Summer 2017 Astron 9 Week 5 Final Version

RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE RICHARD ANANTUA PREDESTINATION (MON JUL 31) REVIEW OF JUL 24, 26 & 28

• Flying atomic clocks, muon decay, light bending around stars and gravitational waves provide experimental confirmation of relativity • In 1919, Einstein’s prediction for the deflection of starlight was confirmed by • The 1971 Hafele-Keating experiment confirmed • The 1977 CERN muon experiment confirmed • In 2016 LIGO confirmed the existence of REVIEW OF JUL 24, 26 & 28

• Flying atomic clocks, muon decay, light bending around stars and gravitational waves provide experimental confirmation of relativity • In 1919, Einstein’s prediction for the deflection of starlight was confirmed by Sir Arthur Eddington • The 1971 Hafele-Keating experiment confirmed motional and gravitational time dilation • The 1977 CERN muon experiment confirmed motional time dilation • In 2016 LIGO confirmed the existence of gravitational waves TIME TRAVEL IN RELATIVITY

• Methods of time travel theoretically possible in relativity are • (Apparently) faster than light travel – window into the past? • Closed timelike curved • (Apparent) paradoxes with time travel • Kill your grandfather – past time travel may alter seemingly necessary conditions for the future time traveler to have started the journey • Predestination - apparent cause of past event is in the future SPECIAL RELATIVISTIC VELOCITY ADDITION

• In special relativity, no object or information can travel faster than light, a fact reflected in the relativistic law of composition of velocities

v v vAB v = A vv

vBC B

C • vAB is the (1D) velocity of A with respect to B

• vBC is the (1D) velocity of B with respect to C POP QUIZ - FASTER THAN LIGHT TRAVEL

• A spaceship traveling at v>c is returning from a moving planet (L0 away right before the journey). Which arrives at the Earth observer earlier: information from later times or information from earlier times? Is there a speed at which the ship travels into the past?

t=t t=t3 2 t=t1 v>c

vPE L0 POP QUIZ - FASTER THAN LIGHT TRAVEL

• For the return trip, light emitted from later times arrives earlier at the Earth observer, so the observer sees the spaceship’s travel as a movie running backwards.

t=t t=t3 2 t=t1 v>c

vPE L0 POP QUIZ - FASTER THAN LIGHT TRAVEL

• For the return trip, light emitted from later times arrives earlier at the Earth observer, so the observer sees the spaceship’s travel as a movie running backwards. If the speed increases to make the total trip time negative, ships may arrive before a ship is launched.

vSE

vPE L0

(Δt)=? +? POP QUIZ - FASTER THAN LIGHT TRAVEL

|vSP|=|vSE| v − v v = v v 1 − c

vPE L0 � � + v � v − v (Δt) = + , v = v ≡ v v − v v − v v v 1 − c POP QUIZ - FASTER THAN LIGHT TRAVEL

-vSE

vPE L0 � � + v � v − v (Δt) = + , v = v ≡ v v − v v − v v v 1 − c

� � ⟹ v = + � − 1 v v REVIEW OF JUL 31

• If the spaceship reverses instantly at the planet, what appears to happen? How could mass be conserved? REVIEW OF JUL 31

• Exercise: Faster-than-light travel has unexpected theoretical implications: • If a spaceship makes a faster-than-light journey to a distant planet and back, what appears different about the outbound and return journeys? • The outbound journey appears to run forward while the return journey appears to run backwards in time • If the spaceship reverses instantly at the planet, what appears to happen? How could mass be conserved? • Image pair annihilation: After the spaceship arrives on Earth, an image of the spaceship appears to go backwards until it reaches the outbound spaceship’s image at the planet, then both images disappear. Mass could be conserved if the backwards-moving ship is made of antimatter FASTER THAN LIGHT TRAVEL - WARPDRIVE SPACETIME

• In 1994, Miguel Alcubierre proposed a partially curved spacetime in which long distance travel may appear faster than light travel according to observers in the flat portion of spacetime http://iopscience.iop.org/article/10.1088/0264- 9381/11/5/001/meta;jsessionid=44D4E5916BC9330CA8BBC73C36457298.c2.iopscience.cld.iop.org . WARPDRIVE SPACETIME (GROUP ACTIVITY) • Consider the metric �� = −�� + (�� − � �⃗ − �⃗ (�) v � ) + �� + �� , v (�) = �(�) �⃗(�) = 0 where is the 0 worldline of a spaceship moving right and f is a positive function with f(0)=1, f(rs>R)=0 0 • Show that some future lightcones on an x -x-plane slice have slopes < 1; show vs>c for some portion of the journey. WARPDRIVE SPACETIME (GROUP ACTIVITY) • Consider the metric �� = −�� + (�� − � �⃗ − �⃗ (�) v � ) + �� + �� , v (�) = �(�) �⃗(�) = 0 where is the 0 worldline of a spaceship moving right and f is a positive function with f(0)=1, f(rs>R)=0 0 • Show that some future lightcones on an x -x-plane slice have slopes < 1; show vs>c for some portion of the journey. 0 = �� = −� �� + (�� − � �⃗ − �⃗ � v � ) ⟹ �� = �� − � �⃗ − �⃗(�) v � �� v � ⟹ 1 = − � �⃗ − �⃗ (�) �� v � ⟹ ±1 = − � �⃗ − �⃗ (�) �� ⟹ = ± ⃗⃗ () �� If 0 < � �⃗ − �⃗ � < 1, and the ship moves right � > 0 �� then <1 for the rightmost portion of the lightcone ��

If 0 < � �⃗ − �⃗ � < 1, �� + � �� Warpdrive spacetime for smooth positive function = ±� + � �⃗ − �⃗ (�) v � ⟹ v � = > � �⃗ − �⃗(�) . Confer Hartle p. 144. �� ⃗ − �⃗ � � �⃗ − �⃗ (�) = 1 − � MINKOWSKI METRIC IN SPHERICAL COORDINATES • Minkowski flat space is empty, so it can be equally well represented in coordinates emphasizing spherical symmetry instead of planar symmetry Cartesian Coordinates Spherical Coordinates

�� = �� = −� �� + �� + �� + �� = −� �� + �� + � �� + � sin � ��

• Exercise: What are the lengths of Arc 1 and Arc 2 if the sphere has radius r+dr and �, �, d� and d� are defined as in the diagram? �̂

Arc 1dr �

Arc 2 MINKOWSKI METRIC IN SPHERICAL COORDINATES • Minkowski flat space is empty, so it can be equally well represented in coordinates emphasizing spherical symmetry instead of planar symmetry Cartesian Coordinates Spherical Coordinates

�� = �� = −� �� + �� + �� + �� = −� �� + �� + � �� + � sin � ��

• Exercise: What are the lengths of Arc 1 and Arc 2 if the sphere has radius r+dr and �, �, d� and d� are defined as in the diagram? �̂

Arc 1dr � Arc 1: ��

Arc 2: � sin � �� Arc 2 SCHWARZSCHILD GEOMETRY

• Schwarzschild line element for geometry around stationary black hole

��2= −� 1 − �� + 1 − �� + �(�� + ��) � =

• The metric has a coordinate singularity at r=2M= KERR GEOMETRY

� = • Kerr metric ��2=−� 1 − �� − �� + �� + � ≡ �/�, � ≡ � + � cos �, ∆≡ � − 2�� + � �� + (� + � + ) �� • For a Kerr (rotating) black hole • The (outer) event horizon is slightly within the Schwarzschild radius • The ergosphere drags spacetime so fast all light appears to co-rotate with the black hole • The inner horizon is a 2-way horizon ROBERTSON-WALKER METRICS

• The Universe is observed, to a large extent, to be spatially isotropic and homogeneous

��2=−�� + �(�)�� where the scale factor sets the separation of points in the Universe at different times�(�) FLAT ROBERTSON-WALKER SPACE METRIC

• The RW (homogeneous, isotropic) metric in flat 3D space is ��2=−�� + �(�)(�� + �� + ��) which in polar coordinates

� = � sin � cos � �� =? � = � sin � sin � ⟹ �� =? � = � cos � �� =? takes the form ??? FLAT ROBERTSON-WALKER SPACE METRIC

• The RW (homogeneous, isotropic) metric in flat 3D space is ��2=−�� + �(�)(�� + �� + ��) which in polar coordinates

� = � sin � cos � �� = � cos � cos � �� − � sin � sin � �� + cos � cos � �� = � sin � sin � ⟹ �� = � cos � sin � �� + � sin � cos � �� + sin � sin � �� = � cos � �� = −� sin � �� + cos � �� takes the form

��2=−�� + �(�)(�� + �(�� + sin��)) ROBERTSON-WALKER METRIC IN CURVED SPACES

• Curved 3D spaces can be envisioned by embedding surfaced in higher dimensional space. Take the analogues of the sphere and hyperboloid living in 3-space to their generalizations to 4-space (r �� -space)

• For the unit 3-sphere in a positively curved isotropic, homogeneous space:

��2 = −�� + �(�) �� + sin �( d� + sin � ��) 0 ≤ �, � ≤ �, 0 ≤ � ≤ 2�

• For negatively curved isotropic, homogeneous 3-space:

0 ≤ � ≤ �, 0 ≤ � ≤ 2� ��2 = −�� + �(�) �� + sinh �( d� + sin � ��) 0 ≤ � ≤ ∞ ROBERTSON-WALKER METRICS

• Exercise: Show that for (sin � , 1) (�, �) = (�, 0) (sinh � , −1) the RW metrics ��2 = −�� + �(�) �� + sin �( d� + sin � ��)

��2 = −�� + �(�)(�� + �(�� + sin��))

��2 = −�� + �(�) �� + sinh �( d� + sin � ��)

can be written in the unified form ��2 = −�� + � � + � �� + sin�� ROBERTSON-WALKER METRICS

• Exercise: Show that for (sin � , 1) (�, �) = (�, 0) (sinh � , −1) the RW metrics ��2 = −�� + �(�) �� + sin �( d� + sin � ��)

��2 = −�� + �(�)(�� + �(�� + sin��))

��2 = −�� + �(�) �� + sinh �( d� + sin � ��)

can be written in the unified form ��2 = −�� + � � + � �� + sin��

(�, �) = (sin � , 1) (�, �) = (�, 0) (�, �) = (sinh � , 1) ⇒ �� = cos � �� ⇒ �� = �� ⇒ �� = cosh � �� ⇒ = = �� ⇒ = = �� ⇒ = = �� EINSTEIN FIELD EQUATIONS

• The Einstein field equations

8�� = � − Λ� � 1 � = � − �� govern the dynamics of general relativity by relating the geometry of spacetime 2 (confer Problem Set 3 for the definition of the Ricci tensor � and Ricci scalar R) to the stress-energy- momentum � of spacetime regions REVIEW OF AUG 2

• The Universe is observed on cosmological scales to be both ______and ______.

• Isotropic, homogeneous (3+1)D spacetimes are described by Robertson Walker metrics given by

��2 = −�� + � � + � �� + sin��

, with k = -1,0,1 for ______, ______and ______curvature, • Examples of 2D surfaces with each curvature are ______, ______, and ______. REVIEW OF AUG 2

• The Universe is observed on cosmological scales to be both isotropic and homogeneous.

• Isotropic, homogeneous (3+1)D spacetimes are described by Robertson Walker metrics given by

��2 = −�� + � � + � �� + sin��

, with k = -1,0,1 for negative, flat and positive curvature, • Examples of 2D surfaces with each curvature are saddles, flatlands, and spheres. EINSTEIN FIELD EQUATION SOLUTIONS – IN VACUUM � 8�� • Vacuum solutions of � − � = � − Λ�Λare obtained by setting the stress-energy-momentum tensor to 0: 2 �

� = 0 so

� = −Λ�

• The vacuum solutions of the Einstein field equations are Einstein manifolds, i.e., the Ricci tensor � is proportional to the metric tensor � (confer HW 3). • TERMINOLOGY: For Λ ≠ 0, we have lambdavacuum solutions of the Einstein field equations COSMOLOGICAL CONSTANT

• For lambdavacuum solutions of

� = −Λ�

the vacuum effectively acts as a “source” term when we associate vacuum energy density with Λ

• If Λ>0, spacetime is under negative isotropic pressure • If Λ<0, spacetime is under positive isotropic pressure FRIEDMANN COSMOLOGICAL EQUATIONS

• Exercise: Write equation corresponding to the first two diagonal components of the Einstein field equations � 8�� − � = � − Λ�Λ 2 �

��2 = −�� + � � + � �� + sin�� for the generalized RW metric

�̈ �̈ 2�̇ 2 2� �̈ �̇ � �� where and� = = − , �� = + + ��, � = 6 + + � = � + �/�2 � � + � �, � = (�, 0,0,0) � � �2 �2 �2 �2 FRIEDMANN COSMOLOGICAL EQUATIONS

• Exercise: Write equation corresponding to the first two diagonal components of the Einstein field equations � 8�� − � = � − Λ�Λ 2 �

��2 = −�� + � � + � �� + sin�� for the generalized RW metric

�̈ �̈ 2�̇ 2 2� �̈ �̇ � �� where and� = = − , �� = + + ��, � = 6 + + � = � + �/�2 � � + � �, � = (�, 0,0,0) � � �2 �2 �2 �2

� � = , � = ��, � = ��sin� � = −� = �� ⟹ �= �� 1 − �� ̈ 2�̇ 2� �̈ �̇ � 8�� 3 �̈ �̈ �̇ � 8�� ⟹ + + � − 3� + + = �� − Λ� �� ⟹ − + 3 + + = �� + Λ � � � � � � � � 1 �̈ �̇ 2� �̈ �̇ 2 � 8�� + 2 + − 3 + + = � − Λ �̇ �� 8�� 2� �2�2 �2 �4 ⟹ + − Λ = � �̈ �̇ �� 8�� 3 3 ⟹ 2 + + − Λ� = − � � � � � MORE TESTS OF RELATIVITY

• Hubble’s law • Fate of the Universe • Riess, Schmidt and Perlmutter 2011 Nobel Prize HUBBLE’S LAW

• Hubble observed light from galaxies was far more likely to be redshifted than blueshifted, as they were receding into an expanding universe with speed as a linear function of distance

v = �0� Edwin Hubble 1889-1953 • The current value of the Hubble constant (which is constant over space, not time) is km/s � = 70 0 Mpc indicating that for every megaparsec one looks into the sky, objects recede an additional 70km/s FATE OF THE UNIVERSE (GROUP ACTIVITY)

�� 8�� ̇ � + − Λ = �, � = • Solve the 1st Friedmann equation for the scale factor a(t) i� 3 3 � n an expanding Λ -dominated Universe. FATE OF THE UNIVERSE (GROUP ACTIVITY)

�� 8�� ̇ � + − Λ = �, � = • Solve the 1st Friedmann equation for the scale factor a(t) i� 3 3 � n an expanding Λ -dominated Universe.

� � = � ⇒ 3

�� ̇ = = �� = � � �� 3 � ⇒ � � ~� = � FATE OF THE UNIVERSE

• If the Universe has Ω < 1 , then its expansion will accelerate into a cold, isolated future (Big Freeze). • If the Universe has Ω = 1 , then it will continue expanding indefinitely, but at an ever-slowing rate (Flat Universe). • If the Universe has Ω > 1 , then it will eventually contract under its own gravity into a fiery collapse (Big Crunch).

�� 8�� + − Λ = � st � 3 3 1 Friedmann Equation

�̇ � = � = Ω�, � = Ω�, � = Ω�, � Ω = Ω + Ω + Ω, Ω = 1 km/s � = 70 �, = 9.47x10 0 Mpc REISS, SCHMIDT, PERLUTTER 2011 NOBEL PRIZE – ACCELERATING UNIVERSE

• A. Reiss (JHU), B. Schmidt (Australia National U.) and S. Perlmutter (Berkeley) won the 2011 Nobel Prize in physics for observing the accelerating expansion of the Universe. Popular science summary given in: https://www.nobelprize.org/nobel_prizes/physics/laureates/2011/popular-physicsprize2011.pdf

� = Ω�, � = Ω�, � = Ω�, �� 8�� + − Λ = � Ω = Ω + Ω + Ω, Ω = 1 � 3 3 � = 9.47x10 , �̇ � = � km/s � = 70 0 Mpc Brightness (mag) vs. distance (z) for Type Ia supernovae from observations by Brian Schmidt’s High-z Supernova Search Team (Riess et al. 1998) and Saul Perlmutter’s Supernova Cosmology Project (Perlmutter et al. 1999). Theoretical curves overlay the observations for cosmological models (Ω, Ω) =(1.0,0.0), (0.3,0.0), (0.3,0.7). The best fit is for the Λ–dominated Universe. EINSTEIN FIELD EQUATION SOLUTIONS – CLOSED TIMELIKE CURVES • Spacetimes with closed timelike curves (CTC) may have timelike (ds2<0, i.e., locally slower than c) wordlines that create closed loops on spacetime diagrams. • Any event on a closed timelike curve can equally be regarded as cause and effect of every other event.

ct Event y,z 0

Event Event 00000 00 Closed timelike curves form causal loops x Event Event O 0000 000 CLOSED TIMELIKE CURVES

• Spacetimes with closed timelike curves (CTC) may have timelike (ds2<0, i.e., locally slower than c) wordlines that create closed loops on spacetime diagrams. • Any event on a closed timelike curve can equally be regarded as cause and effect of every other event. • Exercise: Construct closed timelike curves (CTC) � in cylindrical coordinates for a spacetime with line element of the following form ��2 = −� � ��2 + 2� � �� + � � ��2 + �(�)(��2 + ��2) � �� Compute the proper distance along such a CTC.�� = �� = � – � �� CLOSED TIMELIKE CURVES

• Consider a closed worldline � with fixed time t, height z and cylindrical radius s so that ��2 = � � ��2 For C(R)<0, the worldline is timelike, and the proper length is obtained by integrating along the azimuth: 2� �� 2 � = �(�) ⟹ �� = � – � � �� = 2�� – � � , if � � < 0 0 GÖDEL METRIC

• An example of a spacetime supporting closed timelike curves is given by the Gödel metric, where the Einstein field equations have lambda and spinning dust terms (with z-axis of rotation)

��2 � 2 � 2 ��2 = −�2��2 + − �2�� + �2 1 − ��2 + ��2 � 2 � 2� 1 + 2�

• Exercise: What parameter sets scale of the spacetime? • Exercise: In what subregion might closed timelike curves exist? GÖDEL METRIC

• An example of a spacetime supporting closed timelike curves is given by the Gödel metric, where the Einstein field equations have lambda and spinning dust terms (with z-axis of rotation)

��2 � 2 � 2 ��2 = −�2��2 + − �2�� + �2 1 − ��2 + ��2 � 2 � 2� 1 + 2�

• The Gödel parameter � sets the scale of this spacetime • The metric �� component becomes negative past the Gödel radius �� = 2� , beyond which closed timelike curves exist � � 2 • Each point in spacetime rotates at angular speed �� = = 2� �� GÖDEL METRIC – LIGHT CONES AND CTC • The light cones and closed timelike curves of the Gödel metric ��2 � 2 � 2 ��2 = −�2��2 + − �2�� + �2 1 − ��2 + ��2 � 2 � 2� 1 + 2� have been visualized in many works, including Buser et al., 2013. http://iopscience.iop.org/article/10.1088/1367-2630/15/1/013063/pdf • CTCs concentric about the temporal axis (dt=0) may be isometrically transformed into CTCs traversing the origin (cf. Grave F., Buser M., Müller T., Wunner G. and Schleich W. P. 2009 Phys. Rev. D 80 103002) and thus allowing time travel!

Lightcones on a t=0 slice Transformation of a concentric of the Gödel spacetime rotate CTC (blue solid line) to a CTC (green solid line) passing the as we move along the origin illustrated in the radial direction at each (t,x,y)-subspace (a), in the azimuth (Buser et al., (x,y)-subspace (b) and in the 2013). Gödel radius in red (t,r)-subspace (c) (Buser et al., 2013). Gödel radius in red. PREDESTINATION – CAUSAL LOOPS

49:29 Temporal Agent recruits John to chase his nemesis (also Fizzle Bomber) through time

38:45 Jane names baby Jane, becomes 32:30 Mystery lover John, moves to NYC (John) leaves Jane as disgruntled pregnant "Unmarried Mother"

36:35 Jane finds out 38:10 Jane is told she is intersexed she needs surgery and needs to become a man hysterectomy PREDESTINATION – CAUSAL LOOPS

• Causal loops may be related in interesting ways.

57:30 John is recruited by 49:29 Temporal Temporal Agent to Agent recruits John hunt nemesis/Fizzle to chase his nemesis Bomber, believed (also Fizzle Bomber) on his way to meet through time with 1963 Jane

38:45 Jane names baby Jane, 59:26-59:46 1963 becomes John, 32:30 Mystery lover Jane runs into John moves to NYC as (John) leaves Jane (time traveler), and disgruntled pregnant they fall in love "Unmarried Mother"

1:12-1:13 John and 36:35 Jane finds 36:35 Jane finds out 38:10 Jane is told Jane conceive a out she needs she is intersexed she needs surgery baby (themselves), hysterectomy and and needs to become a man Agent steals baby is intersexed hysterectomy PREDESTINATION – POTENTIAL PARADOXES

• 59:26-59:46 In 1963 Jane runs into John (time traveler), and says he is looking for someone, and then John (time traveler) recalls a phrase Jane told/is telling her when he/she was Jane before his sex change. PREDESTINATION – POTENTIAL PARADOXES

• Review in class Mon Aug 7 • Final in class (2.5 hr) Wed Aug 9 • 300 pts (30% of course grade): Part I (100 pts) conceptual, Part II (100 pts) computational, Part III (100 pts) recall of popular science • Closed book, no calculator, no laptop • I may refer to short clips I play in class during the exam • Problems less computationally intensive than most computationally-intensive homework problems • Topics are drawn from entire course (slides, books, movies, articles, e-mails), and may include: • Math methods: Multivariate Calculus, Linear Algebra, Differential Equations • Classical mechanics: Newton’s Laws, Maxwell’s Equations • Special Relativity: Postulates, Lorentz Boosts, Length Contraction, Time Dilation, Relativity of Simultaneity, Experiments • General Relativity: Black Holes, Equivalence Principle, Tensors, Spacetime Diagrams, Worldlines, Observations • Popular Science: Interstellar, Flatland, The Time Machine, Predestination • Unification