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Summer 2017 Astron 9 Week 5 Final Version

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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 = $% %' A !" v$%v%' (& * ) 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)/0/=? +? 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. |vSP|=|vSE| v69 − v97 v67 = v v 1 − 69 97 c< vPE L0 � � + v 4 � 4 97 v − v (Δt) = 4 + 67 97 , v = v ≡ v /0/ v − v v69 − v97 69 67 67 97 v v 1 − 69 97 c< 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 � � + v 4 � 4 97 v − v (Δt) = 4 + 67 97 , v = v ≡ v /0/ v − v v69 − v97 69 67 67 97 v v 1 − 69 97 c< �< A �< ⟹ v = + � < − 1 v97 v97 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? • 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) NO • Consider the metric ��< = −�<��< + (�� − � �⃗ − �⃗ (�) v � )< + ��< + ��< , v (�) = P D D D NQ �D(�) �⃗D(�) = 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) NO • Consider the metric ��< = −�<��< + (�� − � �⃗ − �⃗ (�) v � )< + ��< + ��< , v (�) = P D D D NQ �D(�) �⃗D(�) = 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 = �� = −� �� + (�� − � �⃗ − �⃗D � vD � ) 4 < < ⟹ �� = �� − � �⃗ − �⃗D(�) vD � �� < �� v � ⟹ 1 = − � �⃗ − �⃗ (�) D ��4 D � �� v � ⟹ ±1 = − � �⃗ − �⃗ (�) D ��4 D � NOS ( ⟹ = W X NQ ±(&T U⃗VU⃗ (Q) P P ) �� If 0 < � �⃗ − �⃗ � < 1, and the ship moves right � > 0 D �� ��4 then <1 for the rightmost portion of the lightcone �� If 0 < � �⃗ − �⃗D � < 1, �� + � �� �� Warpdrive spacetime for smooth positive function = ±� + � �⃗ − �⃗ (�) v � ⟹ v � = > � j D D D �⃗ − �⃗D(�) . Confer Hartle p. 144. �� � �⃗ − �⃗D � � �⃗ − �⃗ (�) = 1 − D �j 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 < no < < < < < < < < < < < < < < �� = �no�� = −� �� + �� + �� + �� �� = −� �� + �� + � �� + � 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 < no < < < < < < < < < < < < < < �� = �no�� = −� �� + �� + �� + �� �� = −� �� + �� + � �� + � 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 <t <t V( stz{ ��2= −�< 1 − ��< + 1 − ��< + �<(��< + ���<���<) � = U U v* <st • The metric has a coordinate singularity at r=2M= %u v* KERR GEOMETRY st � = z{ • Kerr metric v* <tU jt}U D~•*€ |* ��2=−�< 1 − ��< − ���� + ��< + < < < < |* |* • � ≡ �/�, � ≡ � + � cos �, < < <tU}* D~•*€ ∆≡ � − 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 � ⟹ ˆ �� = � 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) (�, �) = ˆ (�, 0) (sinh � , −1) the RW metrics ��2 = −�<��< + �<(�) ��< + sin< �( d�< + sin< � ��<) ��2 = −�<��< + �<(�)(��< + �<(��< + sin<���<)) ��2 = −�<��< + �<(�) ��< + sinh<
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