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

Einstein’s assumption: the speed of light is independent of the (constant ) velitlocity, v, of the observer. It forms the bbiasis for speciilal reltiitlativity.

’ ’ ’ ‘ Sppgeed of light = C = |r2 –r1||( / (t2 –t1))| = |r2 –r1 ||( / (t2 –t1 ) = |dr/dt| = |dr’/dt’| C2 = |dr|2/dt2 = |dr’|2 /dt’ 2 Both measure the same speed!

ThiscanberewrittenThis 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. A LtLorentz tftransforma tion reltlates position and time in the two frames. Sometimes it is called a “boost” .

• http://hyperphysics.phy‐astr.gsu.edu/hbase/relativ/ltrans.html#c2 How does one “derive” the transformation? Only need two special cases.

Recall the picture of the two frames measuring the speed of the same light signal.

covariant and contravariant components*

*For more details about contravariant and covariant components see http://web.mst.edu/~hale/courses/M402/M402_notes/M402‐Chapter2/M402‐Chapter2.pdf metric tensor relates components Using indices instead of x, y, z 4‐dimensional dot product

You can think of the 4‐vector dot product as follows: 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 4‐dimensional vector component notation

• xµ stdtands for x0, x1 , x2, x3 for µ=0,1,2,3 ct, x, y, z = (ct, r)

• xµ stands for

x0 , x1 , x2 , x3 for µ=01230,1,2,3 ct, ‐x, ‐y, ‐z = (ct, ‐r) partial derivatives

µ /x µ stands for

(/(ct) , /x, /y, /z)

= ( /(ct) , ) partial derivatives

µ stands for

(/(ct) , ‐/x, ‐ /y, ‐ /z)

= ( /(ct) , ‐) Invariant dot products using 4‐component notation

µ µ xµx = µ=01230,1,2,3 xµx

(repeated index summation implied)

= (ct)2 ‐x2 ‐y2 ‐z2 Invariant dot products using 4‐component notation

µ µ µ = µ=0123µ=0,1,2,3 µ (repeated index summation implied)

= 2/(ct)2 ‐ 2 2 = 2/x2 + 2/y2 + 2/z2 Any four vector dot product has the same value in all frames moving with constant velocity wrtw.r.t. each other.

Examples: µ µ xµx pµx µ µ pµp xµx

µ µ pµx xµA Lorentz Invariance

• Lorentz invariance of the laws of physics is satisfied if the laws are cast in terms of four‐vector dot prodd!ucts! • Four vector dot products are said to be “Lorentz scalars”. • In the relativistic field theories, we must use “Lorentz scalars” to express the iiinteractions.