The Lorentz Transformations

Question 1

What is the equation for γ, the constants in the Lorentz transformations?

Answer 1

γ = (1-(u^2 / c^2))^(-1/2)

Question 2

What are the final version of the Lorentz transformations?

Answer 2

t' = γ(t-(ux/(c^2)))
x' = γ(x-ut)
y' = y
z' = z

Question 3

How do you invert the Lorentz transformations?

Answer 3

Set u = -u.

Question 4

What is length contraction?

Answer 4

When something looks smaller due to its motion in the S’ frame when observing from the S frame.

Question 5

What is the equation for length contraction?

Answer 5

L = L0/γ

Question 6

What is time dilation?

Answer 6

When time appears longer in the S frame compared to the S’ frame.

Question 7

What i the equation for time dilation?

Answer 7

Δt = γΔt0, where t0 is the time measured in the clock which is at rest.

Question 8

What did Rossi and Hall do to prove time dilation?

Answer 8

Measured muon flux at ground level and 1600m up a mountain, and found the muons half life changed in that distance.

Question 9

What is the equation for frequency using the relativistic doppler effect?

Answer 9

f/f0 = 1/(γ*(1+(u/c)) if the light travels away from the observer. If the light travels towards the observer, the equation is the same but with a minus sign.

Question 10

What are the reciprocals of time dilation? (measured in frame S’ and S)

Answer 10

Δt = γΔt’ in S’ and Δt’ = γΔt in S.

Question 11

What is the twin paradox?

Answer 11

One twin travels to a star 30 light years away and back in 1 year, and when they return they find their twin to be 60 years older.

Question 12

Which 3 frames do we need to consider in the twin paradox?

Answer 12

S: Bob’s rest frame (Earth)
S’: Alice’s rest frame, outbound, +u relative to Bob
S’’: Alice’s rest frame, inbound, -u relative to Bob.

Question 13

What are the 3 events in the twin paradox?

Answer 13

1. Alice leaves Earth and x=x’=x’‘=0, t=t’=t’‘=0
2. Alice reaches start, x=L, t=L/u
3. Alice returns to Earth, x=0, t= 2L/u

Question 14

What do you do after determining the 3 events in the twin paradox?

Answer 14

You use the LT to compute t, t’ and t’’ in each event. Then you use these to determine the time elapsed for each twin.

Question 15

What are the findings of the twin paradox? Why is this?

Answer 15

That the time elapsed for bob is much much greater than Alice (factor of γ). This is because Alice switches between inertial frames and bob does not.

Question 16

How would you use relativistic doppler shift for the twin paradox?

Answer 16

Use the equations for frequency when Alice is moving towards/away from Bob.

Question 17

What are the equations for the addition of velocities in S and S’?

Answer 17

v = (v’+u)/((1+uv’/(c^2)))

v’ = (v-u)/(1-(uv/(c^2)))

Question 18

What is the equation for velocity in the y and z axis (vy’ and vz’)?

Answer 18

vy’ = vy/(γ*(1-(uvx/c^2)))

vz’ = vz/(γ*(1-(uvx/c^2)))