March 9, 2011 Special Relativity, continued

Lorentz Transform

Stellar Aberration Discovered by James Bradley in 1728 Bradley was trying to confirm a claim of the detection of stellar parallax, by Hooke, about 50 years earlier Parallax was reliably measured for the first time by Friedrich Wilhelm Bessel in 1838 Refn: A. Stewart: The Discovery of Stellar Aberration, Scientific American, March 1964 Term paper by Vernon Dunlap, 2005

Because of the Earth’s motion in its orbit around the Sun, the angle at which you must point a telescope at a star changes A stationary telescope Telescope moving at velocity v

Analogy of running in the rain

As the Earth moves around the Sun, it carries us through a succession of reference frames, each of which is an inertial reference frame for a short period of time.

Bradley’s Telescope With Samuel Molyneux, Bradley had master clockmaker George Graham (1675 – 1751) build a transit telescope with a micrometer which allowed Bradley to line up a star with cross-hairs and measure its position WRT zenith to an accuracy of 0.25 arcsec. Note parallax for the nearest stars is ~ 1 arcsec or less, so he would not have been able to measure parallax. Bradley chose a star near the zenith to minimize the effects of atmospheric refraction..

The first telescope was over 2 stories high, attached to his chimney, for stability. He later made a more accurate telescope at his Aunt’s house. This telescope is now in the Greenwich Observatory museum. Bradley reported his results by writing a letter to the Astronomer Royal, Edmund Halley. Later, Brandley became the 3 rd Astronomer Royal.

Vern Dunlap sent this picture from the Greenwich Observatory: Bradley’s micrometer

In Bradley measured the star gamma-Draconis. Note scale

Is ~40 arcsec reasonable? The orbital velocity of the Earth is about v = 30 km/s Aberration formula: (small β) (1)

Let Then α is very small, so cosα~1, sinα~α, so (2) Compare to (1): we get Since β~10^4 radians  40 arcsec at most

BEAMING Another very important implication of the aberration formula is relativistic beaming Suppose That is, consider a photon emitted at right angles to v in the K’ frame. Then

So if you have photons being emitted isotropically in the source frame, they appear concentrated in the forward direction.

The Doppler Effect When considering the arrival times of pulses (e.g. light waves) we must consider - time dilation - geometrical effect from light travel time K: rest frame observer Moving source: moves from point 1 to point 2 with velocity v Emits a pulse at (1) and at (2) The difference in arrival times between emission at pt (1) and pt (2) is where

ω` is the frequency in the source frame. ω is the observed frequency Relativistic Doppler Effect term: relativistic dilation classical geometric term

Transverse Doppler Effect : When θ=90 degrees,

Proper Time Lorentz Invariant = quantity which is the same inertial frames One such quantity is the proper time It is easily shown that under the Lorentz transform

is sometimes called the space-time interval between two events dimension : distance For events connected by a light signal:

Space-Time Intervals and Causality Space-time diagrams can be useful for visualizing the relationships between events. ct x World line for light future past The lines x=+/ ct represent world lines of light signals passing through the origin. Events in the past are in the region indicated. Events in the future are in the region on the top. Generally, a particle will have some world line in the shaded area

x ct The shaded regions here cannot be reached by an observer whose world line passes through the origin since to get to them requires velocities > c Proper time between two events: “time-like” interval “light-like” interval “space-like” interval

x ct x’ ct’ x=ct x’=ct’ Depicting another frame In 2D

Superluminal Expansion Rybicki & Lightman Problem One of the niftiest examples of Special Relativity in astronomy is the observation that in some radio galaxies and quasars, and Galactic black holes, in the very core, blobs of radio emission appear to move superluminally, i.e. at v>>c. - When you look in cm-wave radio emission, e.g. with the VLA, they appear to have radio jets emanating from a central core and ending in large lobes. DRAGN = double-lobed radio-loud active galactic nucleus

Superluminal expansion Proper motion μ=1.20 ± 0.03 marcsec/yr  v(apparent)=8.0 ± 0.2 c μ=0.76 ± 0.05 marcsec/yr  v(apparent)=5.1 ± 0.3 c VLBI (Very Long Baseline Interferometry) or VLBA

Another example:

M 87

HST WFPC2 Observations of optical emission from jet, over course of 5 years: v(apparent) = 6c Birreta et al

Recently, superluminal motions have been seen in Galactic jets, associated with stellar-mass black holes in the Milky Way – “micro-quasars”. + indicates position of X-ray binary source, which is a 14 solar mass black hole. The “blobs” are moving with v = 1.25 c. GRS Radio Emission Mirabel & Rodriguez

Most likely explanation of Superluminal Expansion: vΔtvΔt θ v cosθ Δt (1) (2) v sinθ Δt Observer Blob moves from point (1) to point (2) in time Δt, at velocity v The distance between (1) and (2) is v Δt However, since the blob is closer to the observer at (2), the apparent time difference is The apparent velocity on the plane of the sky is then

v(app)/c

To find the angle at which v(app) is maximum, take the derivative of and set it equal to zero, solve for θ max Result: and then When γ>>1, then v(max) >> v

Special Relativity: 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.