Foundations of Special Relativity

Question 1

What is Galileo’s principle of relativity?

Answer 1

The laws of mechanics are the same in all uniformly moving reference frames.

Question 2

What is the standard configuration?

Answer 2

Two inertial frames S and S’, such that S’ moves in the positive x direction at vecocity u relative to S.

Question 3

What are the four Galileon transformations for the standard configuration?

Answer 3

t' = t
x' = x-ut
y' = y
z' = z

Question 4

How do you derive the equation for the velocity v of an object moving at velocity v’ in the x’ direction?

Answer 4

v = dx/dt

Use Galileon transformations and then rearrange to end up with v = v’ + u

Question 5

What is the luminiferous aether?

Answer 5

A medium in which light travels similarly to sound, and longitudinal waves do not exist there. Has to be almost stationary with respect to the Earth.

Question 6

What was the Michelson-Morley experiment?

Answer 6

Considered measuring the Aether wind while travelling perpendicular and parallel to it. Light was sent between these points.

Question 7

What is the time for the parallel trip of light from 0 to A back to 0, if the distance is L1?

Answer 7

t a = L1/c-u + l1/c+u

= L1 ((c+u) + (c-u))/(c^2-u^2). Rearrange this and then take out a factor of (2L1/c)

Question 8

What is the time for the perpendicular trip of light from 0 to B back to 0, if the distance is L2?

Answer 8

t b = 2L2/sqrt(c^2 - u^2), then take out a factor of (2L2/c)
(consider view from the Aethers point of view)

Question 9

What are the 2 time differences between t a and t b?

Answer 9

Simply t a - t b and then rotate the system by 90 degrees so the square root appears on one denominator and disappears on the other.

Question 10

What is the difference between the time differences?

Answer 10

Minus one from the other and rearrange.

Question 11

How can we approximate the answer and what is the final result?

Answer 11

u &laquo_space;c so u^2 &laquo_space;c^2, so the answer is (L1+L2/c)*(u^2/c^2)

Question 12

How did Michelson and Morley experimentally disprove the Aether?

Answer 12

Used interferometry (light source splits at 0 and goes to A and B which have mirrors). They found no shift between the interference patterns of the two light sources returning.

Question 13

What was the Lorentz-Fitzgerald Contraction?

Answer 13

They clung onto the idea of the Aether and suggested that all object contract by a factor of sqrt(1-(u^2/c^2)) along the direction of the aether wind, which would shrink L1.

Question 14

What are the 2 postulates of Special Relativity?

Answer 14

1. The laws of physics are the same in all inertial frames.

2. The speed of light in empty space is the same in all inertial frames.

Question 15

What is the graph for when in 3 equally spaced observers A, B and C, B sends a pulse of light to A and C to synchronise their clocks (all in S’ frame)?

Answer 15

ct’ on y-axis and x’ on x-axis. A,B and C listed on x-axis and 2 45 degree diagonal lines come from B to the same y-axis point at A and C.

Question 16

Now consider the view from S. What does the graph look like now?

Answer 16

ct on y-axis and x on x-axis. The two events in S’ are simultaneous, but in this graph they are not simultaneous - the events appear to happen at a great x-value for A and C, making a wonky V shape and maxing the x’ constant lines diagonal.

Question 17

What does the relativity of simultaneity show?

Answer 17

That t’ =/ t

Question 18

What are the new coordinate transforms after discovering that t’ =/ t?

Answer 18

x' = γ(x-ut)
t' = γ(t + ((1-γ^2)/(γ^2 *u))*x)
y' = y
z' = z
where γ is a constant