March 9, 2011 Special Relativity, continued Lorentz Transform x ( x v t ) y y z z t (t cv2 x ) cos cos 1 cos 1 12 v c 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 3rd Astronomer Royal. Vern Dunlap sent this picture from the Greenwich Observatory: Bradley’s micrometer In 1727-1728 Bradley measured the star gamma-Draconis. Note scale Is ~40 arcsec reasonable? The orbital velocity of the Earth is about v = 30 km/s v 4 10 c Aberration formula: cos cos ' 1 cos (cos )( 1 cos ) 2 2 cos cos cos (small β) 2 (1) cos cos sin Let Then angl of abe cos cos sin sin cos cos( ) α is very small, so cosα~1, sinα~α, so (2) cos cos sin Compare to (1): we get 2 cos cos sin sin Since β~10^4 radians 40 arcsec at most BEAMING Another very important implication of the aberration formula is relativistic beaming sin tan cos cos cos ' 1 cos Suppose Then 2 tan 1 That is, consider a photon emitted at right angles to v in the K’ frame. s in 1 1 F 1 , or si i sma s n 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 d t A t t 1 cos c where 2 t ω` is the frequency in the source frame. ω is the observed frequency 2 t A 1 cos 1 term: relativistic dilation 1 1 cos classical geometric term Relativistic Doppler Effect Transverse Doppler Effect: 1 cos When θ=90 degrees, Proper Time Lorentz Invariant = quantity which is the same inertial frames One such quantity is the proper time c 2d 2 c 2dt2 dx2 dy2 dz2 It is easily shown that under the Lorentz transform d d cd is sometimes called the space-time interval between two events • dimension : distance • For events connected by a light signal: cd 0 Space-Time Intervals and Causality Space-time diagrams can be useful for visualizing the relationships between events. ct future x 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. World line for light Generally, a particle will have some world line in the shaded area 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: ct x 2 ct x 2 2 0 2 ct x 2 2 “time-like” interval “light-like” interval 2 ct x 2 2 2 0 “space-like” interval 2 ct ct’ Depicting another frame x’ x x=ct x’=ct’ In 2D 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 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 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”. 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 Most likely explanation of Superluminal Expansion: (1) v cosθ Δt Blob moves from point (1) to point (2) in time Δt, at velocity v θ vΔt The distance between (1) and (2) is v Δt (2) v sinθ Δt However, since the blob is closer to the observer at (2), the apparent time difference is Observer t app v t 1 cos c The apparent velocity on the plane of the sky is then v app v t sin t app v sin v 1 cos c v app v sin v 1 cos c v(app)/c To find the angle at which v(app) is maximum, take the derivative of v app v sin v 1 cos c and set it equal to zero, solve for θmax Result: then cos MAX v MAX v c and v 1 2 v 2 1 sin MAX 1 2 1 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 0 x ct x1 x x y 2 x3 z Written x 0,1,2,3 4-vectors can be transformed via multiplication by a 4x4 matrix. The Minkowski Metric 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 if 0 Or 1 if 1,2,3 0 if Then the invariant s s c t x y z 2 2 2 2 2 can be written 3 s 2 0 3 x 0 x 2 3 s It’s cumbersome to write 2 0 3 x 0 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: s x x 2 x (1) Now let’s define xμ – with SUBSCRIPT rather than SUPERSCRIPT. Covariant 4-vector: x Contravariant 4-vector: x0 ct x1 x x2 y x3 z More on what this means later. x x 0 ct x1 x x2 y x3 z So we can write x x x x i.e. the Minkowski metric, can be used to “raise” or “lower” indices. s 2 x x Note that instead of writing we could write s x x 2 assume the Minkowski metric. The Lorentz Transformation where 0 0 0 0 0 0 1 0 0 0 0 1 v 1 and c 1 2 Notation: F00 F F 10 F20 F30 F01 F02 F11 F21 F12 F22 F31 F32 F03 F13 F23 F33 Instead of writing the Lorentz transform as x ( x vt ) y y z z v t (t 2 x) c we can write x x 0 0 or 0 0 0 0 1 0 0 ct t x 0 x x x 0 y y 1 z z ct ct x x x ct y y z z We can transform an arbitrary 4-vector Aν A A Kronecker-δ Define Note: (1) 1 0 0 0 0 1 0 0 (2) For an arbitrary 4-vector 0 0 1 0 0 0 0 1 A A A 1 for 0 for ~ Inverse Lorentz Transformation We wrote the Lorentz transformation for CONTRAVARIANT 4-vectors as x x The L.T. for COVARIANT 4-vectors than can be written as ~ x x Since s x x 2 or where ~ is a Lorentz invariant, x x x x ~ x x x x ~ Kronecker Delta General 4-vectors Transforms via A (contravariant) A A A A Covariant version found by Minkowski metric Covariant 4-vectors transform via ~ A A Lorentz Invariants or SCALARS A A Given two 4-vectors and B B SCALAR PRODUCT A A B B This is a Lorentz Invariant since ~ A B A B ~ A B A B A B Note: A A can be positive (space-like) zero (null) negative (time-like) The 4-Velocity dx u d (1) The zeroth component, or time-component, is 0 dx dt 0 u c c u d d and where u 1 u2 1 2 c u u magnitudeof theordinary velocity dx dy dz , , dt dt dt Note: γu is NOT the γ in the Lorentz transform which is 1 v2 1 2 c dx u d The 4-Velocity (2) The spatial components i dx i i u uu d So the 4-velocity is c u u u where u 1 u2 1 2 c u theordinary velocity So we had to multiply by u to make a 4-vector, i.e. something whose square is a Lorentz invariant. dx dy dz , , dt dt dt How does so... u u0 (u 0 u1 ) u1 ( u 0 u1 ) u2 u 2 u or uc (c u u u ) 1 1 1 uu ( c u u u ) u u 3 u u 3 2 u 1 2 c u 1 2 c 2 u 2 2 u u u u u 3 u 3 where u u transform? v 1 2 c 2 1 / 2 1 / 2 1 / 2 where v=velocity between frames Wave-vector 4-vector Recall the solution to the E&M Wave equations: E exp(ik r it ) 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. / c k k Tensors (1) Definitions zeroth-rank tensor first-rank tensor second-rank tensor Lorentz scalar 4-vector 16 components: (2) Lorentz Transform of a 2nd rank tensor: T T T 0,1,2,3 0.1.2.3 (3) T contravariant tensor T covariant tensor related by T T transforms via ~ ~ T T (4) Mixed Tensors one subscript -- covariant one superscript – contra variant T T T T so the Minkowski metric “raises” or “lowers” indices. (5) Higher order tensors (more indices) T T etc (6) Contraction of Tensors Repeating an index implies a summation over that index. result is a tensor of rank = original rank - 2 Example: T is the contraction of T (sum over nu) (7) Tensor Fields A tensor field is a tensor whose components are functions of the space-time coordinates, 0 1 2 x ,x ,x ,x 3 (7) Gradients of Tensor Fields Given a tensor field, operate on it with x for x x 0, x1, x 2, x 3 to get a tensor field of 1 higher rank, i.e. with a new index Example: if scalar x We denote x then is a covariant 4-vector as , Example: T if is a second-ranked tensor T , x where third rank tensor thecomponentsof T (8) Divergence of a tensor field Take the gradient of the tensor field, and then contract. Example: A Given vector Divergence is A , Divergence is T , Example: Tensor T (9) Symmetric and anti-symmetric tensors If If T T T T then it is symmetric then it is anti-symmetric COVARIANT v. CONTRAVARIANT 4-vectors Refn: Jackson E&M p. 533 Peacock: Cosmological Physics x to x Suppose you have a coordinate transformation which relates or x0 , x1, x 2 , x3 x0 , x1, x2 , x3 by some rule. A COVARIANT 4-vector, Bα, transforms “like” the basis vector, or x 0 x1 x 2 x 3 B B0 B1 B2 B3 x x x x or x B B x A CONTRAVARIANT 4-vector transforms “oppositely” from the basis vector x A A x 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 x x cos dx cos dx x x cos dx cos dx dx dx dx dx Peacock gives examples for transformations in normal flat 3-space for non-orthogonal axes where dx dx dx dx 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 A A Where the Minkowski Matrix is 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 So the difference is the sign of the time-like component Example: Show that xμ=(ct,x,y,z) transforms like a contravariant vector: x x ct ct ct x x x x x x x 0 x 1 x 2 x 3 0 x 1 x 2 x 3 x x x x x Let’s let 0 x ct' x 0 x ct 0 x x 1 x x 1 x In SR In GR x x A g A g themetrictensor Gravity treated as curved space. Of course, this type of picture is for 2D space, and space is really 3D Two Equations of Dynamics: 2 d x dx dx 0 2 d d d c d g dx dx 2 where and 2 proper time 1 g g g g 2 x x x = The Affine Connection, or Christoffel Symbol For an S.R. observer in an inertial frame: g 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 And the equation of motion is simply 2 d x 0 2 d Acceleration is zero.