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.

Or The Minkowski Metric Then the invariant s can be written

It’s cumbersome to write 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) (1) becomes:

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

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.

The Lorentz Transformation where

Notation:

Instead of writing the Lorentz transform as we can write

or

We can transform an arbitrary 4-vector A ν

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

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 Sinceis a Lorentz invariant, or Kronecker Delta

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

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

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

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

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.

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

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.

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

(3) contravariant tensor covariant tensor related by transforms via

(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

(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,

(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: ifthen is a covariant 4-vector We denoteas

Example:ifis a second-ranked tensor third rank tensor where

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

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

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

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 But also so x’ y’ x y Peacock gives examples for transformations in normal flat 3-space for non-orthogonal axes where

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

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

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

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

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

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

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

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,

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

The transformation equations are:

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

can be written become 4-vector

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 2 nd rank tensor with 6 independent components:

Can show that

We can re-write Maxwell’s Equations

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

Transformation of E and B: Lorentz transform F μν Let

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 andin other frames

What is invariant? (1) i.e. is an invariant (2) so is an invariant

Fields of a Uniformly Moving Charge If we consider a charge q at rest in the K’ frame, the E and B fields are where

Transform to frame K We’ll skip the derivation: see R&L p The field of a moving charge is the expression we derived from the Lienard-Wiechart potential: We’ll consider some implications 

Consider the following special case: (1) charged particle at x=y=z=0 at t=0 v = (v,0,0) uniform velocity (2) Observer at x = z = 0 and y = b sees

What do E x (t), E y (t) look like?

Ex(t) and Ey(t) (1) max(Ey) >> max (Ex) particularly for gamma >>1 (2) Ey, Bz strong only for 2t 0 (3) As particle goes faster, γ increases, E-field points in y-direction more

Ex(t) and Ey(t) (4) The observer sees a pulse of radiation, of duration (5) When gamma >>1, β~1 so (6) To get the spectrum that the observer sees, take the fourier transform of E(t)  E(w) We can already guess the answer 

The spectrum will be whereis the fourier transform The integral can be written in terms of the modified Bessel function of order one, K 1 The spectrum cuts off for

Rybicki & Lightman give expressions for and some approximate analytic forms -- Eqns 4.74, 4.75

Bessel Functions see Numerical Recipes Bessel functions are useful for solving differential equations for systems with cylindrical symmetry Bessel Fn. of the First Kind J n (z), n integer Bessel Fn. of the Second Kind Y n (z), n integer Jn, Yn are linearly independent solutions of

Modified Bessel Functions: