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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’|
Einstein’s 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 – dx’2 – dy’2 – dz’2
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. 1
Transformation matrix relates differentials
a bf h
a + 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.pdf
The metric tensor, g, relates covariant and contravariant components
covariant components
contravariant components
Using indices instead of x, y, z
covariant components
contravariant components
4-dimensional dot productYou can think of the 4-vector dot product as follows:
covariant components
contravariant components
Why all these minus signs?
• Einstein’s 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 /dt
Four dimensional gradient operator
covariant components
contravariant components
4-dimensional vectorcomponent notation
• xµ ( x0, x1 , x2, x3 ) µ=0,1,2,3
= ( ct, x, y, z ) = (ct, r)
• xµ ( x0 , x1 , x2 , x3 ) µ=0,1,2,3
= ( ct, -x, -y, -z ) = (ct, -r)
contravariant components
covariant components
partial derivatives
/xµ µ
= (/(ct) , /x , /y , /z)
= ( /(ct) , )
3-dimensional gradient operator
4-dimensionalgradient operator
partial derivatives
/xµ µ
= (/(ct) , -/x , - /y , - /z)
= ( /(ct) , -)
Note this is not
equal to
They differ by a
minus sign.
Invariant dot products using4-component notation
xµ xµ = µ=0,1,2,3 xµ xµ
(repeated index one up, one down) summation)
xµ xµ = (ct)2 -x2 -y2 -z2
Einstein summation notation
covariant
contravariant
Invariant dot products using4-component notation
µµ = µ=0,1,2,3 µµ (repeated index summation )
= 2/(ct)2 - 2
2 = 2/x2 + 2/y2 + 2/z2
Einstein summation notation
Any four vector dot product has the same valuein all frames moving with constant
velocity w.r.t. each other.
Examples:
xµxµ pµxµ pµpµ µµ
pµµ µAµ
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 Lorentz
transformation.
Summation
over implied
4x4 Lorentz
transformation.
For the graduate students: dx = [ x/x ] dx
= L dx
/x = [ x/x ] /x
= L /x
Invariance: 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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