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

ASTR 1040 Recitation: Relativity Part III

Ryan Orvedahl

Department of Astrophysical and Planetary Sciences

March 31 & April 2, 2014

This Week

Night Observing: Wednesday April 2 (8:30 pm) Day Observing: Thursday Afternoon

Use Heliostat and Hα filters to view the Sun

• R. Orvedahl (CU Boulder)

Mass Transfer Mar 31 & Apr 2 2 / 15

Today’s Schedule

Past/Current Homework Questions? Past/Current Lecture Questions? More Special Relativity Group Projects

• R. Orvedahl (CU Boulder)

Mass Transfer Mar 31 & Apr 2 3 / 15

Special Relativity Reminder

Speed of light is constant for everyone Time Dilation Length Contraction

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Mass Transfer Mar 31 & Apr 2 4 / 15

Special Relativity: Time Dilation

Train car moving to the right at speed v Person in train sends laser pulse from ground to a mirror directly above What does person on the ground see?

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Mass Transfer Mar 31 & Apr 2 5 / 15

SR Time Dilation

Moving clocks run slow: t′ =

t

√

1−(v/c)2

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Mass Transfer Mar 31 & Apr 2 6 / 15

Lorentz Transformations

Traditional variables ct′ =

ct−ux/c

√

1−(u/c)2

x′ =

x−ut

√

1−(u/c)2

y ′ = y z′ = z More compact form ct′ = γ(ct − βx) x′ = γ(x − βct) y ′ = y z′ = z β ≡ u/c & γ ≡ (1 − β2)−1/2

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Mass Transfer Mar 31 & Apr 2 7 / 15

Group Project I: Velocities

A Train moves with speed u with respect to the ground. What do people on the train measure for your a) x velocity, v ′

x

b) y velocity, v ′

y

ct′ = γ(ct − βx) x′ = γ(x − βct) y ′ = y z′ = z vx ≡ x/t vy ≡ y/t β ≡ u/c γ ≡ (1 − β2)−1/2

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Mass Transfer Mar 31 & Apr 2 8 / 15

Group Project I: Answer

γ ≡

• 1 −

u

c

2−1/2 a) v ′

x = vx − u

1 − uvx

c2

b) v ′

y =

vy γ(1 − uvx

c2 )

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Mass Transfer Mar 31 & Apr 2 9 / 15

Group Project II: Introduction

Distant galaxy emits radiation in ˆ y direction in S′ frame (the rest frame

• f the source)

What angle does the light make with the x axis in the S frame, which moves at speed u with respect to S′? π/2, > π/2, < π/2? Why?

• R. Orvedahl (CU Boulder)

Mass Transfer Mar 31 & Apr 2 10 / 15

Group Project II: Introduction

Distant galaxy emits radiation in ˆ y direction in S′ frame (the rest frame

• f the source)

What angle does the light make with the x axis in the S frame, which moves at speed u with respect to S′? π/2, > π/2, < π/2? Why?

• R. Orvedahl (CU Boulder)

Mass Transfer Mar 31 & Apr 2 10 / 15

Group Project II: Doppler Beaming

Galaxy emits light in ˆ y direction in S′ frame. What is sin θ in S frame? θ is measured from x axis. vx =

v′

x+u

1+uv′

x/c2

vy =

v′

y

γ(1+uv′

x/c2)

sin θ = vy/v v = v 2

x + v 2 y

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Mass Transfer Mar 31 & Apr 2 11 / 15

Group Project II: Answer

sin θ = 1 γ vx =

v′

x+u

1+uv′

x/c2 = u

vy =

v′

y

γ(1+uv′

x/c2) = c

• 1 − u2/c2

v = v 2

x + v 2 y = c

⇒ sin θ = vy/v =

• 1 − u2/c2 = 1/γ
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Mass Transfer Mar 31 & Apr 2 12 / 15

What About General Relativity?

Special Relativity seems somewhat simple in the kind of math it uses What about General Relativity? It’s just messy

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Mass Transfer Mar 31 & Apr 2 13 / 15

Real General Relativity

Einstein tensor (Curvature): Gµν ≡ Rµν − 1

2Rgµν

Include cosmological constant (Dark Energy): Λ Include matter/energy: Tµν Full Einstein Equations: Gµν + Λgµν = 8πG

c4 Tµν

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Mass Transfer Mar 31 & Apr 2 14 / 15

Real General Relativity

Gµν + Λgµν = 8πG

c4 Tµν is actually a set of 10 non-linear, partial

differential equations, so very very hard to solve

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Mass Transfer Mar 31 & Apr 2 15 / 15