lorentz transformation

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transformaciones de lorentz en relatividad adfkaldsggnadfjnvjkb


<p>Slide 1</p> <p>Relativistic Invariance(Lorentz invariance)The laws of physics are invariant under a transformation between two coordinate frames moving at constant velocity w.r.t. each other.(The world is not invariant, but the laws of physics are!)</p> <p>Review: Special Relativity</p> <p>Speed of light = C = |r2 r1| / (t2 t1) = |r2 r1 | / (t2 t1) = |dr/dt| = |dr/dt| Einsteins assumption: the speed of light is independent of the (constant ) velocity, v, of the observer. It forms the basis for special relativity. This can be rewritten: d(Ct)2 - |dr|2 = d(Ct)2 - |dr|2 = 0 </p> <p> d(Ct)2 - dx2 - dy2 - dz2 = d(Ct)2 dx2 dy2 dz2</p> <p> d(Ct)2 - dx2 - dy2 - dz2 is an invariant! It has the same value in all frames ( = 0 ).</p> <p> |dr| is the distance light moves in dt w.r.t the fixed frame.C2 = |dr|2/dt2 = |dr|2 /dt 2 Both measure the same speed!http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html#c2</p> <p>A Lorentz transformation relates position and time in the two frames. Sometimes it is called a boost .How does one derive the transformation?Only need two special cases.</p> <p>Recall the pictureof the two framesmeasuring thespeed of the samelight signal.</p> <p>Next step: calculate right hand side of Eq. 1 using matrix result for cdt and dx.Eq. 1Transformation matrix relates differentialsabfha+ bf+ h</p> <p>= 0= 1= -1</p> <p>c[- + v/c] dt =0 = v/c</p> <p>But, we are not going to need the transformation matrix!</p> <p>We only need to form quantities which are invariant underthe (Lorentz) transformation matrix.</p> <p>Recall that (cdt)2 (dx)2 (dy)2 (dz)2 is an invariant.</p> <p> It has the same value in all frames ( = 0 ). This is a special invariant, however. Suppose we consider the four-vector: (E/c, px , py , pz )</p> <p> (E/c)2 (px)2 (py)2 (pz)2 is also invariant. In the center of mass of a particle this is equal to (mc2 /c)2 (0)2 (0)2 (0)2 = m2 c2</p> <p>So, for a particle (in any frame)</p> <p> (E/c)2 (px)2 (py)2 (pz)2 = m2 c2 </p> <p> covariant and contravariant components*</p> <p>*For more details about contravariant and covariant components seehttp://web.mst.edu/~hale/courses/M402/M402_notes/M402-Chapter2/M402-Chapter2.pdfThe metric tensor, g, relates covariant and contravariant components</p> <p>covariant componentscontravariant componentsUsing indices instead of x, y, z</p> <p>covariant componentscontravariant components4-dimensional dot productYou can think of the 4-vector dot product as follows: </p> <p>covariant componentscontravariant componentsWhy all these minus signs?Einsteins assumption (all frames measure the same speed of light) gives : d(Ct)2 - dx2 - dy2 dz2 = 0 From this one obtains the speed of light. It must be positive! C = [dx2 + dy2 + dz2]1/2 /dt16Four dimensional gradient operator</p> <p>covariant componentscontravariant components4-dimensional vectorcomponent notation x ( x0, x1 , x2, x3 ) =0,1,2,3 = ( ct, x, y, z ) = (ct, r)</p> <p> x ( x0 , x1 , x2 , x3 ) =0,1,2,3 = ( ct, -x, -y, -z ) = (ct, -r)</p> <p>contravariant componentscovariant componentspartial derivatives /x = (/(ct) , /x , /y , /z) = ( /(ct) , )3-dimensional gradient operator4-dimensionalgradient operatorpartial derivatives /x = (/(ct) , -/x , - /y , - /z) = ( /(ct) , -)Note this is not equal to They differ by a minus sign.Invariant dot products using4-component notation x x = =0,1,2,3 x x </p> <p> (repeated index one up, one down) summation)</p> <p> x x = (ct)2 -x2 -y2 -z2 Einstein summation notationcovariantcontravariant Invariant dot products using4-component notation = =0,1,2,3 (repeated index summation ) = 2/(ct)2 - 2</p> <p> 2 = 2/x2 + 2/y2 + 2/z2 Einstein summation notationAny four vector dot product has the same valuein all frames moving with constantvelocity w.r.t. each other. Examples: xx px pp p A For the graduate students: Consider ct = f(ct,x,y,z) Using the chain rule: d(ct) = [f/(ct)]d(ct) + [f/(x)]dx + [f/(y)]dy + [f/(z)]dz </p> <p> d(ct) = [ (ct)/x ] dx = L 0 dx </p> <p> dx = [ x/x ] dx = L dx First row of Lorentztransformation. Summationover implied 4x4 Lorentztransformation. For the graduate students: dx = [ x/x ] dx = L dx /x = [ x/x ] /x = L /xInvariance: dx dx = L dx L dx = (x/x)( x/x)dx dx = [x/x ] dx dx = dx dx = dx dx</p> <p>Lorentz Invariance Lorentz invariance of the laws of physics is satisfied if the laws are cast in terms of four-vector dot products!Four vector dot products are said to be Lorentz scalars.In the relativistic field theories, we must use Lorentz scalars to express the interactions. </p>