RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE RICHARD ANANTUA x 0 - PowerPoint PPT Presentation

RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE RICHARD ANANTUA x 0 : 7:30p EST x i : Flatiron Institute, Center for Computational Astrophysics

RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE: TALK OUTLINE 1. Popular Science Jeopardy 2. Unpopular Science (lots of tensors): a) Electromagnetism b) Special Relativity c) General Relativity 3. Fate of the Universe References: • Interstellar (2014) - Directed by Christopher Nolan, Interstellar portrays a bleak future of a climate-change- • ravaged Earth and a daring mission led by Cooper’s team to chart out escape plans in a planetary system around the supermassive black hole Gargantua. The Time Machine (1895) – Authored by H. G. Wells, this classic tome portrays a gripping class conflict nearly a • million years into the future of Victorian England, where our ancestors split into two unrecognizable classes— with both upper and lower appearing completely devoid of humanity

JEOPARDY Are We Alone? Time Travel Astronomy Astrology Cosmology or To Infinity and Cosmetology? Beyond! X X X X X X X X X X X X 300 X X X X X X X 400 X X X X 500 X X X X . .

JEOPARDY – TIME TRAVEL

JEOPARDY – ARE WE ALONE •

JEOPARDY – ASTRONOMY •

JEOPARDY – FINAL JEOPARDY

MATH METHODS – VECTOR CALCULUS Vectors in 3D may be multiplied using • a 3 ⃗ Dot product: • , = . # , # + . " , " + . ! , ! = ., cos 7 Cross product: • a×⃗ , = (. " , ! − , " . ! )0 * + (. ! , # − , ! . # )0 ' + (. # , " − , # . " ) ̂ ) A vector field F (x,y,z) is specified by assigning a vector to each point in space. At each point we may • calculate Divergence (whether the vector field looks like a source or a sink) • &$ $ = &$ &) + &$ ! " # ! " ⃗ &' + &* Curl (whether a paddlewheel would rotate in the vector field) • &$ &' − &$ " ! &) &$ &) − &$ !× ⃗ # ! $ = &* &$ &* − &$ " " z &* z z

ELECTROMAGNETISM – MAXWELL’S EQUATIONS Maxwell’s equations describe the dynamics of electric and magnetic fields • v Lorentz Force Law: ⃗ ⃗ $ = 8(+ + v×.) $ ! . ! " + = , × q Gauss’s Law - $ Now that we know these Gauss’s Law ! " . = 0 basic relationships between for Magnetism + and . … Let there be light! Poynting flux (power per unit area): !×+ = − &. Faraday’s Law 5 = 1 &2 ⃗ +×. 3 $ Let’s also see Maxwell’s Eqs. in Ampère Law %& !×. = 3 $ ⃗ 4 + 3 $ - $ integral form %'

ELECTROMAGNETISM – MAXWELL’S EQUATIONS Rewriting Maxwell’s equations in integral form • Differential Form Integral Form = = > ()* ! " + = , ; + " < ⃗ Gauss’s Law - $ - $ Gauss’s Law ∯ . " < ⃗ = = 0 ! " . = 0 for Magnetism A = − &Φ + !×+ = − &. @ + " <⃗ Faraday’s Law &2 &2 &Φ & Ampère Law @ . " <⃗ %& A = 3 $ C ()* + 3 $ - $ !×. = 3 $ ⃗ 4 + 3 $ - $ &2 %'

SPECIAL THEORY OF RELATIVITY – ON THE ELECTRODYNAMICS OF MOVING BODIES • In 1905 Albert Einstein (1879-1955) wrote observations from a series of thought experiments and observations in On the Electrodynamics of Moving Bodies: • Relative, not absolute motion, establishes current in magnet-conductor system Simultaneity depends on an observer’s state of motion, or reference frame • • Clocks synchronize if the difference in times (measured locally) from when light is emitted at A to when light arrives at B equals the corresponding difference for the reverse journey

RELATIVITY OF E AND B - MAGNET AND CONDUCTOR In the frame of the conductor, the moving magnet creates a changing magnetic flux that results in an • electric field in the conductor, and subsequently a current I v A = − &Φ + + @ + " <⃗ S N &2 In the frame of the magnet…? •

RELATIVITY OF E AND B - MAGNET AND CONDUCTOR In the frame of the conductor, the moving magnet creates a changing magnetic flux that results in an • electric field in the conductor, and subsequently a current I v A = − &Φ + + @ + " <⃗ S N &2 In the frame of the magnet, the moving conductor’s free charges are deflected by the magnetic field • I ⃗ ⃗ $ ! = 8v×. $ ! N S . v B ,

POSTULATES OF SPECIAL RELATIVITY • Postulate 1: “The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion” (Einstein, 1905, p.4). The laws of physics are the same in any two inerWal frames • Postulate 2: “Any ray of light moves in the ‘stationary’ system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body” (Einstein, 1905, p.4). The speed of light in a vacuum is the constant c regardless of the moWon of the E = 299,792,458 m/s emiXng frame

REFERENCE FRAMES Frame S is a set of space and time coordinates with Origin O at (0,0,0,0). • At other points r in S, clocks at rest in S are synchronized with the origin by accounting for light travel • time r/c (simultaneous events at O and r register r/c later on clocks at r ). z S r y x O

TRANSFORMATIONS – VECTOR ROTATIONS " (1 − cos &) cos & + ( ! ( ! ( # 1 − cos & − ( $ sin & ( ! ( $ 1 − cos & + ( # sin & A( P ) = r ’=A r • At t=0, imagine rotating 3-vector r into r’ : " (1 − cos &) ( # ( ! 1 − cos & + ( $ sin & cos & + ( # ( # ( $ 1 − cos & − ( ! sin & " (1 − cos &) ( $ ( ! 1 − cos & − ( # sin & ( $ ( # 1 − cos & + ( ! sin & cos & + ( $ 1 • This can be written Q′ - = S = -. Q . = 20 = = −P = = ⊺ z ./0 S r y Length invariant under rotation: Q| 4 = =⃗ Q ⊺ =⃗ Q ⊺ ⃗ Q ⊺ = ⊺ = ⃗ Q ⊺ = 20 = ⃗ Q| 4 |=⃗ Q = ⃗ Q = ⃗ Q = ⃗ Q = |⃗ r’ P x O

LORENTZ TRANSFORMATIONS AND INERTIAL FRAMES Inertial frames move relative to each other at constant speed, without acceleration • In relativity, space and time coordinates of an event observed in different inertial frames are related by • Lorentz transformations t->t’(t,x,y,z), x->x’(t,x,y,z), y-> y’(t,x,y,z), z->z’(t,x,y,z) Frame S’ is said to be “boosted” with velocity (v,0,0) with respect to S, and has Origin O’ at (t’,vt’,0,0). • z z’ S S’ r y v Vector r’ in S’ is related y’ to r in S via Lorentz transformation r’ x, x’ O O’

LORENTZ TRANSFORMATIONS Space and time components are mixed by Lorentz boost L µ n in the direction Y X =(n x ,n y ,n z ) to another • inertial (non-accelerating) frame, where µ = 0,1,2,3; n = 0,1,2,3 −VWX " −VWX # V −VWX ! −VWX " −VWX # V −VWX ! 7 0 W = 8 ; V = 4 4 (V − 1)X ! X " (V − 1)X ! X # −VWX ! 1 + (V − 1)X ! (V − 1)X ! X " (V − 1)X ! X # −VWX ! 1 + (V − 1)X ! 5 = 6 = 02 !" Λ 5 Λ 6 4 4 1 + (V − 1)X " (V − 1)X " X # #" −VWX " (V − 1)X " X ! 1 + (V − 1)X " (V − 1)X " X # −VWX " (V − 1)X " X ! X Y = 1 −VWX # (V − 1)X # X ! 4 −VWX # (V − 1)X # X ! (V − 1)X # X " 1 + (V − 1)X # 4 (V − 1)X # X " 1 + (V − 1)X # The Lorentz transformation of 4-vector x / into (x’) / takes the form • 5 ' 6 ' 5 9 = Λ 6 In Einstein summation notation, an index repeated upstairs/downstairs (or vice versa) indicates a sum over that index

FOUR VECTORS Spacelike Timelike Part Part 9 % = 2:, 9 ⃗ ; % = <9 % <= = > <9 % <: = >2, > ⃗ ? 0 % = 1 2 , 4 5 % = 6 2 , 8 ⃗ @ % = 2A, ⃗ B

LORENTZ TRANSFORMATIONS - EXAMPLE For a boost of velocity v = (v,0,0) of Frame S’ relative to Frame S , what is the Lorentz transformation • tensor, and how does the 4-displacement (' $ , ' 0 , ' 4 , ' 1 ) transform? V −VW 0 0 −VW V 5 = 0 0 Λ 6 0 0 1 0 0 0 0 1 (' $ )′ = V' $ − VW' 0 (' 0 )′ = −VW' $ + V' 0 5 ' 6 ' 5 ′ = Λ 6 (' 4 )′ = ' 4 (' 1 )′ = ' 1

THE TIME MACHINE- SPACE VS. TIME What are fundamental differences between space and time dimensions? Consider p. 6: • Unlike other spacetime dimensions, time has a fixed direction (arrow of time) in which events progress, • as seen in thermodynamics

WORLDLINES, WORLDSHEETS, WORLDVOLUMES The worldline of a particle is the path generated by its trajectory in space and time (times the speed of • light x 0 = ct) The worldsheet is the generalization of a worldline to trajectories of 1-D objects, developed by Leonard • Susskind to describe open and closed strings. Worldvolumes are higher dimensional generalizations. ct S y,z The worldlines of one particle at rest in S, and another accelerating in the positive, then negative x-direction x O

TIME DILATION Compare a light clock that undergoes a tick (round trip of a photon) in Frames S and S’ • Special relativity postulates the speed of light is the same in both stationary and moving clock • During a tick Δ2 9 of the moving clock, the path length of light’s worldline is longer in S’ than in S • z z’ v S’ S $% Δ9 = & y y’ $ % ! (( C *+ " /$) ! Δ9 ' = & 1 h ⟹ Δ2′ = VΔ2, V = 1 − v 4 /E 4 x’ x O’ O