1. Special Relativity

2. Galileo’s Principle of Relativity
The laws of mechanics must be the same in all inertial reference frames
Time is the same across all reference frames
Simple addition of velocities between frames

3. Einstein’s Principle of Relativity
The laws of physics must be the same in all inertial reference frames
The speed of light in vacuum has the same value in all inertial frames, regardless of the velocity of the observer or the velocity of the source emitting the light

4. An observer inside a moving train views a ray of light as follows:
𝑐= 2𝑑 ∆𝑡 c∆𝑡= 2d
mirror d v

5. ∆ 𝑡 ′ = 2 𝑑 2 + 𝑣 2 𝑡 2 4 𝑐 = ∆𝑡 1− 𝑣 2 𝑐 2 (cΔt´)2=4 𝑑 2 + 𝑣 2 𝑡 2 4
An observer outside a moving train, on the ground, at rest views a ray of light as follows: d v
(cΔt´)2=4 𝑑 2 + 𝑣 2 𝑡 2 4
∆ 𝑡 ′ = 2 𝑑 2 + 𝑣 2 𝑡 𝑐 = ∆𝑡 1− 𝑣 2 𝑐 2

6. Length contraction
Imagine a spaceship travelling at speed v between two stars a distance Δx apart (as measured in the rest frame of the stars)
An observer at rest relative to the stars sees the trip take a time Δt=Δx/v
The observer in the spaceship also sees Δt´ = Δx´/v where v is the same for both frames
We know from time dilation that Δt´= ∆𝑡 1− 𝑣 2 𝑐 2
Thus the measured lengths must be different as well! The observer on the spaceship sees a contracted length Δx´ = Δx √(1-v2/c2)
The factor − 𝑣 2 𝑐 is so useful in relativity it’s given it’s own symbol γ
For time dilation Δt´ = γΔt and for length contraction Δx´ = Δx/γ

7. Momentum & Energy
Changing your frame of reference will create an observable change in momentum and energy, but they are both still conserved.
Invariant mass energy
𝐸 2 = 𝑝 2 𝑐 2 +( 𝑚 𝑐 2 ) 2 ( 𝑚 𝑐 2 ) 2 = 𝐸 2 − 𝑝 2 𝑐 2
Not affected by reference frame
Energy and momentum are not conserved separately, but as a combination
This implies that for a particle at rest: 𝐸=𝑚 𝑐 2 !!!!

8. Momentum & Energy 𝜌 =𝛾𝑚 𝑣 𝐾𝐸= 𝛾−1 𝑚 𝑐 2 There is energy in mass
Consider a person weighing 100 kg:
𝐸=(100𝑘𝑔) (3𝑥 𝑚 𝑠 ) 2
𝐸=9𝑥 𝐽
A few kilograms of fuel of a nuclear source is worth a few tons of fuel of a chemical source
A useful relationship for relativistic momentum and energy:
𝜌 =𝛾𝑚 𝑣
𝐾𝐸= 𝛾−1 𝑚 𝑐 2

9. ct
An object sitting at rest, whose position (x) stays the same x ct
An object with some velocity, v x

10. 𝑐= 𝑥 𝑡 Θ = 45º when the object is moving at the speed of light 𝑐𝑡=𝑥
A real object with mass cannot travel at the speed of light
Light has no mass
So, a real object cannot go beyond 45º from the y axis, in either direction θ

11. Elsewhere
FUTURE E L S W H R E L S W H R
PAST

12. Strict Causality
C=3x 10 8 m/s t X
100 ly
Planet X is 100 ly away.
t= 𝑥 𝑐 =100𝑙𝑦 9.46𝑥 𝑚 𝑠 1 𝑙𝑦 ÷3𝑥 𝑚 𝑠 =100 𝑦𝑒𝑎𝑟𝑠
If a signal were sent from earth, at the speed of light, it would take Planet X 100 years to be able to recognize the signal

13. ct
ct’
If an object, ct’, is moving and another object, ct is at rest, relative to ct’, it looks like ct is moving
Tilted Axis
ct’
x=ct
Spacetime distance between 2 points:
s2= x2 + y2 + z2 - (ct)2
The speed of light is the same for all axis θ x’

14. Consequences The pole in the barn effect v
The pole and barn both see each other as being less than 100 m by the gamma factor.
At some instance the doors of the barn could be closed and opened, trapping the pole in the barn.
The pole sees the doors of the barn close one at a time, and open one at a time v
100 m
100 m

15. 2. Twin Paradox
Twin B Leaves on a rocket to mars.
Twin A sees twin B as moving close to the speed of light; he sees twin B as 21 and himself as 25.
When the rocket turns around, twin 2 sees himself as 25 and twin A as 21. A B
20 years old
20 years old
Once things accelerate, they are no longer in the inertial reference frame.
General Relativity deals with accelerated reference frames.

16. General Relativity a=9.8m/ 𝑠 2 g=9.8m/ 𝑠 2
A person in an elevator ascending at 9.8m/ 𝑠 2 would feel no different than a person in an elevator at rest.
They could not determine whether the force they were feeling was from the acceleration of the elevator or the force due to gravity.

17. Adding Velocities
A person standing on an island wants to know how fast a canon ball is coming toward them. Classical version: v1 + v2 = v’ Relativistic version: 𝒗 𝟏 + 𝒗 𝟐 𝟏+ 𝒗 𝟏 𝒗 𝟐 𝒄 𝟐 =𝒗′ If the canon were a ‘photon torpedo’ from the starship enterprise moving at the speed of light: 𝒗 𝟏 +𝒄 𝟏+ 𝒗 𝟏 𝒄 𝒄 𝟐 = 𝒗 ′ = 𝒗 𝟏 +𝒄 𝟏+ 𝒗 𝟏 𝒄 = 𝒗 𝟏 +𝒄 𝒄 𝒄 + 𝒗 𝟏 𝒄 =𝒄 𝒗 𝟏 +𝒄 𝒗 𝟏 +𝒄 =𝒄 v2 v1

18. Equations to Know 𝐾𝐸= 𝛾−1 𝑚 𝑐 2
Special Relativity: 𝛾= 1 1− 𝑣 2 𝑐 2 Time Dilation: ∆ 𝑡 ′ =𝛾∆𝑡 Length Contraction: ∆ 𝑥 ′ = ∆𝑥 𝛾
Momentum & Energy: 𝜌 =𝛾𝑚 𝑣
𝐾𝐸= 𝛾−1 𝑚 𝑐 2
Adding Velocities: 𝒗 𝟏 + 𝒗 𝟐 𝟏+ 𝒗 𝟏 𝒗 𝟐 𝒄 𝟐 =𝒗′