Relativistic Invariance (Lorentz invariance)

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

contravariant components

covariant components

= g

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.