChangeClr("activecaption","threedhighlight") } function Deselect() { ChangeClr("threedface","menutext") } function Perform() { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) e.Action() else PPTSld.event.cancelBubble=3Dtrue } function ChangeClr( bg,clr ) { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) { e.style.backgroundColor=3Dbg e.style.color=3Dclr } } function M_HasPrevSld() { return( parent.HasPrevSld() ) } function M_GoNextSld() { if( gIsEndShow ) M_End(); else GoToNextSld() } function M_GoPrevSld() { if( gIsEndShow ) { history.back(); PPTSld.event.ca= ncelBubble=3Dtrue; } else GoToPrevSld() } function M_True() { return true } function M_End() { window.close( self ) } function CreateMenuItem( node,text,action,eval ) { var e=3DCreateItem( node ) e.type=3D"menuitem" e.Action=3Daction e.IsActive=3Deval e.innerHTML=3Dtext if( !e.IsActive() ) e.style.color=3D"threedshadow" e.onclick=3DPerform e.onmouseover=3DHighlight e.onmouseout=3DDeselect s=3De.style; s.fontFamily=3DsFont s.fontSize=3D"9pt" s.paddingLeft=3D2 } function CreateSeparator( node ) { var sObj=3DCreateItem( node ) SetCMBorder(sObj,"menu","menu") var s=3DsObj.style s.borderTopColor=3D"threedshadow" s.borderBottomColor=3D"threedhighlight" s.height=3D1 s.fontSize=3D"0px" } function CreateItem( node ) { var elem=3DPPTSld.document.createElement("DIV") node.insertBefore( elem ) return elem } function SetCMBorder( o,ltClr,rbClr ) { var s=3Do.style s.backgroundColor=3D"menu" s.borderStyle=3D"solid" s.borderWidth=3D1 s.borderColor=3DltClr+" "+rbClr+" "+rbClr+" "+ltClr } ------=_NextPart_01C7992A.2795BFB0 Content-Location: file:///C:/1E4AD011/Relat1_1_files/fullscreen.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; 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charset="us-ascii" Corso di Relatività Generale II Parte ------=_NextPart_01C7992A.2795BFB0 Content-Location: file:///C:/1E4AD011/Relat1_1_files/outline.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii"
Slide Show
Outline
<= /a>= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
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Light like
 These curves lie on the <= span lang=3DIT style=3D'font-family:"Times New Roman";color:#333399;mso-color-i= ndex: 5'>hyperboloid , are straight <= span lang=3DIT style=3D'font-family:"Times New Roman";color:#333399;mso-color-i= ndex: 5'>lines and are characterized by a first integral of the motion which is the angle <= span lang=3DIT style=3D'font-family:"Times New Roman";color:#333399;mso-color-i= ndex: 5'>shift   a
&#= 13;
 Light like geodesics are conserved=
  under conformal transformations