Relativistic Invariance (Lorentz invariance)

1. 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!)

2. Review: Special Relativity Einstein’s assumption: the speed of light is independent of the (constant ) velocity, v, of the observer. It forms the basis for special relativity. Speed of light = C = |r2 – r1| / (t2 –t1) = |r2’ – r1’ | / (t2’ –t1‘) = |dr/dt| = |dr’/dt’|

3. C2 = |dr|2/dt2 = |dr’|2 /dt’ 2 Both measure the same speed! 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 dtw.r.t the fixed frame.

4. A Lorentz transformation relates position and time in the two frames. Sometimes it is called a “boost” . • http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html#c2

5. 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. Eq. 1 Transformation matrix relates differentials a b f h a + b Next step: calculate right hand side of Eq. 1 using matrix result for cdt’ and dx’. f + h

6. = -1 = 1 = 0

8. c[- + v/c] dt =0  = v/c

10. But, we are not going to need the transformation matrix! We only need to form quantities which are invariant under the (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.

11. 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

12. 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

13. The metric tensor, g, relates covariant and contravariant components covariant components contravariant components

14. Using indices instead of x, y, z covariant components contravariant components

15. 4-dimensional dot product You can think of the 4-vector dot product as follows: covariant components contravariant components

16. 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

17. Four dimensional gradient operator contravariant components covariant components

18. 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

19. partial derivatives /xµ µ = (/(ct) , /x, /y , /z) = ( /(ct) , ) 4-dimensional gradient operator 3-dimensional gradient operator

20. partial derivatives /xµ  µ = (/(ct) , -/x, - /y , - /z) = ( /(ct) , -) Note this is not equal to  They differ by a minus sign.

21. Invariant dot products using4-component notation xµxµ = µ=0,1,2,3 xµxµ (repeated index one up, one down)  summation) xµxµ = (ct)2 -x2 -y2 -z2 contravariant covariant Einstein summation notation

22. Invariant dot products using4-component notation µµ = µ=0,1,2,3µµ (repeated index summation ) = 2/(ct)2 - 2 2 = 2/x2 + 2/y2 + 2/z2 Einstein summation notation

23. Any four vector dot product has the same valuein all frames moving with constantvelocity w.r.t. each other. Examples: xµxµ pµxµ pµpµµµ pµµ µAµ

24. For the graduate students: Considerct =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 = L0dx dx  = [ x/x ]dx = L  dx Summation over implied First row of Lorentz transformation. 4x4 Lorentz transformation.

25. 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

26. 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.