ASTR 1040 Recitation: Relativity Ryan Orvedahl Department of - - PowerPoint PPT Presentation

▶

Sep 26, 2023 137 likes •468 views

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

SLIDE 1

ASTR 1040 Recitation: Relativity

Ryan Orvedahl

Department of Astrophysical and Planetary Sciences

February 17 & 19, 2014

SLIDE 2

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

SLIDE 3

Today’s Schedule

Special Relativity General Relativity Black Holes

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 3 / 27

SLIDE 4

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

SLIDE 5

History of Relativity

Maxwell’s E & M: Laws of physics set speed

f light at ∼ 3 × 108 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

SLIDE 6

History of Relativity

Michelson-Morley Experiment: Find the universal rest frame

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 6 / 27

SLIDE 7

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

SLIDE 8

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

SLIDE 9

Special Relativity

Time Dilation: t = γτp Length Contraction: L = Lp

Lorentz Gamma Factor: γ =

1− v2

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 9 / 27

SLIDE 10

Special Relativity

Rest Frame Coords: t, x, y, z Moving Frame Coords: t′, x′, y ′, z′ ct′ =

ct−vx/c

√

1−v2/c2

x′ =

x−vt

√

1−v2/c2

y ′ = y z′ = z

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 10 / 27

SLIDE 11

Special Relativity

How do velocities add? Is the speed of light held constant? Two frames moving at velocity v with respect to one another u′ = x′

t′ =

x−vt

√

1−v2/c2 t−vx/c2

√

1−v2/c2

⇒ u′ =

x−vt t−vx/c2

u′ =

x−vt t−vx/c2

⇒ u′ =

t(x/t−v) t(1−vx/(tc2))

⇒ u′ = u − v 1 − vu/c2 If u = c: u′ =

c−v 1−vc/c2

⇒ u′ =

c−v 1−v/c

⇒ u′ = c(c−v)

c−v

= c

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 11 / 27

SLIDE 12

General Relativity

Geometry you didn’t learn in High School Constant in any reference frame: ds2 = dx2 + dy 2 + dz2 Constant in any reference frame: ds2 = −c2dt2 + dx2 + dy 2 + dz2

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 12 / 27

SLIDE 13

General Relativity

Thinking of Time as an Extra Dimension

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 13 / 27

SLIDE 14

General Relativity

Special Relativity (1905) Laws of physics are same for all inertial reference frames Speed of light is constant in all intertial reference frames Does not explain how forces work General Relativity (1915) Laws of physics are same for ALL reference frames Gravity is caused by massive objects “bending” space-time No difference between gravity and an accelerating ref frame

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 14 / 27

SLIDE 15

General Relativity

Equivalence Principle: stationary in gravity = accelerating w/o gravity

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 15 / 27

SLIDE 16

General Relativity

Equivalence Principle: freefall in gravity = constant velocity w/o gravity Constant velocity v, if u = v u′ =

u−v 1−vu/c2

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

SLIDE 17

General Relativity

This leads to some strange results: Curved space-time Black holes

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 17 / 27

SLIDE 18

Curvature

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 18 / 27

SLIDE 19

Curvature

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 19 / 27

SLIDE 20

Curvature

Matter tells space how to curve

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 20 / 27

SLIDE 21

Black Holes

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 21 / 27

SLIDE 22

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

SLIDE 23

Black Holes

Spinning Black Holes

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 23 / 27

SLIDE 24

Black Holes

Frame Dragging

R. Orvedahl (CU Boulder)

Relativity Feb 17 & 19 24 / 27

SLIDE 25