# lorentz transformation

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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!)

Review: Special Relativity

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

d(Ct)2 - dx2 - dy2 - dz2 = d(Ct)2 dx2 dy2 dz2

d(Ct)2 - dx2 - dy2 - dz2 is an invariant! It has the same value in all frames ( = 0 ).

|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

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.

Recall the pictureof the two framesmeasuring thespeed of the samelight signal.

Next step: calculate right hand side of Eq. 1 using matrix result for cdt and dx.Eq. 1Transformation matrix relates differentialsabfha+ bf+ h

= 0= 1= -1

c[- + v/c] dt =0 = v/c

But, we are not going to need the transformation matrix!

We only need to form quantities which are invariant underthe (Lorentz) transformation matrix.

Recall that (cdt)2 (dx)2 (dy)2 (dz)2 is an invariant.

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 )

(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

So, for a particle (in any frame)

(E/c)2 (px)2 (py)2 (pz)2 = m2 c2

covariant and contravariant components*

*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

covariant componentscontravariant componentsUsing indices instead of x, y, z

covariant componentscontravariant components4-dimensional dot productYou can think of the 4-vector dot product as follows:

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

covariant componentscontravariant components4-dimensional vectorcomponent notation x ( x0, x1 , x2, x3 ) =0,1,2,3 = ( ct, x, y, z ) = (ct, r)

x ( x0 , x1 , x2 , x3 ) =0,1,2,3 = ( ct, -x, -y, -z ) = (ct, -r)

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

(repeated index one up, one down) summation)

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

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

d(ct) = [ (ct)/x ] dx = L 0 dx

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

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.

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