ASTR 1040 Recitation: Relativity Ryan Orvedahl Department of Astrophysical and Planetary Sciences February 17 & 19, 2014
This Week Fiske Planetarium: Thurs Feb 20 (9:30 am) Observing Session: Thurs Feb 20 (7:30 pm) R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 2 / 27
Today’s Schedule Special Relativity General Relativity Black Holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 3 / 27
History of Relativity Newton’s Relativity: Laws of physics are the same for all inertial frames Time is the same for everyone Apply a constant force to an object, it will accelerate forever R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 4 / 27
History of Relativity Maxwell’s E & M: Laws of physics set speed of light at ∼ 3 × 10 8 m/s How can all reference frames measure same light going same speed? Idea of a universal rest frame ⇒ “aether” R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 5 / 27
History of Relativity Michelson-Morley Experiment: Find the universal rest frame R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 6 / 27
History of Relativity Einstein’s Relativity: Laws of physics (including speed of light) are same for all inertial frames Measuring things like time, length and mass depends on your reference frame R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 7 / 27
Special Relativity As objects travel faster and faster ... Their relative time slows down (time dilation) The object becomes shorter (length contraction) R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 8 / 27
Special Relativity Time Dilation: t = γτ p L = L p Length Contraction: γ 1 Lorentz Gamma Factor: γ = � 1 − v 2 c 2 R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 9 / 27
Special Relativity Rest Frame Coords: t , x , y , z Moving Frame Coords: t ′ , x ′ , y ′ , z ′ ct ′ = ct − vx / c √ 1 − v 2 / c 2 x ′ = x − vt √ 1 − v 2 / c 2 y ′ = y z ′ = z R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 10 / 27
Special Relativity How do velocities add? Is the speed of light held constant? Two frames moving at velocity v with respect to one another x − vt √ u ′ = x ′ ⇒ u ′ = 1 − v 2 / c 2 x − vt t ′ = t − vx / c 2 t − vx / c 2 √ 1 − v 2 / c 2 u − v ⇒ u ′ = u ′ = u ′ = t ( x / t − v ) x − vt ⇒ t − vx / c 2 t (1 − vx / ( tc 2 )) 1 − vu / c 2 If u = c : u ′ = ⇒ u ′ = ⇒ u ′ = c ( c − v ) c − v c − v = c 1 − vc / c 2 1 − v / c c − v R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 11 / 27
General Relativity Geometry you didn’t learn in High School Constant in any Constant in any reference frame: reference frame: ds 2 = − c 2 dt 2 + dx 2 + dy 2 + dz 2 ds 2 = dx 2 + dy 2 + dz 2 R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 12 / 27
General Relativity Thinking of Time as an Extra Dimension R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 13 / 27
General Relativity General Relativity (1915) Special Relativity (1905) Laws of physics are same Laws of physics are same for ALL reference frames for all inertial reference frames Gravity is caused by massive objects “bending” Speed of light is constant space-time in all intertial reference frames No difference between gravity and an Does not explain how accelerating ref frame forces work R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 14 / 27
General Relativity Equivalence Principle: stationary in gravity = accelerating w/o gravity R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 15 / 27
General Relativity Equivalence Principle: freefall in gravity = constant velocity w/o gravity Constant velocity v , if u = v u ′ = u − v 1 − vu / c 2 Objects are not moving with respect to you ⇒ think astronauts floating in the Space Station (freefall) R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 16 / 27
General Relativity This leads to some strange results: Curved space-time Black holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 17 / 27
Curvature R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 18 / 27
Curvature R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 19 / 27
Curvature Matter tells space how to curve R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 20 / 27
Black Holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 21 / 27
Black Holes No Hair Theorem: To an external observer, the black hole is completely described by 3 parameters Mass Electric Charge Spin/Angular Momentum R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 22 / 27
Black Holes Spinning Black Holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 23 / 27
Black Holes Frame Dragging R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 24 / 27
Black Holes How do we find Black Holes? R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 25 / 27
Black Holes How do we find Black Holes? Gravitational Lensing R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 25 / 27
Black Holes How do we find Black Holes? Gravitational Lensing Gravitational Radiation R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 25 / 27
Black Holes How do we find Black Holes? Gravitational Lensing Gravitational Radiation Look for lots of mass in a little space R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 25 / 27
Black Holes How do we find Black Holes? Gravitational Lensing Gravitational Radiation Look for lots of mass in a little space Look for its disk R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 25 / 27
Black Holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 26 / 27
Black Holes R. Orvedahl (CU Boulder) Relativity Feb 17 & 19 27 / 27
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