Special Relativity

Question 1

What is an inertia frame of reference ?

Answer 1

• One in which newtons laws of motion hold

- One in which the frame of reference is either at constant velocity or rest

Question 2

True or False:

Newtons’s laws of motion are invariant under Galilean transformations

Answer 2

True

Question 3

What is gamma ?

Answer 3

gamma = 1/sqrt(1-v^2/c^2)

Question 4

Describe what side of the light line you would find time like events and space like events

Answer 4

• You would find time like events in the upper part of the spacetime diagram (left)
• You would find the space like events in the lower part of the spacetime diagram (right)

Question 5

What is the equation for time dilation and describe how you would go about deriving it

Answer 5

• delta t’ = gamma * delta t
• You would go about deriving the equation by setting up a Pythagoras triangle with (v delta t’ )/2 at the base, the height h and the hypotenuse (c delta t’)/2 and isolate the (c delta t’) term

Question 6

What is the definition of the proper time and proper length of an event ?

Answer 6

• Proper time is the shortest time recorded for an event (the frame s , which is the moving clock rather than the stationary one)
• Proper length is the longest length of an object and that will also be in its own frame of reference (frame s)

Question 7

What is the equation for length contraction ?

Answer 7

L’ = L/gamma

Question 8

What are the Lorentz transformations for spacetime four vector?

Answer 8

• ct’ = gamma(ct-bx)
• x’ gamma(x-bct)
• y’ = y
• z’ = z
  where b=v/c
  The primes and signs can be swapped to interchange between frames

Question 9

What is the space time interval and describe a property about it with regards to relativity

Answer 9

• (delta s)^2 = (c delta t)^2 - (delta x)^2 - (delta y)^2 -(delta z)^2
• The space time interval is invariant under Lorentz transformations

Question 10

What is the equation for the proper time ?

Answer 10

Delta t = gamma . delta tao

Question 11

What is the equation for the relativistic Doppler shift with respect to frequency?

Answer 11

f’ = f sqrt( (1+b)/(1-b) )
where b = v/c
The plus or minus values will change depending if the object is moving closer too or further away from the observer

Question 12

Which side of the light line will simultaneous events take place and which side will events in the same location take place?

Answer 12

• Simultaneous events will take place on the right hand side of the light line
• Events in the same location will take place on the left hand side of the light line

Question 13

What is the relativistic momentum equation ?

Answer 13

P= gamma m v
It is derived by taking the derivative of the normal momentum with respect to the proper time rather than time in general

Question 14

How do you go about deriving the equation for velocity addition?

Answer 14

• Start with v’ = delta x’ / delta t ‘
• Sub in the equivalent Lorentz transformations for each of these
• Multiply both top and bottom by 1/ delta t and rearrange

Question 15

How would you go about deriving the relativistic force?

Answer 15

Take the derivative of relativistic momentum with respect to time

Question 16

Is this the correct equation for the rest energy ?

E=mc^2

Answer 16

No

It is E(0) = mc^2

Question 17

What equation must we use to conserve energy between all reference frames?

Answer 17

E = gamma m c^2

Question 18

Is this statement true?

The kinetic energy is simply the difference between the total energy and the rest energy

Answer 18

Yes

Question 19

What are the Lorentz transformations for energy and momentum ?

Answer 19

E' = gamma (E - b P(x) c)
P'(x) c = gamma (P(x) c - b E)
where 
b=v/c
Again these can be swapped between frames by swapping primes and signs

Question 20

What are the Lorentz in-variance equations for space - time and momentum - energy ?

Answer 20

Space-Time
(c delta t)^2 - (delta x)^2=(c delta t’)^2 - (delta x’)^2
=> (delta s)^2 = (delta s’)^2

Momentum-Energy
(E)^2-(P(x) c)^2 = (E’)^2 - (P’(x) c)^2
=> (E)^2 - (P c)^2 = (m c^2)^2

Question 21

What is key to setting up the Compton effect derivation ?

Answer 21

That to make sure energy conservation is conserved, that in the initial energy you take into account the rest energy of the particles