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March 21, 2011

March 21, 2011. Turn in HW 6 Pick up HW 7: Due Monday, March 28 Midterm 2: Friday, April 1. 4-vectors and Tensors. Four Vectors. x,y,z and t can be formed into a 4-dimensional vector with components. Written. 4-vectors can be transformed via multiplication by a 4x4 matrix.

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March 21, 2011

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  1. March 21, 2011 Turn in HW 6 Pick up HW 7: Due Monday, March 28 Midterm 2: Friday, April 1

  2. 4-vectors and Tensors

  3. Four Vectors x,y,z and t can be formed into a 4-dimensional vector with components Written 4-vectors can be transformed via multiplication by a 4x4 matrix.

  4. The Minkowski Metric Or Then the invariant s can be written

  5. It’s cumbersome to write (1) So, following Einstein, we adopt the convention that when Greek indices are repeated in an expression, then it is implied that we are summing over the index for 0,1,2,3. (1) becomes:

  6. Now let’s define xμ – with SUBSCRIPT rather than SUPERSCRIPT. Covariant 4-vector: Contravariant 4-vector: More on what this means later.

  7. So we can write i.e. the Minkowski metric, can be used to “raise” or “lower” indices. Note that instead of writing we could write assume the Minkowski metric.

  8. The Lorentz Transformation where

  9. Notation:

  10. Instead of writing the Lorentz transform as we can write

  11. or

  12. We can transform an arbitrary 4-vector Aν

  13. Kronecker-δ Define (1) Note: (2) For an arbitrary 4-vector

  14. Inverse Lorentz Transformation We wrote the Lorentz transformation for CONTRAVARIANT 4-vectors as The L.T. for COVARIANT 4-vectors than can be written as where Since is a Lorentz invariant, or Kronecker Delta

  15. General 4-vectors (contravariant) Transforms via Covariant version found by Minkowski metric Covariant 4-vectors transform via

  16. Lorentz Invariants or SCALARS Given two 4-vectors SCALAR PRODUCT This is a Lorentz Invariant since

  17. Note: can be positive (space-like) zero (null) negative (time-like)

  18. The 4-Velocity (1) The zeroth component, or time-component, is where and Note: γu is NOT the γ in the Lorentz transform which is

  19. The 4-Velocity (2) The spatial components where So the 4-velocity is So we had to multiply by to make a 4-vector, i.e. something whose square is a Lorentz invariant.

  20. How does transform? so... or where where v=velocity between frames

  21. Wave-vector 4-vector Recall the solution to the E&M Wave equations: The phase of the wave must be a Lorentz invariant since if E=B=0 at some time and place in one frame, it must also be = 0 in any other frame.

  22. Tensors (1) Definitions zeroth-rank tensor Lorentz scalar first-rank tensor 4-vector second-rank tensor 16 components: (2) Lorentz Transform of a 2nd rank tensor:

  23. (3) contravariant tensor covariant tensor related by transforms via

  24. (4) Mixed Tensors one subscript -- covariant one superscript – contra variant so the Minkowski metric “raises” or “lowers” indices. (5) Higher order tensors (more indices) etc

  25. (6) Contraction of Tensors Repeating an index implies a summation over that index.  result is a tensor of rank = original rank - 2 Example: is the contraction of (sum over nu) (7) Tensor Fields A tensor field is a tensor whose components are functions of the space-time coordinates,

  26. (7) Gradients of Tensor Fields Given a tensor field, operate on it with to get a tensor field of 1 higher rank, i.e. with a new index Example: if then is a covariant 4-vector We denote as

  27. Example: if is a second-ranked tensor third rank tensor where

  28. (8) Divergence of a tensor field Take the gradient of the tensor field, and then contract. Example: Divergence is Given vector Example: Divergence is Tensor

  29. (9) Symmetric and anti-symmetric tensors If then it is symmetric then it is anti-symmetric If

  30. COVARIANT v. CONTRAVARIANT 4-vectors Refn: Jackson E&M p. 533 Peacock: Cosmological Physics Suppose you have a coordinate transformation which relates or by some rule. A COVARIANT 4-vector, Bα, transforms “like” the basis vector, or or A CONTRAVARIANT 4-vector transforms “oppositely” from the basis vector

  31. For “NORMAL” 3-space, transformations between e.g. Cartesian coordinates with orthogonal axes and “flat” space  NO DISTINCTION Example: Rotation of x-axis by angle θ y’ y x x’ But also so Peacock gives examples for transformations in normal flat 3-space for non-orthogonal axes where

  32. Now in SR, we add ct and consider 4-vectors. However, we consider only inertial reference frames: - no acceleration - space is FLAT So COVARIANT and CONTRAVARIANT 4-vectors differ by Where the Minkowski Matrix is So the difference is the sign of the time-like component

  33. Example: Show that xμ=(ct,x,y,z) transforms like a contravariant vector: Let’s let

  34. In SR In GR Gravity treated as curved space. Of course, this type of picture is for 2D space, and space is really 3D

  35. Two Equations of Dynamics: where and = The Affine Connection, or Christoffel Symbol

  36. For an S.R. observer in an inertial frame: And the equation of motion is simply Acceleration is zero.

  37. Covariance of Electromagnetic Phenomena

  38. Covariance of Electromagnetic Phenomena 4-current and 4-potentials Define the 4-current where = 3-vector, current = charge density Recall the equation for consersvation of charge: In tensor notation

  39. Let’s look at all this another way: Consider a volume element (cube) with dimensions containing N electrons Charge in the cube = N e Charge density Suppose in the K’ frame, the charges are at rest, so that the current What is the current in the K frame? Assume motion with velocity v, parallel to the x-direction

  40. The volume in K will be length contraction in one direction The number of electrons in the volume must be the same in both the K and K’ frames. Thus,

  41. Similarly, for the current: Now, in analogy to the expression for proper time: we can write

  42. The transformation equations are:

  43. 4-Potential Recall the vector potential and scalar potential which satisfied and the Lorentz guage Define the 4-potential

  44. can be written become 4-vector

  45. Recall that E and B are related to A and φ by E and B have 6 independent components, so we’ll write the Electro-magnetic force as an anti-symmetric 2nd rank tensor with 6 independent components:

  46. Can show that

  47. We can re-write Maxwell’s Equations

  48. More Notation: Instead of writing Write where [ ] means: permute indices even permutations  + sign odd permutations  - sign E.G.:

  49. Transformation of E and B: Lorentz transform Fμν Let

  50. or, for v = velocity in x-direction NOTE: The concept of a pure electric field (B=0) or a pure magnetic field (E=0) is NOT a Lorentz invariant. if B=0 in one frame, in general and in other frames

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