March 9, 2011

1. March 9, 2011 Special Relativity, continued

2. Lorentz Transform

3. 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

4. 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

5. Analogy of running in the rain

6. 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.

7. 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. .

8. 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 3rd Astronomer Royal.

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

11. In 1727-1728 Bradley measured the star gamma-Draconis. Note scale

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

13. 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

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

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

16. 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

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

18. Transverse Doppler Effect: When θ=90 degrees,

19. 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

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

21. Space-Time Intervals and Causality Space-time diagrams can be useful for visualizing the relationships between events. ct 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. future x World line for light past Generally, a particle will have some world line in the shaded area

22. 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 x Proper time between two events: “time-like” interval “light-like” interval “space-like” interval

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

24. Superluminal Expansion Rybicki & Lightman Problem 4.8 - 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

25. Superluminal expansion VLBI (Very Long Baseline Interferometry) or VLBA 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

26. Another example:

27. M 87

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

29. Recently, superluminal motions have been seen in Galactic jets, associated with stellar-mass black holes in the Milky Way – “micro-quasars”. GRS 1915+105 Radio Emission + indicates position of X-ray binary source, which is a 14 solar mass black hole. The “blobs” are moving with v = 1.25 c. Mirabel & Rodriguez

30. Most likely explanation of Superluminal Expansion: (1) 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 vΔt θ v cosθΔt (2) v sinθΔt Observer The apparent velocity on the plane of the sky is then

31. v(app)/c

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

33. Special Relativity: 4-vectors and Tensors

34. 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.

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

36. 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:

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

38. 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.

39. The Lorentz Transformation where

40. Notation:

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

42. or

43. We can transform an arbitrary 4-vector Aν

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

45. 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

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

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

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

49. 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

50. 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.